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J Optim Theory Appl (2011) 149: 474–493 DOI 10.1007/s10957-011-9807-5

The Maximum Principle for Optimal Control Problems with State Constraints by R.V. Gamkrelidze: Revisited A.V. Arutyunov · D.Y. Karamzin · F.L. Pereira

Published online: 26 January 2011 © Springer Science+Business Media, LLC 2011

Abstract A maximum principle in the form given by R.V. Gamkrelidze is obtained, although without a priori regularity assumptions to be satisfied by the optimal trajectory. After its formulation and proof, we propose various regularity concepts that guarantee, in one sense or another, the nondegeneracy of the maximum principle. Finally, we show how the already known first-order necessary conditions can be deduced from the proposed theorem. Keywords Optimal control · Maximum principle · State constraints

Communicated by B. Mordukhovich. This research was supported by the Russian Foundation for Basic Research, project 09-01-00619, by a grant from the President of the Russian Federation, MK-119.2009.1, by the Russian Federal Program “Scientific and pedagogical staff of innovative Russia in 2009–2013” (contract 16.740.11.0426 of 11/26/2010), and by the Foundation for Science and Technology (Portugal), support of R&D unit 147 (ISRP), and projects PTDC/EEA-CRO/104901/2008, and SFRH/BPD/26231/2006. A.V. Arutyunov Peoples’ Friendship University of Russia, Moscow, Russia e-mail: [email protected] D.Y. Karamzin () Dorodnicyn Computing Centre, Russian Academy of Sciences, Moscow, Russia e-mail: [email protected] D.Y. Karamzin University of Porto, Porto, Portugal F.L. Pereira Faculty of Engineering, University of Porto, Institute for Systems & Robotics, Porto, Portugal e-mail: [email protected]

J Optim Theory Appl (2011) 149: 474–493

475

1 Introduction In this paper, we investigate necessary conditions of optimality for optimal control problem with state constraints in the form of the Pontryagin’s Maximum Principle (for short MP) [1]. For problems with state constraints these conditions were first obtained by R.V. Gamkrelidze in 1959 [2, 3] and subsequently published in the classic monograph [1, Chap. 6]. This MP was obtained under a certain regularity assumption on the optimal trajectory. Somewhat later, in 1963, A.Ya. Dubovitskii and A.A. Milyutin proved another MP for problems with state constraints, [4]. In contrast with the MP of R.V. Gamkrelidze, this MP was obtained without a priori regularity assumptions, and, thus, it degenerates in many cases of interest [5, 6]. Therefore, in subsequent studies [5–11], under an additional assumption of controllability of the optimal trajectory, other versions of the MP have been obtained so that they no longer degenerate. In this paper, we prove a MP in the form proposed by R.V. Gamkrelidze (see Theorem 3.1 below) without any a priori regularity assumptions on the optimal trajectory. This MP is obtained directly from the one of [6] by considering a new adjoint variable. However, without a priori regularity assumptions, the MP of Theorem 3.1 may degenerate. Therefore, in Theorems 4.1 and 4.2, we prove that, under certain additional conditions of controllability relatively to the state constraints at the end-points, or regularity of the control process, degeneracy will not occur, since a stronger (than in Theorem 3.1) non-triviality condition will be satisfied. This article is organized as follows. After presenting the problem formulation in next section, in Sect. 3 we introduce Maximum Principle and its proof without a priori regularity assumption. Then, the ensuing section discusses the nondegeneracy issues, being followed by another one in which the links with the results of R.V. Gamkrelidze are addressed. Finally, before presenting some brief conclusions, we consider the case of problems with the data measurable in time.

2 Problem Formulation Consider the following optimal control problem ⎧ t ⎪ Minimize K0 (p) + t12 f0 (x, u, t)dt ⎪ ⎪ ⎪ ⎪ ⎨subject to x˙ = f (x, u, t), t ∈ [t1 , t2 ], t1 < t2 , ⎪ G(x, t) ≤ 0, R(x, u, t) ≤ 0, ⎪ ⎪ ⎪ ⎪ ⎩ K1 (p) = 0, K2 (p) ≤ 0, p = (x1 , x2 , t1 , t2 ).

(1)

Here, the given vector functions G, R, and Ki , i = 1, 2, take values in the arithmetic spaces of dimensions, respectively, d(G), d(R), and d(Ki ), functions K0 , and f0 are scalar, x˙ = dx dt , t ∈ [t1 , t2 ] (here, times t1 and t2 are not assumed to be fixed a priori), x is the state variable taking values in n-dimensional arithmetic space Rn , and u ∈ Rm is the control parameter. The end-point vector is denoted by p ∈ Rn × Rn × R × R. As for the class of admissible controls, we consider the set of measurable, essentially bounded, functions u(·). Let u(t), t ∈ [t1 , t2 ], be the admissible control, x(t),

476


t ∈ [t1 , t2 ], the corresponding trajectory, and p the associated end-point vector. Then, the triplet (p, x, u) will be called an admissible control process iff it satisfies the following (1) end-point constraints: K1 (p) = 0, K2 (p) ≤ 0, (2) mixed constraints: u(t) ∈ U (x(t), t), a.e. in [t1 , t2 ], and (3) state constraints: G(x(t), t) ≤ 0, ∀t ∈ [t1 , t2 ]. Here U (x, t) := {u : R(x, u, t) ≤ 0}. Next, we assume that all functions involved in the formulation of the problem are continuously differentiable, and that the function G is twice continuously differentiable. Let us introduce the following properties on the constraints. We say that: • End-point constraints are regular at p = (x1 , x2 , t1 , t2 ): K1 (p) = 0, K2 (p) ≤ 0, iff ∂K1 (p) = d(K1 ), and ∂p j ∂K2 ∂K1 ∃d ∈ ker (p) : (p), d > 0, ∂p ∂p

rank

j

∀j : K2 (p) = 0.

(Here and in the sequel, the upper indices denote the coordinates of a vector or vector function.) • Mixed constraints are regular, iff, for any (x, u, t) such that R(x, u, t) ≤ 0, there exists a vector q = q(x, u, t) satisfying

∂R j (x, u, t), q > 0, ∂u

∀j : R j (x, u, t) = 0.

• State constraints are regular, iff, for any (x, t) such that G(x, t) ≤ 0, there exists a vector z = z(x, t) satisfying

∂Gj (x, t), z > 0, ∂x

∀j : Gj (x, t) = 0.

• State constraints are compatible with the end-point constraints at a point p ∗ iff ∃ε > 0 such that {p ∈ R2n+2 : |p ∗ − p| ≤ ε, K1 (p) = 0, K2 (p) ≤ 0} ⊆ {p : G(x1 , t1 ) ≤ 0, G(x2 , t2 ) ≤ 0}. The compatibility assumption between the end-point and the state constraints is not burdensome. Note that, by adding to problem (1) the additional constraints G(x1 , t1 ) ≤ 0,

and G(x2 , t2 ) ≤ 0,

(2)

the original problem can be reduced to a new problem for which the state constraints are already compatible with the end-point constraints. Nevertheless, for a


477

given process (p, x, u), this reduction is meaningful only when the end-point constraints of the original problem (1), supplemented by the constraints (2), remain regular. Let (p ∗ , x ∗ , u∗ ) be an admissible process for the problem (1) and p ∗ = (x1∗ , x2∗ , ∗ t1 , t2∗ ) the optimal end-point vector. Let us denote by T , J (t), and Q(x, u, t), respectively, [t1∗ , t2∗ ], {j : Gj (x ∗ (t), t) = 0}, and Q(x, u, t) =

∂G ∂G (x, t)f (x, u, t) + (x, t). ∂x ∂t

Definition 2.1 We say that the point p ∗ satisfies the controllability condition relatively to the state constraints iff, for s = 1, 2, ∃fs ∈ conv f (xs∗ , U (xs∗ , ts∗ ), ts∗ ) : j

∂G ∂Gj ∗ ∗ s ∗ ∗ (−1) (x , t ), fs + (xs , ts ) > 0, ∂x s s ∂t

∀j : Gj (xs∗ , ts∗ ) = 0.

Here, conv denotes the convex hull of the set. Definition 2.2 The admissible process (p ∗ , x ∗ , u∗ ) is called regular iff there exist a number ε0 > 0 and bounded functions q1 , and q2 , qi : T → Rm , i = 1, 2 such that, for all t ∈ T , and, for almost all θ ∈ T satisfying |θ − t| ≤ ε0 , the following holds for s = 1, 2, ∂Qj ∗ (−1)s (3) (x (θ ), u∗ (θ ), θ ), qs (θ ) ≥ ε0 , ∀j ∈ J (t), ∂u

∂R j ∗ (x (θ ), u∗ (θ ), θ ), qs (θ ) ≥ ε0 , ∂u

∀j : R j (x ∗ (θ ), u∗ (θ ), θ ) ≥ −ε0 .

(4)

Definition 2.2 is a generalization of the concept of regular trajectory given in [2] to encompass the case of measurable controls and of arbitrary behavior of the trajectory at the boundary of the state constraints. Remark 2.1 Let the process (p ∗ , x ∗ , u∗ ) be regular. Then, there exists ε > 0 such that the set J (t) in Definition 2.2 can be replaced by the larger set Jε (t) = {j : Gj (x ∗ (t), t) ≥ −ε}. This follows from the compactness of segment T . Remark 2.2 Let (p ∗ , x ∗ , u∗ ) be an admissible process. Then, by Lemma 3 of [11], the controllability condition from Definition 2.1 holds with the strict inequalities replaced by non-strict ones. Namely, for s = 1, 2, ∃fs ∈ conv f (xs∗ , U (xs∗ , ts∗ ), ts∗ ) : j

∂G ∂Gj ∗ ∗ (−1)s (xs∗ , ts∗ ), fs + (xs , ts ) ≥ 0, ∂x ∂t

∀j ∈ J (ts∗ ).

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Controllability relatively to the state constraints, and regularity of the admissible process are related to each other as follows. Lemma 2.1 Suppose that an admissible process (p ∗ , x ∗ , u∗ ) be regular. Then, the point p ∗ satisfies the controllability condition relatively to the state constraints. The proof is given in Sect. 4.

3 Maximum Principle Consider the extended Pontryagin function H¯ (x, u, ψ, μ, λ0 , t) := ψ, f (x, u, t) − μ, Q(x, u, t) − λ0 f0 (x, u, t), and the small Lagrangian l(p, λ) := λ0 K0 (p) + λ1 , K1 (p) + λ2 , K2 (p), where λ = (λ0 , λ1 , λ2 ). Definition 3.1 We say that an admissible process (p ∗ , x ∗ , u∗ ) satisfies the Pontryagin Maximum Principle (MP) iff there exist a vector λ = (λ0 , λ1 , λ2 ), with λ0 ∈ R, λ1 ∈ Rd(K1 ) , λ2 ∈ Rd(K2 ) , λ0 ≥ 0, λ2 ≥ 0, and λ2 , K2 (p ∗ ) = 0, an absolutely continuous function ψ : T → Rn , a function μ : T → Rd(G) , and a measurable, essentially bounded function r : T → Rd(R) , such that λ0 , ψ, and μ do not vanish simultaneously, and ∂ H¯ ∂R (t) + r(t) (t), a.e., ∂x ∂x ∂l ∗ ∂G ∗ ψ(ts∗ ) = (−1)s+1 (p , λ) + μ(ts∗ ) (t ), ∂xs ∂xs s ψ˙ = −

max H¯ (u, t) = H¯ (t)

u∈U (t)

(5) s = 1, 2,

a.e.,

∂ H¯ ∂R h˙ = (t) − r(t) (t), a.e., ∂t ∂t ∂l ∗ ∂G ∗ (t ), h(ts∗ ) = (−1)s (p , λ) − μ(ts∗ ) ∂ts ∂t s ∂ H¯ ∂R (t) = r(t) (t), a.e., ∂u ∂u

r(t), R(t) = 0, r(t) ≥ 0, a.e.,

(6) (7) (8)

s = 1, 2,

(9) (10) (11)

where h(t) := maxu∈U (t) H¯ (u, t). In addition, the function h is absolutely continuous on the interval T , and the vector-function μ satisfies the following properties:


479

(a) Each of the functions μj is constant on any time segment S = [s1 , s2 ], in which the optimal trajectory lies in the interior of the set defined by j -th state constraint, i.e., when Gj (s) < 0, ∀s ∈ S; (b) The vector-function μ is left continuous on the interval ]t1∗ , t2∗ [ and μ(t2∗ ) = 0; (c) Each one of the functions μj nonincreasing. The process (p ∗ , x ∗ , u∗ ), that satisfies the MP, is called extreme, and the set (λ, ψ, μ, r) is called the associated set of Lagrange multipliers. Here and in the sequel, the following notation is adopted. First, if some of the arguments of the mappings H¯ , G, R, f , U , etc., and of their derivatives are omitted, then this means that they have been replaced by the optimum values x ∗ (t), and u∗ (t) or Lagrange multipliers ψ(t), μ(t), and λ0 . Second, all Lagrange multipliers or gradients of functions are considered as row vectors, while vector-functions or vectors such as f , x, and u, are considered as column vectors. The elements of the Jacobi i matrix for the mapping F (x) : Rn → Rk are ∂F ∂xj (x), and the rows of this matrix are the gradients of F i . Theorem 3.1 Let the process (p ∗ , x ∗ , u∗ ) be optimal to problem (1). Suppose that the end-point constraints be regular at the point p ∗ , the state and the mixed constraints be regular, and that the state constraints be compatible with the end-point constraints at the point p ∗ . Then, the process (p ∗ , x ∗ , u∗ ) satisfies the MP. Remark 3.1 By using the obvious formula ∂ 2G ∂ 2G d ∂G (t) = (t) = (t)f (t) + dt ∂x ∂x∂t ∂x 2 d ∂G ∂ 2G ∂ 2G (t) = (t)f (t) + 2 (t) = dt ∂t ∂x∂t ∂t

∂Q ∂G ∂f (t) − (t) (t), ∂x ∂x ∂x (12) ∂G ∂f ∂Q (t) − (t) (t) ∂t ∂x ∂t

and the definition of Q, it is easy to show that, along with the Lagrange multipliers (λ, ψ, μ, r), the conditions (5)–(11), (a) and (c) are also satisfied by the set of the Lagrange multipliers ∂G λ, ψ(t) + a (t), μ(t) + a, r(t) , ∂x where a is an arbitrary vector from Rd(G) . Indeed, under this transformation, the new value of Hamiltonian h(t) is its old value minus the function a ∂G ∂t (t), the maximum condition is maintained, and conditions (5), and (8) are true in view of (12). Moreover, by virtue of what was said above, conditions (5)–(11), (a) and (c) can always be trivially met, by taking the following set of Lagrange multipliers: λ = 0,

ψ(t) = a

∂G (t), ∂x

μ(t) = a,

r = 0,

where a is an arbitrary vector. But, in view of the condition (b), μ(t2∗ ) = 0 implies that a = 0, which yields a contradiction since the Lagrange multipliers cannot be all

480


simultaneously zero. This shows that condition (b) is essential. (Without this condition, the MP of Theorem 3.1 can be satisfied in a trivial manner as described above.) Remark 3.2 Next, we explain why the regularity of the state and the mixed constraints is necessary in Theorem 3.1. If the state constraints are not regular (albeit, for ∗ example, the function G is scalar and ∂G ∂x (τ ) = 0 for some τ < t2 ), then the MP is sat∗ isfied trivially by the multipliers λ = 0, ψ(t) = ∂G ∂x (t) ∀t ∈ [t1 , τ ], ψ(t) = 0, ∀t > τ , ∗ μ(t) = 1, ∀t ∈ [t1 , τ ], μ(t) = 0, ∀t > τ , and r = 0. It is clear that the function ψ is absolutely continuous and, in view of (12), it satisfies the theorem (see Remark 3.1). Regularity of the mixed constraints is essential, and, without it, Theorem 3.1 is simply wrong. This is shown by Example 1 of [12]. The assumptions of regularity for the mixed constraints can be relaxed by demanding regularity only along the optimal control process. However, then, the maximum condition takes different and more general form, [12]. Remark 3.3 The compatibility of the state and the end-point constraints is essential, and, without it, Theorem 3.1 is false. This is shown in the following example. Example 3.1 Let n = 2, m = 1, d(G) = 1, x = (x 1 , x 2 ). Consider the problem ⎧ t minimize x 2 (0) + 02 u(t)dt, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨subject to x˙ 1 = u, x˙ 2 = 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

t1 = 0,

x 1 (0) ≤ 0,

x 1 (t) ≥ t − 1,

x 2 (t2 ) = 1,

∀t ∈ [0, t2 ].

Let us show that the process x ∗ = (0, t), u∗ = 0, t2∗ = 1 is optimal. For any admissible process (x(·), u(·), t2 ), we have that

t2 2 x (0) + u(t)dt = 1 − t2 + x 1 (t2 ) − x 1 (0) 0

≥ 1 − t2 + t2 − 1 − x 1 (0) = −x 1 (0) ≥ 0. But, the functional to be minimized vanishes at the process (x ∗ , u∗ , t2∗ ), and hence, the process is optimal. Let us show that, nevertheless, the process (x ∗ , u∗ , t2∗ ) does not satisfy the MP. Assume the contrary. By applying the MP with ψ = (ψ 1 , ψ 2 ) we have H¯ (x, u, ψ, μ, λ0 , t) = ψ 1 u + ψ 2 + μu − μ − λ0 u. By virtue of (5), the function ψ(·) is constant. Denote its coordinates by ψ 1 and ψ 2 . From (10) and condition (a), it follows that μ(t) = λ0 − ψ 1 , ∀t ∈ [0, 1[. Because of the left-end transversality condition, ψ 2 = λ0 and, by virtue of the right-end time transversality condition, and since μ(1) = 0, we have max (ψ 1 u + ψ 2 − λ0 u) = 0.

u∈R1


481

Here, ψ 1 = λ0 , ψ 2 = 0 ⇒ λ0 = 0, ψ = 0, and μ = 0. Thus, a contradiction with the MP is obtained since all the Lagrange multipliers are simultaneously zero. This happened because the compatibility of the end-point with the state constraints is violated in this example. Remark 3.4 If the time end-points t1 and t2 are fixed in problem (1), then the time transversality conditions (9) should be omitted (they are not informative for the fixed time problem), and, if the trajectory end-points x1 and x2 are fixed, then for the same reason, transversality condition (6) should be omitted. Proof of Theorem 3.1 Let us apply Theorem 4.1, from [6, Chap. 2], to problem (1). There exist a simultaneously nonzero vector λ = (λ0 , λ1 , λ2 ), with λ0 ≥ 0, and ˜ a Borel vector meaλ2 ≥ 0, a left-continuous function of bounded variation ψ(t), sure η and a measurable essentially bounded function r(t) ≥ 0, such that

t∗ 2 ∂H ∂R ˜ ˜ (ψ(θ ), θ ) − r(θ ) (θ ) dθ ψ(t) = ∂x ∂x t

∂l ∗ ∂G (θ )dη − − (p , λ), ∀t ∈ T , (13) ∗ ∂x ∂x 2 [t,t2 ] ˜ 1∗ ) = ψ(t

∂l ∗ (p , λ), ∂x1

(14)

˜ ˜ t) = H (ψ(t), t), max H (u, ψ(t),

u∈U (t)

˜ t) = − max H (u, ψ(t),

u∈U (t)

t2∗

t

a.e.,

(15)

∂H ∂R ˜ (ψ(θ ), θ ) − r(θ ) (θ ) dθ ∂t ∂t

∂l ∗ ∂G (θ )dη + (p , λ), ∀t ∈ T , ∂t ∂t2 ∂l ∗ ∗ s+1 ∂l ∗ ∗ (p , λ), ts = (−1)s (p , λ), s = 1, 2, max H xs , u, (−1) ∂xs ∂ts u∈U (ts∗ ) +

[t,t2∗ ]

(16) (17)

supp(ηj ) ⊆ Tj := {t ∈ T : Gj (x ∗ (t), t) = 0},

(18)

∂R ∂H ˜ (ψ(t), t) = r(t) (t), ∂u ∂u

(19)

a.e.,

λ2 , K2 (p ∗ ) = 0, r(t), R(t) = 0, a.e.

(20)

Here, H (x, u, ψ, λ0 , t) = ψ, f (x, u, t) − λ0 f0 (x, u, t) is the classic Pontryagin function. Let

dη, ∀t < t2∗ , with μ(t2∗ ) = 0. μ(t) = [t,t2∗ ]

ˆ ˜ ˆ ∗ ) = − ∂l (p ∗ , λ). Thus, although both functions Let ψ(t) = ψ(t) for t < t2∗ and ψ(t 2 ∂x2 ψˆ and μ might exhibit discontinuities at the end-points t ∗ , and t ∗ , they are left1

2

482


continuous on the interval ]t1∗ , t2∗ [. Obviously, equalities (13), and (16) continue to ˆ We use this below. Let hold for all t < t2∗ when the function ψ˜ is replaced by ψ. ˆ + μ(t) ψ(t) := ψ(t)

∂G (t), ∂x

t ∈ T.

Let us show that the Lagrange multipliers (λ, ψ, μ, r) satisfy the MP. First of all, clearly, that the function μ satisfies the conditions (a)–(c) (condition (a) follows from (18), and the conditions (b) and (c) are guaranteed by construction). Now, we show that the function ψ is absolutely continuous and satisfies (5). By using that ˆ ∗ ), (13), the first formula of (12), and the definition of dη + dμ(t) = 0, ψ(t2∗ ) = ψ(t 2 Q, we obtain, for an arbitrary point t ∈ T , ∂G (t) ∂x

t∗ 2 ∂H ∂R ∂G ∗ ˆ ˆ = ψ(t2 ) + (ψ(θ ), θ ) − r(θ ) (θ ) dθ − (θ )dη ∗ ∂x ∂x t [t,t2 ] ∂x ∂G ∗ ∂G ∗ − μ(t2 ) (t ) − μ(t) (t) ∂x 2 ∂x

t∗ 2 ∂H ∂R ∂G ˆ ), θ ) − r(θ ) (ψ(θ (θ ) dθ − (θ )dη = ψ(t2∗ ) + ∂x ∂x t [t,t2∗ ] ∂x 2

∂G ∂ G ∂ 2G (θ )dμ(θ ) − (θ ) dθ − μ(θ ) (θ )f (θ ) + ∂x∂t ∂x 2 [t,t2∗ ] ∂x [t,t2∗ ]

t∗ 2 ∂H ∂R ˆ ), θ ) − r(θ ) (ψ(θ (θ ) dθ = ψ(t2∗ ) + ∂x ∂x t

t∗

t∗ 2 2 ∂f ∂f ∂G ∂G + (θ ) (θ )dθ − (θ ) (θ )dθ μ(θ ) μ(θ ) ∂x ∂x ∂x ∂x t t

t∗

∗ t2 2 ∂ 2G ∂ 2G − (θ )dθ μ(θ ) 2 (θ )f (θ )dθ − μ(θ ) ∂x∂t ∂x t t

t∗

t∗

t∗ 2 ∂H 2 2 ∂Q ∂R ∗ = ψ(t2 ) + (ψ(θ ), θ )dθ − (θ )dθ − (θ )dθ μ(θ ) r(θ ) ∂x ∂x ∂x t t t

t∗ ¯ 2 ∂H ∂R ∗ = ψ(t2 ) + (θ ) − r(θ ) (θ ) dθ. ∂x ∂x t

ˆ + μ(t) ψ(t) = ψ(t)

Here, we used the fact that the product of two functions of bounded variation is a function of bounded variation, and also the formula of integration by parts for the Stieltjes integral (see Exercise 195 of [13]). Thus, we proved that the function ψ is absolutely continuous and satisfies (5).


483

By carrying out similar reasoning, but, now, for the function h (instead of function ψ ), by using that by definition ∂G ˆ t) − μ(t) h(t) = max H¯ (u, t) = max H (u, ψ(t), (t), ∂t u∈U (t) u∈U (t)

∀t ∈ T ,

from (16) and (12), we get (8), and that h is absolutely continuous. It is easy to see (by virtue of the definitions of functions H¯ and ψ ), that the remaining conditions (6), (7), (9), (10), and (11) follow from the conditions (14), (15), (17), (18), and (20) respectively. It remains to prove that λ0 , ψ , and μ cannot vanish simultaneously. Indeed, assume the contrary. Then, by (10), (11), the regularity of the mixed constraints implies ∂l r = 0. By using the transversality conditions (6) and (9), we have ∂p (p ∗ ) = 0. Hence, by the regularity of the end-point constraints at the point p ∗ and since λ0 = 0, we conclude that λ = 0. But the Lagrange multipliers cannot be all zero by construction. This completes the proof of the theorem. 4 Nondegeneracy Conditions As it was already mentioned above in Remark 3.2, the conditions of Theorem 3.1 will degenerate, if we omit the regularity of the end-point and of the state constraints. Such a degeneracy has a trivial character since it is due to the fact that some derivatives vanish. However, the phenomenon of degeneracy for problems with state constraints can be more pervasive, and the MP may be trivially satisfied even when all regularity of the end-point, state and mixed constraints are present. Let us elucidate this. For simplicity, we consider the case of the autonomous problem with fixed time end-points t1 , and t2 and fixed trajectory end-points x1 , and x2 , scalar state constraint (i.e., when d(G) = 1) and integral functional. Suppose first that the left optimal end-point x1∗ belongs to the boundary of the set B = {x : G(x) ≤ 0}. Then, if ∂G ∗ ∗ max − (21) (t ), f (x1 , u) = 0, ∂x 1 u∈U (t1∗ ) it is easy to see that the conditions of Theorem 3.1 are automatically satisfied by the set of Lagrange multipliers 1, if t = t1∗ , 0 ψ = 0, μ(t) = λ = 0, r = 0. 0, if t > t1∗ , But, then, the maximum condition (7) holds trivially, and, therefore, in this case, the MP of Theorem 3.1 is not meaningful (and, therefore, not suitable for study). Similarly, if the right end-point of the optimal trajectory x2∗ belongs to the boundary of B, and if ∂G ∗ (t2 ), f (x2∗ , u) = 0, max∗ (22) u∈U (t2 ) ∂x

484


then the conditions of Theorem 3.1 (by Remark 3.1) are automatically satisfied by the set of Lagrange multipliers ∂G (t), ψ(t) = ∂x

λ = 0, 0

μ(t) =

1, 0,

if t < t2∗ , if t = t2∗ ,

r = 0.

Note that both maxima in (21) and in (22) are non-negative by virtue of Remark 2.2. Thus, in both cases the MP is not meaningful. At the same time, the following theorem shows that, if the controllability condition (see Definition 2.1) holds, then the MP does not degenerate. Theorem 4.1 Let an admissible process (p ∗ , x ∗ , u∗ ) satisfy the MP. Suppose that the end-point constraints be regular at p ∗ , the state and the mixed constraints be regular, the state constraints be compatible with the end-point constraints at the point p ∗ , and that the point p ∗ satisfies the conditions of controllability relatively to the state constraints. Then, for any Lagrange multipliers (λ, ψ, μ, r) associated with the process (p ∗ , x ∗ , u∗ ) by virtue of the MP, the nontriviality condition ∂G λ0 + meas t ∈ T : ψ(t) − μ(t) (t) = 0 > 0 (23) ∂x holds. Here, meas denotes the Lebesgue measure. In other words, either λ0 > 0, or the maximum condition (7) is meaningful on some set with nonzero Lebesgue measure. Proof Assume that (23) be violated. Then, λ0 = 0 and ψ(t) = μ(t)

∂G (t), ∂x

∀t ∈]t1∗ , t2∗ [.

(24)

This implies that ∂G (t), h(t) = max H¯ (u, t) = −μ(t) ∂t u∈U (t)

∀t ∈]t1∗ , t2∗ [.

(25)

From transversality conditions (6), (9) and definition of H¯ we have: ∂l ∗ s+1 ∂l ∗ ∗ max (−1) (p , λ), f (u, ts ) = (−1)s (p , λ), ∂xs ∂ts u∈U (ts∗ )

s = 1, 2.

(26)

Let 1 = μ(t1∗ +) − μ(t1∗ ) be the jump of μ at point t1∗ , and 2 = μ(t2∗ ) − μ(t2∗ −) its jump at t2∗ . By the continuity of ψ , from (24) and the transversality conditions (6), we obtain ∂G ∗ ∂l ∗ (t ), (p , λ) = s ∂xs ∂x s

s = 1, 2.


485

Similarly, by the continuity of h, from (25) and transversality conditions (9), we get ∂l ∗ ∂G ∗ (t ), (p , λ) = s ∂ts ∂t s

s = 1, 2.

Because of the condition c), s ≤ 0. Therefore, by substituting the obtained expressions into (26), by using the fact that the supremum of a linear function over a closed set coincides with its supremum over the convex hull of this set, and by using the controllability conditions (see Definition 2.1), we find that 1 = 2 = 0. ∂l (p ∗ , λ) = 0. Then, by regularity of the end-point constraints, and Consequently, ∂p ¯

H (t) = 0 a.e., so, by ussince λ0 = 0, we conclude that λ = 0. In view of (24), ∂∂u ing regularity of the mixed constraints, and formulas (10) and (11), we obtain that r(t) = 0 a.e. in T . By repeating a part of the calculations carried out in the proof of Theorem 3.1, and by using (24) and (5) for an arbitrary t ∈ T , we have

∂G (t) ψ(t) = μ(t) ∂x ∂G ∗ ∂G = − μ(t2∗ ) (t2 ) − μ(t) (t) ∂x ∂x 2

t∗

2 ∂G ∂ G ∂ 2G (θ )dμ(θ ) − (θ ) dθ =− μ(θ ) (θ )f (θ ) + ∂x∂t ∂x 2 [t,t2∗ ] ∂x t

t∗

2 ∂f ∂G ∂G (θ )dμ(θ ) + (θ ) (θ )dθ =− μ(θ ) ∗ ∂x ∂x ∂x [t,t2 ] t

t∗ 2 ∂G ∂f ∂ 2G ∂ 2G − (θ ) + (θ ) (θ ) dθ μ(θ ) (θ )f (θ ) + ∂x∂t ∂x ∂x ∂x 2 t

t∗ 2 ∂G ∂Q ∂f =− (θ )dμ(θ ) − (θ )dθ + μ(θ ) ψ(θ ) (θ )dθ ∂x ∂x [t,t2∗ ] ∂x [t,t2∗ ] t

∂G (θ )dμ(θ ) + ψ(t) − ψ(t2∗ ). =− [t,t2∗ ] ∂x But, since λ = 0 and μ(t2∗ ) = 0, then, from the transversality conditions (6) for s = 2, we get ψ(t2∗ ) = 0. Thus, it is proved that

∂G (θ )dμ(θ ) = 0, ∀t ∈ T . ∗ [t,t2 ] ∂x From this identity, by using the regularity of the state constraints, the condition a), and the equality μ(t2∗ ) = 0, it is straightforward to conclude that μ = 0, whence, by (24), ψ = 0. We have obtained a contradiction with the fact that the Lagrange multipliers cannot be simultaneously zero. So, we have (23) and the proof is complete. The following theorem shows that, if the optimal process is regular, then the nontriviality condition (23) can be significantly strengthened.

486


Theorem 4.2 Suppose that an admissible process (p ∗ , x ∗ , u∗ ) satisfies the MP. Assume that the end-point constraints be regular at the point p ∗ , the mixed constraints be regular, the state constraints be compatible with the end-point constraints at the point p ∗ , and that the process (p ∗ , x ∗ , u∗ ) be regular. Then, for any Lagrange multipliers (λ, ψ, μ, r), associated with the process (p ∗ , x ∗ , u∗ ) by virtue of the MP, the following nontriviality condition is true ∂G 0 λ + ψ(t) − μ(t) (27) (t) > 0, ∀t ∈ T . ∂x First we prove Lemma 2.1. Consider the left end-point of the trajectory, x1∗ , and let us prove the controllability there. We assume that J (t1∗ ) = ∅, because, otherwise, the assertion is obvious. Let us show that, for any vector α = (α 1 , . . . , α d(G) ) = 0 satisfying α j ≥ 0, ∀j ∈ J (t1∗ ), and α j = 0, ∀j ∈ / J (t1∗ ), there exists a vector f1 ∈ f (U (t1∗ ), t1∗ ) (dependent on α) such that ∂G ∗ ∂G ∗ α (28) (t ), f1 + α (t ) < 0. ∂x 1 ∂t 1 Indeed, for a given α consider the function g(x, t) = α, G(x, t). Clearly g(t) ≤ 0, ∀t ∈ T and g(t1∗ ) = 0. Now, we show that ∃u1 ∈ U (t1∗ ), z1 ∈ Rm such that ∂g ∗ ∂g ∗ ∗ (t1 ), f (u1 , t1 ) + (t ) ≤ 0, ∂x ∂t 1 ∂Q α (u1 , t1∗ ), z1 ≥ ε0 , ∂u j ∂R (u1 , t1∗ ), z1 ≥ ε0 , ∀j : R j (u1 , t1∗ ) = 0, ∂u

(29)

where ε0 is taken from Definition 2.2. Indeed, consider the measurable set ∂g ∂g (t), f (t) + (t) ≤ 0 . E := t ∈ T : ∂x ∂t Note that meas(E ∩ O) > 0 for every neighborhood O of t1∗ , because, otherwise, for ∂g (t), f (t) + ∂g some neighborhood O, ∂x ∂t (t) > 0 a.e. in O ∩ T , and, then, g(t) = t ∂g ∂g (

(θ ), f (θ ) + (θ ))dθ > 0, ∀t ∈ (O ∩ T ) \ {t1∗ }, which contradicts with the t1∗ ∂x ∂t properties of the function g(t) stated above. We choose a sequence {θi } so that θi ∈ E, θi → t1∗ , at the points θi inequalities (3), (4), |u∗ (θi )| ≤ u∗ L∞ hold and u∗ (θi ) ∈ U (θi ). Put ui = u∗ (θi ), zi = q2 (θi ), where the function q2 is from Definition 2.2. By passing to a subsequence, we have ui → u1 , zi → z1 as i → ∞. By taking the limit in inequalities (3), and (4), and by the definition of the set E, we arrive at (29).


487

∂g ∗ ∗ Let us consider the two cases. If ∂x (t1 ), f (u1 , t1∗ ) + ∂g ∂t (t1 ) < 0, then (28) holds ∂g ∗ ∗ for f1 = f (u1 , t1∗ ). In the second case, when ∂x (t1 ), f (u1 , t1∗ ) + ∂g ∂t (t1 ) = 0, we de∂g ∗ ∂g note φ(u) = ∂x (t1 ), f (u, t1∗ ) + ∂t (t1∗ ). Then φ(u1 ) = 0, and, in view of the obvious identity ∂g ∂g

α, Q(x, u, t) ≡ (x, t), f (x, u, t) + (x, t), ∂x ∂t

and also of the second inequality in (29) we have ∂φ ∂u (u1 ), z1 ≥ ε0 . Hence, by using the last group of inequalities of (29), we find that, for all sufficiently small δ > 0, we have R(u1 − δz1 , t1∗ ) < 0, φ(u1 − δz1 ) < 0 and, hence, f1 = f (u1 − δz1 , t1∗ ) satisfies (28). Thus (28) is proved. Now, we are ready to see the controllability relatively to the state constraints in the left end-point. In the space Rn+1 , we consider the convex cone K, which is the convex ∂Gj hull of the rays generated by the finite number of vectors ∂(x,t) (t1∗ ), j ∈ J (t1∗ ). The cone K is acute (i.e. contains no nonzero subspace) because regularity of the process implies positive linear independence of the vectors generating the cone K. Therefore, int K ◦ = ∅, where K ◦ is the polar cone to K. Consider the set F = (f (U (t1∗ ), t1∗ ), 1) in the space Rn+1 . This is easy to show that

∂G K ◦ = {ξ : ξ, ∂(x,t) (t1∗ ) ≤ 0 ∀j ∈ J (t1∗ )}. Therefore, in order to get the controllability conditions introduced in Definition 2.1, it suffices to prove that j

conv F ∩ int K ◦ = ∅.

(30)

Suppose the contrary. By applying the separability theorem to the disjoint convex sets conv F and int K ◦ , there is a nonzero vector h ∈ Rn+1 , for which h, d ≥ 0, ∀d ∈ conv F and h ∈ K ◦◦ . But, the polyhedral cone K is closed and convex, and hence K ◦◦ = K and this implies that h ∈ K. Therefore, by the definition of the cone K, ∂G there exists a vector α, such that h = α ∂(x,t) (t1∗ ) and α j ≥ 0, ∀j ∈ J (t1∗ ), α j = 0, ∗ ∀j ∈ / J (t1 ). Hence, by virtue of (28), ∃d = (f1 , 1) ∈ F such that h, d < 0. This contradiction proves (30). By carrying out similar reasoning in the right end-point of the trajectory, we obtain the controllability relatively to the state constraints in the right end-point. Lemma 2.1 is proved. Proof of Theorem 4.2 Suppose that (27) be violated. Then, λ0 = 0 and ∃t0 ∈ T such that ψ(t0 ) = μ(t0 ) ∂G ∂x (t0 ). From Remark 3.1, it follows that the Lagrange multipliers ∂G ¯ λ, ψ(t) = ψ(t) − μ(t0 ) (t), μ(t) ¯ = μ(t) − μ(t0 ), r ∂x satisfy all conditions of the MP except condition (b). Consider the number ε > 0 from Remark 2.1. By condition (a), and the fact that x˙ ∗ is essentially bounded, there exists δ0 > 0, such that, if τ ∈ T and μ(τ ¯ ) = 0, then μ¯ j (t) = 0,

∀t ∈ T : |t − τ | ≤ δ0 , ∀j ∈ / Jε (τ ).

(31)

Now, let τ ∈ T and μ(τ ¯ ) = 0. Then, by the condition (c), μ(t) ¯ ≤ 0, ∀t ≥ τ . So, by multiplying, for every t, both sides of (10) by the vector q1 (t) from the regularity

488


condition, and in the light of Remark 2.1, we conclude, by using (11) and (31), the existence of positive numbers δ and c such that ¯ |μ(t)| ¯ + |r(t)| ≤ c|ψ(t)|,

a.e. in T : τ ≤ t ≤ τ + δ,

(32)

for any τ ∈ T for which μ(τ ¯ ) = 0. We now prove that ψ¯ = 0,

r = 0,

μ(t) ¯ = 0,

∀t ∈]t1∗ , t2∗ [.

(33)

¯ 0 ) = 0 and μ(t ¯ 0 ) = 0, then, from (5) and (32), by taking τ = t0 Indeed, since ψ(t ¯ in (32), and by applying Gronwall’s Inequality, we find out that ψ(t) = 0 for all t, and μ(t) ¯ = 0, r(t) = 0 for almost all t in T ∩ [t0 , t0 + δ[. Consider t1 = t0 + 2δ , if ¯ 1 ) = 0, and μ(t ¯ 1 ) = 0. t0 + 2δ ≤ t2∗ , and t1 = t2∗ , otherwise. As already proved, ψ(t ¯ Therefore, by the above arguments but carried out for the point t1 , we get that ψ(t) = 0 for all t, and μ(t) ¯ = 0, and r(t) = 0 for almost all t in T ∩ [t1 , t1 + δ[. By considering t2 = t1 + 2δ , if t1 + 2δ ≤ t2∗ and t2 = t2∗ otherwise, and similarly for ¯ the points t3 , t4 etc., we conclude, after a finite number of steps, that ψ(t) = 0, for all t, and that μ(t) ¯ = 0, and r(t) = 0, for almost all t in [t0 , t2∗ ]. Similar arguments but to the left side from the point t0 , and by using function q2 instead of q1 , we will finally obtain (33). From (33), by virtue of the transversality conditions (6), we obtain (−1)s

∂l ∗ ∂G ∗ (p , λ) = μ(t ¯ s∗ ) (t ), ∂xs ∂xs s

s = 1, 2.

Analogously, by virtue of (9) and the continuity of h, (−1)s

∂l ∗ ∂G ∗ (p , λ) = μ(t ¯ s∗ ) (t ), ∂ts ∂ts s

s = 1, 2.

It is also true the formula (26) derived above. Besides, regularity of the optimal process implies the controllability condition relatively to the state constraints at p ∗ (see Lemma 2.1). ¯ 2∗ ) ≤ 0 by the condition (c) and due to the fact that Note that, μ(t ¯ 1∗ ) ≥ 0 and μ(t ∗ ∗ μ(t) ¯ = 0, ∀t ∈ ]t1 , t2 [. Therefore, by substituting the expressions obtained above in the formula (26), and by using the controllability relatively to the state constraints at ¯ 1∗ ) = μ(t ¯ 2∗ ) = 0. Hence, μ¯ = 0 and, therefore, in view of the point p ∗ , we get that μ(t condition (b), μ(t) = μ(t0 ), ∀t ∈ T , and, thus, μ = 0. Thus, ψ = 0. This contradicts to the fact that all the Lagrange multipliers λ0 , ψ, μ cannot vanish simultaneously. Hence, (27) holds. The theorem is proved. Note that the assumptions of Theorem 4.2 also imply λ0 + max |ψ(t)| > 0. t∈T

(34)

Indeed, suppose that (34) be violated. Then, λ0 = 0 and ψ = 0. From the regularity of the optimal process, the regularity of the mixed constraints, conditions (a)–(c) and formula (10), it follows that μ(t) = 0, ∀t ∈ ]t1∗ , t2∗ ] and, hence, (27) is violated. This contradiction proves (34).


489

5 Connections with the Results of R.V. Gamkrelidze Here, we show how Theorem 4.2 encompasses Theorem 25 of [1]. We shall further assume that the functions f , and R be twice continuously differentiable (despite of the fact that second derivative of f , and R is not involved in the conditions of Theorems 3.1, 4.1, 4.2). In Theorem 25, d(G) = 1, the set T1 = {t ∈ T : G(t) = 0} consists of a finite union of segments, the optimal control u∗ (t) is piecewise smooth, and the optimal process is regular. In addition, the end-points of the trajectory, x1 and x2 , are fixed (and, therefore, the Lagrange multipliers λ1 and λ2 are omitted below). Theorem 25 in [1] was proved in two stages. First, it was proved Theorem 22 of this reference, in which assumed that T1 = T . This corresponds to the case in which the optimal trajectory lies entirely in the state constraints boundary. Next, in Theorem 24 of this reference, the so-called jump conditions1 were obtained. Finally, Theorem 25 is obtained by combining Theorems 22 and 24. First, let us note that in Theorem 4.2, when d(G) = 1, along with (27), and (34), the following non-triviality condition is true λ0 + |ψ(t)| > 0 ∀t ∈ T1 .

(35)

Indeed, assume the opposite, i.e., that λ0 = 0 and ∃ t ∗ ∈ T1 such that ψ(t ∗ ) = 0. In view of (10), the regularity of optimal process, by using Gronwall’s Inequality applied to (5) (see similar arguments in p. 110 of [6]), and condition (a), we arrive to ψ = 0. This contradicts (34). Condition (35) is exactly the non-triviality condition of Theorem 22. Let the collection λ0 , ψ(t), μ(t), r(t) satisfy Theorem 4.2. Represent the vector ∂G ∗ ∗ n ψ(t1∗ ) in the form ψ(t1∗ ) = z + μ∗ ∂G ∂x (x1 ), where z ∈ R such that z, ∂x (x1 ) = 0, ∗ 1 and μ ∈ R is a certain number. By Remark 3.1, the multipliers ∂G ¯ λ0 , ψ(t) (t), μ(t) ¯ = μ(t) − μ∗ , r(t) = ψ(t) − μ∗ ∂x satisfy the conditions (5)–(11), and also (a), (c). We show now that they will also sat¯ ∗ ) = 0 for some point isfy the nontriviality condition (35). Indeed, if λ0 = 0 and ψ(t t ∗ ∈ T1 , then, as it was already done earlier, from (5), (10), due to the Gronwall’s ¯ Inequality and the regularity of the optimal process, we conclude that ψ(t) = 0, ∗ , ∀t ∈ ]t ∗ , t ∗ [, thus μ(t) ¯ = 0, ∀t ∈ ]t1∗ , t2∗ [. Therefore, ψ(t) = μ∗ ∂G (t), μ(t) = μ 1 2 ∂x contradicting (27). The Lagrange multipliers of Theorem 22 were denoted as (ψ(t), λ(t), ν(t)). Let ¯ us check whether the multipliers ψ(t) = (−λ0 , ψ(t)), λ(t) = μ(t), ¯ and ν(t) = r(t) satisfy Theorem 22. From formula (10), by using the regularity and by the fact that all the functions are sufficiently smooth, we deduce that the function μ(t) ¯ is piecewise smooth. It is easy to see that all conditions of Theorem 22 immediately follow 1 This is a condition on the jumps of the function ψ at the points of junction. That is, at such points, where

the trajectory meets the boundary of the state constraint set or leaves it.

490


from (35), (5)–(11), (a), (c). Condition (b) of Theorem 22 is satisfied, since, by con¯ ∗ ) ∈ ker ∂G (x ∗ ). struction, ψ(t 1 ∂x 1 Now, in order to get Theorem 25, in the above arguments, we need, instead of the constant μ∗ , to consider some piecewise-constant function μ∗ (t), such that μ(t), μ (t) = μ∗i , ∗

t : G(t) < 0, t ∈ Si ,

where Si = [si,1 , si,2 ] are the subintervalswhere the optimal trajectory goes along the boundary of the state constraints T1 = i Si , and the numbers μ∗i are selected by ∂G the same rule as above, i.e. ψ(si,1 ) = zi + μ∗i ∂G ∂x (si,1 ), where zi , ∂x (si,1 ) = 0. It is clear that μ(t) ¯ = 0 on any segment on which the optimal trajectory entirely belongs to the interior of the state constraints set, and there is just an ordinary MP there on such segments (as the extended Pontryagin function coincides with the classical one there). On the other hand on each segment Si the MP from Theorem 22 is true. The jump conditions at the junction end-points si,1 , si,2 hold by construction. Thus Theorem 25 is a consequence of Theorem 4.2. Note that, in Theorem 4.2, we got even a stronger statement than those in Theorems 22, 25. Namely, the maximum of the classic Pontryagin’s function H (x, u, ψ) in Theorem 22 is taken over the subset of regular points of the set {u ∈ U (t) : Q(u, t) = 0}. At the same time, in Theorem 4.2, we can obviously use all this set, and even a larger set {u ∈ U (t) : μ(t)Q(u, ¯ t) ≥ 0}. In addition, the condition (c) is stronger than the corresponding one in Theorem 22. It is also interesting to note that, in Theorem 4.2, the function μ is non-negative. This is because μ(t2∗ ) = 0 and μ decreases monotonically. But, this inequality is not observed in Theorem 22 (and, especially, Theorem 25). Indeed, when constructing the Lagrange multipliers for Theorem 22, we subtracted from μ(t) some constant μ∗ and thus, in general, we might lose the sign of λ(t) = μ(t). ¯ A natural question arises: is it possible to ensure that we still have μ(t) ¯ ≥ 0? The answer is negative as it may be observed from the following example. Example 5.1 Let n = m = 2, d(G) = 1. Consider the problem ⎧ π/2 ⎪ Minimize 0 |x|2 dt ⎪ ⎪ ⎪ ⎨ subject to x˙ = u, t ∈ [0, π/2], ⎪ x(0) = (0, 1), x(π/2) = (1, 0), ⎪ ⎪ ⎪ ⎩ 2 |x| ≥ 1, x, u ∈ R2 . It is easy to see that the control u1 (t) = cos(t), u2 (t) = − sin(t) and the trajectory x 1 (t) = sin(t), x 2 (t) = cos(t) define an optimal process. This trajectory x(t) is regular in the sense of the definition given in [1], p. 297. Let us apply the MP from [1]. In the notation of [1], we have H (x, u, ψ) = ψ1 u1 + ψ2 u2 + ψ0 |x|2 ;

p(x, u) = −2 x, u.


491

There are a continuous function ψ(t) = (ψ0 (t), ψ1 (t), ψ2 (t)), where ψ0 (t) = const ≤ 0, and a scalar function λ(t) such that ∂H ∂p (x(t), u(t), ψ(t)) = λ(t) (x(t), u(t)), ∂u ∂u

∀t.

So, ψ1 (t) = −2λ(t)x 1 (t), ψ2 (t) = −2λ(t)x 2 (t). Therefore, λ(·) is continuous. However, in view of condition (b), ψ1 (0)x 1 (0) + ψ2 (0)x 2 (0) = 0, and this implies that λ(0) = 0. Suppose that λ(t) ≥ 0, ∀t. Then, condition (c) and the continuity of the function λ(·) imply that λ(t) ≡ 0. Then, also ψ1 (t) ≡ ψ2 (t) ≡ 0, and, from the equation for the adjoint variable, it follows that ψ0 = 0. This leads to a contradiction. This example clearly shows that it is not possible to strengthen Theorem 22 by requiring non-negative function λ(t) there. On the other hand, we could construct an example in which λ(t) cannot be already non-positive, and, thus, we cannot ensure the sign of λ(t) in Theorem 22. The foregoing fully applies to the jump conditions. In Theorem 25, we cannot know whether the function ψ(t) will “jump” either along the gradient ∂G ∂x (t) or along the opposite direction.

6 The Problem with Measurable Dependence on Time Here, we assume that the functions K0 , K1 , K2 , and G are continuously differentiable, the function G is twice continuously differentiable in x for all t, the functions f , R are continuously differentiable in (x, u) for almost all t, and measurable in t for any fixed (x, u) together with their partial derivatives with respect to (x, u), and on any bounded set, f , and R and their partial derivatives with respect to (x, u) are bounded and continuous in (x, u) uniformly in (x, u, t). In this section, regularity of the mixed constraints is understood in the following sense: for every c > 0, there exists δ > 0 such that, for any (x, u) and for almost all t, for which |x| + |u| + |t| ≤ c,

R j (x, u, t) ≤ δ,

∀j,

there is a vector q = q(x, u, t), such that |q| ≤ 1,

q,

∂R j (x, u, t) ≥ δ, ∂u

∀j : R j (x, u, t) ≥ −δ.

For a measurable scalar function g : R → R and a point τ ∈ R we set, [6], ess lim g(t) = lim t→τ

ess sup g(t),

ε→0+ t∈[τ −ε,τ +ε]

ess lim g(t) = lim t→τ

ess inf

ε→0+ t∈[τ −ε,τ +ε]

g(t).

These expressions are, respectively, called by essential upper and essential lower limits of g at τ . We introduce the following additional assumption.

492


Assumption (B) The sets U (x, t) are uniformly bounded in (x, t) in some neighborhoods of points (xs∗ , ts∗ ), s = 1, 2. Theorem 6.1 Let a process (p ∗ , x ∗ , u∗ ) be optimal for problem (1). Suppose that the end-point constraints be regular at the point p ∗ , the state and the mixed constraints be regular, the state constraints be compatible with the end-point constraints at p ∗ , and that the Assumption (B) be in force. Then, the process (p ∗ , x ∗ , u∗ ) satisfies a weakened MP, i.e. conditions obtained from the MP by removing condition (8) and by replacing the time transversality conditions (9) by the following: ess lim∗ sup H¯ (xs∗ , u, ψ(ts∗ ), μ(ts∗ ), t) t→ts u∈U (t)

≥ (−1)s

∂l ∗ ∂G ∗ (p , λ) − μ(ts∗ ) (t ), ∂ts ∂ts s

s = 1, 2,

ess lim sup H¯ (xs∗ , u, ψ(ts∗ ), μ(ts∗ ), t) t→ts∗ u∈U (t)

≤ (−1)s

∂l ∗ ∂G ∗ (p , λ) − μ(ts∗ ) (t ), ∂ts ∂ts s

s = 1, 2.

Proof of Theorem 6.1 is completely analogous to that of Theorem 3.1. Assumption (B) is essential, as illustrated by the Example 3.2 in Chap. 2 of [6].

7 Conclusion In this article, we derive necessary conditions of optimality for control problems with state constraints, the MP in the form of the one by R.V. Gamkrelidze. The assumptions of controllability relatively to the state constraints and of regularity of control process are used to ensure non-triviality conditions (23) and (27), respectively. These necessary conditions (Theorem 3.1) are easy to derive from the earlier version of the MP in the form of the one by Dubovitskii and Milyutin, [6], and at the same time, they encompass the necessary conditions of [1]. It is important to note that from Theorem 3.1, in turn, the MP of [6] is easy to deduce. Thus, these new conditions are nothing but a reformulation of the MP from [6] under the assumption of higher smoothness for G(x, t). The following question naturally arises. What happens, if, in the proof of Theorem 3.1, instead of the MP from [6], the original version of the Dubovitskii-Milyutin MP (see in [14, 15]) is used? We answer this question now. For the sake of simplicity, let us assume that the time interval end-points are fixed, i.e., T = [0, 1], that f0 = 0, and that the problem is autonomous. If, instead of the MP from [6], we had used MP by Dubovitskii-Milyutin in the proof of Theorem 3.1, then we would have that the function h in Theorem 3.1 is constant on the interval ]0, 1[ only, as it might exhibit discontinuities at the endpoints of this time interval. However, we get somewhat more than that with the help of MP from [6]. Namely, we proved that the function h is absolutely continuous on


493

the interval [0, 1] and therefore, in this case, the function h is constant on the whole segment [0, 1]. This is critically important! Indeed, for deriving the non-triviality condition in Theorem 4.1 we used the formula (26), which in this case takes the form s+1 ∂l ∗ ∗ (p , λ), f (u, ts ) = 0, s = 1, 2. (36) max (−1) ∂xs u∈U (ts∗ ) Formula (36) implies the desired condition of non-triviality (23). But if the function h is discontinuous at the end-points, then from (36) this conclusion can not be obtained. Thus, if, instead of the MP from [6], the Dubovitskii-Milyutin MP is used, then Theorem 4.1 cannot be proved! This fully applies to Theorem 4.2. Our investigation in this article led us to the conclusion that the MP by R.V. Gamkrelidze is a rather natural form of the first-order necessary conditions for problems with state constraints. As a paradigmatic example, we observe that, from this MP, it follows directly the geodesic equation (see [1], the end of Chap. 6).

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