Necessary conditions of the minimum in an impulse optimal control

Journal of Mathematical Sciences, Vol. 139, No. 6, 2006

NECESSARY CONDITIONS OF THE MINIMUM IN AN IMPULSE OPTIMAL CONTROL PROBLEM D. Yu. Karamzin

UDC 517.977.52

Abstract. The paper is devoted to studying the impulse optimal control problem with inequality-type state constraints and geometric control constraints defined by a measurable multivalued mapping. The author obtains necessary optimality conditions in the form of the Pontryagin maximum principle and nondegeneracy conditions for the latter.

CONTENTS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Statement of the Problem and Main Definitions . . . . . . . . . . . . . . . . . . . . . . Some Notation and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lemmas and Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reductions R1 and R2 and the v-Problem . . . . . . . . . . . . . . . . . . . . . . . . . Simplest Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear-Convex Problems. Existence Theorem for Solution. Maximum Principle for the lem with Endpoint Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . State-Constrained Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nondegeneracy Conditions and Completion of the Proof of the Maximum Principle . . General Nonlinear Impulse Control Problem . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.

. . . . . . . . . . . . . . . . . . . . Prob. . . . . . . . . . . . . . . . . . . . . . . .

7087 7093 7096 7107 7109 7112 7119 7130 7132 7146 7148

Statement of the Problem and Main Definitions

The present paper is devoted to the study of an impulse control problem with a vector-valued measure. First, by examining two simple examples, we explain the origin of impulse controls, and, what is most important, we show the principal difference between the vector and scalar cases. We begin with the rocket dynamics problem. The well-known problem of transferring a spacecraft from one orbit to another with minimal fuel expenditure is schematically modeled with simplifications as follows (see [18]): 1 u(t)dt → min,

J(u) =

x˙ = x + u,

x(0) = 0,

x(1) = 1,

u(t) ≥ 0.

(1)

0

Here, the scalar u is the speed of combustion of fuel at the expense of which the traction force is created and x is the position of the material point. The rocket engine of the spacecraft can burn down the fuel with a very high speed, producing a heavy traction force. Therefore, the assumption on the unboundedness from above of the control is realistic. Now, to strictly formalize problem (1), we must describe the class of controls. If, as the class of controls, we take one of the spaces Lp , 1 ≤ p ≤ ∞, then the minimum of functional (1) is not attained in this class, and a minimizing sequence of absolutely continuous trajectories converges to a discontinuous function. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005. c 2006 Springer Science+Business Media, Inc. 1072–3374/06/1396–7087 

7087

Such a phenomenon is explained as follows. On one hand, in Lp , for p > 1, the set of admissible controls in problem (1) is unbounded, and it is bounded only in L1 . On the other hand, in problem (1), we minimize exactly the norm of the control in L1 ; however, the unit ball in L1 is no longer a weakly sequentially compact set. Therefore, we cannot guarantee the existence of a solution in any of the classes Lp , p ≥ 1. In such degenerate cases, one usually performs the procedure of completion or relaxation of the problem, and the completed class of controls is said to be impulsive. In this case, the main problem is: in which metric is the convergence of controls understood, i.e., with respect to which metric is the completion performed? Examining problem (1), we perform such a completion procedure. First of all, we note that every ordinary control u(t) of problem (1), i.e., a measurable Lebesgue dμ = u(t). integrable function can be considered as an element (denoted by μ) of the space C ∗ [0, 1] if we set dt ∗ (C [0, 1] is the topological dual of C[0, 1], and its elements are Borel measures.) Therefore, with each admissible control u(t) in problem (1), we associate an absolutely continuous nonnegative Borel measure μ with density u(t), and, moreover, μT V = uL1 , where μT V is the total variation of the measure. Now, ∗ [0, 1] of as is easily seen, the existence of a solution of problem (1) can be guaranteed in a wider class C+ ∗ admissible controls, the set of Borel nonnegative measures. (This is a consequence of the weak sequential compactness of the ball in C ∗ [0, 1].) Moreover, the set of absolutely continuous Borel measures is weak∗ everywhere dense in C ∗ [0, 1] and, therefore, the completion is well defined: it is performed with respect to the weak∗ topology and leads to the whole set of nonnegative Borel measures (which is essential for the physical interpretation, since with each impulse control, one must associate a certain sequence of ordinary controls). Moreover, the well-posedness means that with each impulse control, one associates a unique state trajectory x(t), which is no longer an absolutely continuous function and can have discontinuities (if, for example, the right-hand side contains the Dirac measure). However, in the exampleconsidered, the control is scalar-valued. Everything substantially changes if in the relaxed (completed) problem, the control is a vector-function. In such a case (if, moreover, the Frobenius condition presented below is violated), the completion procedure described above does not work. It turns out that for vector-valued controls, in the relaxed problem, we cannot restrict ourselves to the consideration of only vector-valued measures, and the impulse control becomes an object having wider properties than a vector-valued measure. The following example from economics yields a problem that cannot be relaxed in the space of vector-valued measures. The model of optimizing the expenditures for advertising two goods has the following form [18]: 1 [α1 u1 (t) + α2 u2 (t)]dt → min,

α1 , α2 ≥ 0,

α1 + α2 = 1,

0

⎧ x1 (0) ∈ (0, 1), ⎨x˙ 1 = a(1  − x1 )u1 − bx1 , u1 ≥ 0, u2 ≥ 0. 1 − x2 ⎩x˙ 2 = c u2 − dx2 , x2 (0) ∈ (0, x1 (0)), x1 Here, u1 and u2 are the current expenditures for advertising goods (money/unit of time) and x1 and x2 are volumes of sale. It turns out that, for this problem, the completion procedure described above is not correct, since it leads to a whole funnel of solutions containing more than one trajectory corresponding to a vector-valued measure. In the present paper, we propose methods that allow one to study the mentioned class of problems. Therefore, we will study the the following impulse optimal control problem: J(p, u, μ, {vr }) = e0 (p) → min, dx = f (x, u, t)dt + g(x, t)dμ, e1 (p) ≤ 0,

e2 (p) = 0,

ϕ(x, t) ≤ 0, 7088

t ∈ [t0 , t1 ],

(2) (3) (4) (5)

u(t) ∈ U (t) ⊂ Rm a.a. t, p = (x0 , x1 , t0 , t1 ),

Range(μ) ⊂ K,

x0 = x(t0 ),

x1 = x(t1 ).

Here, e1 , e2 , and ϕ are vector-functions taking their values in Rkj , j = 1, 2, 3, respectively, g is a matrix with k4 columns and n rows, t ∈ R1 is time, μ is a k4 -dimensional Borel vector-valued measure defined on the interval of time T = [t0 , t1 ] and taking its values in the cone K, {vr } is the set of measurable vectorfunctions considered on the closed interval [0, 1], which depends on the parameter r ∈ T and is connected with the vector-valued measure μ in a certain way (this will be made more precise in what follows), x is the state variable taking its values on the n-dimensional arithmetic space Rn , and the notation “a.a. t” means “for almost all t in the sense of the Lebesgue measure on the real line.” The vector u taking its values in Rm and corresponding to a regular control parameter is called the control. As the class of admissible controls, we consider essentially bounded measurable functions u(t) such that u(t) ∈ U (t) a.a. t. A pair (μ; {vr }) corresponds to an irregular or so-called impulse control . The class of impulse controls will be defined below. The vector p ∈ R2n+2 is called the endpoint vector. The functions e0 , e1 , e2 , ϕ, and g defining the functional being minimized, the endpoint and state constraints, and also the matrix next to the control vector-valued measure are continuously differentiable in the totality of the variables, the vector-function f is continuously differentiable in x a.a. t, measurable in t for any fixed (x, u) together with its partial derivatives in x, and on any bounded set, f and its partial derivatives in x are bounded and continuous in (x, u) a.a. t uniformly in t. The multivalued mapping U (t) is measurable and bounded, i.e., U (t) ⊆ B, where B is bounded. The set U (t) is closed a.a. t. The cone K is convex and closed. Below, we give the definition of a solution of Eq. (3) (see also [18, 32, 33, 36]). Take a vector Δ ∈ K, k4  |Δj |. Then the elements of the set HK (Δ; γ) ⊂ Lk∞4 ([0, 1]) Δ = (Δ1 , . . . , Δk4 ), and a number γ ≥ |Δ| = j=1

are measurable essentially bounded vector-functions v = (v 1 , . . . , v k4 ) defined on the closed interval [0, 1] and taking their values in the cone K such that (1)

k4 

|v j (s)| = γ a.e. in s ∈ [0, 1];

j=1

1

v j (s)ds = Δj , j = 1, . . . , k4 .

(2) 0

It is seen from this that the set HK (Δ; γ) is always nonempty for γ = |Δ|, and for γ > |Δ|, this set can be either empty or nonempty (depending on the chosen cone K and the relation between the numbers γ and |Δ|). If μ is a vector-valued measure, then • |μ| =

k4  j=1

|μj |, where |μj |(B) = sup



|μj (Bk )|, B =



Bk , i.e., the measure |μj | is a variation of

k

the charge μj [27]; • Ds(μ) = {r ∈ T : |μ|({r}) > 0} is the discrete support of μ; • μc is its continuous component, i.e., an atom-free measure whose components are continuous components of components of μ; • μ({r}) is the vector from K equal to the value of μ on the singleton {r}. The following concept plays a key role in the further presentation. A set {vr } of vector-functions depending on the parameter r ∈ T is said to be associated with the vector-valued measure μ if there exists γr ≥ |μ|({r}) in which no more than countably many elements are a set {γr }r∈T of nonnegative numbers  different from zero such that γr < ∞ and vr ∈ HK (μ({r}); γr ) ∀r ∈ T . 7089

An impulse control in the problem (2)–(5) is the pair q = (μ; {vr }), where μ is a vector-valued measure assuming its values in the cone K and {vr } is a certain set of functions associated  with μ. A variation of γr δr . Here δr is the Dirac an impulse control q = (μ; {vr }) is a scalar-valued Borel measure |q| = |μc |+ measure supported at the point r. The definition of the associated set immediately implies Ds(μ) ⊆ Ds(|q|) (note that the inclusion can be strict). Moreover, vr = 0 whenever r ∈ / Ds(|q|). Therefore, in essence, the set {vr } depends on the parameter r ∈ Ds(|q|) and has no more than a countable set of functions different from zero. Also, note that in the case where the cone K is embedded in the positive orthant, the definition of the associated set is simpler. Indeed, in this case, the functions vr are nonnegative, which implies |q| = |μ|, i.e., the variation of the impulse control coincides with that of the vector-valued measure (which is, in general, not true for an arbitrary cone). This implies vr = 0 ⇔ r ∈ Ds(μ) if K ⊆ Rk+4 . The latter assertion is also always true for an arbitrary acute cone. Let r ∈ T , v ∈ Lk∞4 ([0, 1]), and x ∈ Rn . Then the function αr (·) = αr (·; x, v) is defined as a solution of the following set of equations: α˙ r = g(αr , r)v,

s ∈ [0, 1],

αr (0) = x.

We set ξ(x, r, v) = αr (1). Now let us directly pass to the definition of the solution. Fix an arbitrary impulse control q = (μ; {vr }). A function x(t) defined on the closed interval T is called a solution of Eq. (3) corresponding to the triple (x0 , u, q) if 

t x(t) = x0 +

f (x, u, s)ds + t0

+

 

ξ(x(r− ), r, vr ) − x(r− )

g(x, s)dμc

[t0 ,t]

∀t > t0 ,

x(t0 ) = x0 .

(6)

r∈Ds(|q|) r≤t

This definition is correct in the following sense. Let μi be a sequence of absolutely continuous vectorvalued measures weakly converging to a vector-valued measure μ, and let xi be the absolutely continuous trajectory corresponding to μi in accordance with (now ordinary) differential equation (3). Then there exists a set of functions {vr } associated with μ such that the trajectory x(t) corresponding to the impulse control q = (μ, {vr }) in accordance with Eq. (6) is the limit of {xi }. More precisely, passing to a / Ds(|q|) \ {t0 , t1 }. subsequence, xi (t) → x(t) ∀t ∈ Conversely, for any function x(t) that is a solution in the sense of (6), there exists a sequence of absolutely continuous vector-valued measures weakly converging to μ such that the sequence of the corresponding trajectories converges to x(t). The strict formulations and proofs of the facts presented are contained in Sec. 3. Fix μ and start to examine item-by-item all possible sets of functions {vr } associated with μ, every time trying to find a certain new trajectory x(t) in this process using Eq. (6). As a result, we obtain a certain set of solutions, which is denoted by P = P(μ). It follows from what was said above that, first, the constructed set P exhausts any approximation method for μ by absolutely continuous measures, i.e., if for a function x, there exists its absolutely continuous approximation by solutions of Eq. (3), then x ∈ P and the set P contains no “extra” solutions, i.e., those solutions for which there are no approximations. Hence we arrive at the conclusion that the set P is exactly the so-called integral funnel of solutions arising in approximating μ by absolutely continuous measures. Therefore, in general, an arbitrary vector-valued measure μ generates a whole “bag” of trajectories, each of which claims to be a “solution” of Eq. (3). To precisely specify which of the trajectories of the “bag” will be called a solution, it is necessary to extract a branch of the integral funnel, which, in fact, is done by introducing a set of functions {vr } associated with μ. Having fixed a control u, a measure μ, and a set {vr } associated with it, according to the triple (u, μ, {vr }) and the initial value x0 , we uniquely find the trajectory x(t). As we see, the 7090

associated family {vr } reflects an approximation of μ by absolutely continuous measures and, in a certain sense, is the “interaction scheme of components of the vector measures on the discontinuity trajectory of the system.” It becomes clear from what was said above that such an “interaction scheme” (as well as the vector-valued measure) must be included in the control parameter, and this corresponds to the concept of impulse component presented above. Further, we point out the case where the so-called Frobenius condition holds; it is a particular but very important case for applications. We say that the Frobenius condition holds for system (3) if the vector fields g j pairwise commute (here g j are columns of the matrix g), i.e., gxj (x, t)g i (x, t) ≡ gxi (x, t)g j (x, t) ∀i, j. In early works on impulse controls, this assumption was assumed to a priori hold (see, e.g., [18, 50]). What gives the Frobenius condition? It turns out that, in such a case, the integral funnel P(μ) of solutions described above degenerates and is a single trajectory for any vector-valued measure μ (a consequence of [50, Lemma 4.5, p. 159]). Then, as is easily understood, the impulse control is a vector-valued measure itself, and the introduction of the set associated with it is extra. Thus, in the Frobenius case the situation is considerably simplified, and the definition of the solution given above transforms into the classical well-known definition, which can be found in [18, 50]. If there is no Frobenius condition, then, in general, the integral funnel P(μ) consists of a “large number” of trajectories, which shows the following example. Example 1.1. Let n = 1, k4 = 2, K = R2 , and T = [0, 1]. Consider the following dynamical system with a vector-valued measure μ = (μ1 , μ2 ): dx = xdμ1 + x2 dμ2 ,

x(0) = 1.

Take μ = 0 and show that the integral funnel P(0) contains at least two different trajectories (certainly it contains many more trajectories, a whole continuum of them). One trajectory is trivially found: it is the trajectory x(t) ≡ 1 corresponding to the zero associated set vr ≡ 0 ∀r. Let us construct the second trajectory x ˜(·). We set vr ≡ 0 ∀r > 0, γ0 = 2, and  

⎧ ⎧ 1 1 ⎪ ⎪ ⎪0, ⎪2, s ∈ 0, , s ∈ 0, , ⎪ ⎪ ⎪ ⎪ 4 4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪     ⎨ ⎨ 1 1 1 1 1 2 v0 (s) = 0, , , , , s∈ s∈ v0 (s) = −2, ⎪ ⎪ 4 2 4 2 ⎪ ⎪ ⎪ ⎪     ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ⎩1, ⎩ ,1 , ,1 . s∈ −1, s∈ 2 2 Let us find a solution α(s) of the system associated with the point 0. It is easy to see that

  √ −2s 1 1 1 1 α(s) = , s ∈ 0, , . and α(s) = 2 ee , s ∈ 1 − 2s 4 4 2 √ Now note that α(1/2) = 2/ e = 1, while α = 1 is a stationary point of the system on the remaining closed interval [1/2, 1]. This implies that α(1) = c = 1, and hence x ˜(t) = c for t > 0, i.e., the trajectory corresponding to the associated set is not identically equal to unity, which is what we were required to show. The above example is also interesting because it shows how with a measure having no atom component, one can associate a discontinuous trajectory according to Eq. (3) (this turns out to be possible, since the cone K is not acute in the example considered). This once again points to the fact that the impulse control in a problem with a vector-valued measure is a concept wider than the vector-valued measure itself. 7091

To complete the statement of the problem, it remains to give the definition of a trajectory satisfying the state constraints (5). If φ(x, t) is a continuous scalar-valued function, then we set1     gc sup φ(x, t) = max max φ αt s; x(t− ), vt , t . t∈T

t∈T s∈[0,1]

Now inequality (5) should be understood in the generalized sense: ϕ(x, t) ≤ 0 ⇔ gc sup ϕj (x, t) ≤ 0, j = 1, . . . , k3 . This means that the trajectory x(t) satisfies the state constraints (5) iff (1) ϕ(x(t), t) ≤ 0 ∀t ∈ [t0 , t1 ]; (2) ϕ(αr (s), r) ≤ 0 ∀s ∈ [0, 1] ∀r ∈ Ds(|q|) (see also [18, 32, 33]). The triple (p, u, q), q = (μ, {vr }), is called a control process if ∃ x(t): x(t0 ) = x0 , x(t1 ) = x1 , and x, u, and q satisfy (3). A process is said to be admissible if all constraints of the problem considered hold for it. An admissible process (p∗ , u∗ , q∗ ) is said to be optimal if for any admissible process (p, u, q), the inequality e0 (p∗ ) ≤ e0 (p) holds. For problem (2)–(5), it is required to obtain necessary optimality conditions in the form of a nondegenerate (informative) Pontryagin maximum principle (MP) [40]. Let us introduce the main definitions and assumptions, which will be used below. Definition 1.1. Endpoint constraints are said to be regular if for any vector p = (x0 , x1 , t0 , t1 ) satisfying (4), the following conditions hold: ∂ej2 (p), j = 1, . . . , k2 , are linearly independent; (1) the vectors ∂p (2) there exists a vector p¯ ∈ R2n+2 such that   ∂ej1 ∂e2 (p)¯ p = 0, (p), p¯ > 0 ∀j : ej1 (p) = 0. ∂p ∂p Definition 1.2. State constraints are said to be regular if for any (x, t) satisfying (5), there exists a vector q = q(x, t) ∈ Rn such that   j ϕx (x, t), q > 0 ∀j : ϕj (x, t) = 0. Definition 1.3. Let a point p∗ = (x∗0 , x∗1 , t∗0 , t∗1 ) satisfy the endpoint constraints (4), and let the inequalities ϕ(x∗k , t∗k ) ≤ 0, k = 0, 1, hold. We will say that at the point p∗ , the state constraints are in concordance with endpoint constraints if there exists ε > 0 such that {p ∈ R2n+2 : |p − p∗ | ≤ ε, e1 (p) ≤ 0, e2 (p) = 0} ⊆ {p : ϕ(xk , tk ) ≤ 0, k = 0, 1}. For more details on the concepts introduced in Definitions 1.1–1.3, see [1, p. 91–94]. Now let us introduce into consideration the smoothness assumption. Assumption (S). The function f is continuously differentiable in the totality of the variables, and the multivalued mapping U (·) is constant, i.e., U (t) ≡ U for a certain compact set U . Definition 1.4. Let Assumption (S) hold. An admissible trajectory x(t), t ∈ [t0 , t1 ], is said to be controllable at the endpoints (with respect to the state constraints) if there exist vectors uk ∈ U and mk ∈ K such that  j  ϕx (xk , tk ), f (xk , uk , tk ) + g(xk , tk )mk + ϕjt (xk , tk ) < 0 ∀j : ϕj (xk , tk ) = 0, where xk = x(tk ), k = 0, 1. If g ≡ 0 or μ belongs to the class of absolutely continuous vector-valued measures with densities from L∞ , then Definition 1.4 transforms into the known definition of the controllability for the ordinary optimal control problem related to (2)–(5) [1, p. 112]. 1

gc sup = graph completion supremum.

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2.

Some Notation and Definitions

Let us introduce certain notation and definitions, which will be used in what follows. Let T = [t0 , t1 ] be a given interval on the real axis; then C(T ) denotes the Banach space of functions f : T → R1 continuous on T with the usual norm f C = max |f (t)|; V (T ) is the space of functions t∈T

of bounded variations on T that are right-continuous on the interval (t0 , t1 ); V n (T ) is the space of ndimensional vector-functions x(t) = (x1 (t), . . . , xn (t)) such that xj ∈ V (T ), j = 1, . . . , n; C ∗ (T ) is the space topologically dual to C(T ). It is known that elements of C ∗ (T ) are Borel charges on T , i.e., countably-additive set functions given on σ(T ), where σ(T ) is the σ-algebra of Borel subsets of T . More precisely, every continuous linear functional on C(T ) has the form  S(f ) = f (t) dμ, T

where μ is a Borel charge on T and, moreover, S = |μ|(T ) [26]. Here, |μ| is a variation of the charge μ. The variation of a Borel charge μ is the Borel measure (i.e., a nonnegative charge) |μ| = μ+ + μ− , where μ+ and μ− are disjoint Borel measures from the Jordan decomposition of the charge μ: μ = μ+ − μ− . The total variation of the charge μ is μ = |μ|(T ), i.e., the norm of the element μ in C ∗ (T ) [27]. Also, it is known that C ∗ (T ) is isomorphic to V0 (T ) = {g ∈ V (T ) : g(t0 ) = 0} with the norm g = Var |tt10 [g] [26]; here, Var |ba [g] denotes the variation of the function g on the closed interval [a, b] [26]. Further, denote by ∗ (T ) the set of Borel measures given on T and by C ∗ (T ) the set of k -vector-valued Borel measures C+ 4 K taking their values in a closed convex cone K. The latter means that μ(B) ∈ K ∀B ∈ σ(T ). The notation ∗ (T ), means that μj − μj ∈ C ∗ (T ), j = 1, . . . , k . μ1 ≤ μ2 , where μ1 , μ2 ∈ CK 4 + 2 1 Let g ∈ V (T ), and let s ∈ T . We set g(s+ ) = lim g(t) and g(s− ) = lim g(t) (the right and left limits t→s+

t→s−

of the function at the point, respectively). By c, c1 , c2 , . . . ,. etc. we denote positive constants. If μ ∈ C ∗ (T ) is a Borel charge on T = [t0 , t1 ], then its distribution function F (t; μ) is defined by the formula  F (t; μ) =

dμ = μ([t0 , t]),

t ∈ (t0 , t1 ],

F (t0 ; μ) = 0.

[t0 ,t] ∗ (T ), then F (t; μ) is The countable additivity of the Borel charge μ implies F (t; μ) ∈ V0 (T ). If μ ∈ C+ a monotone, nondecreasing, right-continuous function on (t0 , t1 ), and, moreover, F (t1 ; μ) = μ. Conversely, any function g ∈ V (T ) defines a Borel charge ϑ[g] by the following formula [26, Theorem 6, p. 29]: ϑ[g]([t0 , t]) = g(t+ ) − g(t0 ), t ∈ T.

Therefore, ϑ[F (t; μ)] = μ. The measure generated by the length is denoted by L. By definition, L = ϑ[t]. The Dirac measure at a point r is denoted by δ(r). If x ∈ V n (T ), then ϑ[x] is a vector-valued measure ∗ (T ), then F (t; μ) is a whose components are charges ϑ[xj ], j = 1, . . . , n, and, respectively, if μ ∈ CK vector-function whose components are distribution functions of the components of μ. A variation of a vector-valued measure (vector-function) is defined as the sum of variations of its components. When it is clear which functions are being discussed, we will write F (t) instead of F (t; μ) and Fi (t) instead of F (t; μi ) for brevity. Functions f ∈ V (T ) are assumed to be defined on the whole real axis. The values of f outside the closed interval T are defined to be equal constants by continuity: f (t) = f (t1 ) if t > t1 , and f (t) = f (t0 ) if t < t0 . Analogously, Borel charges μ ∈ C ∗ (T ) are extended to the whole real line by setting μ(B) = μ(B ∩ T ) for an arbitrary Borel set B ⊂ R1 . ∗ (T ); then the components of the vector-valued measures μ and μ are, respectively, Let μ ∈ CK c d ∗ (T ). The support continuous and discrete components of μ. By definition, μ = μc + μd ; μc , μd ∈ CK supp(μ) of a vector-valued measure is defined as the support of the variation: supp(μ) = supp(|μ|). In turn, the support of a Borel measure on a closed interval is defined as the set of points of growth for its distribution function. Introduce the following notation: Ds(μ) = {r ∈ T : |μ|({r}) > 0} is the 7093

discrete support of the vector-valued measure, Cont(μ) = [T \ Ds(μ)] ∪ {t0 } ∪ {t1 } is the set of continuity points of the distribution function F (t; μ) including the endpoints of the closed interval [t0 , t1 ]; clearly, Cont(μ) ∪ Ds(μ) = T . The weak∗ convergence in C ∗ (T ) is defined as follows: a sequence of charges μi ∈ C ∗ (T ) weakly w converges to a charge μ ∈ C ∗ (T ) (denoted as μi → μ) if   f (t) dμi → f (t) dμ ∀f ∈ C(T ). T

T

A sequence of vector-valued measures converges if all its components weakly converge. The notation w w (μi , |μi |) → (μ, ν) means that, along with the weak convergence of the measures μi → μ themselves, we w have the weak convergence of variations |μi | → ν. Let us present a number of known assertions, which will be used in what follows. w

∗ (T ) and μ → μ, then (1) μ ∈ C ∗ (T ) (this is directly verified) and (2) μ  ≤ const ∀i • If μi ∈ CK i i K (Banach–Steinhaus theorem, [41]). w ∗ (T ) and (μ , |μ |) → (μ, ν), then (1) |μ| ≤ ν (see [18, p. 238]) and (2) F (t; μi ) → F (t; μ) • If μi ∈ CK i i ∀t ∈ Cont(ν) [1, Lemma 7.1, p. 134]. • Let fi ∈ V (T ), fi (t) → f˜(t) ∀t ∈ T . Then if Var |T [fi ] ≤ c ∀i, it follows that ∃f ∈ V (T ) : w Var |T [f ] ≤ c, f (t) = f˜(t) a.e., and ϑ[fi ] → ϑ[f ] (first Helly theorem, [27, p. 359]). ∗ • In the space C (T ), the closed unit ball B1 = {μ ∈ C ∗ (T ) : μ ≤ 1} is weakly∗ sequentially compact [27, Theorem 3, p. 199]. • Let fi ∈ V (T ) and let Var |T [fi ] ≤ c ∀i. Then there exists a subsequence {fik } converging at each point of the closed interval T (second Helly theorem, [27, p. 360]). Consider two functions f, g ∈ V (T ). The function f is Borel,2 and hence it is measurable with respect to the Borel measure |ϑ[g]|. In addition, f is bounded. Then there exists the Lebesgue–Stieltjes integral  f (t) dϑ[g] T

of the function f with respect to the charge ϑ[g]. Now assume that Ds(ϑ[f ])∩Ds(ϑ[g]) = ∅. In such a case, the Lebesgue–Stieltjes integral of the function f (t) with respect to the function g(t), t1 f (t) dg(t), t0

is also defined, and its value coincides with that of the Lebesgue–Stieltjes integral of [26, p. 41]. We can integrate the Lebesgue–Stieltjes integral by parts from [26, Problem 224]: t1

t1 f (t) dg(t) = f (t1 )g(t1 ) − f (t0 )g(t0 ) −

t0

g(t) df (t). t0

Thus, the integration-by-parts formula also holds for the Lebesgue–Stieltjes integral   f (t)dϑ[g] = f (t1 )g(t1 ) − f (t0 )g(t0 ) − g(t)dϑ[f ] T

T

whenever the functions f and g have no common discontinuity points on the closed interval [t0 , t1 ]. Let fi ∈ V (T ): fi (t) → f (t) ∀t ∈ Cont(ϑ[f ]). Then fi → f a.e. with respect to the measure |ϑ[g]|. Indeed, the set Ds(ϑ[f ]) is of zero |ϑ[g]|-measure, since the discrete supports of the Borel charges ϑ[f ] and ϑ[g] 2

Indeed, f is represented as the sum of two monotonic functions, which, in turn, are Borel by definition.

7094

are disjoint. If sup |fi (t)| ≤ c ∀i, then by the Lebesgue theorem [26, p. 43], we pass to the limit under the t∈T

integral sign:



 fi (t) dϑ[g] → T

 Let x(t) = [t0 ,t]

f (t) dϑ[g]. T

f (s) dμ, t > t0 , x(t0 ) = 0, where f ∈ V (T ), and μ ∈ C ∗ (T ). In this case, we say that

the function x is generated by the charge μ. Then x ∈ V (T ), and the charge ϑ[x] is absolutely continuous with respect to the measure |μ|. Indeed, the following inequality holds for the function x: sup |x(s) − x(a)| ≤ Var |ba [x] ≤ sup |f (s)| × |μ|([a, b]) ∀a, b ∈ T,

s∈[a,b]

a ≤ b,

(7)

s∈[a,b]

which follows from the properties of the Riemann–Stieltjes integral [27, p. 356]. ∗ (T ) if Var |b [x] ≤ We say that the variation of a function x ∈ V (T ) is majorized by a measure ν ∈ C+ a cν([a, b]) ∀a, b ∈ T , a ≤ b, or, in other words, |ϑ[x]| ≤ cν. For example, if the function x is generated by the charge μ, then the variation of x is majorized by |μ|. Let |ϑ[x]| ≤ cν. Then the following conditions hold: (a) the function x is continuous at each point of the set (t0 , t1 ) ∩ Cont(ν); (b) if g : R1 → R1 is a Lipschitzian function, then g(x) is also majorized by ν. w

∗ (T ), and let ν → ν. We say that a sequence of variations ˜(t) ∀t ∈ T , νi ∈ C+ Let xi ∈ V (T ), xi (t) → x i x] ≤ cν([a, b]) of functions xi is majorized by a sequence of measures {νi } if |ϑ[xi ]| ≤ cνi ∀i. Then Var |ba [˜ ˜. It is easy to see that ∀a ≤ b,3 and the continuity points of the function F (t; ν) are those of the function x there exists a function x ∈ V (T ) such that x(t) = x ˜(t) ∀t ∈ Cont(ν) and |ϑ[x]| ≤ cν, i.e., xi (t) → x(t) ∀t ∈ Cont(ν), where x is no longer an arbitrary function but an element of the space V (T ). If a function ϕ : T → R1 is nonnegative, continuous, and

t ϕ(t) ≤ c

ϕ(s)ds + b,

b ≥ 0,

t ∈ T,

t0

then the following Gronwall inequality holds [45, p. 450]: ϕ(t) ≤ bec . m a.a. t, |U | ≤ c, and q = (μ; {v }), where μ ∈ C ∗ (T ) and {v } is the Let u ∈ Lm r r ∞ (T ), u(t) ∈ U ⊆ R K set of functions associated with μ (for the definition, see Sec. 1). Consider the equation of the impulse control problem: (8) dx = f (x, u, t)dt + g(x, t)dq, t ∈ T, x(t0 ) = x0 ∈ Rn . To prove the assertions presented in Sec. 3, the requirements on the functions f and g presented in Sec. 1 are extra strong, and they can be weakened. Assume that in Eq. (8) the function f is Lipschitzian in x a.e t in each ball uniformly in (x, u, t), is measurable in t for any fixed (x, u), is continuous in (x, u) a.e. t, and for any bounded set B ⊂ Rn × U , the function sup |f (x, u, t)| is Lebesgue integrable in T . The (x,u)∈B

function g is Lipschitzian in (x, t) in every ball. Let x ∈ V n (T ). We set D(t) = Ds(|q|) ∩ [t0 , t] and define the function  [αr (1) − x(r− )], Φ(x, t) =

(9)

r∈D(t)

where α˙ r = g(αr , r)vr , s ∈ [0, 1], and αr (0) = x(r− ). 3

This is a consequence of the inequality Var |ba [˜ x] ≤ lim inf Var |ba [xi ]. i→∞

7095

Since x(t) is bounded, series (9) absolutely converges at each point of the closed interval T and is a bounded function by the requirement on g imposed above. The function x ∈ V n (T ) is called a solution of Eq. (8) corresponding to the triple (x0 , u, q) if t x(t) = x0 +

 f (x, u, s) ds +

g(x, s) dμc + Φ(x, t),

t ∈ (t0 , t1 ],

x(t0 ) = x0 .

[t0 ,t]

t0

In the next section, on the set of all impulse controls, we introduce a metric with respect to which correct passages to the limit in Eq. (8) will be performed. Here, not considering the question on the metric, we mention the following useful property, which follows from Lemma 3.2 proved in Sec. 3. Consider a sequence of impulse controls qi such that qi  ≤ const4 ; let ui → u a.e., x0,i → x0 , and xi (t) ∈ V n (T ) be a solution of the equation t ∈ T,

dxi = f (xi , ui , t)dt + g(xi , t)dqi ,

xi (t0 ) = x0,i .

Then if gc sup |xi (t)| ≤ const, it follows that there exists an impulse control q such that passing to a t∈T

subsequence, we have xi (t) → x(t) ∀t ∈ Cont(|q|), where x(t) is the trajectory corresponding to the triple (x0 , u, q). In this case, we cannot assert that sup |xi (t)| → sup |x(t)|, i → ∞, but the following convergence holds: t∈T

gc sup |xi (t)| → gc sup |x(t)|. t∈T

t∈T

t∈T

3.

Lemmas and Propositions

Lemma 3.1. Assume that there exists a weakly convergent sequence of charges μi ∈ C ∗ (T ) such that w (μi , |μi |) → (μ, ν) and a sequence of functions fi ∈ V (T ) such that the following conditions hold : w

∗ (T ), η → η, such that Ds(ν) ∩ (1) there exists a weakly convergent sequence of measures ηi ∈ C+ i Ds(η) = ∅ and |ϑ[fi ]| ≤ cηi ∀i; (2) fi (t) → f (t) ∀t ∈ Cont(η), where f ∈ V (T ).

Then



 fi (t) dμi →

Ai = T

f (t) dμ. T

Proof. Before proving the assertion of the lemma itself, we note the following two simple properties. w 1. Let μi ∈ C ∗ (T ), (μi , |μi |) → (μ, ν), ti → s ∈ T . Then max{F (s− ; μ), F (s+ ; μ)} − ν({s}) ≤ lim inf F (ti ; μi ) −

ti →s +

≤ lim sup F (ti ; μi ) ≤ min{F (s ; μ), F (s ; μ)} + ν({s}). ti →s

(10)

Indeed, it is known that Fi (t) = F (t; μi ) ∈ V0 (T ) and Fi (t) → F (t) = F (t; μ) ∀t ∈ Cont(ν). The latter property is easily deduced from [1, Lemma 7.1, p. 134] and the representation of the function F as the sum of two monotonic functions, one of which is its variation. Further, for definiteness, let F (s− ) ≤ F (s+ ). The sequence {Fi (ti )} is bounded. Extract a subsequence Fi (ti ) → a. Assume that a > F (s− ) + ν({s}). Fix ε > 0 so that s − ε ∈ Cont(ν). For sufficiently large i, we have ti > s − ε. Hence Fi (s − ε) − Fi (ti ) + |μi |([s − ε, ti ]) ≥ 0. Passing to the limit as i → ∞, we obtain F (s − ε) + ν([s − ε, s]) ≥ a. Now, letting ε tend to zero, we obtain the following contradiction: F (s− ) + ν({s}) ≥ a > F (s− ) + ν({s}). Other cases are considered analogously. 4

Here and in what follows, q = |q|(T ).

7096

w

2. Let μi ∈ C ∗ (T ), (μi , |μi |) → (μ, ν), ε ≥ 0, and let K be a closed subset of T such that ν({t}) ≤ ε ∀t ∈ K, i.e., K contains no atoms of measure ν whose value is greater than ε. Then lim sup sup |F (t; μi ) − F (t; μ)| ≤ ε. i→∞

(11)

t∈K

Indeed, construct a sequence of points ti such that 1 |Fi (ti ) − F (ti )| + ≥ sup |Fi (t) − F (t)|. i t∈K Using the compactness of the set K and passing to a subsequence, we have ti → s, where s ∈ K. Now (11) follows from (10). Let us directly pass to the proof of the lemma. Here, for simplicity, we assume that Ds(μi ) ∩ Ds(ηi ) = ∅. This assumption does not lead to a loss of generality. Indeed, by the condition Ds(ν) ∩ Ds(η) = ∅ and the weak convergence, we have ∀ε > 0 ∃ N = N (ε): |μi |(Ds(μi ) ∩ Ds(ηi )) ≤ ε ∀i ≥ N . Let us show that there exists a sequence of absolutely continuous charges μ ˆi such that w μi |) → (μ, ν); (1) (ˆ μi , |ˆ    ˆ ˆ   fi (t)dˆ μi . (2) Ai − Ai → 0 as i → ∞, where Ai = T

For each i, consider a sequence of absolutely continuous measures {μi,τ } weakly converging to μi : w

(μi,τ , |μi,τ |) → (μi , |μi |),

τ → ∞.

Let {φk } be a countable everywhere dense set of functions in C 2 (T ). For each i, choose a number τi such that          i        φk (t) d(μi,τ , |μi,τ |) − φk (t) d(μi , |μi |) + Ai − fi (t) dμi,τ  ≤ 1 . (12) i i i     i    k=1  T

T

T

Let us show that such a number τi always exists. Integrating by parts, for every (i, τ ), we have             Ai − fi (t) dμi,τ  ≤ fi (t1 )[F (t1 ; μi,τ ) − Fi (t1 )] + F (t; μi,τ ) − Fi (t) dϑ[fi ].     T

T

w

Since (μi,τ , μi,τ ) → (μi , |μi |), it follows that F (t; μi,τ ) → Fi (t) ∀t ∈ Cont(μi ) as τ → ∞. But, by the assumption made, the set Ds(μi ) is of zero |ϑ[fi ]|-measure. Then F (t; μi,τ ) → Fi (t) |ϑ[fi ]|-a.e. Now, using the Lebesgue theorem, it is easy to find a number τi such that (12) holds.   w μi , |ˆ μi |) → (μ, ν) and Aˆi − Ai  → 0 as i → ∞. We set μ ˆi = μi,τi . Then it follows from (12) that (ˆ From the triangle inequality, we have                       fi (t) dˆ   μi − f (t) dμ ≤  fi (t) d(ˆ μi − μ) +  [fi (t) − f (t)] dμ . (13)        T

T

T

T

The second summand on the right-hand side tends to zero by the Lebesgue theorem. Indeed, fi (t) → f (t) μ-a.e. Moreover, the fi are uniformly bounded, since their variations are majorized by a weakly convergent sequence of measures.  fi (t)d(ˆ μi − μ) → 0. We set Fˆi (t) = F (t; μ ˆi ). Integrating by parts, we Thus, it remains to prove that deduce that



T



fi (t) d(ˆ μi − μ) = fi (t1 )[Fˆi (t1 ) − F (t1 )] − T

[Fˆi (t) − F (t)] dϑ[fi ]. T

The weak convergence implies Fˆi (t1 ) → F (t1 ). Let us prove that

 T

[Fˆi (t) − F (t)]dϑ[fi ] → 0. 7097

We have the following two possibilities: Ds(ν) = ∅ and Ds(ν) = ∅. Let Ds(ν) = ∅. Then the measure μ is continuous, and the consequence of inequality (11) with ε = 0 and K = T completes the proof. Let Ds(ν) = ∅. Fix a sufficiently small ε > 0. There exists a finite nonempty set of points sj ∈ Ds(ν), j = 1, . . . , N , N = N (ε) ≥ 1, such that ν({sj }) ≥ ε. Since Ds(ν) ∩ Ds(η) = ∅, it follows that ∀j ≤ N ∃δj > 0 : η(Cj ) ≤ εN −1 , where Cj = [sj − δj , sj + δj ], Cj are pairwise disjoint and sj ± δj ∈ Cont(η).  Let Oj be a neighborhood of the point sj such that Oj ⊂ Cj . Then O = N j=1 Oj is an open set. Hence N K = T \ O is closed. Moreover, O ⊂ C, where C = j=1 Cj and η(O) ≤ η(C) ≤ ε. This implies              [Fˆi (t) − F (t)] dϑ[fi ] ≤ Fˆi (t) − F (t) d|ϑ[fi ]| + Fˆi (t) − F (t) d|ϑ[fi ]|.     T

K

C

Using (11) and the condition |ϑ[fi ]| ≤ cηi ∀i, we arrive at the estimate           ˆ  lim sup  [Fi (t) − F (t)] dϑ[fi ] ≤ lim sup supFˆi (t) − F (t) |ϑ[fi ]|(K) i→∞  i→∞ t∈K  T     + lim sup supFˆi (t) − F (t) |ϑ[fi ]|(C) ≤ εcη + 2εcν ≤ c1 ε. i→∞

t∈C

But ε > 0 is arbitrary, and hence the right-hand side of (13) tends to zero. The lemma is proved. Remark. Under the conditions of Lemma 3.1, we cannot omit the requirement that the variations of the functions fi be majorized by a weakly convergent sequence of measures, say, replacing it by a weaker condition of the uniform boundedness with respect to the variation. Example 3.1. Let T = [0, 1], and let μi = δ(ti ) be a sequence of the Dirac measures supported at the 1 points ti = . Consider the following sequence of functions {fi } (small humps): i ⎧ ⎪ t ∈ [0, i−1 ], ⎨it, fi (t) = 2 − it, t ∈ [i−1 , 2i−1 ], ⎪ ⎩ 0, t ∈ [2i−1 , 1].   w fi (t)dμi = 1 ∀i, whereas f (t)dμ = 0. Clearly, μi → μ = δ(0) and fi (t) → 0 ∀t ∈ [0, 1]. However, [0,1]

[0,1]

The fact is that, for the sequences {μi } and {fi } constructed in the example, there is no sequence of w measures {ηi } satisfying the conditions of Lemma 3.1. Indeed, |ϑ[fi ]| → η = 2δ(0), but Ds(μ) ∩ Ds(η) = {0} = ∅. Proposition 3.1. Let all the conditions of Lemma 3.1 hold. We set   fi (s) dμi , x(t) = f (s) dμ, t ∈ (t0 , t1 ], xi (t0 ) = x(t0 ) = 0, xi (t) = [t0 ,t]

i ∈ N,

[t0 ,t]

and consider a convergent sequence of points {ti }, ti → s ∈ T . If s ∈ / Ds(ν), then xi (ti ) → x(s). Proof. Fix ε > 0 such that s ± ε ∈ Cont(ν) ∩ Cont(η) and consider the difference  fi (t) dμi = Ai,ε . xi (s + ε) − xi (ti ) = [ti ,s+ε]

It follows from (7) that |Ai,ε | ≤ c|μi |([ti , s + ε]) ∀i. Passing to the limit, we have −cν([s, s + ε]) ≤ lim inf [xi (s + ε) − xi (ti )] ≤ lim sup[xi (s + ε) − xi (ti )] ≤ cν([s, s + ε]). i→∞

7098

i→∞

By Lemma 3.1, xi (s + ε) → x(s + ε) as i → ∞. Therefore, x(s + ε) − cν([s, s + ε]) ≤ lim inf xi (ti ) ≤ lim sup xi (ti ) ≤ x(s + ε) + cν([s, s + ε]). i→∞

i→∞

Letting ε → 0, we complete the proof. Proposition 3.2. Consider a vector-function x(t) : x ∈ V n (T ), |ϑ[x]| ≤ cν, and a scalar function φ : Rn × R1 → R1 . If φ is Lipschitzian, then the variation of the superposition φ(x, ·) is majorized by the measure ν + L. Proof. By the definition of the variation of a function on a closed interval, N   b  φ(x(sk ), sk ) − φ(x(sk−1 ), sk−1 ). Var a [g(x, t)] = sup SN k=1

Here, the least upper bound is taken over all possible finite partitions of the closed interval [a, b]. Let us use the Lipschitzian property of φ and the property that |ϑ[x]| ≤ cν: N    φ(x1 (sk ), . . . , xn (sk ), sk ) − φ(x1 (sk−1 ), . . . , xn (sk−1 ), sk−1 ) k=1

⎛

≤ c1 ⎝

n 

⎞ b j Var  [x ] + b − a⎠ ≤ const[ν([a, b]) + L([a, b])]. a

j=1

The proposition is proved. Proposition 3.3. If, for a certain fixed triple (x0 , u, q), q = (μ, {vr }), there exists a solution of Eq. (8), then it is unique. Proof. Assume that there exist functions x, y ∈ V n (T ), x = y, and x and y satisfy (8). By the definition of solution,  t         g(x, s) − g(y, s) d|μc | + Φ(x, t) − Φ(y, t) ∀t ∈ T. x(t) − y(t) ≤ f (x, u, s) − f (y, u, s) ds + [t0 ,t]

t0

Using the Gronwall inequality, we arrive at the estimate         cx(s) − y(s)(d|μc | + ds) + c|q|({s})ec|q|({s}) x(s− ) − y(s− ) sup x(s) − y(s) ≤ s∈[t0 ,t]

s∈D(t)

[t0 ,t]

   ≤ c sup x(s) − y(s) × (|μc |([t0 , t]) + t − t0 ) + c1 |q|({s})As , s∈[t0 ,t]

s∈D(t)

   where As = x(s− )−y(s− ), s ∈ D(t), D(t) = Ds(|q|)∩[t0 , t], and c1 = cecq . Denote φ(t) = sup x(s)− s∈[t0 ,t]  y(s) and γ(t) = c(|μc |([t0 , t]) + t − t0 ). Enumerate the atoms of the measure |q| on [t0 , t] in decreasing order of their values. We obtain the sequence aj = |q|({sj }), sj ∈ D(t), j ∈ N. In this case, it can happen that the measure |q| has no or only finitely many atoms sj , j = 1, . . . , p, p ≥ 1, on the closed interval [t0 , t]. In the first case, we set aj = 0 ∀j, and in the second case, we set aj = 0 ∀j > p. By construction, ∞  aj ≤ q. Now, in the new notation, the obtained inequality aj+1 ≤ aj , aj → 0+, j → ∞, and j=1

becomes φ(t) ≤ φ(t)γ(t) + c1

∞ 

aj Asj .

(14)

j=1

7099

Our goal is to estimate the series

∞ 

aj Asj .

j=1

Fix ε > 0. Since Asj ≤ φ(t) ∀j, there exists a natural number N = N (ε) such that

∞ 

aj Asj ≤ εφ(t).

j=N +1

Let us estimate the partial sum SN =

N 

aj Asj . In this case, without loss of generality, we can assume

j=1

that sj < sj+1 , j = 1, . . . , N − 1.5 It follows from (14) that As1 ≤ φ(t)γ(t) + c1 εφ(t) = Q(t), As2 ≤ Q(t) + a1 As1 ≤ Q(t)(1 + a1 ), As3 ≤ Q(t) + a1 As1 + a2 As2 ≤ Q(t) + a1 Q(t) + a2 (1 + a1 )Q(t) = Q(t)(1 + a1 )(1 + a2 ), ........................ and so on. Finally, AsN ≤ Q(t)

N −1 

(1 + aj ).

j=1

Therefore, SN ≤ Q(t)

N  j=1

The product

∞ 

aj

j−1 

(1 + ai ).

i=1

(1 + aj ) converges, since the numerical series

j=1

p. 58]. Then SN ≤ Q(t)

log(1 + aj ) ≤

j=1 N  j=1

This implies

∞ 

∞ 

aj ≤ q converges [25,

j=1

aj eq ≤ Q(t)qeq . Now it follows from (14) that   φ(t) ≤ φ(t)γ(t) + c1 εφ(t) + Q(t)qeq .    φ(t) ≤ φ(t) γ(t) + εc1 1 + c1 qeq .

Letting ε → 0, we finally obtain   φ(t) ≤ c 1 + cqec(1+q) (|μc |([t0 , t]) + t − t0 )φ(t)

∀t ∈ T.

(15)

Now  follows from the continuity of the Borel measure |μc |. Indeed, there exists s1 > t0 ,  the uniqueness c(1+q) (|μc |([t0 , s1 ]) + s1 − t0 ) < 1. Then φ(s1 ) = 0, and on the closed interval [t0 , s1 ], the c 1 + cqe solutions x(t) and y(t) coincide. On the closed interval [s1 , t1 ], the conditions of the proposition hold. Applying the described procedure to the closed interval [s1 , t1 ], we also find s2 such that x(t) = y(t) ∀t ∈ [s1 , s2 ], and so on. It is easy to see that for finitely many steps, we can pass through the whole closed interval T . Then x(t) ≡ y(t). This is what was required to be proved. The proposition is proved. Proposition 3.4. Let x(t) be a solution of (8), and, moreover, let the functions f and g be Lipschitzian with constant c > 0 in x uniformly in (x, u, t) ∈ Rn × U × T . Then |x(t)| ≤ k1 |x0 | + k2 5

∀t ∈ T,

(16)

In other words, enumerating once more the finite set of numbers aj , sj , j = 1, . . . , N − 1, we obtain what was required. It is easy to see that such a new notation does not have an influence on the process of the proof.

7100

where k1 = bp1 , k2 =

p 

bj−1 1 b2 , p = [2(q + 1)b3 ] + 1,

j=1

  b1 = 2 1 + cqec(1+q) ,

  b2 = b1 c1 q 1 + ecq ,

b3 =

cb1 , 2

c1 = max |g(0, s)|. s∈T

    Proof. We set φ(t) = sup |x(s)| and γ(t) = c |μc | [t0 , t] +t−t0 . Arguing in the same way as in proving s∈[t0 ,t]

Proposition 3.3, we have φ(t) ≤ |x0 | + φ(t)γ(t) + c1 q + c1 qe

cq

cq

+ ce

∞ 

aj Asj ,

j=1

  where As = |x(s− )| and aj and sj were introduced above. We set c2 = |x0 | + c1 q 1 + ecq . Fix ∞ N   ε > 0; then ∃N = N (ε) : aj Asj ≤ εφ(t). The partial sum of the series satisfies SN = aj Asj ≤ j=N +1

q

j=1

Q(t)qe , where Q(t) = φ(t)γ(t) + c2 which is analogous to (15):

+ cecq εφ(t).

Letting ε → 0, we obtain the following inequality,

    φ(t) ≤ b3 φ(t) |μc | [t0 , t] + t − t0 + c3 ,

c3 =

b1 |x0 | + b2 c2 b1 = . 2 2

  Now choose s1 so that |μc | [t0 , s1 ] + s1 − t0 = (2b3 )−1 . If there is no such s1 , then we take s1 = t1 . Then |x(t)| ≤ b1 |x0 | + b2 ∀t ∈ [t0 , s1 ]. Applying the described procedure to the closed interval [s1 , t1 ], we estimate x(t) on the next closed interval [s1 , s2 ], and so on. The whole closed interval [t0 , t1 ] will be exhausted in p steps. Thus, there exist numbers k1 and k2 satisfying (16). The proposition is proved. ∗ (T ) such Proposition 3.5. Consider a sequence of absolutely continuous vector-valued measures μi ∈ CK w that μi → μ, a sequence of controls ui → u a.e., a sequence of vectors x0,i → x0 ∈ Rn , and the sequence of solutions xi ∈ V n (T ) corresponding to them, which satisfy



t xi (t) = x0,i +

f (xi , ui , s) ds + t0

g(xi , s) dμi ,

t ∈ T.

(17)

[t0 ,t]

Let |xi (t)| ≤ const. Denote by P({μi }) the set of limit trajectories corresponding to the approximation {μi }, i.e., the set of functions x(t) from V n (T ) such that, from {xi }, it is possible to choose a subsequence converging to x(t) a.a. t. Then P({μi }) = ∅, and for every trajectory x(t) from P({μi }), there exists a set {vr } associated with μ such that x(t) is a solution of (8) corresponding to the triple (x0 , u, q), where q = (μ; {vr }). Moreover, passing to a subsequence, we have xi (t) → x(t) ∀t ∈ Cont(|q|) and max |xi (t)| → gc sup |x(t)|. t∈T

t∈T

Proof. Since the μi weakly converge, it follows that μi  ≤ const. Passing to a subsequence, we have w ∗ (T ). Further, since the x are uniformly bounded, (7) implies that the sequence of |μi | → ν ∈ C+ i variations of xi is majorized by the weakly convergent sequence {|μi | + L}. Proposition 3.2 implies ˜(t) ∀t ∈ T (second Helly theorem). |ϑ[g(xi , t)]| ≤ c(|μi | + L). Passing to a subsequence, we have xi (t) → x ˜(t) ∀t ∈ Cont(ν). The function x(t) exists, since Let us find a function x ∈ V n (T ) such that x(t) = x Var |ba [˜ x] ≤ c(ν + L)([a, b]) ∀a ≤ b. Let us show that x(t) satisfies (8) for a certain set of functions 7101

associated with μ. Here, without loss of generality, we assume that |mi | > 0 a.e., where mi denotes the density of μi .6 Let us show that there exist a sequence of absolutely continuous measures {¯ μi } and a sequence of w w natural numbers ki ≥ i such that (1) (¯ μi , |¯ μi |) → (μd , νd ) and (2) |μki − μ ¯i | → νc as i → ∞. If Ds(ν) = ∅, then we set μ ¯i = 0, ki = i ∀i. Let Ds(ν) = ∅. Consider a chain of sets Di , i ∈ N, ordered with respect to inclusion such that  1 ν({r}) ≤ ; here, D0 = ∅. Define the sets Sr,i = [r − hi , r + hi ], r ∈ Di , Di−1 ⊆ Di ⊆ Ds(ν) and i r∈Ds(ν)\Di

as sets of closed pairwise disjoint neighborhoods of points r such that (1) hi > 0 and hi → 0 as i → ∞;    μ(Sr,i ) − μ({r}) ≤ 1 , where Si =  Sr,i ; ν({r}) + (2) ν(Si ) − i r∈Di r∈Di

r∈Di

(3) r ± hi ∈ Cont(ν). The existence of such a set Si follows from the regularity of μ and ν. Further, choose the number      |μk |(Sr,i ) − ν(Sr,i ) + μk (Sr,i ) − μ(Sr,i ) ≤ 1 . This is possible because of the ki ≥ i so that i i i r∈Di

weak convergence. We set μ ¯i (B) = μki (B ∩ Si ) for any Borel set B ⊂ R1 . It is easy to verify that w w (¯ μi , |¯ μi |) → (μd , νd ). Hence (μki − μ ¯i , |μki − μ ¯i |) → (μc , νc ). Using the constructed numbers ki , from the initial sequence of triples (xi , ui , μi ), we choose a subsequence denoted by the same index (i). Rewrite Eq. (17) in the form t xi (t) = x0,i +



 g(xi , s) d(μi − μ ¯i ) +

f (xi , ui , s) ds + [t0 ,t]

t0

We set = x0,i +

t ∈ T,

∀i.

[t0 ,t]



t xci (t)

g(xi , s) d¯ μi ,

g(xi , s) d(μi − μ ¯i );

f (xi , ui , s) ds + t0

[t0 ,t]



xdi (t) =

g(xi , s) d¯ μi ,

t ∈ T,

∀i.

[t0 ,t]

xci (t) 

Thus, xi (t) =  t f (x, u, s)ds + t0

+

[t0 ,t]

xdi (t)

∀t ∈ T . By the Lebesgue theorem and Lemma 3.1, xci (t) → xc (t) = x0 +

g(x, s)dμc ∀t ∈ T , i → ∞. We set xd = x − xc . Clearly, xdi (t) → xd (t) ∀t ∈ Cont(ν).

Let us show that xd = Φ(x, t) for a certain set {vr } associated with μ. Indeed, let t ∈ Cont(ν). Fix ε > 0. Choose a number N = N (ε) so that case,



ν({r}) ≤ ε. In this

r∈Ds(ν)\DN

          d g(xi , s)mi (s) ds − xi (t) ≤ const ε. lim sup    i→∞ r∈D(N,t)Sr,i 

6

(18)

Otherwise, we consider the sequence of measures μ ˜ i = μi + i−1 mLi , where m ∈ K, dLi = dt ∀t : |mi (t)| = 0; it is equal to 0 otherwise, and we transform (17) into the form t

t [f (xi , ui , s) − pi (s)] ds +

xi (t) = x0,i + t0

where pi (s) = i−1 g(xi , s)m (then pi (s) → 0 uniformly on T ).

7102

g(xi , s) d˜ μi , t0

Here and in what follows, D(N, t) = {r ∈ DN : r ≤ t} and the vector mi (s) is the density of the measure μi . Denote ri− = r − hi . For r ∈ D(N, t), on the closed interval Sr,i consider the equation xdi (s)

s

xdi (ri− )

=

+

s ∈ Sr,i .

(19)

r ∈ D(N, t).

(20)

g(xi , τ )mi (τ ) dτ, ri−

For sufficiently large i, define the functions7 πr,i (τ ) =

F (τ ; |μi |) − F (ri− ; |μi |) , |μi |(Sr,i )

The function πr,i transforms the closed interval Sr,i into the closed interval [0, 1] and is absolutely conk4  |mji (τ )| dπr,i j=1 = > 0. Therefore, there exists the inverse function tinuous and strictly increasing: dτ |μi |(Sr,i ) θr,i : [0, 1] → Sr,i , θr,i = (πr,i )−1 . Making the change of variables ω = πr,i (τ ) in (19) and transforming, we arrive at the equation αr,i (s) =

xdi (ri− )

s + xci (θr,i (s)) +

g(αr,i (ω), θr,i (ω))vr,i (ω)dω,

s ∈ [0, 1],

r ∈ D(N, t),

0 j (ω) = where αr,i (ω) = xi (θr,i (ω)), vr,i

mji (θr,i (ω))|μi |(Sr,i ) , j = 1, . . . , k4 . It is easy to see that θr,i (ω) → r k4  |mki (θr,i (ω))| k=1

uniformly on [0, 1], and, by Proposition 3.1, xci (θr,i (ω)) → xc (r) uniformly on [0, 1] as i → ∞. By w compactness arguments, passing to a subsequence, we have vr,i → vr weakly in Lk24 ([0, 1]) and αr,i → αr uniformly in C n ([0, 1]). Moreover, the functions vr are associated with μ. From the latter equation, as i → ∞, we obtain s − αr (s) = x(r ) + g(αr , r)vr dω, s ∈ [0, 1], r ∈ D(N, t). 0

xdi (ri− )

(Here, we use the property that → xd (r− ) by construction.) This and (18) imply        − xd (t) − [αr (1) − x(r )] ≤ const ε =⇒ xd (t) = Φ(x(t), t).    r∈D(N,t) It remains to verify that max |xi (t)| → gc sup |x(t)|. Indeed, the inequality t∈T

t∈T

lim inf max |xi (t)| ≥ gc sup |x(t)| i→∞

t∈T

t∈T

is obvious. The other inequality lim sup max |xi (t)| ≤ gc sup |x(t)| is deduced by using Proposition 3.1. The proposition is proved. 7

i→∞

t∈T

t∈T

The numbers i must be taken starting from a certain i0 so that t ∈ / Sr,i ∀i ≥ i0 , r ∈ D(N, t).

7103

∗ (T ) be the set of scalar-valued Borel measures 3.1. The set I(T ) and the metric ρw . Let VK (μ) ⊆ C+ w ∗ (T ) : (μ , |μ |) → ν such that ∃μi ∈ CK (μ, ν). Note that |μ| ∈ VK (μ) and ν ≥ |μ| ∀ν ∈ VK (μ).8 An impulse i i control is the triple q = (μ; ν; {vr }), where ν ∈ VK (μ) and vr ∈ HK (μ({r}); ν({r})). The scalar-valued measure ν is called the variation of the impulse control q and is denoted by |q|. The set of all impulse controls on the closed interval T is denoted by I(T ). Let q = (μ; ν; {vr }). For measures μ and ν, for each i, let us find the sets Di , Si , and Sr,i , r ∈ Di , and also the numbers hi and ri− as in Proposition 3.5.  1  vr (2hi )−1 (t − ri− ) if t ∈ Sr,i , r ∈ Di , and mA / Si . Let μA We set mA d,i (t) = d,i (t) = 0 if t ∈ d,i be 2hi w A A an absolutely continuous vector-valued measure with density mA d,i . Obviously, (μd,i , |μd,i |) → (μd , νd ). A For each i, consider the absolutely continuous vector-valued measure μA c,i with density mc,i such that w A A A A A A μA c,i (Si ) = 0, |mc,i | > 0 a.e. on T \ Si , and (μc,i , |μc,i |) → (μc , νc ), i → ∞. Take μi = μc,i + μd,i . The A obtained set of absolutely continuous vector-valued measures μA i = μi [q] is said to be approximating for the impulse control q. Let q ∈ I(T ) and let {μA i [q]} be an arbitrary approximation family for q. By definition, for each i, the A −1 |) strictly increases on T . We set πi (t) = F (t; |μA function F (t; |μA i i |)μi  , i ∈ N, t ∈ T . The function πi is absolutely continuous and strictly increases on the closed interval T , mapping it onto the closed interval [0, 1]. Moreover, π˙ i > 0 a.e. Hence there exists the inverse function θi : [0, 1] → T , which also strictly increases and is absolutely continuous [34, p. 256, Problem 13]. Let Fi (s) = F (θi (s); μA i ). It is easy to k 4 show that the functions Fi uniformly converge in C ([0, 1]) to a certain k4 -vector-valued function, which is denoted by F.9 It follows from the construction that F j (t) is Lipschitzian, F j (0) = 0, F j (1) = F (t1 , μj ),  k4    dF j  j   j = 1, . . . , k4 , and  dt (t) = q. The function F is, in essence, a deformed distribution function of j=1

the measure “expanded” and “contracted” on the parts of rapid and small growth of the function F j (t; μ), respectively. Moreover, with each atom r of the measure |q| on the closed interval [0, 1], we associate a certain closed interval [ar , br ] ⊆ [0, 1] on which the function F is equal to the function vr associated with μ at the point r with accuracy up to a certain contraction coefficient. Note that the length of the closed interval [ar , br ] is equal to |q|({r})q−1 , and the point, e.g., br , is exactly |q|([t0 , r])q−1 . Therefore, the numbers ar and br depend only on |q| and r and are independent of the set {vr }. Therefore, to each impulse control q = (μ; ν; {vr }), we put in correspondence the pair of vector-functions (F, F), where F (t) = F (t; μ ˜) is the distribution function of the vector-valued measure μ ˜ = (μ, ν) and the function F(t) = F(t; q) is constructed in the way shown above. It is easy to see that such a correspondence is one-to-one: in the pair (F, F), according to the function F , we find the vector-valued measure (μ, ν) and hence the closed intervals [ar , br ] mentioned above, whereas thefunction F defines the set {vr } associated with μ by its values on [ar , br ]. On the remaining set [0, 1] \ r [ar , br ], the function F is obtained from F (t; μ) by a discontinuous change of time; its values on this set are completely defined by the function F (t; μ ˜). Let q1 = (μ1 ; ν1 ; {vr,1 }) and q2 = (μ2 ; ν2 ; {vr,2 }) be two elements from I(T ). Define the distance between q1 and q2 by the formula t1 ρw (q1 , q2 ) = max |F1 (s) − F2 (s)| + s∈[0,1]

|F1 (t) − F2 (t)|ds + |˜ μ1 (T ) − μ ˜2 (T )|, t0

˜k ), μ ˜k = (μk , νk ), and Fk (t) = F(t; qk ), k = 1, 2. It is easily verified that the function where Fk (t) = F (t; μ ρw has all the properties of a metric and, therefore, Mw = (I(T ); ρw ) is a metric space. 8

For certain concrete cones K, we can say more about the set VK (μ). For example, if K is contained in one of the orthants, then VK (μ) = {|μ|}. If K is a half-space or the whole space, then VK (μ) = {ν : ν ≥ |μ|}. Finally, if K is acute, i.e., if ∃h ∈ Rk4 , |h| = 1, ∃α > 0: h, x ≥ α|x| ∀x ∈ K, then we can assert that ν ≤ α−1 |μ| ∀ν ∈ VK (μ). 9 It is easy to note that the limit function F is independent of the sequence approximating q and is uniquely defined.

7104

The following assertion holds. Statement. The space Mw is a complete metric space. More precisely, Mw is the completion of the set of absolutely continuous vector-valued measures (μ, |μ|) given on T in the metric ρw . The proof is performed in the standard way and directly follows from the definitions. The convergence of elements in the metric ρw will be denoted in the same way as the weak convergence w w of measures by the symbol →. Therefore, the notation qi → q means that ρw (qi , q) → 0. Each absolutely continuous vector-valued measure on μ can be considered as an element of I(T ) if we set ν = |μ|. In this w w connection, the notation μi → q, where μi are absolutely continuous, means that (μi , |μi |) → q. Let us mention some simple properties of the metric ρw : w • for any sequence approximating q, we have μA i [q] → q; w w • if qi = (μi ; νi ; {vr,i }) and qi → q = (μ; ν; {vr }), then (μi , νi ) → (μ, ν), i.e., the function ρw is a metrization of the weak convergence of vector-valued measures; • the set Bc = {q : q ≤ c} is compact in the metric ρw . The following property of the metric ρw is its main property. Supplement to Proposition 3.5. Assume that under the conditions of Proposition 3.5, the sequence of absolutely continuous vector-valued measures μi ρw -converges to a certain element q ∈ I(T ). Then the set P({μi }) consists of exactly one trajectory — the solutions of (8) corresponding to the impulse control q. Therefore, the ρw -convergence of absolutely continuous vector-valued measures implies the convergence of the corresponding trajectories at each point of the set Cont(|q|). This assertion directly follows from the proofs of Propositions 3.3 and 3.5 and the definition of the metric ρw . Theorem 3.1. If the functions f and g satisfy the conditions of Proposition 3.4, then the solution of Eq. (8) corresponding to the triple (x0 , u, q) exists and is unique. w

A A Proof. Consider an element q ∈ I(T ). Take a sequence μA i = μi [q] approximating q. Since μi → q, the assertion of the theorem follows from Propositions 3.3–3.5 (supplement) and also from the existence theorem for ordinary differential equations [20]. w

Lemma 3.2. Let qi → q, ui → u a.e., x0,i → x0 , and xi (t) be a solution of (8) corresponding to the triple (x0,i , ui , qi ). If gc sup |xi (t)| ≤ const, then xi (t) → x(t) ∀t ∈ Cont(|q|), where x(t) is a solution of t∈T

(8) with the data (x0 , u, q). Moreover, gc sup |xi (t)| → gc sup |x(t)|. t∈T

t∈T

Proof. Since the xi are uniformly bounded, (7) implies that the sequence of variations of xi is majorized by a weakly convergent sequence {|qi | + L}. Passing to a subsequence, we have xi (t) → x(t) ∀t ∈ Cont(|q|). Let us show that x(t) satisfies Eq. (8) with the data (x0 , u, q). For each number i, consider a set of absolutely continuous vector-valued measures {μA i,τ [qi ]} ap{tk }, k ∈ N, be a countable everywhere dense set of points in T such that proximating qi . Let  tk ∈ X = Cont(|q|) ∩ [ ∞ i=1 Cont(|qi |)]. Such an everywhere dense set exists, since T \ X is countable. Let xi,τ be a solution of the equation t xi,τ (t) = x0,i +

 f (xi,τ , ui , s)ds +

t0

g(xi,τ , s)dμA i,τ [qi ],

t ∈ T.

(21)

[t0 ,t]

For each i and for sufficiently large τ , this solution exists by the requirements for f and g and Propositions 3.3–3.5. Indeed, let c = gc sup |xi (t)| + 1. Let us “cut off” the functions f and g. We set   g(x, t), |x| ≤ c, f (x, u, t), |x| ≤ c, g˜(x, t) = f˜(x, u, t) = |x| > c, |x| > c, g(xp , t), f (xp , u, t), 7105

where xp is the projection of the vector x on the sphere |x| = c. Now, f˜ and g˜ satisfy the condition of Proposition 3.4, and hence, by Theorem 3.1, for each τ , there exists a solution xi,τ of Eq. (21) in which the functions f and g are replaced by f˜ and g˜. By Proposition 3.4, all solutions xi,τ are uniformly bounded in τ . Then by Propositions 3.3 and 3.5, for large τ , we have |xi,τ (t)| ≤ c, and, therefore, there exists a solution of (21). Using Proposition 3.5, for each i, we choose a number τi such that i    1   |xi,τi (tk ) − xi (tk )| + max |xi,τi (t)| − gc sup |xi (t)| + ρw (μA i,τi [qi ], qi ) ≤ . t∈T i t∈T k=1

w

We set μ ˆi = μA ˆi = xi,τi . By construction, μ ˆi → q. According to Proposition 3.5, there exists i,τi and x ˆi (t) → x ˆ(t) a solution x ˆ of Eq. (8) on the closed interval T corresponding to the triple (x0 , u, q) and x ∀t ∈ Cont(|q|). But, by construction, x ˆi (tk ) → x(tk ), i → ∞, k = 1, 2, . . . .. Then the inequality μi | + L) implies x ˆi (t) → x(t) ∀t ∈ Cont(|q|). Hence x ˆ = x, and x(t) is a solution of (8) |ϑ[ˆ xi ]| ≤ const(|ˆ corresponding to the triple (x0 , u, q). Now, the fulfillment of the lemma for the whole sequence xi follows from the uniqueness theorem (Proposition 3.3). 3.2. The operator Pr, the function , and the criterion of ρw -convergence in I(T ). Along with I(T ), we will consider the set of 2k4 -dimensional impulse controls ID (T ) whose elements will be denoted by qD . By definition, qD “is twice as long” as q ∈ I(T ), i.e., qD = (μD ; νD ; {vD,r }), where the measure μD is of dimension 2k4 . We set D(T ) = I(T ) × I(T ). Each element of the set D(T ) can be automatically considered as an element of the set ID (T ). Indeed, if (q1 , q2 ) ∈ D(T ), q1 = (μ1 ; ν1 ; {vr,1 }), and q2 = (μ2 ; ν2 ; {vr,2 }), then the element qD = ((μ1 , μ2 ); ν1 + ν2 ; {(vr,1 , vr,2 )}) belongs to ID (T ). Thus, we have the strict inclusion D(T ) ⊂ ID (T ). Let us introduce into consideration the projection operator Prk : ID (T ) → I(T ), k = 1, 2. For an ∗ (T ), of dimension 2k , we set absolutely continuous vector-valued measure μD = (μ1 , μ2 ), μ1 , μ2 ∈ CK 4 Prk (μD , |μD |) = (μk , |μk |), k = 1, 2, i.e., as the value of the operator, we take either the first or last k4 components. Further, the operator Prk is extended to the whole space ID (T ) by continuity as follows. Let qD ∈ ID (T ). Consider an arbitrary sequence of absolutely continuous vector-valued measures μD,i = (μ1,i , μ2,i ) w (of dimension 2k4 ) such that μD,i → qD . It is easy to prove that the sequence of absolutely continuous vector-valued measures Prk (μD,i , |μD,i |) = (μk,i , |μk,i |) converges in the metric ρw . Denote by qk its limit. It is easy to see that this limit is independent of the method for approximating the element qD by absolutely continuous measures. We set Prk (qD ) = qk . By construction, the operator Prk is ρw w w continuous, i.e., if qD,i → qD as i → ∞, then Prk (qD,i ) → Prk (qD ). There are two more properties of the operator Prk : (1) |qD | = | Pr1 (qD )| + | Pr2 (qD )|; (2) if qD ∈ D(T ) and qD = (q1 , q2 ), then Prk (qD ) = qk , k = 1, 2. Introduce into consideration the function  : ID (T ) → R+ . We set 

2

|y1 − y2 | d|qD | +

(qD ) = [t0 ,t1 ]

Here, qD ∈ ID (T ),

t1



d(y1 , y2 ) = EdqD , dzk = d| Prk (qD )|,

|z1 − z2 |2 dt + |z1 (t1 ) − z2 (t1 )|2 .

t0

yk (t0 ) = 0, zk (t0 ) = 0,

k = 1, 2,

where E is the identity matrix of order 2k4 × 2k4 and the notation (y1 , y2 ) means the 2k4 -dimensional vector whose first k4 components are the vector y1 and whose last k4 components are the vector y2 . The 7106

integration with respect to the measure |qD | is performed according to the rules given above. For example, if x(t) is the solution of (3) corresponding to the triple (p, u, q), then, by definition, 

 g0 (x, t) d|q| =

φ(q) = [t0 ,t1 ]

g0 (x, t) dνc + [t0 ,t1 ]

1  

g0 (αr , r)ν({r}) ds,

ν = |q|.

r∈Ds(ν) 0 w

Moreover, the functional φ(q) is ρw -continuous, i.e., if qi → q, then φ(qi ) → φ(q) (see the proofs of Proposition 3.5 and Lemma 3.2). Now we present one very convenient criterion of ρw -convergence, which will be used everywhere in w what follows. Consider a sequence of controls qD,i ∈ D(T ), qD,i = (q1,i , q2,i ). Assume that (1) q2,i → q; w (2) (qD,i ) → 0 as i → ∞. Then q1,i → q as i → ∞. Indeed, using the properties of the ρw -compactness w of impulse controls and passing to a subsequence, we have qD,i → qD . Applying Lemma 3.2, we have ˜ ) for a certain q, q (qi,D ) → (qD ) = 0. From the properties of the function , we deduce that qD = (˜ ˜ ∈ I(T ), and hence qD ∈ D(T ). Then the property of ρw -continuity of the operator Pr implies qD = q (q, q), which is what was required to be proved. Note that this criterion can be slightly generalized: if

w

Pr(qD,i ) → q k

and (qD,i ) → 0,

w

then qD,i → (q, q) ∈ D(T ).

w

Remark. If qi → q, then, in general, it is not true that (qD,i ) → 0, where qD,i = (qi , q) ∈ D(T ), i.e., the function  does not yield a necessary condition for the ρw -convergence. Indeed, for example, the Dirac measure cannot be approximated by absolutely continuous measures by using the function . Here, there arises an analogy with the strong convergence of measures; however, we note that, in general, the condition (qD,i ) → 0 does not imply the convergence of the measures |qi | in the norm of variation. For K = [0, +∞), the corresponding example is constructed as follows. We set mi (t) = 2 if t ∈ [si,k−1 , si,k ] k for even k, where si,k = t0 + (t1 − t0 ), k = 0, . . . , i, and mi (t) = 0 otherwise. Let μi be a measure with i w density mi . Then, obviously, μi → L, whence, taking into account that |mi | ≤ 2, we have (μi , L) → 0. But μi − L → t1 − t0 > 0, i.e., there is no strong convergence. Let us summarize the arguments of Secs. 3.1 and 3.2. On the set I(T ) of impulse controls, we have introduced the metric ρw . With respect to this metric, we carried out the correct passage to the limit in the equation with a vector-valued measure. Using the function , we have found a convenient criterion for the ρw -convergence. The set of absolutely continuous vector-valued measures (μ, |μ|) is ρw -everywhere dense in I(T ). With accuracy up to an isometry, an impulse control is a Cauchy sequence of absolutely continuous vector-valued measures in the ρw -metric.

4.

Reductions R1 and R2 and the v-Problem

In this section, we present two reductions, which will be used at various steps in proving the maximum principle (MP). The reduction R1 is formulated for the scalar case k4 = 1 and μ ≥ 0 and allows us to obtain transversality conditions in time for the problem in which the dependence of f and U on t is only measurable. The reduction R2 (v-problem) is formulated for an arbitrary cone K and is used for proving the nondegeneracy of the MP. Also, using it, we can deduce the transversality conditions in time, although it applies only under smoothness conditions.

7107

Reduction R1 . Let K = [0, +∞). Along with the initial problem (2)–(5), which will be denoted by (P ), we consider the following problem (P1 ): ⎧ ⎪ e0 (p) → min, ⎪ ⎪ ⎪ ⎪ ⎪ dx = f (x, u, t)dt + g(x, t)dμ, ⎪ ⎪ ⎪ ⎪ dαk = g(αk , θk )dυk , αk (t1−k ) = xk , ⎪ ⎪ ⎪ ⎨dθ = 0, θ = t , k = 0, 1, t ∈ [t , t ], 0 1 k k k (P1 ) : ⎪ (p) ≤ 0, e (p) = 0, e 2 ⎪ ⎪ 1 ⎪ ⎪ ⎪ ϕ(x, t) ≤ 0, ϕ(αk , θk ) ≤ 0, k = 0, 1, ⎪ ⎪ ⎪ ⎪ u(t) ∈ U (t) a.a. t, μ, υk ≥ 0, k = 0, 1, ⎪ ⎪ ⎪ ⎩p = (ξ , ξ , t , t ), x = x(t ), ξ = α (t ), k = 0, 1. 0 1 0 1 k k k k k Proposition 4.1. Problems (P ) and (P1 ) are equivalent. This means that for each admissible process (p, u, μ) of problem (P ), there exists an admissible process (˜ p, u ˜, μ ˜, υ0 , υ1 ) of problem (P1 ) such that e0 (p) = p), and vice versa. e0 (˜ Proof. Let (p, u, μ) be an admissible process of problem (P ). Let us construct  the admissible process (˜ p, u ˜, μ ˜, υ0 , υ1 ) of problem (P1 ) corresponding to it. We set u ˜ = u, μ ˜ = μ− μ({tk })δ(tk ), x ˜0 = x(t+ 0 ), k=0,1

−1 ˜ x ˜1 = x(t− p, u ˜, μ ˜, υ0 , υ1 ) is an 1 ), υk = μ({tk })(t1 − t0 ) L, and tk = tk , k = 0, 1. It is easy to see that (˜ p). admissible process of (P1 ) and e0 (p) = e0 (˜  ˜, μ = μ ˜+ υk δ(tk ), Conversely, let (˜ p, u ˜, μ ˜, υ0 , υ1 ) be an admissible process of (P1 ). Take u = u k=0,1

xk , tk , (−1)k+1 υk ), and tk = t˜k , k = 0, 1. The process (p, u, μ) is admissible for (P ), xk = αk (tk ) = ξ(˜ p). and e0 (p) = e0 (˜ The proposition is proved. The reduction R1 allows us to restrict ourselves to the consideration of the problem in which the control measure has no atoms at the endpoints of the interval of time. Reduction R2 . The reduction R2 is the reduction to the so-called v-problem [1]. Assume that Assumption (S) holds. Introduce into consideration the following problem (P2 ) (v-problem): ⎧ ⎪ e0 (p) → min, ⎪ ⎪ ⎪ ⎪ dx = (v + 1)f (x, u, χ)dt + g(x, χ)dq, t ∈ [t0 , t1 ], ⎪ ⎪ ⎪ ⎨dχ = (v + 1)dt, (P2 ) : ⎪ e1 (p) ≤ 0, e2 (p) = 0, ϕ(x, χ) ≤ 0, ⎪ ⎪ ⎪ ⎪ ⎪ p = (x0 , x1 , χ0 , χ1 ), u(t) ∈ U, |v(t)| ≤ 1/2 a.a. t, ⎪ ⎪ ⎩ q = (μ; {vr }), Range(μ) ⊂ K. Proposition 4.2. Problems (P ) and (P2 ) are equivalent. ˜1 , χ0 , χ1 , t˜0 , t˜1 , u ˜, v, μ ˜, {˜ vr }) be admissible in (P2 ). Since 3/2 ≥ χ˙ ≥ 1/2 > 0 a.e., Proof. Let a process (˜ x0 , x it follows that χ strictly increases on the closed interval [t˜0 , t˜1 ]. Hence there exists the inverse function π(t) = χ−1 (t), t ∈ [χ(t˜0 ), χ(t˜1 )], which also strictly increases and is absolutely continuous [34]. We set ˜0 , u(t) = u ˜(π(t)), F (t; μ) = F (π(t); μ ˜), and vr = v˜π(r) , r ∈ T . The function t0 = χ(t˜0 ), t1 = χ(t˜1 ), x0 = x u(t) is measurable, since χ(t) is absolutely continuous and, therefore, transforms measurable sets into measurable sets [34]. Let us show that x(t) = x ˜(π(t)). Indeed, making the change of the variable of 7108

integration τ = π(s), we have π(t)  x ˜(π(t)) = x ˜0 + (v + 1)f (˜ x, u ˜, χ) dτ +

g(˜ x, χ) d(˜ μ; {˜ vπ(r) })

[t˜0 ,π(t)]

t˜0



t = x0 +



g(x, s) d(μ; {vr }) = x(t),

f (x, u, s) ds +

t ∈ [t0 , t1 ].

[t0 ,t]

t0

p). This implies that (p, u, μ, {vr }) is an admissible process for (P ) and e0 (p) = e0 (˜ Conversely, let (p, u, q) be an admissible process for (P ). We set v = 0. Then the process (p, t0 , t1 , u, 0, q) is an admissible process for (P2 ). Moreover, the value of the function e0 (p) being minimized does not change. The proposition is proved. Under the smoothness assumption, the reduction R2 allows us to reduce the initial problem (P ) to an autonomous problem with free time. 5.

Simplest Problem

In this section, on a fixed closed interval of time T = [t0 , t1 ], we study the following problem without constraints: J(p, u, q) = e0 (p) → min, dx = f (x, u, t)dt + g(x, t)dq, u(t) ∈ U (t) ⊂ R p = (x0 , x1 ), Introduce into consideration the functions

m

a.a. t,

Range(μ) ⊂ K,

x0 = x(t0 ),

H(x, u, ψ, t) = f (x, u, t), ψ ,

q = (μ; {vr }),

(22)

x1 = x(t1 ).

Q(x, ψ, t) = ψ  g(x, t).

The function H is called the Pontryagin function. By definition, the vector-function Q is a row, its derivative in x is the matrix with columns gxj (x, t)ψ, j = 1, . . . , k4 , and its derivative in ψ coincides with the matrix g(x, t). Theorem 5.1. Let (p∗ , u∗ , q∗ ), q∗ = (μ∗ ; {vr∗ }), be a solution of problem (22). Then there exists a function ψ ∈ V n (T ) such that (1) the pair (x∗ , ψ) is a solution of the following Hamiltonian system with a vector-valued measure:  dx∗ = Hψ (t)dt + Qψ (t)dq∗ (23) dψ = −Hx (t)dt − Qx (t)dq∗ , t ∈ T ; (2) the following transversality conditions hold : ∂e0 ∗ ∂e0 ∗ (p ), ψ1 = − (p ); ∂x0 ∂x1 (3) the following maximum condition holds for ordinary controls: ψ0 =

max H(u, t) = H(t) a.a. t;

u∈U (t)

(4) the following maximum condition holds for the impulse controls: Q(t) ∈ K ◦

∀t ∈ T,

Qr (s) ∈ K ◦ ∀s ∈ [0, 1],    Q(t), dq∗ = 0.

∀r ∈ Ds(|q∗ |),

T

7109

Here and in what follows, we use the following notation: ψk = ψ(tk ), k = 0, 1, H(t, u) = H(x∗ (t), u, ψ(t), t), Q(t) = Q(x∗ (t), ψ(t), t),

H(t) = H(t, u∗ (t)),

Hx (t) = Hx (x∗ (t), u∗ (t), ψ(t), t), . . . ,

and so on. In other words, if the functions H and Q or their partial derivatives in some of the arguments x, ψ, and u are omitted, then we substitute x∗ (t), ψ(t), and u∗ (t) for them. In condition (4), the subscript r of the function Q means that we substitute the values αr∗ (s), σr (s), and r for its arguments, respectively. Here, the functions (αr∗ , σr ) and r ∈ Ds(|q∗ |) correspond to the solution of system (23) on the discontinuity at the point r. By definition, this means that the pair (αr∗ , σr ) satisfies the following autonomous Hamiltonian system: ⎧ ∗ ∗ ∗ ∗ ∗ − ⎪ ⎪ ⎨α˙ r = g(αr , r)vr , αr (0) = x (r ), k4  (24) ⎪ σ˙ = − gxj (αr∗ , r)σr vr∗j , σr (0) = ψ(r− ), s ∈ [0, 1], r ∈ Ds(|q∗ |). ⎪ ⎩ r j=1

Further, the cone K ◦ in Condition (4) is the cone normal to K at zero or its polar cone, which is defined as follows:   K ◦ = NK (0) = {y ∈ Rk4 : x, y ≤ 0 ∀x ∈ K}. Remark. On the discontinuity trajectory of the system, condition (4) can be written in the following more traditional form of the maximum condition:     max Qr (s), v = Qr (s), vr∗ (s) = 0 a.e. in s ∈ [0, 1], (25) v∈Sr

where Sr = K ∩ {v : |v| = γr } and γr = |q∗ |({r}).10 Proof. Let (p∗ , u∗ , q∗ ) be an optimal process. Without loss of generality, assume that e0 (p∗ ) = 0. Consider w ¯i → q∗ . Denote by m ¯ i the a sequence of absolutely continuous vector-valued measures {¯ μi } such that μ density of the measure μ ¯i . For sufficiently large i, on the closed interval T there exists a solution (denoted ¯i ), and |¯ xi (t)| ≤ const ∀i. We set p¯i = (x∗0 , x ¯i (t1 )). By by x ¯i ) corresponding to the triple (x∗0 , u∗ , μ ∗ ∗ xi (t)| → gc sup |x∗ (t)|. Then p¯i → p∗ . It follows from Lemma 3.2, x ¯i (t) → x (t) ∀t ∈ Cont(|q |) and max |¯ t∈T

t∈T

pi ) ≥ 0. Denote εi = e0 (¯ pi ). Then εi → 0+ as i → ∞. the optimality that e0 (¯ ¯ i (t)| + i}, c = gc sup |x∗ (t)| + 1, and for each i ∈ N, consider the We set Ki (t) = K ∩ {m : |m| ≤ |m t∈T

following ordinary optimal control problem: ⎧ t1 t1 ⎪ ⎪  2 ⎪ ⎪ ⎪ ¯ i |) dt + |z − z¯i |2 dt + |z1 − z¯1,i |2 → min, φi (p, u, m) = e0 (p) + y − y¯i  (|m| + |m ⎪ ⎪ ⎪ ⎪ ⎪ t0 t0 ⎪ ⎪ ⎪ ⎨x˙ = f (x, u, t) + g(x, t)m, ¯ i , y0 = y¯0,i = 0, ⎪y˙ = m, y¯˙ i = m ⎪ ⎪ ⎪ ˙ ¯ i |, z0 = z¯0,i = 0, ⎪ ⎪z˙ = |m|, z¯i = |m ⎪ ⎪ ⎪ u(t) ∈ U (t), m(t) ∈ Ki (t) a.a. t, ⎪ ⎪ ⎪ ⎪ ⎩max |x(t)| ≤ c, p = (x0 , x1 ).

(26) t ∈ T,

t∈T

10

Here and everywhere in what follows, we accept the following convention about the norms in the finite-dimensional space R . If x ∈ Rk , then the notation |x| means the sum of modules of coordinates, and |x|2 denotes the sum of their squares. k

7110

Let Xi be the set of triples a = (p, u, m) satisfying the conditions of problem (26). The set Xi is nonempty, pi , u∗ , m ¯ i ). On Xi , let us introduce the metric since it contains the triple a ¯i = (¯ t1 ρ(a1 , a2 ) = |p1 − p2 | +

t1 |u1 − u2 | dt +

t0

|m1 − m2 | dt. t0

It is easy to see that Xi is a complete metric space (since it is a closed subset of the Banach space k4 R2n × Lm 1 × L1 ). For any admissible process (p, u, m) of problem (26), we have φi (p, u, m) ≥ 0, and, moreover, φi (¯ ai ) = ε2i . Apply the Ekeland variational principle [14, p. 246]. For each i ∈ N, there exists a process ai = (pi , ui , mi ) such that ai ) = ε2i , (1) φi (ai ) ≤ φi (¯ ¯ i ) ≤ εi , (2) ρ(ai , a (3) the process (pi , ui , mi ) is a solution of the problem ⎧ ! " t1 ⎪ ⎪   ⎪ ⎪ ⎪ |u − ui | + |m − mi | dt → min, φi (p, u, m) + εi |p − pi | + ⎪ ⎪ ⎪ ⎪ ⎪ t0 ⎪ ⎪ ⎪ ⎨x˙ = f (x, u, t) + g(x, t)m, (27) ¯ i , y0 = y¯0,i = 0, y˙ = m, y¯˙ i = m ⎪ ⎪ ⎪ ⎪ ¯ i |, z0 = z¯0,i = 0, t ∈ T, z˙ = |m|, z¯˙i = |m ⎪ ⎪ ⎪ ⎪ ⎪ u(t) ∈ U (t), m(t) ∈ Ki (t) a.a. t, ⎪ ⎪ ⎪ ⎪ ⎩max |x(t)| ≤ c, p = (x0 , x1 ). t∈T

For large i, problem (27) is an ordinary optimal control problem without constraints and with a nondifferentiable cost functional εi |p − pi |. Apply the nonsmooth variant of the maximum principle proved in [14, p. 195] to problem (27). For each i ∈ N, there exist absolutely continuous functions ψi , ζi , and ςi and also vectors hk,i , k = 0, 1, |hk,i | → 0 as i → ∞, such that ∂Qi ∂Hi (t) − (t)mi , t ∈ T, ψ˙ i = − ∂x ∂x ∂e0 ∂e0 (pi ) + h0,i , ψi (t1 ) = − (pi ) + h1,i , ψi (t0 ) = ∂x0 ∂x1 ¯ i |), ζi (t1 ) = 0, ζ˙i = 2(yi − y¯i )(|mi | + |m ς˙i = 2(zi − z¯i ), ςi (t1 ) = −2(z1,i − z¯1,i ),   max Hi (u, t) − εi |u − ui (t)|) = Hi (t) a.a. t,

(28)

    m, Qi (t) + ζi (t) + |m| ςi (t) − |yi (t) − y¯i (t)|2 − εi |m − mi (t)| m∈Ki (t)     = mi (t), Qi (t) + ζi (t) + |mi (t)| ςi (t) − |yi (t) − y¯i (t)|2 a.a. t.

(29)

max



u∈U (t)

Here, the subscript i of the functions H and Q and their partial derivatives means that for the arguments x, u, and ψ, we substitute the values xi (t), ui (t), and ψi (t), respectively; the triple (xi , yi , zi ) is an optimal trajectory of problem (27). The measure with density mi will be denoted by μi . The conditions of the variational principle imply that pi → p∗ and ui → u∗ strongly in L1 , and also    |yi − y¯i |2 (|mi | + |m ¯ i |) + |zi − z¯i |2 dt + |z1,i − z¯1,i |2 → 0 as i → ∞. T w

Then it follows from the criterion of ρw -convergence that μi → q∗ (see Sec. 3.2). 7111

By the constraints of problem (27) and Proposition 3.4, we have |xi (t)| + |ψi (t)| ≤ const ∀i. By Lemma 3.2, ψi (t) → ψ(t) ∀t ∈ Cont(|q∗ |). Moreover, ψi (t0 ) → ψ0 , ψi (t1 ) → ψ1 . Conditions (1) and (2) of the theorem are proved. Passing to the limit in (28), we obtain condition (3). Let us prove condition (4). We have from Lemma 3.2 that Qi (t) → Q(t) ∀t ∈ Cont(|q∗ |) and |ζi | + |ςi | → 0 uniformly in C(T ). Consider the maximum condition (29). Let us use a consequence of the sharp penalty method [14, p. 55]. Then (29) implies Qi (t) + hi ∈ Ni (t) a.a. t, where Ni (t) is the normal cone to the set Ki (t) at the point mi (t) and the vector hi from Rk4 is such that ¯ i (t)| + i}. For the closed convex cone K, we have the |hi | → 0 as i → ∞. We set Ai = {t : |mi (t)| = |m inclusion NK (x) ⊆ NK (0) ∀x ∈ K. Hence Ni (t) ⊆ K ◦ ∀t ∈ T \ Ai . On the other hand, L(Ai ) → 0 as μi  → 2q∗  < ∞). Whence, in the limit, we have Q(t) ∈ K ◦ ∀t ∈ (t0 , t1 ) (since Qi i → ∞ (since μi  + ¯ converges to Q almost everywhere and the components of Q(t) are right-continuous functions on (t0 , t1 )). Substituting the value m = 0 in (29), we arrive at the inequality     mi (t), Qi (t) + ζi (t) + |mi (t)| ςi (t) − |yi (t) − y¯i (t)|2 ≥ −εi |mi (t)| a.a. t. Integrating and passing to the limit (using Lemma 3.2 and taking into account that mi L1 ≤ const), as i → ∞, we have    Q(t), dq∗ ≥ 0. (30) T

Relation (25) for every r ∈ Ds(|q∗ |) is obtained by passing to the limit from condition (29). In this case, (29) must be rewritten in the form of the integral maximum principle, and under the integral sign, we need to make the following change of variable (see Sec. 3.1): πi (t) =

F (t; |μi | + |¯ μi |) , μi  + ¯ μi 

t ∈ T,

(the function πi strictly increases, since the vector-valued measure μ ¯i can always be taken so that |m ¯ i (t)| > 0 a.a. t). The maximum condition (25) can also be easily obtained by directly applying the increment method using a needle variation of the associated control vr∗ , as is done in the case of the ordinary problem [1]. Now, in the standard way, from the properties of the Hamiltonian  system, (25), and the Dem’yanov  theorem [15, Theorem 3.1, p. 35],11 we deduce that max Qr (s), v ≡ const. Taking into account that v∈Sr

Q(t) ∈ K ◦ a.e., we deduce from this that Qr (s) ∈ K ◦ ∀s ∈ [0, 1] ∀r ∈ Ds(|q∗ |), whence Q(t) ∈ K ◦ ∀t ∈ T . Finally, combining the result obtained with (30), we arrive at condition (4). The theorem is proved. 6. Linear-Convex Problems. Existence Theorem for Solution. Maximum Principle for the Problem with Endpoint Constraints Let the function f0 : Rn × Rm × R1 → R1 satisfy the same conditions as the function f does. Consider the problem t1 (31) J(p, u) = e0 (p) + f0 (x, u, t) dt → min t0

under constraints (3)–(5). In this and the following two sections, we assume that the following additional assumptions LC hold: (a) for any fixed (x, t), the vector-function f is linear (affine) in the variable u and the function f0 is convex in u; 11

Or see, e.g., the arguments in [40].

7112

(b) for almost all t, the set U (t) is closed and convex. The problems for which assumptions LC hold are said to be linear-convex (LC problems). In addition, let us impose on problem (31) a restriction of the form |p| ≤ c,

gc sup |x(t)| ≤ c,

|U (t)| ≤ c

a.a. t,

q ≤ c,

(32)

where c is a certain positive number. We set Tc = [−c, c]. Theorem 6.1. If in the LC problem (31), (32) there exists an admissible process, then there also exists an optimal process. Proof. Let {(pi , ui , qi )} be a minimizing sequence for problem (31). Using constraints (32), the compactness of the ball in a finite-dimensional space, the weak sequential compactness of the ball in a Hilbert space, and the ρw -compactness of the set Bc = {q ∈ I(Tc ) : q ≤ c}, we pass to a subsequence and find a triple (p, u, q) such that (1) pi → p = (x0 , x1 , t0 , t1 ); w (2) ui → u weakly in Lm 2 ([t0 , t1 ]); w (3) qi → q ∈ I(Tc ). By Lemma 3.2,12 xi (t) → x(t) ∀t ∈ Cont(|q|) and gc sup ϕj (xi , t) → gc sup ϕj (x, t) as i → ∞ ∀j, where x(t) is the solution of (3) corresponding to the triple (p, u, q). Hence the pair (x, p) satisfies constraints (4) and (5). The Mazur theorem [24, p. 177] implies u(t) ∈ U (t) a.e., and the functional J(p, u) is lower semicontinuous in u, i.e., J(p, u) ≤ lim inf J(pi , ui ). i→∞

Thus, the process (p, u, q) is admissible for problem (31), and the functional attains the minimum value at it. In what follows, all linear-convex problems will be studied under the following additional assumption: k4 = 1, K = [0, +∞), i.e., for the case where the impulse control is a scalar-valued nonnegative Borel measure. Before directly passing to the study of LC problems, we make one necessary supplement to Theorem 5.1 and prove the transversality conditions in time. Proposition 6.1. Let K = [0, +∞), let (p∗ , u∗ , q∗ ) be a solution of problem (22) with nonfixed time, and let p∗ = (x∗0 , x∗1 , t∗0 , t∗1 ). Then, along with the conditions of Theorem 5.1 (which hold on the closed interval T ∗ = [t∗0 , t∗1 ]), we have the following transversality conditions in time: ⎧ # 1 $ ⎪   ⎪ ∂e 0 ∗ ⎪ ⎪ gt (αk∗ , t∗k ), σk ds − (p ) ≥ 0, ess lim sup max H(βk∗ , u, γk , t) + (−1)k+1 Δ∗k ⎪ ⎪ ∂tk ⎨ t→t∗k u∈U (t) 0 k = 0, 1, (33) # 1 $ ⎪ ⎪   ∂e ⎪ 0 ∗ ⎪ ⎪ max H(βk∗ , u, γk , t) + (−1)k+1 Δ∗k (p ) ≤ 0, gt (αk∗ , t∗k ), σk ds − ess lim inf ⎪ ⎩ t→t∗k u∈U (t) ∂tk 0

max H(x∗k , u, ψk , t) + (−1)k ess lim inf ∗ t→tk

u∈U (t)

∂e0 ∗ (p ) ≤ 0, ∂tk

k = 0, 1.

(34)

Here and in what follows, we use the following notation: Δ∗k = μ∗ ({t∗k }), ψk = ψ(t∗k ), k = 0, 1, β0∗ = x∗ (t∗+ 0 ),

β1∗ = x∗ (t∗− 1 ),

γ0 = ψ(t∗+ 0 ),

γ1 = ψ(t∗− 1 ).

The functions (αk∗ , σk ), k = 0, 1, in condition (33) satisfy (24) for r = t∗k . 12

Obviously, Lemma 3.2 holds iff the controls weakly converge and f is linear in u (the proof is analogous).

7113

Proof. We will prove the transversality conditions in time using [1]. Let us prove (33). First, assume that the measure μ∗ has no atoms at the points t∗k , i.e., Δ∗k = 0, k = 0, 1. We will prove (33) for k = 1. Fix ε > 0 so that t∗1 − ε ∈ Cont(μ∗ ); let Tε = [t∗1 − ε, t∗1 ]. On the closed interval [t∗0 , t∗1 − ε], we define the measure με := μ∗ + μ∗ (Tε )δ(t∗1 − ε). We set pε = (x∗0 , xε (t∗1 − ε), t∗0 , t∗1 − ε), where xε is the trajectory corresponding to the triple (x∗0 , u∗ , με ) (it exists for sufficiently small ε). The optimality of the process (p∗ , u∗ , μ∗ ) implies & % ∂e0 ∗ ∂e0 ∗ ∗ e0 (pε ) − e0 (p ) ≥ 0 ∀ε > 0 ⇒ (p ), Δxε − ε (p ) + o(|Δxε |) + o(ε) ≥ 0, (35) ∂x1 ∂t1 where Δxε = xε (t∗1 − ε) − x∗1 . We set ξε∗ = ξ(x∗ (t∗1 − ε), t∗1 − ε, μ∗ (Tε )); let Δxε = Δx1ε + Δx2ε , where     1 ∗ ∗ 2 ∗ ∗ ∗ ∗ ∗ Δxε = ξε − x (t1 − ε) + g(x , t) dμ . Δxε = − f (x , u , t) dt, Tε

Tε

The estimate |Δx1ε | ≤ const ε is obvious. Let us show whenever μ∗ (Tε ) = 0, we assume that μ∗ (Tε ) > 0 ∀ε.

that

|Δx2ε |

≤ const εμ∗ (Tε ). Since |Δx2ε | = 0

Let us construct a sequence of absolutely continuous measures {μi } with densities mi > 0 a.e. defined w on the closed interval Tε such that μi → μ∗ on Tε and μi (Tε ) = μ∗ (Tε ). Let xi be a continuation of the solution x∗ to the closed interval Tε corresponding to the measure μi (such a continuation exists for sufficiently large i). By Lemma 3.2, xi (t) → x∗ (t) ∀t ∈ Cont(μ∗ ). Then Δxε,i = xε (t∗1 − ε) − xi (t∗1 ) → Δxε , Δx1ε,i → Δx1ε , and Δx2ε,i → Δx2ε as i → ∞, where     Δx2ε,i = ξε∗ − x∗ (t∗1 − ε) + g(xi , t)mi dt . Δx1ε,i = − f (xi , u∗ , t) dt, Tε

To estimate

|Δx2ε,i |, 

Here, x1,i (t) =

Tε

consider the equation x˙ 2,i = g(x1,i + x2,i , t)mi ,

t t∗1 −ε

t ∈ Tε ,

x2,i (t∗1 − ε) = x∗ (t∗1 − ε).

(36)

f (xi , u∗ , τ )dτ and xi = x1,i + x2,i .

Consider the functions

F (t; μi ) − F (t∗1 − ε; μi ) , t ∈ Tε . μ∗ (Tε ) dπi The function πi : Tε → [0, 1] is absolutely continuous and > 0. Therefore, there exists the inverse dt −1 function πi : [0, 1] → Tε , which is also absolutely continuous and strictly increases. Making the change of variable s = πi (t) in (36) and setting αi (s) = x2,i (πi−1 (s)), we arrive at the equation πi (t) =

α˙ i = g(x1,i (πi−1 (s)) + αi , πi−1 (s))μ∗ (Tε ),

s ∈ [0, 1],

αi (0) = x∗ (t∗1 − ε).

By definition, ξε∗ = α(1), where α(s) satisfies α˙ = g(α, t∗1 − ε)μ∗ (Tε ), s ∈ [0, 1], α(0) = x∗ (t∗1 − ε).   s   ∗ ∗ c |αi (τ ) − α(τ )| + |f (xi , u , θ)| dθ + ε dτ ∀s. This implies |αi (s) − α(s)| ≤ μ (Tε ) 0

Tε

By the Gronwall inequality, |Δx2ε,i | = |αi (1) − α(1)| ≤ const εμ∗ (Tε ), and, therefore, |Δx2ε | ≤ const εμ∗ (Tε ) = o(ε). The latter holds, since μ∗ (Tε ) → 0 as ε → 0. 7114

By the obtained estimates and also by the proved transversality conditions (condition (2) of Theorem 5.1), inequality (35) becomes    ∂e0 ∗ ∗ ∗ (p ) + o(ε) ≥ 0. ψ1 , f (x , u , t) dt − ε ∂t1 Tε

From this, we deduce

∗

t1

max H(u, ψ1 , t)dt − ε

t∗1 −ε

u∈U (t)

∂e0 ∗ (p ) + o(ε) ≥ 0, ∂t1

and, dividing the inequality by ε > 0, as ε → 0 we obtain ess lim sup max H(x∗1 , u, ψ1 , t) − t→t∗1

u∈U (t)

∂e0 ∗ (p ) ≥ 0, ∂t1

which corresponds to the first inequality in (33). Let us obtain the second inequality. We set μ∗ ([t∗1 , +∞)) = 0. Extend the control u∗ to the right beyond the point t∗1 in an admissible way so that u∗ (t) ∈ Argmax H(x∗1 , u, ψ1 , t), u∈U (t)

t > t∗1 .

Let us continue the solution of (3) corresponding to the constructed u∗ to the closed interval [t∗1 , t∗1 + δ] for a sufficiently small δ > 0. We set pε = (x∗0 , x∗ (t∗1 + ε), t∗0 , t∗1 + ε), where ε ∈ (0; δ]. Now, arguing as above, but not to the left of the point t∗1 but to the right of it, we obtain condition (33) of the theorem for k = 1. The case k = 0 is considered analogously. Now let us show that conditions (33) hold for Δ∗k > 0, k = 0, 1. For this purpose, we use the reduction R1 (see Sec. 4) of the initial problem (P ) and consider the reduced problem (P1 ).13 If the process (p∗ , u∗ , μ∗ ) is a solution of (P ), then the process (β0∗ , β1∗ , t∗0 , t∗1 , u∗ , Δ∗0 , Δ∗1 , μ ˜) is a solution 1  ˜ = μ∗ − Δ∗k δ(t∗k ), and β0∗ and β1∗ are introduced above. The measure μ ˜ has no atoms of (P1 ). Here, μ k=0

at the points t∗k , k = 0, 1, and the MP for such a problem was just obtained. Applying the MP to problem (P1 ), we conclude that there exists a function ψ R ∈ V n (T ∗ ) such that x, ψ R , t)dt − Qx (˜ x, ψ R , t)d˜ μ, dψ R = −Hx (˜ ⎧ R  ∗ ∗ ∗ ∂e0 ∗ ⎪ ⎪ ⎨ψ0 = ξx (β0 , t0 , −Δ0 ) ∂x (p ),

t ∈ T ∗,

0

⎪ ⎪ ⎩ψ1R = −ξx (β1∗ , t∗1 , Δ∗1 ) ∂e0 (p∗ ), ∂x1 % & ∂e 0 ∗ ∗ k+1 ∗ ∗ ξΔ (βk , tk , (−1) Δk ), (p ) = 0, k = 0, 1, ∂xk

(37)

(38)

x(t), u, ψ R (t), t) = H(˜ x(t), u∗ (t), ψ R (t), t) a.e., max H(˜

u∈U (t)

Q(˜ x(t), ψ R (t), t) ≤ 0,

t ∈ T ∗,

13

Here, (P ) and (P1 ) are the problems considered in Sec. 4 but without state constraints. More precisely, if problem (P ) has no constraints, then (P1 ) is deduced as follows (equivalent definition): ⎧ ⎪ ⎨e0 (p) → min, (P1 ) : dx = f (x, u, t)dt + g(x, t)dμ, dΔk = 0, Δk ≥ 0, k = 0, 1, ⎪ ⎩ p = (ξ0 , ξ1 , t0 , t1 ), ξk = ξ(xk , tk , (−1)k+1 Δk ).

7115

Q(˜ x(t), ψ R (t), t) = 0 ∀t ∈ supp(˜ μ), ⎧

% & ∂e ∂e ⎪ 0 0 ∗ R k ∗ ∗ ∗ k+1 ∗ ∗ ⎪ max H(βk , u, ψk , t) + (−1) (p ) + ξt (βk , tk , (−1) Δk ), (p ) ≥ 0, ⎪ ⎨ess lim sup ∂tk ∂xk t→t∗k u∈U (t) k = 0, 1.

% & ⎪ ∂e0 ∗ ∂e0 ∗ ⎪ ∗ R k ∗ ∗ k+1 ∗ ⎪ max H(βk , u, ψk , t) + (−1) (p ) + ξt (βk , tk , (−1) Δk ), (p ) ≤ 0, ⎩ess lim inf t→t∗k u∈U (t) ∂tk ∂xk Here, x ˜ is an optimal trajectory in problem (P1 ), the matrix function ξx and the vector-functions ξΔ and ξt are the derivatives of ξ in x, Δ, and t, respectively. ˜(t∗k ) = βk∗ , k = 0, 1. Take ψ(t) = ψ R (t) ∀t ∈ (t∗0 , t∗1 ), ψ(t∗k ) = Clearly, x ˜(t) = x∗ (t) ∀t ∈ (t∗0 , t∗1 ) and x ∂e0 ∗ (−1)k (p ), k = 0, 1, and show that ψ satisfies conditions (1)–(4) of Theorem 5.1 and condition (33). ∂xk Obviously, it suffices to verify them only at two points t∗k , k = 0, 1. Let k = 1 (the case k = 0 is analogous), and let (α1∗ , σ1 ) be a solution of (24) for r = t∗1 . It is known that the functions (ξx , ξΔ , ξt ) are defined as solutions of the following systems of ordinary differential equations: (ξx , ξΔ , ξt )(β1∗ , t∗1 , Δ∗1 ) = (ξ¯x , ξ¯Δ , ξ¯t )(1), where (ξ¯x , ξ¯Δ , ξ¯t )(0) = (E, 0, 0) (E is the n-dimensional identity matrix), and for s ∈ [0, 1], ⎧ ∗ ∗ ¯ ∗ ¯˙ ⎪ ⎨ξx (s) = gx (α1 (s), t1 )ξx (s)Δ1 , ξ¯˙Δ (s) = gx (α1∗ (s), t∗1 )ξ¯Δ (s)Δ∗1 + g(α1∗ (s), t∗1 ), ⎪ ⎩ ¯˙ ξt (s) = gx (α1∗ (s), t∗1 )ξ¯t (s)Δ∗1 + gt (α1∗ (s), t∗1 )Δ∗1 . We conclude from (37) that σ1 (1) = ψ1 . Therefore, conditions (1)–(3) are proved. To prove condition (4), let us show that σ1 , g(α1∗ , t∗1 ) ≡ 0. d Indeed, taking into account that σ1 , g(α1 , t∗1 ) = 0 ∀s ∈ [0, 1], by a direct calculation we obtain ds d σ1 , ξ¯Δ  = σ1 , g(α1∗ , t∗1 ) = const ∀s ∈ [0, 1]. ds On the other hand, by the definition of ξΔ , we find from (38) that σ1 (0), ξ¯Δ (0) = σ1 (1), ξ¯Δ (1) = 0. Hence σ1 , ξ¯Δ  ≡ 0; then σ1 , g(α1∗ , t∗1 ) ≡ 0 on [0, 1], i.e., condition (4) is proved. Condition (33) follows from the relation d σ1 , ξ¯t  = σ1 , gt (α1∗ , t∗1 )Δ∗1 . ds Inequalities (34) are deduced in exactly the same way as their analog in (33). The proposition is proved. Remark. In deducing inequality (34), the condition K = [0, +∞) was not used, and, therefore, the transversality conditions in time in the form (34) are proved for the case of an arbitrary cone K. In the remaining part of this section, we will study the LC problem (2)–(4), i.e., the problem with endpoint but without state constraints. Introduce the notation 2    ej (p), λj , l(p, λ) = j=0

The function l is called the small Lagrangian. 7116

λ = (λ0 , λ1 , λ2 ).

Theorem 6.2. Let (p∗ , u∗ , μ∗ ) be an optimal process in the LC problem (2)–(4). Then there exist a number λ0 ≥ 0, vectors λj ∈ Rkj , j = 1, 2, and a function ψ ∈ V n (T ∗ ) such that the following conditions hold : (39) dψ = −Hx (t)dt − Qx (t)dμ∗ , t ∈ T ∗ ; ∂l ∗ ∂l ∗ (p , λ), ψ1 = − (p , λ); (40) ψ0 = ∂x0 ∂x1   λ1 ≥ 0, λ1 , e1 (p∗ ) = 0; (41) max H(u, t) = H(t) a.a. t;

(42)

u∈U (t)

Q(t) ≤ 0 ∀t, Q(t) = 0 ∀t ∈ supp(μ∗ ); ⎧

1  ⎪   ⎪ ∂l ∗ ⎪ ∗ k+1 ∗ ∗ ∗ ⎪ gt (αk , tk ), σk ds − − Δk (p , λ) ≥ 0, ess lim sup max H(βk , u, γk , t) + (−1) ⎪ ⎪ ∂tk ⎨ t→t∗k u∈U (t) 0

1  ⎪ ⎪   ∂l ∗ ⎪ ∗ k+1 ∗ ∗ ∗ ⎪ ⎪ gt (αk , tk ), σk ds − − Δk (p , λ) ≤ 0, inf max H(βk , u, γk , t) + (−1) ⎪ess lim ⎩ t→t∗ u∈U (t) ∂t k

(43)

k = 0, 1,

k

0

(44) (45)

|λ| = 1. Moreover, inequalities (44) can be replaced by max H(x∗k , u, ψk , t) + (−1)k ess lim inf ∗ t→tk

u∈U (t)

∂l ∗ (p , λ) ≤ 0, ∂tk

k = 0, 1.

(46)

However, we cannot guarantee that both conditions (44) and (46) hold simultaneously, i.e., we have two theorems, two different versions of the MP. Proof. We perform the proof using the penalty method [1]. Let a process (p∗ , u∗ , μ∗ ) be a solution of the LC problem (2)–(4). First, we assume that Δ∗k = 0, k = 0, 1. Let us apply the penalty method. For this purpose, take an arbitrary natural i and define the function e0,i by the formula   2 2 e0,i (p) = e0 (p) + |p − p∗ |2 + i |e+ 1 (p)| + |e2 (p)| . Here, for an arbitrary vector a = (a1 , . . . , ar ), by a+ we denote the vector with coordinates (aj )+ = max(0, aj ), j = 1, . . . , r. Let F (t) = F (t; μ∗ ) be the distribution function of the optimal measure. Take c = gc sup |x∗ (t)| + μ∗  + |p∗ | + 1 and an arbitrary ε > 0. For each i ∈ N define the penalty problem ⎧ t1 ⎪ ⎪   ⎪ ∗ 2 ⎪ |u − u∗ (t)|2 + |y − F (t)|2 dt → min, ⎪ ⎨Ji (p, u, μ) = e0,i (p) + |y1 − F (t1 )| + ε t0

⎪ ⎪ dx = f (x, u, t)dt + g(x, t)dμ, dy = dμ, y0 = 0, t ∈ [t0 , t1 ], ⎪ ⎪ ⎪ ⎩max{|p|, gc sup |x|, y } ≤ c, u(t) ∈ U (t) a.a. t, μ ≥ 0. 1

t∈T ∗

(47)

This problem will be called the i-problem. By Theorem 6.1, each of the i-problems has a solution. Denote this solution by (pi , ui , μi ); let (xi , yi ) be optimal trajectories of the i-problem. Using the compactness arguments and passing to a subsequence, w w ∗ ∗ we find a process (p, u, μ) such that pi → p, ui → u weakly in Lm 2 (Tc ), and μi → μ weakly in C (Tc ). (Here, Tc = [−c, c], the controls ui are extended to Tc in an arbitrary admissible way, and the measures μi are extended by zero.) By Lemma 3.2, xi (t) → x(t) and yi (t) → y(t) ∀t ∈ Cont(μ). Let us prove that p = p∗ , u = u∗ , and μ = μ∗ . 7117

Let us show that (p, u, μ) is an admissible process in problem (2)–(4). Indeed, by the imposed constraints, we have const 2 2 . |e+ 1 (pi )| + |e2 (pi )| ≤ i Passing to the limit in this inequality as i → ∞, we obtain that p satisfies (4). Therefore, (p, u, μ) is an admissible process, and hence e0 (p) ≥ e0 (p∗ ). Further, Ji (pi , ui , μi ) ≤ Ji (p∗ , u∗ , μ∗ ) = e0 (p∗ ) ⇒ e0 (pi ) + |pi − p∗ |2 + |y1,i − F (t∗1 )|2 + ε

t1,i 

 |ui − u∗ (t)|2 + |yi − F (t)|2 dt ≤ e0 (p∗ ).

t0,i

Let us pass to the limit in the obtained inequality as i → ∞ using the weak lower semicontinuity of the left-hand side in u. We have t1   ∗ 2 ∗ 2 |u − u∗ (t)|2 + |y − F (t)|2 dt ≤ e0 (p∗ ). e0 (p) + |p − p | + |y1 − F (t1 )| + ε t0

Since it is already known that e0 (p) ≥ e0 (p∗ ), we deduce from this that p = p∗ , u = u∗ , and μ = μ∗ . Moreover, ui − u∗ L2 → 0. Passing to a subsequence, we obtain ui → u∗ a.e. For large i, all the inequality-type constraints of problem (47) are strict. Therefore, the necessary conditions of Theorem 5.1 can be applied to the i-problem. By Theorem 5.1 and Proposition 6.1, there exist a function ψi ∈ V n (Ti ), an absolutely continuous function ζi ∈ C(Ti ), and a number λ0,i > 0 for which the following conditions hold: |λi | + gc sup |ψi (t)| = 1, t∈Ti

dψi = −Hxi (t)dt − Qix (t)dμi ,

t ∈ Ti , ∂l ψ1,i = − (pi , λi ) − 2λ0,i (x1,i − x∗1 ), ∂x1 ζi (t1,i ) = −2λ0,i (y1,i − F (t∗1 )),

∂l (pi , λi ) + 2λ0,i (x0,i − x∗0 ), ∂x0 dζi = 2λ0,i ε[yi − F (t)]dt,   max Hi (u, t) − λ0,i ε|u − u∗ (t)|2 = Hi (t) − λ0,i ε|ui (t) − u∗ (t)|2 a.a. t,

ψ0,i =

u∈U (t)

Qi (t) + ζi (t) ≤ 0 ∀t,

Qi (t) + ζi (t) = 0

∀t ∈ supp(μi ),

ess lim sup max [H(βk,i , u, γk,i , t) − λ0,i ε|u − u∗ (t)|2 ]

k

+(−1)

t→tk,i u∈U (t)

 ∂l ∗ (pi , λi ) + 2λ0,i (tk,i − tk ) ≥ O(ε) + O(Δk,i ), ∂tk

k = 0, 1,

ess lim inf max [H(βk,i , u, γk,i , t) − λ0,i ε|u − u∗ (t)|2 ]

+(−1)k Here,

t→tk,i u∈U (t)

 ∂l (pi , λi ) + 2λ0,i (tk,i − t∗k ) ≤ O(ε) + O(Δk,i ), ∂tk

λ1,i = 2iλ0,i e+ 1 (pi ), β0,i =

Ti + xi (t0,i ),

λ2,i = 2iλ0,i e2 (pi ),

= [t0,i , t1,i ], β1,i =

Δk,i − xi (t1,i ),

λi = (λ0,i , λ1,i , λ2,i );

= μi ({tk,i }), γ0,i =

k = 0, 1.

k = 0, 1,

ψi (t+ 0,i ),

γ1,i = ψi (t− 1,i ).

The subscript i of the functions H and Q (and also of their partial derivatives) means that instead of a part of the variables x, u, and ψ, we substitute the values xi (t), ui (t), and ψi (t), respectively. 7118

The family {Qi } is equicontinuous. Indeed, the functions Qi are Lipschitzian with the same constant for all i. Extracting a subsequence, we obtain that Qi (t) → Q(t) uniformly. Also, it is obvious that ζi → 0 uniformly as i → ∞. Using Lemma 3.2 and also the condition Δ∗k = 0, k = 0, 1, which implies that Δk,i → 0 as i → ∞, we first pass to the limit in the necessary conditions for the i-problem as i → ∞ and then (repeating the procedure) as ε → 0; as a result, we obtain conditions (39)–(45). The theorem is proved for the case where Δ∗k = 0, k = 0, 1. If Δ∗k > 0, then using the reduction R1 (see Sec. 4), we arrive at the transversality conditions in time (44). Independently of the values of Δ∗k , conditions (46) are proved in the same way as above and are obtained by passing to the limit from inequalities (34) written for the i-problem. In this case, we need not use the reduction R1 . The theorem is proved. 7.

State-Constrained Problem

Introduce the notation

  W j (x, t) = ϕjx (x, t), g(x, t) , j = 1, . . . , k3 . In proving the transversality conditions in time in the state-constrained problem, we will need a certain monotonicity assumption. Assumption (M). Let x∗ (t) be the optimal trajectory under consideration, which is defined on the ∗− ∗ ∗ ∗ closed interval [t∗0 , t∗1 ]. Denote E0∗ = (x∗+ 0 , t0 ) and E1 = (x1 , t1 ). For every E = (x, t) and Δ > 0, on the closed interval [0, 1] consider the function α(s) = α(s; E, Δ), which is a solution of the corresponding associated system, i.e., α˙ = g(α, t)Δ, α(0) = x. Let ϕj (Ek∗ ) = W j (Ek∗ ) = 0 for certain j = 1, . . . , k3 and k = 0, 1. Then we require that the function W j (α(s), t) be monotonically nondecreasing on the closed interval [0, 1] for all E and Δ such that |E − Ek∗ | + Δ ≤ δ for a certain fixed δ > 0. If W j (Ek∗ ) = 0, then the number δ is chosen so that W j (α(s), t) = 0 ∀s ∈ [0, 1] ∀E : |E − Ek∗ | + Δ ≤ δ. Let us decode the monotonicity of the function W j in Assumption (M) in terms of derivatives. For this purpose, assume that the function ϕ is twice continuously differentiable. Then Assumption (M) holds if there exists δ > 0 such that ˙ j (E) ≥ 0 ∀E : |E − E ∗ | ≤ δ ∀j, k : ϕj (E ∗ ) = W j (E ∗ ) = 0, (48) W k k k     ˙ j = gx g, ϕjx + g, ϕjxx g . where W Let us analyze condition (48). Condition (48) holds for those superscripts j for which ϕj (Ek∗ ) = 0 but j ˙ j (E ∗ ) > 0. Condition (48) also holds for problems in which the state constraints are W (Ek∗ ) = 0 or W k convex on the trajectories of the associated system. The following problems can serve as an example of such a problem: • the linear problem: g is independent of x and ϕ linearly depends on x; ˙ j ≡ 0 in both cases); • the ordinary problem: g ≡ 0 (W • the problem in which the function g is independent of x and the matrix ϕjxx is positive-semidefinite. Theorem 7.1. Let (p∗ , u∗ , μ∗ ) be a solution of the LC problem (2)–(5), and let the state constraints be in concordance with the endpoint constraints. Then there exists a number λ0 ≥ 0, vectors λ1 ∈ Rk1 , λ1 ≥ 0, ∗ (T ∗ ) and λ2 ∈ Rk2 , a vector-function ψ ∈ V n (T ∗ ), and a vector-valued measure η = (η 1 , . . . , η k3 ), η j ∈ C+ such that Ds(μ∗ ) ∩ Ds(η j ) = ∅ ∀j, and for each point r ∈ Ds(μ∗ ), there exist its own vector-function ∗ ([0, 1]), j = 1, . . . , k , such σr ∈ V n ([0, 1]) and its own vector-valued measure ηr = (ηr1 , . . . , ηrk3 ), ηrj ∈ C+ 3 that (1) the functions ψ and σr , r ∈ Ds(μ∗ ), satisfy the equations 

t ψ(t) = ψ0 −

Hx (s) ds − t∗0

[t∗0 ,t]

Qx (s) dμ∗c +



∗ ϕ x (x , s) dη + Σ(ψ, t),

t ∈ (t∗0 , t∗1 ],

[t∗0 ,t]

7119



Σ(ψ, t) =

[σr (1) − ψ(r− )],

r∈Ds(μ∗ ) r≤t

⎧ ∗ ∗ ∗ ⎪ ⎨dαr = g(αr , r)Δr ds, ∗ s ∈ [0, 1], dσr = −gx (αr∗ , r)σr Δ∗r ds + ϕ x (αr , r)dηr , ⎪ ⎩ ∗ ∗ − − ∗ ∗ αr (0) = x (r ), σr (0) = ψ(r ), Δr = μ ({r}); (2) the following transversality condition holds: ψ0 =

∂l ∗ (p , λ), ∂x0

ψ1 = −

(3) for each atom r of the measure μ∗ , we have   g(αr∗ (s), r), σr (s) = 0

∂l ∗ (p , λ); ∂x1

∀s ∈ [0, 1],

supp(ηrj ) ⊆ {s ∈ [0, 1] : ϕj (αr∗ (s), r) = W j (αr∗ (s), r) = 0}

∀j;

(4) the following complementary nonrigidity condition holds:   λ1 , e1 (p∗ ) = 0, ϕj (x∗ (t), t) = 0

η j -a.e.

∀j;

(5) the following maximum condition holds for the regular Hamiltonian: max H(u, t) = H(t) a.a. t;

u∈U (t)

(6) the following maximum condition holds for the impulse Hamiltonian: Q(t) ≤ 0

∀t,

Q(t) = 0

μ∗ -a.e.;

(7) the following inequality holds: ess lim inf max H(x∗k , u, ψk , t) + (−1)k ∗ t→tk

u∈U (t)

∂l ∗ (p , λ) ≤ 0, ∂tk

k = 0, 1;

(8) the following nontriviality condition holds: |λ| + η +



ηr  = 1.

r∈Ds(μ∗ )

If, moreover, the Assumption (M) holds and Δ∗k = 0, k = 0, 1,14 then the inequalities in item (7) can be replaced by the following more complete transversality conditions in time: ⎧ ∂l ∗ ⎪ ⎪ max H(x∗k , u, ψk , t) + (−1)k (p , λ) ≥ 0, ⎨ess lim sup ∗ ∂tk t→tk u∈U (t) k = 0, 1. (49) ⎪ ∗ k ∂l ∗ ⎪ max H(xk , u, ψk , t) + (−1) (p , λ) ≤ 0, ⎩ess lim inf t→t∗k u∈U (t) ∂tk And, once again, as in Theorem 6.2, we cannot assert that the conditions in item (7) and (49) hold simultaneously, i.e., once again, we have two variants of the MP. Before proving the theorem, we prove the following assertion. 14

The case Δ∗k > 0 can be studied by using the reduction R1 .

7120

w

∗ ([0, 1]), η → η, and, moreover, let Proposition 7.1. Let fi ∈ C([0, 1]), let fi → f uniformly, let ηi ∈ C+ i supp(η) ⊆ W0 = {t : f (t) = 0}. We set  fi (s)dηi , t > 0, xi (0) = 0. xi (t) = [0,t]

1 Then if the function f is Lipschitzian, it follows that Var 0 [xi ] → 0. Proof. Fix an arbitrary ε > 0. Consider the set S(W0 , ε) that is an ε-neighborhood of W0 . The Lipschitzian property of f implies |f (t)| ≤ const ε ∀t ∈ S(W0 , ε). The weak convergence implies ηi (T \ S(W0 , ε)) → 0 as i → ∞. The uniform convergence implies fi − f C ≤ ε for large i. Then, for any a < b,            fi (s)dηi  ≤ c εηi ([a, b)) + ηi ([a, b) ∩ [T \ S(W0 , ε)]) .      [a,b) 1 This and the definition of the variation of a function on a closed interval imply lim sup Var 0 [xi ] ≤ const ε. i→∞ 1 But ε > 0 is arbitrary, and hence Var 0 [xi ] → 0. The proposition is proved.

Proof of Theorem 7.1. First, let us prove the most difficult second variant of the theorem, i.e., that with the transversality conditions in time (49). Let (p∗ , u∗ , μ∗ ) be a solution of problem (2)–(5). Take δ > 0 from Assumption (M) so small that Definition 1.3 holds simultaneously. According to this δ, using the continuity of μ∗ at the points t∗k , k = 0, 1, we find points t0,δ > t∗0 , t1,δ < t∗1 , and a small ε > 0 such that (μ∗ +L)([t∗0 , t0,δ +ε]∪[t1,δ −ε, t∗1 ]) < δ/2. Define the function ϕ+ δ by the formula ⎧ ⎨ϕ+ (x, t), t0,δ < t < t1,δ , + ϕδ (x, t) = ⎩0 otherwise. The components of ϕ+ δ are nonnegative and lower semicontinuous. We set F (t) = F (t; μ∗ ) and c = gc sup |x∗ | + |p∗ | + μ∗  + 1, take arbitrary i, A ∈ N, and consider the penalty problem ⎧ 't1 ⎪ ⎪ ⎪Ji,A (p, u, μ) = e0 (p) + |p − p∗ |2 + |y1 − F (t∗1 )|2 + A |ϕ+ (x, t)|2 dt ⎪ ⎪ t0 ⎪ ⎪ ⎪ t1 ⎪  ' ' ⎪ ⎪ +A |ϕ+ (x, t)|2 dμ + i−1 |u − u∗ (t)|2 + |y − F (t)|2 dt → min, ⎪ δ ⎪ ⎪ t0 ⎪ [t0 ,t1 ] ⎨ dx = f (x, u, t)dt + g(x, t)dμ, dy = dμ, y0 = 0, t ∈ [t0 , t1 ], (50) ⎪ ⎪ ⎪ ⎪ e1 (p) ≤ 0, e2 (p) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ |p − p∗ | ≤ δ/2, μ([t0 , t0,δ + ε) ∪ (t1,δ − ε, t1 ]) ≤ δ/2, ⎪ ⎪ ⎪ ⎪ ⎩ max{gc sup |x|, y1 } ≤ c, u(t) ∈ U (t) a.a. t, μ ≥ 0. The functional Ji,A is weakly lower semicontinuous in (u, μ). Therefore, by Theorem 6.1, for every i, A ∈ N, problem (50) has a solution. Denote this solution by (pi,A , ui,A , μi,A ); let (xi,A , yi,A ) be the corresponding optimal trajectories. Using compactness arguments, for a fixed i, we pass to a subsequence and find an w w ∗ admissible process (p, u, μ) such that pi,A → p, ui,A → u weakly in Lm 2 (Tc ), and μi,A → μ weakly in ∗ C (Tc ) as A → ∞. (Here, Tc = [−c, c]; see Sec. 6.) Moreover, xi,A (t) → x(t) ∀t ∈ Cont(μ). Let us prove that p = p∗ , u = u∗ , and μ = μ∗ (for any fixed i). 7121

Let us show that (p, u, μ) is an admissible process. Indeed, by the imposed constraints, we have   const + 2 2 . |ϕ (xi,A , t)| dt + |ϕ+ δ (xi,A , t)| dμi,A ≤ A Ti,A

Ti,A

Passing to the limit here (using Lemma 3.2), as A → ∞, we obtain t1

+



2

2 |ϕ+ δ (x, t)| dμ = 0.

|ϕ (x, t)| dt + [t0 ,t1 ]

t0

Using the concordance of the state and endpoint constraints and Assumption (M), we deduce from this that gc sup ϕj (x, t) ≤ 0, j = 1, . . . , k3 . Therefore, (p, u, μ) is an admissible process. Hence e0 (p) ≥ e0 (p∗ ). Further, Ji,A (pi,A , ui,A , μi,A ) ≤ Ji,A (p∗ , u∗ , μ∗ ) = e0 (p∗ )  ∗ 2 ∗ 2 |ϕ+ (xi,A , t)|2 dt ⇒ e0 (pi,A ) + |pi,A − p | + |y1,i,A − F (t1 )| + A 

2 |ϕ+ δ (xi,A , t)| dμi,A +

+A Ti,A



1 i

Ti,A



 |ui,A − u∗ (t)|2 + |yi,A − F (t)|2 dt ≤ e0 (p∗ ).

Ti,A

Let us pass to the limit in this inequality as A → ∞ using the lower semicontinuity in u of the left-hand side. We obtain #  $  2 |ϕ+ (xi,A , t)|2 dt + A |ϕ+ e0 (p) + lim sup A δ (xi,A , t)| dμi,A A→∞

∗ 2

Ti,A

+|p − p | + |y1 −

F (t∗1 )|2

Ti,A

1 + i

t1 

 |u − u∗ (t)|2 + |y − F (t)|2 dt ≤ e0 (p∗ ).

t0

(p∗ ).

This implies p = p∗ , u = u∗ , μ = μ∗ , and But, along with this, we have e0 (p) ≥ e0   + 2 2 |ϕ (xi,A , t)| dt + A |ϕ+ A → ∞ ∀i. A δ (xi,A , t)| dμi,A → 0, Ti,A

Ti,A

Taking into account what was proved above, for each i, choose a number Ai such that |pi,Ai − p∗ |2 + ui,Ai − u∗ 2L2 + yi,Ai − F (t)2L2 +   1 + 2 2 +Ai |ϕ (xi,Ai , t)| dt + Ai |ϕ+ δ (xi,Ai , t)| dμi,Ai ≤ i . Ti,Ai

Ti,Ai

Denote the chosen diagonal sequence by (pi , ui , μi ), i.e., shortening the notation, we set pi = pi,Ai ,

ui = ui,Ai ,

μi = μi,Ai . w

(51)

Passing to a subsequence, we have pi → p∗ , ui → u∗ a.e., and μi → μ∗ as i → ∞. Moreover, xi (t) = xi,Ai (t) → x∗ (t) ∀t ∈ Cont(μ∗ ). In (50), replace an arbitrary number A by the specially chosen number Ai . The obtained problem will be called the i-problem. 7122

For large i, all additional inequality-type constraints of the i-problem are strict. Thus, the necessary conditions of Theorem 6.2 can be applied to the i-problem.15 By Theorem 6.2, for each i, there exist functions ψi ∈ V n (Ti ) and ζi ∈ C(Ti ), a number λ0,i ≥ 0, and vectors λ1,i and λ2,i such that dψi = −Hxi (t)dt − Qix (t)dμi + ϕ x (xi , t)dηi , ψ0,i =

t ∈ Ti ,

∂l ∂l (pi , λi ) + 2λ0,i (x0,i − x∗0 ), ψ1,i = − (pi , λi ) − 2λ0,i (x1,i − x∗1 ), ∂x0 ∂x1 2λ0,i [yi − F (t)]dt, ζi (t1,i ) = −2λ0,i [y1,i − F (t∗1 )], dζi = i   λ1,i ≥ 0, e1 (pi ), λ1,i = 0,

max [Hi (u, t) − λ0,i i−1 |u − u∗ (t)|2 ] = Hi (t) − λ0,i i−1 |ui (t) − u∗ (t)|2 a.a. t,

u∈U (t)

2 Qi (t) + ζi (t) − λ0,i Ai |ϕ+ δ (xi (t), t)| ≤ 0

∀t,

2 Qi (t) + ζi (t) − λ0,i Ai |ϕ+ δ (xi (t), t)| = 0 ∀t ∈ supp(μi ),

(52) (53)

(54) (55) (56)

ess lim sup max [H(βk,i , u, γk,i , t) − λ0,i i−1 |u − u∗ (t)|2 ] − λ0,i Ai |ϕ+ (βk,i , tk,i )|2 t→tk,i u∈U (t)

k

+(−1)

 ∂l ∗ (pi , λi ) + 2λ0,i (tk,i − tk ) ≥ 1(i), ∂tk

k = 0, 1,

ess lim inf max [H(βk,i , u, γk,i , t) − λ0,i i−1 |u − u∗ (t)|2 ] − λ0,i Ai |ϕ+ (βk,i , tk,i )|2 t→tk,i u∈U (t)

 k ∂l ∗ +(−1) (pi , λi ) + 2λ0,i (tk,i − tk ) ≤ 1(i), k = 0, 1, ∂tk |λi | + gc sup |ψi (t)| + ηi  = 1.

(57)

(58)

t∈Ti

Here, Ti = [t0,i , t1,i ]; λi = (λ0,i , λ1,i , λ2,i ); ηi = (ηi1 , . . . , ηik3 ) is the vector-valued measure whose components are defined through their own distribution functions as follows:   j j + [ϕ (xi , t)] ds + 2λ0,i Ai [ϕjδ (xi , t)]+ dμi , j = 1, . . . , k3 ; F (t; ηi ) = 2λ0,i Ai [t0,i ,t]

β0,i = xi (t+ 0,i ),

[t0,i ,t]

β1,i = xi (t− 1,i ),

γ0,i = ψi (t+ 0,i ),

γ1,i = ψi (t− 1,i );

the expression 1(i) denotes a sequence of numbers converging to zero as i → ∞; the subscript i of the functions H and Q (and also their partial derivatives) means that for a part of the omitted variables x, u, and ψ in them, we substitute xi(t), ui (t), and ψi (t), respectively. By (58), ζi → 0 uniformly. Using the compactness arguments and passing to a subsequence, we have w from (58) that λi → λ, ηij → η˜j , j = 1, . . . , k3 , i → ∞. We set η(B) = η˜(B) − η˜(B ∩ Ds(μ∗ )) for η 1 , . . . , η˜k3 ). Thus, η = (η 1 , . . . , η k3 ) is a vector-valued measure such that any Borel B ⊆ R1 , η˜ = (˜ ∗ Ds(μ ) ∩ Ds(η) = ∅. It follows from (7) and (58) that the variations of ψi are uniformly bounded in i. Passing to a subsequence, we have ψi (t) → ψH (t) ∀t ∈ T ∗ , i → ∞ (second Helly theorem). Let us find a function ψ ∈ V n (T ∗ ) such that ψ(t) = ψH (t) ∀t ∈ Cont(μ∗ ) ∩ Cont(η). The existence of such a function ψ(t) η | + L)([t1 , t2 ]) ∀t1 ≤ t2 , which is obtained from the follows from the estimate Var |tt21 [ψH ] ≤ c(μ∗ + |˜ inequality |ϑ[ψi ]| ≤ c(μi + |ηi | + L) by passing to the limit as i → ∞. Since ϕ+ δ is a discontinuous function, the i-problem is not standard. However, Theorem 6.2 is analogously proved for it. In this case, conditions (52)–(58) hold independently of whether the measure μi has atoms at the points tk,δ or not. This 2 is a consequence of the lower semicontinuity of the function [ϕ+ δ ] and the fact that the zero set of the derivative of such a function contains the zero set of the function itself. 15

7123

Let us prove that ψ(t) satisfies the conditions of the theorem. For this purpose, construct a family of absolutely continuous measures {ˆ μi }, i ∈ N , approximating all necessary optimality conditions in the i-problem (except for the transversality conditions in time (57)) as follows. Let {tk },# k ∈ N, be a $countable everywhere dense in T ∗ set of points such that tk ∈ X = Cont(μ∗ ) ∩ ∞ ( Cont(μi ) (such a countable everywhere dense set exists, since T ∗ \ X is countable as Cont(η) ∩ i=1

a countable union of countable sets), and let {φk } be a countable everywhere dense in C(Tc ) set of w functions. For each i, there exists a sequence of absolutely continuous measures μi,τ such that μi,τ → μi w w on Ti , μi,τ → μi on [t0,i , t0,δ ], and μi,τ → μi on [t1,δ , t1,i ] as τ → ∞, and the distribution function F (t; μi,τ ) strictly increases on Tc for all τ . Let a pair xi,τ , ψi,τ satisfy the system16 ⎧ dxi,τ = f (xi,τ , ui , t)dt + g(xi,τ , t)dμi,τ , ⎪ ⎪ ⎪ ⎨ dψi,τ = −Hx (xi,τ , ui , ψi,τ , t)dt − Qx (xi,τ , ψi,τ , t)dμi,τ + ϕ x (xi,τ , t)dηi,τ , ⎪ ⎪ ⎪ ⎩ xi,τ (t0,i ) = xi (t0,i ), ψi,τ (t0,i ) = ψi (t0,i ), j where the measure ηi,τ is defined by its distribution function   j j + [ϕ (xi,τ , t)] ds + 2λ0,i Ai [ϕjδ (xi,τ , t)]+ dμi,τ , F (t; ηi,τ ) = 2λ0,i Ai [t0,i ,t]

j = 1, . . . , k3 .

[t0,i ,t]

Using Lemma 3.2, we choose a number τi such that the following inequality holds: ⎡ ⎡   ⎤ k3  i      ⎢ ⎣ φk (t) d(μi,τi − μi ) +  φk (t) d(η j − η j )⎦ i,τi i  ⎢   ⎢ j=1 k=1  Ti Ti ⎢ ⎢  i     ⎢ ψi,τ (tk ) − ψi (tk ) + gc sup |ψi (t)| − max |ψi,τ (t)| ⎢ + i i ⎢ t∈Ti t∈Ti ⎢ k=1  ⎢    ⎢  ⎢ + Q ˜ i (xi,τ , ψi,τ , t) dμi,τ − Q ˜ i (xi , ψi , t) dμi  i i i ⎢   ⎢ ⎢ Ti  Ti  ⎢  k3    ⎢   j j j j ⎢ + ϕ (x , t) dη − ϕ (x , t) dη  i,τ i i i,τi i  ⎢ ⎢ j=1  ⎢ Ti Ti ⎣    1  ˜ ˜  + max Qi (xi,τi , ψi,τi , t) − Qi (xi , ψi , t) + ψi,τi (t1,i ) − ψi (t1,i ) ≤ . t∈Ti i

(59)

˜ i (x, ψ, t) = Q(x, ψ, t) − λ0,i Ai |ϕ+ (x, t)|2 . Here, Q δ ˆi = xi,τi , μ ˆi = μi,τi , and ηˆi = ηi,τi . Passing to a subsequence, we have ψˆi (t) → ψA (t) We set ψˆi = ψi,τi , x ∀t ∈ T ∗ . Let us show that ψA (t) = ψ(t) ∀t ∈ Cont(μ∗ ) ∩ Cont(η). Indeed, since |ϑ[ψi ]| ≤ c(μi + |ηi | + L), it follows that the function ψ(t) is continuous on the set Cont(μ∗ ) ∩ Cont(η) ∩ (t∗0 , t∗1 ). The same is true for the function ψA . Then (59) implies ψA (t) = ψ(t) ∀t ∈ Cont(μ∗ ) ∩ Cont(η). Moreover, obviously, w w ˆi → μ∗ , ηˆij → η˜j , j = 1, . . . , k3 , as i → ∞. x ˆi (t) → x∗ (t) ∀t ∈ Cont(μ∗ ), μ Let us prove condition (1). 16

For each i and large τ , there exists a solution of the system (see analogous arguments in proving Lemma 3.2).

7124

We will show that there exist sequences of absolutely continuous measures {¯ μi } and {¯ ηij }, j = 1, . . . , k3 , w w ¯i → μ∗d and η¯ij → η˜j − η j as i → ∞ ∀j; and also a sequence ki ≥ i of natural numbers such that (1) μ ¯i and ηˆkj i ≥ η¯ij ∀i, j. (2) μ ˆ ki ≥ μ ¯i = η¯ij = 0, ki = i ∀i, j. If Ds(μ∗ ) = ∅, then we set μ ∗ Let Ds(μ ) = ∅. Consider a chain of sets Di ordered with respect to inclusion such that D0 = ∅,  1 μ∗ ({r}) ≤ , i ∈ N. Define the sets Sr,i = [r − hi , r + hi ], r ∈ Di , as Di−1 ⊆ Di ⊆ Ds(μ∗ ), and i ∗ r∈Ds(μ )\Di

a system of closed pairwise disjoint neighborhoods of the points r such that (1) hi > 0, hi → 0 as i → ∞;     k3    .  j   1  ∗ ∗ j η˜ (Si ) − μ ({r}) + η˜ ({r}) ≤ , where Si = Sr,i ; (2) μ (Si ) −  i j=1

r∈Di

r∈Di

(3) r ± hi ∈ Cont(μ∗ ) ∩ Cont(η).

r∈Di

The existence of such a set Si follows from the regularity of the Borel measures μ∗ , η˜j , j = 1, . . . , k3 . Choose a sequence ki ≥ i such that    k3     j   ∗ j μ    ≤ 1. (S ) − μ (S ) (S ) − η ˜ (S ) ˆ η ˆ + r,i r,i r,i r,i k  i  ki   i j=1

r∈Di

ˆki (B ∩ Si ) and This can be done because of the weak convergence of the measures. We set μ ¯i (B) = μ w k3 1 1 η¯i (B) = ηˆki (B ∩ Si ) for any Borel set B ⊆ R , η¯i = {¯ ηi , . . . , η¯i }. It is easy to see that μ ¯i → μ∗d and w w w η¯ij → η˜j − η j as i → ∞ (and hence μ ˆ ki − μ ¯i → μ∗c and ηˆkj i − η¯ij → η j ), j = 1, . . . , k3 . Also, it is obvious ∗ (T ) and η ∗ (T ) ∀i, j. that μ ˆ ki − μ ¯i ∈ C+ ˆkj i − η¯ij ∈ C+ c c Using the constructed numbers ki , from the sequences {ˆ μi }, {ˆ ηi }, {ˆ xi }, {ψˆi }, {ui }, and {Ti }, we choose subsequences denoted (as the initial ones) by the same subscript i. Rewrite Eqs. (3) and (52) in the following form: ⎧   t ⎪ ⎪ ⎪ ⎪ ⎪ f (ˆ xi , ui , s) ds + g(ˆ xi , s) d(ˆ μi − μ ¯i ) + g(ˆ xi , s) d¯ μi , x ˆi (t) = x0,i + ⎪ ⎪ ⎪ ⎪ ⎪ t0,i [t0,i ,t] [t0,i ,t] ⎪ ⎪ ⎪ ⎨ t   ˆ xi (s) ds − ˆ ix (s) d(ˆ ˆ ix (s) d¯ μi − μ ¯i ) − μi H Q Q ψˆi (t) = ψ0,i − ⎪ ⎪ ⎪ ⎪ t0,i [t0,i [t0,i ,t] ⎪  ,t] ⎪ ⎪ ⎪ ⎪   ⎪ + ϕx (ˆ xi , s) d(ˆ ηi − η¯i ) + ϕx (ˆ xi , s) d¯ ηi , t ∈ Ti . ⎪ ⎪ ⎪ ⎩ [t0,i ,t]

[t0,i ,t]

Here and in what follows, the superscript i and the “hat” of the functions H and Q (and also their partial derivatives) mean that for a part of the omitted variables x, u, and ψ in them, we substitute the values x ˆi (t), ui (t), and ψˆi (t), respectively. We set 

t x ˆci (t)

= x0,i +

g(ˆ xi , s)d(ˆ μi − μ ¯i ),

f (ˆ xi , ui , s) ds + [t0,i ,t]

t0,i

t ψˆiη (t)

 x ˆdi (t)

= ψ0,i − t0,i

[t0,i ,t]

g(ˆ xi , s) d¯ μi , [t0,i ,t]

 ˆ xi (s) ds − H

=

 ˆ ix (s) d(ˆ Q μi − μ ¯i ) +

ϕ xi , s) d(ˆ ηi − η¯i ), x (ˆ

[t0,i ,t]

7125





ψˆid (t) = −

ϕ xi , s) d¯ ηi , x (ˆ

ˆ ix (s) d¯ Q μi +

[t0,i ,t]

t ∈ Ti .

[t0,i ,t]

ˆci (t) + x ˆdi (t) and ψˆi (t) = + ψˆid (t) ∀t ∈ Ti . Taking into account that x ˆi (t) → x∗ (t) Thus, x ˆi (t) = x ∗ ∀t ∈ Cont(μ ), by the Lebesgue theorem and Lemma 3.1 we have ψˆiη (t)

x ˆci (t)

→

x∗c (t)

=

x∗0

t +

∗



∗

f (x , u , s) ds + t∗0

g(x∗ , s) dμ∗c

∀t ∈ T ∗ ,

i → ∞.

[t∗0 ,t]

Analogously, Lemma 3.1 implies 

t ψˆiη (t)

→ ψη (t) = ψ0 −

Hx (s) ds − t∗0

Qx (s) dμ∗c +

[t∗0 ,t]



∗ ϕ x (x , s) dη

[t∗0 ,t]

∀t ∈ Cont(η), i → ∞. We set ψd = ψ − ψη . Then ψˆid (t) → ψd (t) ∀t ∈ Cont(μ∗ ) ∩ Cont(η). Let us show that ψd = Σ(ψ, t). Indeed, let t ∈ Cont(μ∗ ) ∩ Cont(η). Fix ε > 0. Choose a number N = N (ε) such that    ∗ η |({r}) ≤ ε. μ ({r}) + |˜ r∈Ds(μ∗ )\DN

In this case,

  ⎡ ⎤        ⎢ ⎥   ˆ ix (s) d¯ −Q μi + ϕ xi , s) d¯ ηi ⎦ − ψˆid (t) ≤ const ε. lim sup  ⎣ x (ˆ   i→∞  r∈D(N,t) Sr,i Sr,i

Here and in what follows, D(N, t) = {r ∈ DN : r ≤ t}. We set ri− = r − hi , and, for r ∈ D(N, t), on the closed interval Sr,i , consider the system  ⎧ d (s) = x d (r − ) + ⎪ ⎪ ˆ g(ˆ xi (τ ), τ ) d¯ μi , x ˆ ⎪ i i i ⎪ ⎪ ⎨ [ri−,s]  s ∈ Sr,i . ⎪ −   d d ˆ ˆ ˆ ⎪ ψ (s) = ψ (r ) − g (ˆ x (τ ), τ ) ψ (τ ) d¯ μ + ϕ (ˆ x (τ ), τ ) d¯ η , ⎪ i i i i i x x i i i ⎪ ⎪ ⎩ − − [ri ,s]

(60)

(61)

[ri ,s]

For sufficiently large i, define the functions17 πr,i (τ ) =

F (τ ; μ ¯i ) − F (ri− ; μ ¯i ) , μ ¯i (Sr,i )

r ∈ D(N, t), τ ∈ Sr,i .

The function πr,i maps the closed interval Sr,i onto the closed interval [0, 1] and is an absolutely continuous and strictly increasing function. Therefore, there exists a strictly increasing inverse function θr,i : [0, 1] → Sr,i , θr,i = (πr,i )−1 . By the theorem from [34, p. 244], making the change ω = πr,i (τ ) of the variable under the integral sign in (61) and transforming, we arrive at the system αr,i (s) =

x ˆdi (ri− )

s +

x ˆci (θr,i (s))

+μ ¯i (Sr,i )

g(αr,i (ω), θr,i (ω)) dω, 0

17

The numbers i must be taken starting from a certain i0 so that t ∈ / Sr,i ∀i ≥ i0 , r ∈ D(N, t).

7126

s

σr,i (s) = ψˆid (ri− ) + ψˆiη (θr,i (s)) − μ ¯i (Sr,i )  +

gx (αr,i (ω), θr,i (ω))σr,i (ω) dω

0

ϕ x (αr,i (ω), θr,i (ω)) dηr,i ,

s ∈ [0, 1],

r ∈ D(N, t),

[0,s]

where ˆi (θr,i (ω)), αr,i (ω) = x

j F (ω; ηr,i ) = F (θr,i (ω); η¯ij ),

σr,i (ω) = ψˆi (θr,i (ω)),

j = 1, . . . , k3 .

ˆci (θr,i (ω)) → x∗c (r), It is easy to see that θr,i (ω) → r uniformly on [0, 1], and by Proposition 3.1, x ψˆiη (θr,i (ω)) → ψη (r) uniformly on [0, 1] as i → ∞. It follows from the Gronwall inequality that αr,i is a Cauchy sequence of functions with respect to i in C n ([0, 1]). Its limit is exactly the function αr∗ . The compactness arguments imply that there exist a measure ηrj and a function σr ∈ V n ([0, 1]) such that, after j w j → ηr , j = 1, . . . , k3 , σr,i (s) → σr (s) ∀s ∈ Cont(ηr ). Passing to the passage to a subsequence, we have ηˆr,i the limit in the latter system as i → ∞, we have s ∗ ∗ − ∗ αr (s) = x (r ) + μ ({r}) g(αr∗ , r) dω, 0

σr (s) = ψ(r− ) − μ∗ ({r})

s

gx (αr∗ , r)σr dω +

0



∗ ϕ x (αr , r) dηr ,

s ∈ [0, 1],

r ∈ D(N, t).

[0,s]

x ˆdi (ri− )

→ x∗ (r− ) − x∗c (r) and ψˆid (ri− ) → ψd (r− ).) This (Here, we have used the fact that, by construction, and (60) imply        −  ψd (t) − [σ (1) − ψ(r )] r  ≤ const ε.    r∈D(N,t) But ε > 0 is arbitrary; by the definition of the function Σ, this means that ψd (t) = Σ(ψ, t). Conditions (1) and (2) of the theorem are proved. Condition (8) is obtained by a simple argument ηi  → 0, then, obviously, max |ψˆi (t)| → 0 as i → ∞ (a consequence assuming the contrary: if |λi | → 0 and ˆ t∈Ti

of the Gronwall inequality), and, therefore, (58) is violated for large i. Let us prove condition (3). Fix r ∈ Ds(μ∗ ), j = 1, . . . , k3 . The penalty method and (59) imply   Ai |ϕ+ (ˆ xi , t)|2 dt + Ai |ϕ+ xi , t)|2 dˆ μi → 0. δ (ˆ Ti

Ti

Further,



+

2

Ti



≥ 2λ0,i Ai ⎝

ϕj (ˆ xi , t)[ϕj (ˆ xi , t)]+ dt +

Ti

|ϕ+ xi , t)|2 dˆ μi δ (ˆ

Ti



⎞

ϕjδ (ˆ xi , t)[ϕjδ (ˆ xi , t)]+ dˆ μi ⎠ =

Ti



ϕj (ˆ xi , t)dˆ ηij

Then (62) implies



|ϕ (ˆ xi , t)| dt + 2Ai

2Ai ⎛

(62)

 ϕj (ˆ xi , t) dˆ ηij ≥ 0. Ti

→ 0 as i → ∞. As above, making the change s = πr,i (t) of the variable

Sr,i



1

under the integral sign and passing to the limit, we obtain 0

ϕj (αr∗ , r)dηrj = 0. Taking into account that

ϕj (αr∗ (s), r) ≤ 0, we have from this that ϕj (αr∗ (s), r) = 0 ∀s ∈ supp(ηrj ). 7127

Let us show that Q(t) = Q(x∗ (t), ψ(t), t) ≤ 0 ∀t ∈ T ∗ . Indeed, ζi → 0 uniformly on T ∗ . Denote xi (t), t)|2 . By (62), κ ˆ i (t) → 0 strongly in L1 (T ∗ ); passing to a subsequence, we have κ ˆ i (t) = λ0,i Ai |ϕ+ δ (ˆ κ ˆ i (t) → 0 a.a. t. It follows from (55), (59), and the right continuity of the function Q(t) on the interval (t0,δ , t1,δ ) that Q(t) ≤ 0 ∀t ∈ T ∗ .   1   ˆ i (t)dˆ Q It follows from (56) and (59) that g(αr∗ , r), σr ds = 0. It is easy to μi → 0, whence 0

Ti

obtain from (55), (59), (62), the property  and  that Q(t) ≤ 0 ∀t (using arguments analogous to those ∗ given above) that qr (s) = g(αr (s), r), σr (s) ≤ 0 ∀s ∈ [0, 1]. Hence qr (s) = 0 ∀s ∈ (0, 1). Let us show that qr (0) = qr (1) = 0. Indeed, if ηr {0} = 0, then qr (0) = 0. Let ηrj {0} > 0 for a certain j = 1, . . . , k3 . Then ϕj (αr∗ (0), r) = 0 and W j (αr∗ (0), r) ≤ 0, since otherwise the state constraint is violated. But, in such a case, qr (0) ≥ 0. Along with this, we have the opposite inequality. Hence qr (0) = W j (αr∗ (0), r) = 0. Analogously, qr (1) = 0. Condition (3) is proved. xi , t)]| ≤ c(ˆ μi + L), Lemma 3.1 implies Let us prove condition (4). Since |ϑ[ϕj (ˆ   ϕj (ˆ xi , t) d[ˆ ηij − η¯ij ] → ϕj (x∗ , t) dη j , i → ∞, j = 1, . . . , k3 . Ti



On the other hand, by (62) we have Ti

T∗

ϕj (ˆ xi , t)d[ˆ ηij − η¯ij ] → 0. Now condition (4) follows from the

nonpositivity of the functions ϕj (x∗ , t), j = 1, . . . , k3 . Passing to the limit in (54), we obtain condition (5). Let us show how condition (6) can be proved by using Lemma 3.1. From what was already proved, 1 it follows that Ds(ϑ[Q]) ∩ Ds(μ∗ ) = ∅. Let us show that Var 0 [qr,i ] → 0 as i → ∞ ∀r ∈ Ds(μ∗ ), where   qr,i (s) = σr,i (s), g(αr,i (s), θr,i (s)) , s ∈ [0, 1]. Indeed, k3  

s

s

qr,i (s) = qr,i (0) +

j W j (αr,i , θr,i ) dηr,i

j=1 0 k3  1 1 This implies Var 0 [qr,i ] ≤ Var 0

+

  σr,i , gt (αr,i , θr,i ) dθr,i ,

s ∈ [0, 1].

0



s

 j + 1(i). W j (αr,i , θr,i ) dηr,i

0 j=1 j j ∗ But W (αr,i , θr,i ) → W (αr , r) uniformly on j w j ηr,i → ηr : supp(ηrj ) ⊆ {s : W j (αr∗ (s), r) = 0} ∀j.

have proved that

[0, 1] as i → ∞, and what was proved above implies 1 Then Proposition 7.1 implies Var 0 [qr,i ] → 0. Hence we

ˆ i ]| ≤ c([ˆ μi − μ ¯i ] + |ˆ ηi − η¯i | + L) + 1(i). |ϑ[Q Using Lemma 3.1, (56), (59), and (62), we deduce that     ∗ ˆ μi + κ ˆ i (t) dˆ μi → Q(t)dμ ⇒ Q(t) dμ∗ = 0. 1(i) = Qi (t) dˆ Ti

Ti

T∗

T∗

Taking into account that Q(t) ≤ 0 ∀t, we obtain condition (6). Let us prove the transversality conditions in time (49) for k = 0. Return to the initial sequence of solutions (51) and consider conditions (57). Note ∗ that Δ0,i = μi ({t0,i }) → 0, since μ∗ is continuous at the point t∗0 . This implies that x+ 0,i = β0,i → x0 + 2 and ψ0,i = γ0,i → ψ0 as i → ∞. Denote κi = λ0,i Ai [ϕ+ (x+ 0,i , t0,i )] and show that κi → 0 as i → ∞. There exist the following two possibilities: (1) the sequence Δ0,i contains a subsequence consisting only of zeros; (2) Δ0,i > 0 for all i greater than a certain i0 . In the first case, passing to a subsequence, we have x+ 0,i = x0,i and κi = 0 by the concordance of the state and endpoint constraints. Now let us turn to the second case. 7128

For large i, the function Qi (t) is absolutely continuous in a certain neighborhood of the point t0,i , and there its derivative is equal to Q˙ i =

k3 

2λ0,i Ai [ϕj (xi , t)]+ W j (xi , t) + ωi (t),

(63)

j=1

where ωi (t) is a measurable essentially bounded function, ωi L∞ ≤ const ∀i. Since Δ0,i > 0, it follows from (56) that Qi (t0,i ) + ζ0,i = 0. Using (55) and integrating (63), by the Newton–Leibnitz formula, we obtain from this that t  +Δ k3 0,i  2λ0,i Ai [ϕj (xi , s)]+ W j (xi , s) ds ≤ const Δ.

(64)

j=1 t 0,i

Dividing both sides of (64) by Δ > 0 and letting the latter tend to zero, we deduce that k3 

+ j + λ0,i Ai [ϕj (x+ 0,i , t0,i )] W (x0,i , t0,i ) ≤ const .

(65)

j=1

Assumption (M) and the concordance of the state and endpoint constraints imply that if W j (x+ 0,i , t0,i ) ≤ 0, + = 0. Then (65) implies then [ϕj (x+ , t )] 0,i 0,i + j + λ0,i Ai [ϕj (x+ 0,i , t0,i )] W (x0,i , t0,i ) ≤ const, 2 Denote κji = λ0,i Ai [ϕj+ (x+ 0,i , t0,i )] . Clearly, κi =

k3 

j = 1, . . . , k3 .

(66)

κji . If W j (x+ 0,i , t0,i ) ≤ 0, then, as was noted above,

j=1

κji = 0. Let κji > 0; then W j (x+ 0,i , t0,i ) > 0. Further, #

2 + κji = λ0,i Ai [ϕ+ (x+ 0,i , t0,i )] = λ0,i Ai ϕ (x0,i , t0,i ) + Δ0,i

$2

1 h[ϕj (α0,i , t0,i )]W j (α0,i , t0,i )ds

,

0

where h(t) is the Heaviside function: h(t) = 0 for t ≤ 0 and h(t) = 1 for t > 0. By the concordance of the state and endpoint constraints and Assumption (M), the latter relation implies 2 κji ≤ λ0,i Ai Δ20,i [W j (x+ 0,i , t0,i )] .

Now, it follows from (66) that κji ≤ const

λ0,i Ai Δ20,i

2 [λ0,i Ai ϕj+ (x+ 0,i , t0,i )]

= const

Δ20,i κji

.

This implies

(κji )2 ≤ const Δ20,i → 0, which is what was required to be proved. Now, passing to the limit in (57), we obtain condition (49) of the theorem. Now note that Assumption (M) and the restriction Δ∗k = 0, k = 0, 1, are necessary only for the proof of the transversality conditions in time (49) and can be omitted in proving conditions  (1)–(8) of the

2 theorem. Let us show how this can be done. In (50), we replace the penalty summand [ϕ+ δ (x, t)] dμ by T  + + 2 + [ϕ (x, t)] dμ. It is easy to verify that the replacement of the function ϕδ by ϕ has no influence on T

the arguments proving conditions (1)–(8) of Theorem 7.1. It is important to note that, in this case, the concordance condition of the state and endpoint constraints remains essential as before and cannot be 7129

omitted: it is used, first, in deducing the inequalities in condition (7), and, second, in proving the impulse maximum condition, more precisely, in proving the inequalities Q(t∗k ) ≤ 0. The theorem is proved. 8.

Nondegeneracy Conditions and Completion of the Proof of the Maximum Principle

Here, we prove the nondegenerate MP, which is formulated under the smoothness conditions. In this connection, we will need the following remark. Remark. Under assumption (S), condition (7) of Theorem 7.1 becomes ∂l ∗ (p , λ) ≤ 0, k = 0, 1. max H(x∗k , u, ψk , t∗k ) + (−1)k u∈U ∂tk

(67)

Theorem 8.1. Let (p∗ , u∗ , μ∗ ) be a solution of the LC problem (2)–(5), let Assumption (S) hold, let the state constraints be in concordance with the endpoint constraints, let the state and endpoint constraints be regular, and let the optimal trajectory be controllable at the endpoints with respect to the state constraints. Then there exist a number λ0 ≥ 0, vectors λj ∈ Rkj , λ1 ≥ 0, a vector-function ψ ∈ V n (T ∗ ), a scalar∗ (T ∗ ): Ds(μ∗ ) ∩ Ds(η j ) = ∅ function φ ∈ V (T ∗ ), a vector-valued measure η = (η 1 , . . . , η k3 ), η j ∈ C+ ∗ ∀j, and for each point r ∈ Ds(μ ), there exist its own vector-function σr ∈ V n ([0, 1]), scalar-function ∗ ([0, 1]), j = 1, . . . , k , such that θr ∈ V ([0, 1]), and vector-valued measure ηr = (ηr1 , . . . , ηrk3 ), ηrj ∈ C+ 3 t ψ(t) = ψ0 −

 Hx (s) ds −

t∗0



Qx (s) dμ∗c

[t∗0 ,t]

+



Σ(ψ, t) =

∗ ϕ x (x , s) dη + Σ(ψ, t),

t ∈ (t∗0 , t∗1 ],

[t∗0 ,t]

[σr (1) − ψ(r− )],

r∈Ds(μ∗ ) r≤t



t φ(t) = φ0 +

Ht (s) ds + t∗0

Qt (s) dμ∗c

[t∗0 ,t]

 −



Θ(φ, t) =

∗ ϕ t (x , s) dη + Θ(φ, t),

t ∈ (t∗0 , t∗1 ],

(68)

[t∗0 ,t]

[θr (1) − φ(r− )],

r∈Ds(μ∗ ) r≤t

⎧ ⎪ dαr∗ = g(αr∗ , r)Δ∗r ds, ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ⎨dσr = −gx (αr∗ , r)σr Δ∗r ds + ϕ x (αr , r)dηr ,   ∗ ⎪ ⎪ dθr = gt (αr∗ , r), σr Δ∗r ds − ϕ s ∈ [0, 1], ⎪ t (αr , r)dηr , ⎪ ⎪ ⎪ ⎩ ∗ αr (0) = x∗ (r− ), σr (0) = ψ(r− ), θr (0) = φ(r− ), Δ∗r = μ∗ ({r}); ∂l ∗ ∂l ∗ (p , λ), ψ1 = − (p , λ); ψ0 = ∂x0 ∂x1 ∂l ∗ ∂l ∗ (p , λ), φ1 = (p , λ); φ0 = − ∂t0 ∂t1   g(αr∗ (s), r), σr (s) = 0 ∀s ∈ [0, 1], supp(ηrj ) ⊆ {s ∈ [0, 1] : ϕj (αr∗ (s), r) = W j (αr∗ (s), r) = 0} ∀j;   λ1 , e1 (p∗ ) = 0, ϕj (x∗ (t), t) = 0 η j -a.e. ∀j; max H(u, t) = φ(t) ∀t ∈ (t∗0 , t∗1 ); max H(u, t) = H(t) a.a. t, u∈U

u∈U

Q(t) ≤ 0 7130

∀t,

Q(t) = 0

μ∗ -a.e.;

(69)



  λ0 + L {t : |ψ(t)| > 0} +

  L {t : |σr (t)| > 0} Δ∗r > 0.

(70)

r∈Ds(μ∗ )

Remark. The conditions of Theorem 8.1 imply the following transversality conditions in time: # 1    ∗ ∗ k+1 ∗ ∗ gt (αk∗ , t∗k ), σk dt − Δ∗k ϕ max H(βk , u, γk , tk ) + (−1) t (αk , tk ) dηk u∈U

0 ∗ ∗ ∗ −ϕ t (xk , tk )η({tk }) −

[0,1]

$

∂l ∗ (p , λ) = 0, ∂tk

k = 0, 1.

Here and in Theorem 8.1, we have used the following notation: ψk = ψ(t∗k ), β0∗ = x∗ (t∗+ 0 ),

φk = φ(t∗k ),

Δ∗k = μ∗ ({t∗k }),

β1∗ = x∗ (t∗− 1 ),

γ0∗ = ψ(t∗+ 0 ),

k = 0, 1, γ1∗ = ψ(t∗− 1 ),

αk∗ , σk , and ηk are elements of the associated system at the point t∗k , k = 0, 1. Proof. Let us use the scheme suggested in [1]. In this case, the reference to Theorem 7.1 in the proof means its first variant, i.e., conditions (1)–(8). Formulas (68) and (69) are obtained by applying Theorem 7.1 to the v-problem (reduction R2 , Sec. 4) under the smoothness condition. Its proof is completely analogous to the proof presented in [1, p. 165]. Consider the case Δ∗k = 0, k = 0, 1 (the case Δ∗k > 0 is analogous), and prove (70). Assume that (70) is violated. Then λ0 = 0, ψ(t) = 0 ∀t ∈ B ∗ , B ∗ = (t∗0 , t∗1 ), since the function ψ is right-continuous on this interval. This implies σr (s) = 0 ∀s ∈ [0, 1] ∀r ∈ Ds(μ∗ ). Using the regularity of the state constraints, we deduce that η(B ∗ ) = 0, ηr = 0 ∀r ∈ Ds(μ∗ ) and also that ∗ ∗ ∗ ϕ x (xk , tk )η({tk }) = −

∂l ∗ (p , λ), ∂xk

k = 0, 1.

Analogously, from (68) we obtain   ∂l ∗ ϕt (x∗k , t∗k ), η({t∗k }) = − (p , λ), ∂tk

k = 0, 1.

Substituting the obtained expressions in inequality (67), we have     ∗ ∗ ∗ ∗ k+1 max H x∗k , u, (−1)k+1 ϕ ϕt (x∗k , t∗k ), η({t∗k }) ≤ 0, x (xk , tk )η({tk }), tk + (−1) u∈U

k = 0, 1.

Exactly in the same way, substituting them in the inequality Q(t∗k ) ≤ 0, we obtain   ∗ ∗ ∗ ∗ ∗ k = 0, 1. g(xk , tk ), (−1)k+1 ϕ x (xk , tk )η({tk }) ≤ 0, Now the controllability condition of the trajectory x∗ (t) at the endpoints implies that η({t∗k }) = 0, ∂l ∗ (p , λ) = 0. It follows from the regularity of the endpoint constraints that λ = 0. Therefore, k = 0, 1 ⇒ ∂p we obtain a contradiction with the nontriviality of Theorem 7.1. The theorem is proved. Remark. For a certain class of problems, the controllability assumption (Definition 1.4) can be weakened. Let the problems under consideration be such that all state constraints are convex on the trajectories of the associated system. We say that the optimal trajectory x∗ (t) is controllable if there exist points sk ∈ [0, 1], k = 0, 1, such that Definition 1.4 holds in (αk∗ (sk ), t∗k ). Indeed, in this case, we use the reduction R1 , but in the reduced problem (P1 ), we do not impose additional state constraints (this is by the assumption on the convexity of the state constraints on discontinuities of the system made above). The optimal trajectory 7131

in the reduced problem is controllable. Then, arguing in the same way as in proving Theorem 8.1, we can obtain a more strict nontriviality condition:      L {t : |σr (t)| > 0} Δ∗r > 0. (71) λ0 + L {t : |ψ(t)| > 0} + r∈Ds(μ∗ )\{t∗0 ,t∗1 }

The latter holds, since in the case studied, the functions σk (i.e., the variables adjoint to αk∗ in the reduced problem) are solutions of linear systems. A linear system (g is independent of x and ϕ is linear in x) can serve as an example of the problem with the convexity properties mentioned above. Moreover, for the linear system, condition (71) becomes   λ0 + L {t : |ψ(t)| > 0} > 0. 9.

General Nonlinear Impulse Control Problem

In this section, we remove the linear-convexity assumptions under which Theorems 6.2 and 7.1 were obtained and prove the MP for the general nonlinear impulse control problem (2)–(5) with a vector-valued measure. Here, we first consider this problem on a fixed closed interval of time T = [t0 , t1 ]. In this case, the endpoint vector p becomes p = (x0 , x1 ), p ∈ R2n . Theorem 9.1. Let (p∗ , u∗ , q∗ ) be a solution of problem (2)–(4). Then there exists a number λ0 ≥ 0, vectors λj ∈ Rkj , j = 1, 2, and a function ψ ∈ V n (T ) such that the following conditions hold : dψ = −Hx (t)dt − Qx (t)dq∗ , t ∈ T, ∂l ∗ ∂l ∗ (p , λ), ψ1 = − (p , λ), ψ0 = ∂x0 ∂x1   λ1 ≥ 0, λ1 , e1 (p∗ ) = 0, max H(u, t) = H(t) a.a. t, u∈U (t)   gc sup Q(t), m ≤ 0 ∀m ∈ K,    Q(t), dq∗ = 0, T

|λ| = 1.

Proof. Let (p∗ , u∗ , q∗ ), q∗ = (μ∗ , {vr∗ }), be a solution of problem (2)–(4). Without loss of generality, we ∗ (T ) containing elements q ∈ I (T ) such that Pr (q ) = q∗ . assume that e0 (p∗ ) = 0. Consider the set ID 2 D D D ∗ (T ) is ρ -closed. Take an arbitrary impulse By the ρw -continuity of the projection operator, the set ID w ∗ (T ) and consider the system control qD ∈ ID ⎧ ∗ ∗ ⎪ ⎨d(y, y ) = EdqD , y(t0 ) = y (t0 ) = 0, (72) dz = d|qD |, z(t0 ) = 0, ⎪ ⎩ ∗ 2 ∗ 2 ∗ 2 d = |y − y | d|qD | + |z − z | dt, (t0 ) = |z(t1 ) − z (t1 )| . Here, E is the identity matrix of dimension 2k4 × 2k4 ; the notation (y, y ∗ ) means the 2k4 -dimensional function whose first k4 components are the function y and whose last k4 components are the function y ∗ ; z ∗ (t) = F (t; |q∗D |), q∗D = (q∗ , q∗ ). We set c = gc supt∈T |x∗ (t)| + q∗D  + 1 and consider the Hilbert space R2n+1 × Lm 2 (T ) consisting of triples (p, d, u). Denote by X the subset of its elements (p, d, u), p = (x0 , x1 ), satisfying the following ∗ (T ) such that conditions: there exists an impulse control qD ∈ ID (1) qD  ≤ c; (2) on the closed interval [t0 , t1 ], there exists a solution x(t) of Eq. (3) corresponding to the triple (x0 , u, qD ) such that x1 = x(t1 ), gc supt∈T |x(t)| ≤ c, and u(t) ∈ U (t) a.e.; (3) d = (t1 ), where (t) is the solution of (72) corresponding to the element qD . 7132

Now the solution of Eq. (3) is understood in the sense of the expanded system (72), i.e., as the solution of the following equation, equivalent to the initial equation:   dx = f (x, u, t)dt + g(x, t), O dqD , where O is zero matrix of dimension k4 × n. The set X is nonempty, since it contains the triple (p∗ , 0, u∗ ) corresponding to the element q∗D . By the imposed constraints, Lemma 3.2, the ρw -compactness of impulse controls, and the ρw -continuity of the operator Pr, it follows that the set X is closed and hence is itself a complete metric space with metric induced by the norm |p|2 + |d|2 + u2 , (p, d, u) ∈ X.18  + Fix εi = i−1 , i ∈ N, and set e0,i (p, d) = e0 (p) + d + ε3i . On the nonnegative orthant, let us define the following lower semicontinuous function: ⎧ ⎨ ω1 , ω1 + ω2 > 0, Ω(ω1 , ω2 ) = ω1 + ω2 ⎩0, ω1 = ω2 = 0. In X, consider the functional k1 k2   j+     2 2 φi (p, d) = e0,i (p, d) + Ω [e1 (p)] , [e0,i (p, d)] + Ω [ej2 (p)]2 , [e0,i (p, d)]2 . j=1

j=1

The functional φi is lower semicontinuous and nonnegative. Moreover, φi (p∗ , 0) = ε3i . Let us apply the smooth variational principle [23]. For each i, there exist a triple ai = (pi , di , ui ), a sequence of triples ai,k = (pi,k , di,k , ui,k ) converging to ai in the metric of X as k → ∞, and also a sequence of numbers ci,k : 0 ≤ ci,k ≤ 2−k , k ∈ N, such that: first, |p∗ − pi,k |2 + |di,k |2 + u∗ − ui,k 2 ≤ const ε2i ε3i ;

∀k ∈ N;

(73)

∗ (T ) ID

third, there exists an impulse control qD,i ∈ such that the process (pi , 1,i = second, φi (pi , di ) ≤ di , ui , qD,i ) is a solution of the problem ⎧ ! " t1 ⎪ ⎪ ⎪ ⎪ ⎪ φi (p, 1 ) + εi χ1,i (p, 1 ) + χ2,i (u, t) dt → min, ⎪ ⎪ ⎪ ⎪ ⎪ t0 ⎪   ⎪ ⎪ ⎪ dx = f (x, u, t)dt + g(x, t), O dqD , ⎪ ⎪ ⎪ ⎨ d(y, y ∗ ) = EdqD , y0 = y0∗ = 0, (74) ⎪ ⎪ |, z = 0, dz = d|q 0 D ⎪ ⎪ ⎪  2 ⎪ ∗ 2 ⎪ ⎪ d = |y − y | d|qD | + |z − z ∗ |2 dt, 0 = z1 − q∗D  , t ∈ T, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪|p − pi | + |1 − di | ≤ δi , max{gc sup |x(t)|, qD } ≤ c, ⎪ ⎪ ⎩p = (x , x ), u(t) ∈ U (t), q ∈ I ∗ (T ). 0 1 D D Here, χ1,i (p, 1 ) =

∞ 

  ci,k |p − pi,k |2 + |1 − di,k |2 ,

k=1

ε3i ,

χ2,i (u, t) =

∞ 

ci,k |u − ui,k (t)|2 .

k=1

it follows that e0,i (pi , di ) > 0 for εi < 1. Indeed, Let us explain how δi is chosen. Since φi (pi , di ) ≤ e0 (pi ) < 0, and hence some of the constraints (4), say, ej1 , is violated for j = 1. if e0,i (pi, di ) = 0, then  2 Then Ω [e1+ 1 (pi )] , 0 = 1, which is impossible for εi < 1. Since e0,i (pi , di ) > 0, by the continuity of the function e0,i (p, d) there exists δi > 0 such that e0,i (p, d) > 0 ∀(p, d): |p − pi | + |d − di | ≤ δi . 18

0 Here and in what follows, u =

t1 t0

|u(t)|2 dt.

7133

Denote by xi , yi , zi , and i the optimal trajectories in problem (74). Passing to the limit, from (73) we have pi → p∗ , ui → u∗ a.e. and di → 0 for i → ∞. The latter means that (qD,i ) → 0. By the w ρw -convergence criterion, qD,i → (q∗ , q∗ ). Then xi (t) → x∗ (t) ∀t ∈ Cont(|q∗ |), gc sup |xi | → gc sup |x∗ | as i → ∞ (Lemma 3.2), and for large i, all the inequality-type constraints of problem (74) are strict. Therefore, necessary conditions of Theorem 5.1 are applied to it. By Theorem 5.1, there exist functions ψi ∈ V n (T ), ζi ∈ V k4 (T ), ςi , θi ∈ C(T ) and a number λ0,i > 0 for which19 |λi | + gc sup |ψi (t)| = 1, t∈T



 dψi = −Hxi (t)dt − Qix (t), O dqD,i , (−1)k ψk,i =

t ∈ T,

∂χ1,i ∂l ∂e0 (pi , λi ) − κi (pi ) + λ0,i εi (pi , 1,i ), ∂xk ∂xk ∂xk dζi = −2θi (yi − yi∗ )d|qD,i |, ζi (t1 ) = 0,

dςi = −2θi (zi − z ∗ )dt,

k = 0, 1,

ςi (t1 ) = 2θi (z1,i − q∗D ),

∂χ1,i θi (t1 ) = −λ0,i − λ0,i εi (pi , 1,i ), ∂1   max Hi (u, t) − λ0,i εi χ2,i (u, t) = Hi (t) − λ0,i εi χ2,i (ui (t), t) a.e., dθi = 0,

u∈U (t)

  gc sup Qi (t), m ≤ ci |m| t∈T



∀m ∈ K,

(76)

  (Qi (t), 0), dqD,i + ci ≥ 0.

T

Here,

2 e+ 1 (pi )e0,i (pi ) λ1,i = 2λ0,i  2 , 2 2 [e+ 1 (pi )] + e0,i (pi )

λi = (λ0,i , λ1,i , λ2,i ),

e2 (pi )e20,i (pi )

2 , 2 2 [e+ 2 (pi )] + e0,i (pi )     + κi = e−1 0,i (pi ) λ1,i , e1 (pi ) + λ2,i , e2 (pi ) ; λ2,i = 2λ0,i 

the numbers ci ≥ 0, ci → 0 as i → ∞; the subscript i of the functions H and Q (and also of their partial derivatives) means that in them, for a part of the variables x, u, and ψ, we substitute the values xi (t), ui (t), and ψi (t), respectively; the symbol 0 denotes the vector consisting of k4 zeros. |e+ (pi )| |e2 (pi )| , → 0. This implies κi → 0 as i → ∞. Since φi (pi , di ) → 0 as i → ∞, it follows that 1 e0,i (pi ) e0,i (pi ) Using this property and passing to the limit with the use of Lemma 3.2 in the necessary conditions of the i-problem, we obtain the assertion of the theorem. 19

Problem (74) is not standard, since (a) its dynamics contains the measure |qD | and (b) the last k4 coordinates of the vector qD are fixed: Pr2 (qD ) = q∗ . However, Theorem 5.1 for such a problem is proved in exactly the same way. Let us explain in more detail from where the numbers ci ≥ 0, ci → 0 appear in the right-hand side of inequality (76). Let ∗ )}). We set bi = θi |yi − yi∗ |2 + ςi , hi = gc sup[|bi | + |ζi |]. It is easy to see that hi → 0 as i → ∞. qD,i = (μD,i ; νD,i ; {(vr,i , vr,i Following the proof of Theorem 5.1, we obtain the following conditions:   Qi (t), m ≤ hi |m| ∀m ∈ K ∀t ∈ (t0 , t1 ), % &    vr,i (s) Mr,i (s) = max Qr,i (s) + ζr,i (s), v + |v|br,i (s) = Qr,i (s) + ζr,i (s), (75) + br,i (s) v∈Sr |vr,i (s)| + for a.a. s ∈ Γ+ r,i , r ∈ Ds(|qD,i |). Here, Sr = {v ∈ K : |v| = 1}, Γr,i = {s ∈ [0, 1] : |vr,i (s)| > 0}. By the Dem’yanov theorem [15], from the properties of the Hamiltonian system, relation (75), using the condition α˙ r,i (s) =  ζr,i  + |br,i L∞ ≤ 1(i) → 0 as i → ∞, we deduce that σ˙ r,i (s) = 0 for a.a. s ∈ [0, 1] \ Γ+ r,i , and also the condition   L∞  ≤ 1(i) (uniformly in r). Since ∃k ∈ {0, 1}: Qr,i (k), m ≤ hi |m| ∀m ∈ K, it follows from this that ∃ci ≥ 0, ci → 0: M   r,i L∞  Qr,i (s), m ≤ ci |m| ∀s ∈ [0, 1] ∀m ∈ K ∀r ∈ Ds(|qD,i |). Therefore, (76) is true.

7134

Theorem 9.2. Let (p∗ , u∗ , q∗ ), q∗ = (μ∗ ; {vr∗ }), be a solution of problem (2)–(5), and let the state and endpoint constraints be in concordance. Then there exist a number λ0 ≥ 0, vectors λj ∈ Rkj , j = ∗ (T ): 1, 2, λ1 ≥ 0, a vector-function ψ ∈ V n (T ), a vector-valued measure η = (η 1 , . . . , η k3 ), η j ∈ C+ ∗ ∗ n Ds(|q |) ∩ Ds(η) = ∅, and for each point r ∈ Ds(|q |), there exist its own vector-function σr ∈ V ([0, 1]) ∗ ([0, 1]), such that and its own vector-valued measure ηr = (ηr1 , . . . , ηrk3 ), ηrj ∈ C+ 

t ψ(t) = ψ0 −

Qx (s) dμ∗c

Hx (s) ds − t0

[t0 ,t]

 +

∗ ϕ x (x , s) dη + Φ(ψ, t),

t ∈ (t0 , t1 ],

[t0 ,t]

Φ(ψ, t) =



[σr (1) − ψ(r− )],

(77)

r∈Ds(|q∗ |) r≤t

⎧ dαr∗ = g(αr∗ , r)vr∗ ds, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k4 ⎪  ⎪ ∗ ⎨dσ = − gxj (αr∗ , r)σr vr∗j ds + ϕ s ∈ [0, 1], r x (αr , r)dηr , j=1 ⎪ ⎪ ⎪ ⎪ αr∗ (0) = x∗ (r− ), σr (0) = ψ(r− ), ⎪ ⎪ ⎪ ⎪ ⎩ supp(ηrj ) ⊆ {s : ϕj (αr∗ (s), r) = 0}, j = 1, . . . , k3 , r ∈ Ds(|q∗ |); ∂l ∗ ∂l ∗ (p , λ), ψ1 = − (p , λ); ∂x0 ∂x1   λ1 , e1 (p∗ ) = 0, ϕj (x∗ (t), t) = 0 η j -a.e. ψ0 =

(79) ∀j;

max H(u, t) = H(t) a.e.;

u∈U (t)

  gc sup Q(t), m ≤ 0

 ∀m ∈ K,

|λ| + η +





(78)

 Q(t), dq∗ = 0;

(80) (81) (82)

T

ηr  = 1.

(83)

r∈Ds(|q∗ |)

Remark. Here, the operation gc sup and the integration in (82) are understood according to the adjoint dμ∗c be the Radon–Nikodym derivative of the vector-valued measure μ∗c with system (78). Let m∗c = d|μ∗c | respect to the measure |μ∗c |. Then (82) means the following: ◦ 1] ∀r ∈ Ds(|q∗ |); (1) Q(t) ∈ K ◦ ∀t ∈ T and Qr (s) ∈ K  ∀s ∈ [0,  ∗ ∗ ∗ (2) Q(t), mc (t) = 0 |μc |-a.e. and Qr (s), vr (s) = 0 a.e. in s ∈ [0, 1] ∀r ∈ Ds(|q∗ |).

Before proving the theorem, we consider the following two auxiliary assertions. Proposition 9.1. On the closed interval [0, 1], consider a sequence of continuous nonnegative functions fi ∈ C([0, 1]): fi = 0, fi (0) = fi (1) = 0 ∀i, uniformly converging to a Lipschitzian function f = 0. Let a sequence of positive numbers yi converge to 0. Then 1 Ai = 0

fi (t) dt → ∞. [fi (t)]2 + yi 7135

Proof. Denote by c > 0 the Lipschitz constant of the function f . Since f = 0 and f (0) = 0, there exist a point s ∈ [0, 1) and a number δ > 0 such that f (s) = 0, f (t) > 0 ∀t ∈ (s, s + δ]. Fix ε: 0 < ε < δ. By the uniform convergence, s+δ s+δ   fi (t) 1 dt. dt → 2 [fi (t)] + yi f (t) s+ε

s+ε



s+δ

By the Lipschitzian property, f (t) ≤ c|t − s|. But the integral

1 dt diverges. Then Ai → ∞. c|t − s|

s

The proposition is proved.

Proposition 9.2. On the closed interval [0, 1], consider a sequence of absolutely continuous nonnegative functions fi ∈ C([0, 1]): fi L∞ ≤ c, fi (0) = fi (1) = 0, converging to zero. Let yi > 0, and let 1 Ai = If yi → 0, then [fi (t)]2 e

− y1

[fi (t)]2 + e

0 1 yi

fi (t)

dt → 0.

i

→ 0 uniformly as i → ∞. − 2y1

Proof. Assume the contrary. Then ∃ε > 0, ∃t˜i ∈ (0, 1): fi (t˜i ) ≥ 2εe − 2y1

are continuous, it follows that ∃ti ∈ (0, 1): fi (ti ) = 2εe − 2y1 i

∀i that εe

− 2y1 i

≤ fi (t) ≤ 3εe 1 Ai ≥ 2

∀i. But since fi (0) = 0 and fi

∀i. We deduce from the condition |fi (t)| ≤ c − 2y1

whenever t ∈ O(ti ) = {t : |t − ti | ≤ εc−1 e

 O(ti )

i

i

− 2y1

εe

− y1

9ε2 e

− y1

i

i

− y1

+e

i

dt =

ε2 c−1 e − y1

9ε2 e

i

i

− y1

+e

i

→

i

}. This implies

ε2 c−1 > 0. 9ε2 + 1

We obtain a contradiction. The proposition is proved. Proof of Theorem 9.2. Let (p∗ , u∗ , q∗ ), q∗ = (μ∗ , {vr∗ }), be a solution of problem (2)–(5). Without loss ∗ (T ) consisting of of generality, assume that e0 (p∗ ) = 0. As in proving Theorem 9.1, introduce the set ID ∗ elements qD ∈ ID (T ) such that Pr2 (qD ) = q . By the ρw -continuity of the projection operator, the set ∗ (T ) is ρ -closed. ID w Let ε > 0 be the number from the definition of concordance of the state and endpoint constraints. We set c = gc sup |x∗ (t)| + q∗D  + 1, q∗D = (q∗ , q∗ ), and consider the Hilbert space R2n+1 × Lm 2 (T ) consisting t∈T

of triples a = (p, d, u). Denote by X the subset of its elements (p, d, u), p = (x0 , x1 ), for which there exists ∗ (T ) such that an impulse control qD ∈ ID

(1) qD  ≤ c, e1 (p) ≤ 0, e2 (p) = 0, |p − p∗ | ≤ ε; (2) on the closed interval [t0 , t1 ], there exists a solution x(t) of Eq. (3) corresponding to the triple (x0 , u, qD ) such that x1 = x(t1 ), gc sup |x(t)| ≤ c, and u(t) ∈ U (t) a.e.; t∈T

(3) d = (t1 ), where (t) is the solution of (72) corresponding to the element qD . For each triple a ∈ X, denote the set of elements qD corresponding to conditions (1)–(3) by QD (a). The set X is nonempty, since it contains the triple a∗ = (p∗ , 0, u∗ ). Indeed, this is true, since the set QD (a∗ ) is nonempty: q∗D ∈ QD (a∗ ). By the imposed constraints, Lemma 3.2, the ρw -compactness of impulse controls, and the ρw -continuity of the operator Pr, the set X is closed, and hence is itself a complete metric space with metric induced by the norm |p|2 + |d|2 + u2 , (p, d, u) ∈ X.  + Fix εi = i−1 , i ∈ N, and set e0,i (p, d) = e0 (p) + d + ε3i . Let a = (p, u, d) ∈ X, qD ∈ QD (a); then x(t; a, qD ) is the trajectory corresponding to the triple (p, u, qD ) by Eq. (3). Shortening the notation, 7136

we denote ω j (t) = ϕj+ (x(t; a, qD ), t), γi = e0,i (p, d), and for each j = 1, . . . , k3 , consider the following functional on the space X: ⎧    [ω j ]2 ⎪ ⎪ ⎪ dt + d|qD | , γi > 0, min ⎪ 1 ⎪ − ⎨qD ∈QD (a) [ω j ]4 + e γi T Ωj (a) = 1, gc sup ω j > 0, γi = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0, gc sup ω j = 0, γ = 0. i Note that in the definition of Ωj , the minimum is always attained for γi > 0, since the set QD (a) is compact in the ρw -metric. Let us show that Ωj is lower semicontinuous on X. For this purpose, we perform certain A additional constructions. Let qD ∈ QD (a), a ∈ X, x = x(t; a, qD ), and let μA D,i = μD,i [qD ] be an arbitrary family approximating the impulse control qD . We set20 πi (t) =

t − t0 + F (t; |μA D,i |) t1 − t0 + μA D,i 

,

i ∈ N,

t ∈ T.

The function πi is absolutely continuous, strictly increases on the closed interval T , and maps it onto the closed interval [0, 1]. Hence there exists the inverse function θi : [0, 1] → T , which also strictly increases. ˜i ), where a ˜i = (x0 , u, μA ˜i uniformly We set x ˜i (s) = x(θi (s); a D,i ). It is easy to prove that the functions x n ˜. It is easy to verify that this function converge in C ([0, 1]) to a certain n-vector-function denoted by x is independent of the choice of the approximating sequence for qD and is uniquely defined. The function x ˜(s) = ˜(s;  a, qD ) is defined on the closed interval [0, 1] and is absolutely continuous  x   d˜ x on it and, moreover,   ≤ const, where the constant const depends only on the number c defined dt above. Therefore, to each discontinuous trajectory x = x(t; a, qD ) of the impulse system (3), using the transformation described above and denoted by F, we put in correspondence a certain ordinary absolutely continuous trajectory x ˜ = F[x]. In what follows, we will need the following two properties of 21 the transformation F : • for any continuous function φ : Rn × R1 → R1 , we have gc sup φ(x, t) = F[φ(x, t)]C ; w • if x0,i → x0 , ui − u → 0, and qD,i → qD , then, as follows from Lemma 3.2, xi (t) = x(t; x0,i , ui , qD,i ) → x(t) = x(t; x0 , u, qD )

∀t ∈ Cont(|qD |).

For the transformed functions, the uniform convergence F[xi ] − F[x]C → 0 holds as i → ∞. ˜ j (t) = By the concordance of the state and endpoint constraints, we have ω ˜ j (k) = 0, k = 0, 1, where ω j j F[ω (t; a, qD )], j = 1, . . . , k3 , whenever a ∈ X. Now the lower semicontinuity of Ω directly follows from the definition and Proposition 9.1. Let us consider the following functional on X: φi (a) = e0,i (p, d) +

k3 

Ωj (a).

j=1

The functional φi is lower semicontinuous and nonnegative. Moreover, φi (a∗ ) = ε3i . Let us apply the smooth variational principle [23]. For each i, there exist a process ai = (pi , di , ui ), a sequence of triples ai,k = (pi,k , di,k , ui,k ) converging in the metric of X to ai as k → ∞, and also a sequence of numbers ci,k : 0 ≤ ci,k ≤ 2−k , k ∈ N, such that, first, |p∗ − pi,k |2 + |di,k |2 + u∗ − ui,k 2 ≤ const ε2i

∀k ∈ N,

(84)

20

The so-called “discontinuous change of time” (see [18, 32, 36, 42, 50]). Its proof is easy and directly follows from the definitions. See, e.g., the arguments of Sec. 3 and similar arguments in [36, 42, 43]. 21

7137

∗ (T ) such that the process second, φi (ai ) ≤ ε3i , and, third, there exists an impulse control qD,i ∈ ID (pi , 1,i = di , ui , qD,i ) is a solution of the problem ⎧ " ! t1 k3  ⎪ j+ (x, t)]2  ⎪   [ϕ ⎪ ⎪ dt + d|qD | + εi χ1,i (p, 1 ) + χ2,i (u, t)dt → min, e0,i (p, 1 ) + ⎪ ⎪ − γ1 ⎪ j+ 4 ⎪ j=1 T [ϕ (x, t)] + e ⎪ t0 ⎪   ⎪ ⎪ ⎪ , dx = f (x, u, t)dt + g(x, t), O dq ⎪ D ⎪ ⎪ ⎪ ⎪d(y, y ∗ ) = Edq , y = y ∗ = 0, ⎪ 0 D ⎪ 0 ⎪ ⎪ ⎨ dz = d|qD |, z0 = 0, (85) ⎪d = |y − y ∗ |2 d|q | + |z − z ∗ |2 dt,  = |z − q∗ |2 , ⎪ 0 1 D ⎪ D ⎪ ⎪ ⎪ ⎪ ⎪ = e (p,  ), t ∈ T, dγ = 0, γ 0 0,i 1 ⎪ ⎪ ⎪ ⎪ ⎪ e1 (p) ≤ 0, e2 (p) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ |p − pi | + |1 − di | ≤ δi , |p − p∗ | ≤ ε, max{gc sup |x(t)|, qD } ≤ c, ⎪ ⎪ ⎪ ⎩ ∗ (T ). p = (x0 , x1 ), u(t) ∈ U (t) a.e., qD ∈ ID

Here, z ∗ (t) = F (t; |q∗D |), χ1,i (p, 1 ) =

∞ 

  ci,k |p − pi,k |2 + |1 − di,k |2 ,

k=1

χ2,i (u, t) =

∞ 

ci,k |u − ui,k (t)|2 .

k=1

ε3i ,

it follows that e0,i (pi , di ) > 0 for εi < 1. Indeed, if Let us explain how δi is chosen. Since φi (ai ) ≤ e0,i (pi , di ) = 0, then e0 (pi ) < 0, and hence some of the constraints (5), say, ϕj for j = 1, is violated. Then Ωj (ai ) = 1, which is impossible for εi < 1. Since e0,i (pi , di ) > 0, by the continuity of the function e0,i (p, d), there exists δi > 0 : e0,i (p, d) > 0 ∀(p, d): |p − pi | + |d − di | ≤ δi . Denote by xi , yi , zi , i , and γi the optimal trajectories in problem (85). Passing to a subsequence, we have from (84) that pi → p∗ , ui → u∗ a.e., and di → 0 as i → ∞. The latter means that (qD,i ) → 0. By w the ρw -convergence criterion, qD,i → q∗D . Then xi (t) → x∗ (t) ∀t ∈ Cont(|q∗ |) and gc sup |xi | → gc sup |x∗ | as i → ∞ (Lemma 3.2). Therefore, for large i, all the inequality-type constraints of problem (85) are strict. Therefore, the necessary conditions of Theorem 9.1 are applied to it. By Theorem 9.1, there exist functions ψi ∈ V n (T ), ζi ∈ V k4 (T ), ξi ∈ V (T ), and ςi , θi ∈ C(T ), a number λ0,i ≥ 0, and vectors λj,i ∈ Rkj , j = 1, 2, for which22 k3     |λi | + gc sup |ψi (t)| + nji (t) dt + d|qD,i | = 1, (86) t∈T

j=1 T

k3        ϕjx (xi , t) 1 − hji (t) nji (t) dt + d|qD,i | , dψi = −Hxi (t)dt − Qix (t), O dqD,i +

t ∈ T,

(87)

j=1

∂χ1,i ∂l (−1)k ψk,i = (pi , λi ) + λ0,i εi (pi , 1,i ), k = 0, 1, ∂xk ∂xk dζi = −2θi (yi − y ∗ )d|qD,i |, ζi (t1 ) = 0,

(88)

dςi = −2θi (zi − z ∗ )dt, dθi = 0, 22

ςi (t1 ) = 2θi (z1,i − q∗D ), ∂χ1,i θi (t1 ) = −λ0,i − λ0,i εi (pi , 1,i ), ∂1

Problem (85) is not standard, since (a) its dynamics contains the measure |qD |, (b) the last k4 coordinates of the vector qD are fixed: Pr2 (qD ) = q∗ . However, Theorem 9.1 for such a problem is proved in exactly the same way. For the appearance of the numbers ci ≥ 0, ci → 0 in the right-hand side of (90), see analogous arguments in proving Theorem 9.1.

7138

dξi =

k3  ϕj+ (xi , t)

2γi2

j=1

  nji (t) dt + d|qD,i | ,

ξi (t1 ) = 0,



 e1 (pi ), λ1,i = 0,   max Hi (u, t) − λ0,i εi χ2,i (u, t) = Hi (t) − λ0,i εi χ2,i (ui (t), t) a.e., λ1,i ≥ 0,

u∈U (t)

   gc sup Qi (t), m − κi (t)|m| ≤ ci |m| ∀m ∈ K,    (Qi (t), 0), dqD,i + ci ≥ 0.

(89) (90) (91)

T 1

Here, λi = (λ0,i , λ1,i , λ2,i ), ci ≥ 0, ci → 0; hji (x, t) = [ϕj+ (x, t)]4 e γi , −1  − 1 −2 nji (x, t) = 2λ0,i ϕj+ (x, t)e γi [ϕj+ (x, t)]4 + e γi , κi (x, t) = λ0,i

k3 

[ϕj+ (x, t)]2

j=1

[ϕj+ (x, t)]4 + e

− γ1

j = 1, . . . , k3 ,

;

i

the omitted argument x of the functions hi , ni , and κi means that it is replaced by xi (t); the subscript i of the functions H and Q (and also of their partial derivatives) means that a part of the variables x, u, and ψ are replaced by xi (t), ui (t), and ψi (t), respectively; the symbol 0 denotes the vector consisting of k4 zeros. Also, we set hi (x, t) = (h1i (x, t), . . . , hki 3 (x, t)) and ni (x, t) = (n1i (x, t), . . . , nki 3 (x, t)). w From (86) and the condition qD,i → q∗D , we deduce that gc sup |ζi | + ςi C → 0 as i → ∞. Let j = 1, . . . , k3 . From the variational principle, we know that    [ωij ]2 dt + d|qD,i | → 0, (92) 1 −γ j 4 i [ω ] + e i T ˜ ij = F[ωij ]. Condition (92) for where ωij (t) = ϕj+ (xi , t). Now let us consider the transformed functions ω them becomes 1 [˜ ωij ]2 dt → 0. − γ1 j 4 i [˜ ω ] + e i 0 The concordance of the state and endpoint constraints implies ω ˜ ij (k) = ωij (tk ) = 0, k = 0, 1. Applying j 2 ωi ] , we obtain gc sup |hi | → 0 as i → ∞. From this and (86), we Proposition 9.2 to the functions fi = [˜ deduce that gc sup |ξi | → 0. ∗ (T ), j = 1, . . . , k , are defined as follows: We set ηi = (ηi1 , . . . , ηik3 ), where the Borel measures ηij ∈ C+ 3 t F (t; ηij )

=

  nji (s) ds + d|qD,i | ,

t ∈ (t0 , t1 ],

F (t0 ; ηij ) = 0.

t0 w

Passing to a subsequence, we obtain from (86) and the compactness arguments that λi → λ, ηi → η˜, i → ∞. We set η(B) = η˜(B) − η˜(B ∩ Ds(|q∗ |)) ∀B ∈ σ(T ). Thus, η = (η 1 , . . . , η k3 ) is a vector-valued measure such that Ds(|q∗ |) ∩ Ds(η) = ∅. It follows from (7) and (86) that the variations of ψi are uniformly bounded. Using the second Helly theorem and passing to a subsequence, we have ψi (t) → ψH (t) ∀t ∈ T . Let us find a function ψ ∈ V n (T ) such that ψ(t) = ψH (t) ∀t ∈ Cont(|q∗ |) ∩ Cont(η). The existence of such a function ψ(t) follows from the estimate Var |ba [ψH ] ≤ c(|q∗ | + |˜ η | + L)([a, b]) ∀a ≤ b, which is obtained from the inequality |ϑ[ψi ]| ≤ c(| Pr1 (qD,i )| + |ηi | + L) as i → ∞. 7139

Let us prove that ψ(t) satisfies the conditions of the theorem. For this purpose, we construct a family of absolutely continuous measures {ˆ μi }, i ∈ N, approximating all necessary optimality conditions in the i-problem as follows. Let {tk }, k ∈ N, be a countable set of points everywhere dense in T such that tk ∈ X = Cont(|q∗ |) ∩ ∞ 

( Cont(|qi |) . For each i, consider a sequence of absolutely continuous 2k4 -vector-valued Cont(η) ∩ i=1

w

measures μD,i,τ = (μi,τ , μ∗i,τ ) → qD,i , τ → ∞, approximating the element qD,i . Let a pair (xi,τ , ψi,τ ) satisfy the system ⎧ ⎪ dxi,τ = f (xi,τ , ui , t)dt + g(xi,τ , t)dμi,τ , ⎪ ⎪ ⎪ k3 ⎨    j j dψi,τ = −Hx (xi,τ , ui , ψi,τ , t)dt − Qx (xi,τ , ψi,τ , t)dμi,τ + ϕj x (xi,τ , t) 1 − hi (xi,τ , t) dηi,τ , ⎪ ⎪ j=1 ⎪ ⎪ ⎩ xi,τ (t0 ) = xi (t0 ), ψi,τ (t0 ) = ψi (t0 ). 1 , . . . , η k3 ) and Here, ηi,τ = (ηi,τ i,τ

t F (t; ηi,τ ) = 2λ0,i

  ni (xi,τ , s) ds + d|μi,τ | .

t0

Using Lemma 3.2, choose the number τi such that  ⎡    1  ) + F (t; ηi,τi ) − F (t; ηi ) dt + ηi,τi (T ) − ηi (T ) ≤ ; ρ (q , μ ⎢ w D,i D,i,τi i ⎢ T ⎢ i ⎢     ⎢ ψi,τ (tk ) − ψi (tk ) + gc sup |ψi (t)| − max |ψi,τ (t)| ≤ 1 ; ⎢ i i t∈Ti ⎢ i t∈Ti ⎢ k=1 ⎢    ⎢      ⎢  Qi (xi,τi , ψi,τi , t), dμi,τi − (Qi (xi , ψi , t), 0), dqD,i  ⎢  ⎢   ⎢ T T ⎢    ⎢   1   ⎢  ⎢ + κi (t) dt + d|qD,i | − κi (xi,τi , t)(dt + d|μD,i,τi |) ≤ ; ⎢   i ⎢ T ⎣  T  hi (xi,τ , t)C − gc sup |hi |≤ 1 . i i Moreover, let    gc sup Qi (xi,τi , ψi,τi , t), m − κi (xi,τi , t)|m| ≤ 1(i)|m| ∀m ∈ K.

(93)

ˆi = xi,τi , μ ˆi = μi,τi , μ ˆD,i = μD,i,τi , and ηˆi = ηi,τi . Passing to a subsequence, we have We set ψˆi = ψi,τi , x ψˆi (t) → ψA (t) ∀t ∈ T . Let us show that ψA (t) = ψ(t) ∀t ∈ Cont(|q∗ |)∩Cont(η). Indeed, since |ϑ[ψi ]| ≤ c(| Pr1 (qD,i )|+|ηi |+L), the function ψ(t) is continuous on the set Cont(|q∗ |) ∩ Cont(η) \ {t0 , t1 }. The same is true for the function ψA . Then (93) implies ψA (t) = ψ(t) ∀t ∈ Cont(|q∗ |) ∩ Cont(η). Moreover, (93), obviously implies w w w ˆi = Pr1 (ˆ μD,i ) → q∗ , ηˆi → η˜, and then x ˆi (t) → x∗ (t) ∀t ∈ Cont(|q∗ |) as i → ∞. μ ˆD,i → (q∗ , q∗ ) ⇒ μ Let us prove (77) and (79). Let ν ∗ = |q∗ |. Let us show that there exist sequences of absolutely continuous vector-valued measures ηi }, and also a sequence of natural numbers ki ≥ i such that {¯ μi } and {¯ w

w

μi |) → (μ∗d , νd∗ ), η¯i → η˜ − η for i → ∞; (1) (¯ μi , |¯ w (2) |ˆ μki − μ ¯i | → νc∗ , i → ∞, ηˆki ≥ η¯i ∀i. 7140

If Ds(ν ∗ ) = ∅, then we set μ ¯i = η¯i = 0, ki = i ∀i. Let Ds(ν ∗ ) = ∅. Consider an ordered (with respect to inclusion) chain of sets Di such that D0 = ∅,  1 ν ∗ ({r}) ≤ , i ∈ N. Define the sets Sr,i = [r − εi , r + εi ], r ∈ Di , as Di−1 ⊆ Di ⊆ Ds(ν ∗ ), and i ∗ r∈Ds(ν )\Di

a system of closed pairwise disjoint neighborhoods of the points r such that (1) εi > 0, εi → 0 as i → ∞;   1 ν ∗ (Sr,i ) − ν ∗ ({r}) + |μ∗ (Sr,i ) − μ∗ ({r})| + |˜ η (Sr,i ) − η˜({r})| ≤ ; (2) i r∈Di

(3) r ± εi ∈ Cont(ν ∗ ) ∩ Cont(η). The existence of the set Sr,i follows from the regularity of the measures ν ∗ and η˜. The sequence ki ≥ i is chosen so that        1 |ˆ ˆki (Sr,i ) − μ∗ (Sr,i ) + ηˆki (Sr,i ) − η˜(Sr,i ) ≤ . μki |(Sr,i ) − ν ∗ (Sr,i ) + μ i r∈Di

ˆki (B ∩ Si ) and η¯i (B) = This can be done because of the weak convergence of . measures. We set μ ¯i (B) = μ w 1 Sr,i . It is easy to see that (¯ μi , |¯ μi |) → (μ∗d , νd∗ ), ηˆki (B ∩ Si ) for any Borel set B ⊆ R . Here, Si = w

w

r∈Di

w

μki − μ ¯i | → νc∗ , ηˆki − η¯i → η). Also, it is obvious that ηˆki ≥ η¯i ∀i. η¯i → η˜ − η as i → ∞ (and hence |ˆ Using the constructed numbers ki , from the sequences {ˆ μi }, {ˆ ηi }, {ˆ xi }, {ψˆi }, and {ui }, we choose subsequences and denote them (as the initial sequences) by the same subscript i. Rewrite Eqs. (3) and (87) in the following form23 : ⎧   t ⎪ ⎪ ⎪ ⎪ xi , ui , s) ds + g(ˆ xi , s) d(ˆ μi − μ ¯i ) + g(ˆ xi , s) d¯ μi , x ˆi (t) = x0,i + f (ˆ ⎪ ⎪ ⎪ ⎪ ⎪ t0 [t0 ,t] [t0 ,t] ⎪ ⎪ ⎪ ⎨   t ˆ xi (s) ds − ˆ ix (s) d(ˆ ˆ ix (s) d¯ Q Q μi − μ ¯i ) − μi ψˆi (t) = ψ0,i − H ⎪ ⎪ ⎪ ⎪ t0 [t0 ,t] [t0 ,t] ⎪  ⎪ ⎪ ⎪ ⎪   ⎪ + ϕx (ˆ xi , s) d(ˆ ηi − η¯i ) + ϕx (ˆ xi , s) d¯ ηi + 1(t; i), t ∈ T. ⎪ ⎪ ⎩ [t0 ,t]

[t0 ,t]

Here and in what follows, the subscript i and the “hat” of the functions H and Q (and also of their partial derivatives) mean that for a part of the omitted variables x, u, and ψ, we substitute x ˆi (t), ui (t), and ψˆi (t), respectively. We set t x ˆci (t)

= x0,i +

 g(ˆ xi , s) d(ˆ μi − μ ¯i ),

f (ˆ xi , ui , s) ds + [t0 ,t]

t0

t ψˆiη (t)



 ˆ ix (s) d(ˆ μi − μ ¯i ) + Q

[t0 ,t]

ψˆid (t) = −



ˆ ix (s) d¯ μi + Q

[t0 ,t]

=

g(ˆ xi , s) d¯ μi , [t0 ,t]

 ˆ xi (s) ds − H

= ψ0,i − t0

23

 x ˆdi (t)

ϕ xi , s) d(ˆ ηi − η¯i ) + 1(t; i), x (ˆ

[t0 ,t]

ϕ xi , s) d¯ ηi , x (ˆ

t ∈ T.

[t0 ,t]

Here, 1(t; i) → 0 uniformly in C n (T ) as i → ∞.

7141

Thus, x ˆi (t) = x ˆci (t) + x ˆdi (t) and ψˆi (t) = ψˆiη (t) + ψˆid (t) ∀t ∈ T . Taking into account that x ˆi (t) → x∗ (t) ∀t ∈ Cont(ν ∗ ), by the Lebesgue theorem and Lemma 3.1, we have x ˆci (t)

→

x∗c (t)

=

x∗0

t +

∗



∗

f (x , u , s) ds +

g(x∗ , s) dμ∗c

∀t ∈ T,

i → ∞.

[t0 ,t]

t0

Analogously, from Lemma 3.1, as i → ∞, we have t ψˆiη (t) → ψη (t) = ψ0 −

 Hx (s) ds −

Qx (s) dμ∗c

[t0 ,t]

t0

 +

∗ ϕ x (x , s) dη

∀t ∈ Cont(η).

[t0 ,t]

We set ψd = ψ − ψη . Then ψˆid (t) → ψd (t) ∀t ∈ Cont(ν ∗ ) ∩ Cont(η). ∗ Let us show that ψd = Σ(ψ,

t). Indeed, let t ∈ Cont(ν ) ∩ Cont(η). Fix ε > 0. Choose a number  ν ∗ ({r}) + |˜ η |({r}) ≤ ε. In such a case, N = N (ε) so that r∈Ds(ν ∗ )\DN

  ⎡ ⎤         ⎢ ⎥ ˆd (t) ≤ const ε. ˆ ix (s) d¯ −Q μi + ϕ (ˆ x , s) d¯ η − ψ lim sup  ⎣ ⎦ i i x i  i→∞  r∈D(N,t) Sr,i  Sr,i Here and in what follows, D(N, t) = {r ∈ DN : r ≤ t}. We set ri− = r − εi and, for r ∈ D(N, t), on the closed interval Sr,i , consider the system  ⎧ − d d ⎪ ⎪ ˆi (ri ) + g(ˆ xi (τ ), τ ) d¯ μi , ˆi (s) = x ⎪x ⎪ ⎪ ⎨ [ri−,s]  ⎪ d d −  ˆ ˆ ˆ ⎪ (s) = ψ (r ) − g (ˆ x (τ ), τ ) ψ (τ ) d¯ μ + ϕ xi (τ ), τ ) d¯ ηi . ψ ⎪ i i i i i i x x (ˆ ⎪ ⎪ ⎩ − − [ri ,s]

(94)

(95)

[ri ,s]

For sufficiently large i, define the functions24 πr,i (τ ) =

F (τ ; |¯ μi |) − F (ri− ; |¯ μi |) , |¯ μi |(Sr,i )

r ∈ D(N, t),

τ ∈ Sr,i .

The function πr,i maps the closed interval Sr,i into the closed interval [0, 1] and is absolutely continuous dπr,i |m ¯ i (τ )| = > 0 a.e. (m ¯ i (τ ) is the density of μ ¯i ). Therefore, there exists and strictly increasing: dτ |¯ μi |(Sr,i ) the inverse function θr,i : [0, 1] → Sr,i , θr,i = (πr,i )−1 , which is also absolutely continuous [34]. Making the change ω = πr,i (τ ) of the variable under the integral sign in (95) and transforming, we arrive at the system s d − c ˆi (ri ) + x ˆi (θr,i (s)) + g(αr,i (ω), θr,i (ω))vr,i (ω) dω, αr,i (s) = x σr,i (s) = ψˆid (ri− ) + ψˆiη (θr,i (s)) −  +

0 s   k4

j gxj (αr,i (ω), θr,i (ω))σr,i (ω)vr,i (ω) dω

0

j=1

ϕ x (αr,i (ω), θr,i (ω)) dηr,i ,

s ∈ [0, 1],

r ∈ D(N, t),

[0,s] 24

The numbers i must be taken starting from a certain i0 such that t ∈ / Sr,i ∀i ≥ i0 , r ∈ D(N, t).

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where j F (ω; ηr,i ) = F (θr,i (ω); η¯ij ),

σr,i (ω) = ψˆi (θr,i (ω)),

ˆi (θr,i (ω)), αr,i (ω) = x

j = 1, . . . , k3 ,

m ¯ j (θr,i (ω))|¯ μi |(Sr,i ) j , j = 1, . . . , k4 . (ω) = i vr,i |m ¯ i (θr,i (ω))|  s  s w w The condition μ ˆi → q∗ implies vr,i dω → vr∗ dω uniformly in C k4 ([0, 1]). Whence vr,i → vr∗ weakly 0

0

in Lk24 ([0, 1]). Further, it is easy to see that θr,i (ω) → r uniformly on [0, 1], and by Proposition 3.1, x ˆci (θr,i (ω)) → x∗c (r) and ψˆiη (θr,i (ω)) → ψη (r) uniformly on [0, 1] as i → ∞. The Gronwall inequality implies that αr,i is a Cauchy sequence of functions in i in C n ([0, 1]). Its limit is exactly the function αr∗ .25 By compactness arguments, there exist a vector-valued measure ηr and a function σr ∈ V n ([0, 1]) such w that after the passage to a subsequence, ηˆr,i → ηr and σr,i (s) → σr (s) ∀s ∈ Cont(ηr ). Passing to the limit in the latter system as i → ∞, we have s αr∗ (s) = x∗ (r− ) + g(αr∗ , r)vr∗ dω, 0

σr (s) = ψ(r− ) −

s  k4 0

(Here, we use the fact that from this and (94) that

gxj (αr∗ , r)σr vr∗j dω +

j=1

x ˆdi (ri− )



∗ ϕ x (αr , r) dηr ,

s ∈ [0, 1],

r ∈ D(N, t).

[0,s]

→

x∗ (r− )

−

x∗c (r)

and ψˆid (ri− ) → ψd (r− ) by construction.) It follows

       −  ψd (t) − [σr (1) − ψ(r )] ≤ const ε.    r∈D(N,t)

But ε > 0 is arbitrary; by the definition of the function Σ, this means that ψd (t) = Σ(ψ, t). By (93), we have |ψˆi (t1 ) − ψi (t1 )| → 0. Then (79) is obtained from (88) as i → ∞. Conditions (77) and (79) are proved. Let us prove (78) and (80). By condition,  ϕ(ˆ xi , t) dˆ ηi ≥ 0. T

Passing to the limit in this inequality as i → ∞ (using the arguments presented above), we have    ∗ ϕ(x , t)dη + ϕ(αr∗ , s) dηr ≥ 0. r∈Ds(ν ∗ )[0,1]

T

Taking into account that gc sup ϕj (x∗ , t) ≤ 0, j = 1, . . . , k3 , we arrive at (78) and (80). Passing to the limit in (89), we obtain (81). Let us prove (82). It follows from (91) and (93) that    ˆ i (t), dˆ Q μi ≥ 1(i) → 0. T



Passing to the limit as i → ∞ here, we have

    Q(t), dq∗ ≥ 0. Let us show that gc sup Q(t), m ≤ 0

T

xi , t) → 0 strongly in Lk14 (T ) and κi (αr,i , s) → 0 ∀m ∈ K. Indeed, it follows from (92) and (93) that κi (ˆ xi , t) → 0 a.e. and κi (αr,i , s) → 0 a.e. Then, strongly in Lk14 ([0, 1]). Choosing a subsequence, we have κi (ˆ 25

This is a consequence of the uniqueness of the solution.

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obviously, (90) and (93) imply Q(t) ∈ K ◦ a.e. in t ∈ T and Qr (s) ∈ K ◦ a.e. in s ∈ [0, 1], whence, by the right continuity of the components of the vector-function Q on (t0 , t1 ), we deduce that Q(t) ∈ K ◦ ∀t ∈ (t0 , t1 ) and, analogously, Qr (s) ∈ K ◦ ∀s ∈ (0, 1). By the concordance of the state and endpoint constraints, we have κi (tk ) = 0, k = 0, 1 ∀i ⇒ Q(tk ) ∈ K ◦ , k = 0, 1. Combining the obtained conditions, we arrive at (82). ηi  → 0, then, obviously, Condition (83) is simply obtained by assuming the contrary: if |λi | → 0, ˆ ˆ max |ψi (t)| → 0 as i → ∞ (a consequence of the Gronwall inequality), and, therefore, (86) is violated for t∈T

large i. The theorem is proved. 9.1. Nondegeneracy conditions. In this subsection, we call attention to conditions that guarantee the nondegeneracy of the MP proved in Theorem 9.2. For this purpose, we will consider problem (2)–(5) on a nonfixed interval of time, i.e., now, as in Secs. 6–8, the endpoint vector is p = (x0 , x1 , t0 , t1 ). Theorem 9.3. Let (p∗ , u∗ , q∗ ) be a solution of problem (2)–(5), let assumption (S) hold, let the state and endpoint constraints be in concordance, let the state and endpoint constraints be regular, and let the optimal trajectory be controllable at the endpoints with respect to the state constraints. Then there exist a number λ0 ≥ 0, vectors λj ∈ Rkj , j = 1, 2, λ1 ≥ 0, a vector-function ψ ∈ V n (T ∗ ), a scalar-function ∗ (T ∗ ): Ds(|q∗ |) ∩ Ds(η) = ∅, and for φ ∈ V (T ∗ ), and a vector-valued measure η = (η 1 , . . . , η k3 ), η j ∈ C+ each point r ∈ Ds(|q∗ |), there exist its own vector-function σr ∈ V n ([0, 1]), scalar-function V ([0, 1]), and ∗ ([0, 1]), such that vector-valued measure ηr = (ηr1 , . . . , ηrk3 ), ηrj ∈ C+ t ψ(t) = ψ0 −

 Hx (s) ds −

t∗0

Qx (s) dμ∗c +

[t∗0 ,t]

Σ(ψ, t) =



∗ ϕ x (x , s) dη + Σ(ψ, t),

t ∈ (t∗0 , t∗1 ],

[t∗0 ,t]



[σr (1) − ψ(r− )],

r∈Ds(|q∗ |) r≤t



t φ(t) = φ0 +

Ht (s) ds + t∗0

Qt (s) dμ∗c

[t∗0 ,t]

Θ(φ, t) =

 −



∗ ϕ t (x , s) dη + Θ(φ, t),

t ∈ (t∗0 , t∗1 ],

[t∗0 ,t]

[θr (1) − φ(r− )],

r∈Ds(|q∗ |) r≤t

⎧ ⎪ dαr∗ = g(αr∗ , r)vr∗ ds, ⎪ ⎪ ⎪ k4 ⎪  ⎪ ⎪ ∗ ⎪ dσ = − gxj (αr∗ , r)σr vrj∗ ds + ϕ ⎪ x (αr , r)dηr , ⎨ r   j=1∗ ∗ ⎪ ⎪ s ∈ [0, 1], dθr = gt (αr , r)vr∗ , σr ds − ϕ t (αr , r)dηr , ⎪ ⎪ ⎪ ∗ ∗ − − ⎪ αr (0) = x (r ), σr (0) = ψ(r ), θr (0) = φ(r− ), ⎪ ⎪ ⎪ ⎩supp(η j ) ⊆ {s : ϕj (α∗ (s), r) = 0}, j = 1, . . . , k , r ∈ Ds(|q∗ |); r 3 r ∂l ∗ ∂l ∗ (p , λ), ψ1 = − (p , λ); ψ0 = ∂x0 ∂x1 ∂l ∗ ∂l ∗ φ0 = − (p , λ), φ1 = (p , λ); ∂t0 ∂t1   ϕj (x∗ (t), t) = 0 η j -a.e. ∀j; λ1 , e1 (p∗ ) = 0, max H(u, t) = φ(t) ∀t ∈ (t∗0 , t∗1 ); max H(u, t) = H(t) a.a. t, u∈U

7144

u∈U

  gc sup Q(t), m ≤ 0 t∈T ∗

  λ0 + L {t : |ψ(t)| > 0} +

 ∀m ∈ K,

  Q(t), dq∗ = 0;

T∗



  L {t : |σr (t)| > 0} |q∗ |({r}) > 0,

r∈Ds(|q∗ |)

and inequalities (67) hold. Remark. The conditions of Theorem 9.3 imply the following transversality conditions in time: # 1    ∗ ∗ k+1 ∗ ∗ ∗ ∗ ∗ ϕ gt (αk , tk )vk , σk dt − max H(βk , u, γk , tk ) + (−1) t (αk , tk )dηk u∈U

0 ∗ ∗ ∗ −ϕ t (xk , tk )η({tk }) −

$

∂l ∗ (p , λ) = 0, ∂tk

[0,1]

k = 0, 1.

Here and in Theorem 9.3, we use the following notation: T ∗ = [t∗0 , t∗1 ], β0∗ = x∗ (t∗+ 0 ),

ψk = ψ(t∗k ),

β1∗ = x∗ (t∗− 1 ),

φk = φ(t∗k ), γ0∗ = ψ(t∗+ 0 ),

k = 0, 1, γ1∗ = ψ(t∗− 1 ),

αk∗ , vk∗ , σk , and ηk are the elements of the associated set at the point t∗k , k = 0, 1.

Proof. For the problem with a nonfixed time, the MP proved in Theorem 9.2 can be completed by inequalities (46). Indeed, in the case of the simplest problem, these inequalities are already proved (see the remark to Proposition 6.1). For the general problem, conditions (46) are obtained by simple passages ∗ (T ) should be replaced by to the limit. In this case, in the proofs of Theorems 9.1 and 9.2, the set ID ∗ the ρw -closure of the set ID (Tc ) consisting of those elements qD ∈ ID (Tc ), Tc = [−c, c], for which there exists a closed interval S = S(qD ) ⊆ Tc : supp(| Pr2 (qD )|) ⊆ S, and the restriction of q∗ to S is equal to the projection qD : Pr2 (qD ) = q∗ |S . This is necessary so that for the ε-problem in Theorems 9.1 and 9.2 one can prove inequalities (46). The fact is that the variation applied for proving (46) in Proposition 6.1 ∗ (T ). It is easy to see that such becomes admissible in the ε-problem when it is considered on the set cl ID c ∗ ∗ a replacement of the set ID (T ) by cl ID (Tc ) has no influence on the other arguments proving Theorems 9.1 and 9.2. Now, having inequalities (46) and also the conditions Q(t∗k ) ∈ K ◦ , k = 0, 1, we prove the theorem in exactly the same way as Theorem 8.1 (literally repeating the arguments presented in Sec. 8). In concluding of this section, we present an example showing that if the controllability conditions (Definition 1.4) are violated, then the MP proved in Theorem 9.2 can degenerate (see also [1, Example 4.1, p. 111]). Example 9.1. Consider the problem t ∈ [0, 1]; x0 = 0, x1 = 1;  2 1 dμ → min . x≥t ,

dx = 2tdμ,

[0,1]

Let us show that the minimum in this problem is attained at the Lebesgue measure μ∗ = L, and, correspondingly, the optimal trajectory is the parabola x∗ (t) = t2 . Indeed, for this purpose, it suffices to prove the inequality F (t; μ) ≥ t ∀t ∈ [0, 1] for any admissible measure μ. We have  2t dμ ≥ t2 . x(t) ≥ t2 ⇒ [0,t]

7145

Integrating by parts, we obtain 2F (t; μ)t ≥ t2 + 2

 F (s; μ) ds ⇒ F (t; μ) ≥

[0,t]

1 t + 2 t

 F (s; μ) ds.

(96)

[0,t]

This implies F (t; μ) ≥ t/2. Substituting the obtained inequality in the right-hand side of (96), we arrive at a sharper estimate, F (t; μ) ≥ 3t/4. Repeating the procedure, at the nth step, we have F (t; μ) ≥ (2n − 1)t/2n . Letting n → ∞, we finally arrive at the estimate F (t; μ) ≥ t. Therefore, μ∗ = L. But the MP for this problem degenerates. Indeed, condition (82) yields 2tψ(t) − λ0 = 0 a.a. t ∈ [0, 1]. From this, taking into account that the function ψ(t) is bounded, as t → 0 we have that λ0 = 0. Hence ψ(t) = 0 ∀t ∈ (0, 1). That is, condition (70) is violated. 10.

Conclusions

Impulse control theory began in the early 1960s and, at present, covers a fairly wide field of science. Contributions to its development were made by the works of A. V. Arutyunov, M. I. Gusev, V. A. Dykhta, A. T. Zavalishin, N. N. Krasovskii, A. B. Kurzhanskii, B. M. Miller (in Russia; see [10, 11, 18, 22, 28–30, 32, 33, 50]) and by the works of foreign scientists: A. Bressan, F. L. Pereira, F. Rampazzo, G. N. Silva, R. B. Vinter, and V. Jacimovic (see [10, 13, 36, 42, 43, 47]); this list is far from complete and can be continued. Referring to the topic of the present paper — the necessary first-order conditions and the maximum principle — it is most interesting to mention [18, 32, 33, 36, 43, 47]. Let us briefly compare the results obtained here with the results of these authors. First, we note that the MP for the impulse control problem with a vector measure and without the Frobenius condition was already proved earlier [18, 32, 36]. A common feature of all the mentioned works is that their authors do not consider the space of impulse controls and its topology, not giving a precise definition of an impulse control as an element of a certain metric space. The approach to the definition of a solution of Eq. (3) is similar for all of them, and it is formulated approximately as follows: a function x(t) is called a generalized solution if there exists a sequence of absolutely continuous admissible trajectories converging to x(t) (see [18, 32]). A specific feature of the present paper is that (1) we have introduced the concept of impulse control as an element of the completed space, and, respectively, the concept of closeness of impulse controls (i.e., the space of impulse controls is endowed with a metric; see Sec. 3); (2) we have considered a new object, the differential equation with a vector-valued measure, and with respect to the just-constructed metric, now the correct passages to the limit are possible. After that, we introduced the definition of a generalized solution as a solution of the differential equation with the vector-valued measure corresponding to a certain impulse control. Both mentioned approaches to the definition of a solution are equivalent in their meaning. Nevertheless, the possibility of operating with the metric of the space of impulse controls yields certain advantages, which will be demonstrated below by comparing our Theorems 9.2 and 9.3 and [32, Theorem 1 (MP)]. In [32], B. M. Miller uses the so-called method of discontinuous change of time, which allows one to reduce the initial impulse problem to an ordinary optimal control problem in which the solution does exist. Applying the MP to the ordinary control problem and decoding it, Miller obtains the MP for impulse controls. The advantages of such a method are its simplicity and clarity. However, this method also has a number of disadvantages. In particular, the smoothness requirement and the convexity of the vectorgram f (x, U, t) are needed. In Theorem 9.2, we omit these requirements. Moreover, Theorem 1 in [32] is slightly different from Theorems 9.2 and 9.3 in the form of the MP conditions themselves. First of all, note the difference in the impulse conditions of the maximum (see (82)) in the case where one of the endpoints of the trajectory inTheorem 9.2, we have  lies on the boundary of the state constraints: if, for example,  gc sup Q(t), m ≤ 0 ∀m ∈ K, i.e., a pointwise inequality, and, therefore, Q(t∗k ), m ≤ 0 ∀m ∈ K, k = 0, 1 (the inequality also holds at the endpoints), then in [32] the same condition holds only almost everywhere, and hence can be violated at the endpoints (since the so-called function Q(t) can have discontinuities at 7146

them because of the measure η). The reason for such a difference is that in [32] Miller uses the classical optimality conditions in the Dubovitskii–Milyutin form [17], and in his theorem he does not assume the concordance of the state and endpoint constraints. On one hand, the concordance assumption is purely technical (since it is always attained by introducing additional endpoint constraints of inequality type induced by state constraints at the endpoints [1]), and, on the other hand, it is essential for the nondegenerate MP proved in Theorem 9.3. Indeed, let us consider the following example. Example 10.1. Consider the problem

 2 dμ → min,

y(0) + [0,t1 ]

dx = dμ, x(0) ≤ 0,

dy = dt,

x(t1 ) ≥ 0,

y(t1 ) = 1,

x(t) ≥ t − 1 ∀t ∈ [0, t1 ]. Obviously, here the (local) solution is the process t∗1 = 1, μ∗ = 0, x(t) ≡ 0. Let us show that Theorem 9.3 does not hold for it. In fact, we have  ψx (t) = ψx (0) − dη, t > 0, ψy (t) ≡ λ0 . [0,t]

Since H(t) = ψy (t), it follows from (46) that λ0 = 0. Then (82) implies ψx (t) − 2λ0 ≤ 0 ⇒ ψx (0) ≤ 0. But the transversality conditions at the left endpoint imply ψx (0) ≥ 0 ⇒ ψx (t) = 0 ∀t ∈ [0, 1). Now the transversality conditions in time (see the remark to Theorem 9.3) imply η({1}) = 0. Therefore, ψx ≡ 0 and all the Lagrange multipliers vanish; a contradiction. In the example studied, the nontriviality condition (70) is violated, since the phase constraint is not in concordance with the endpoint constraint for x = 0 and t = 1. In contrast to our Theorem 9.3, the MP proved in [32] degenerates.26 Indeed, in his theorem, Miller does not assume the controllability and concordance conditions, each of which, as was shown above (Examples 9.1 and 10.1), is essential, and without them, the MP degenerates. Therefore, in some cases (for example, always for the autonomous problem with fixed endpoints lying on the boundary of the state constraint), the theorem in [32] becomes a trivial assertion, and, in general, is not applicable. The same remark refers to the remaining works of other authors devoted to impulse controls, which study the state-constrained problems. The nondegeneracy problem of the MP is an important and difficult problem that occurs in ordinary, as well as impulse optimal control problems with state constraints. The nondegeneracy problem of the MP for state constrained problems was probably first posed and partially studied in [4] as an independent problem, where it was observed that the MP in the Dubovitskii–Milyutin form [17] is not informative for certain classes of control problems and holds trivially. Subsequently, the nondegeneracy problem of the MP for ordinary (nonimpulse) problems was studied by A. V. Arutyunov [1, 3–7], A. M. Aseev [8, 9], V. I. Blagodatskikh [8], A. Ya. Dubovitskii and V. A. Dubovitskii [16], A. S. Matveev [31], N. T. Tynuanskii [4], and M. M. A. Ferreira and R. B. Vinter [48]. Also, note that the first variant of the MP for the stateconstrained problem obtained by R. V. Gamkrelidze in 1959 (see [40]) is nondegenerate in advance and is always informative. However, it is difficult to compare this MP with the subsequent ones (for example, with that in the Dubovitskii–Milyutin form), since they are proved under different assumptions and have different conditions. The author does not know of works in which the nondegeneracy for the state constrained impulse problems is studied. In many respects, the approach to the nondegeneracy problem in the present paper is based on the methods suggested in [1]. 26

Here, we say that the MP degenerates if condition (70) is violated.

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Completing the comparison with the results of [32], we note that in [32], Miller has studied a more complicated and general case where in the problem being relaxed, the matrix function g next to the control impulse function depends not only on x and t but on the ordinary measurable control u, which, without doubt, is one of the advantages of [32]. In conclusion, the author expresses his gratitude to Prof. A. V. Arutyunov, his teacher, for the statement of the problem and for his the attention. Acknowledgments. This work was supported by the Russian Foundation for Basic Research, Grant Nos. 05–01–00193 and 05–01–00275. REFERENCES 1. A. V. Arutyunov, Necessary Extremum Conditions [in Russian], Faktorial, Moscow (1997). 2. A. V. Arutyunov, “Relaxations and perturbations of optimal control problems,” Tr. Mat. Inst. Ross. Akad. Nauk, 220, 27–34 (1998). 3. A. V. Arutyunov, “Perturbations of constrained extremal problems and necessary optimality conditions,” In: Progress in Science and Technology, Series on Mathematical Analysis [in Russian], Vol. 27, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR , Moscow (1989), pp. 147–235. 4. A. V. Arutyunov and N. T. Tynyanskii, “On the maximum principle in the state-constrained problem,” Izv. Akad. Nauk SSSR, Ser. Tekh. Kibern., 4, 60–68 (1984). 5. A. V. Arutyunov, “On necessary optimality conditions in the state constrained problem,” Dokl. Akad. Nauk SSSR, 280, No. 5, 1033–1037 (1985). 6. A. V. Arutyunov, “First-order necessary conditions in the state constrained optimal control problem,” Tr. Inst. Prikl. Mat. Tbilis. Univ., 27, 46–59 (1988). 7. A. V. Arutyunov, “Maximum principle and second-order necessary optimality conditions in the optimal control problem with delays,”Soobshch. Akad. Nauk Grus.SSR, 122, No. 2, 265–268 (1986). 8. A. V. Arutyunov, S. M. Aseev, and V. I. Blagodatskikh, “First-order necessary conditions in the optimal control problem for a differential inclusion with state constraints,” Mat. Sb., 184, No. 6, 3–32 (1993). 9. A. V. Arutyunov and S. M. Aseev, “Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints,” SIAM J. Control Optim., 35, No. 3, 930–952 (1997). 10. A. V. Arutyunov, V. Jacimovic, and F. L. Pereira, “Second order necessary conditions of optimality for impulsive control problems,” Int. J. Dyn. Contr. Syst., 9, No. 1, 131–153 (2003). 11. A. V. Arutyunov, V. A. Dykhta, and F. L. Pereira, “Necessary conditions for impulsive nonlinear optimal control problems without a priori normality assumptions,” J. Optim. Theory Appl. (in press). 12. V. I. Blagodatskikh and A. F. Filippov, “Differential inclusions and optimal control,” Tr. Mat. Inst. Akad. Nauk SSSR, 169, 194–252 (1985). 13. A. Bressan and F. Rampazzo, “Impulsive control systems with commutative vector fields,” J. Optim. Theory Appl. 71, 67–83 (1991). 14. F. Clarke, Optimization and Nonsmooth Analysis [Russian translation], Nauka, Moscow (1988). 15. V. F. Dem’yanov, Minimax: Differentiability in Directions [in Russian], LGU, Leningrad (1974). 16. A. Ya. Dubovitskii and V. A. Dubovitskii, “Necessary conditions for the strong minimum in optimal control problems with degeneration of endpoint and state constraints,” Usp. Mat. Nauk, 40, No. 2, 175–176 (1985). 17. A. Ya. Dubovitskii and A. A. Milyutin, “Extremum problems under constraints,” Dokl. Akad. Nauk SSSR, 149, No. 4, 759–762 (1963); Zh. Vychisl. Mat. Mat. Fiz., 5, No. 3, 395–453 (1965). 18. V. A. Dykhta and O. N. Samsonyuk, Optimal Impulse Control with Applications [in Russian], Fizmatlit, Moscow (2000). 7148

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