Necessary Optimality Conditions for a Class of Hybrid Optimal Control

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Necessary Optimality Conditions for a Class of. Hybrid Optimal Control Problems. Vadim Azhmyakov1, Sid Ahmed Attia1,2, Dmitry Gromov1, and Jörg Raisch1,2.

Necessary Optimality Conditions for a Class of Hybrid Optimal Control Problems Vadim Azhmyakov1, Sid Ahmed Attia1,2 , Dmitry Gromov1, and J¨ org Raisch1,2 1

2

Fachgebiet Regelungssysteme, Technische Universit¨ at Berlin, Einsteinufer 17, D-10587 Berlin, Germany {azhmyakov,attia,gromov,raisch}@control.tu-berlin.de http://www.control.tu-berlin.de Systems and Control Theory Group, MPI for Dynamics of Complex Technical Systems, Sandtorstr. 1, D-39106 Magdeburg, Germany

Abstract. In this paper we study a class of Mayer-type hybrid optimal control problems. Using Lagrange techniques, we formulate a version of the Hybrid Maximum Principle for optimal control problems governed by hybrid systems with autonomous location transitions in the presence of additional target constraints.

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Introduction and Problem Formulation

Over the last few years hybrid optimal control theory has been formalized as a natural generalization of classical optimal control theory. For hybrid optimal control problems, the main tool towards the construction of optimal control signals and optimal trajectories is the Hybrid Maximum Principle, which generalizes the classical Pontryagin Maximum Principle (see, e.g., [4]). Several variants of the Hybrid Maximum Principle were developed in [6,7,9,10,3,8]. It is well-known that the standard proof of the Pontryagin Maximum Principle is based on the technique of “needle variations“ [4]. In this paper we derive necessary optimality conditions for a class of hybrid optimal control problems without recourse to the technique of needle variations. Instead, we apply a Lagrange based approach. This approach allows us to obtain necessary conditions for a weak minimum as opposed to the standard Maximum Principle which gives necessary conditions for a strong minimum. In this paper, we consider a hybrid control system H with autonomous location transitions (see, e.g.,[8]). Its state at time t ∈ [0, tf ] is the pair (q, x(t)), where q ∈ Q, x(t) ∈ Rn , and Q is a finite set of locations. While in location q, the temporal evolution of x is determined by x˙ = fq (x, u), where the functions fq are assumed to be continuously differentiable and bounded, the admissible control sets Uq are compact and convex, and Uq := {u(·) ∈ Lm ∞ (0, tf ) : u(t) ∈ Uq , a.e. on[0, tf ]} represent the sets of admissible control signals. Switchings between locations are determined by smooth functions mq : Rn → R, q ∈ Q with nonzero gradients A. Bemporad, A. Bicchi, and G. Buttazzo (Eds.): HSCC 2007, LNCS 4416, pp. 637–640, 2007. c Springer-Verlag Berlin Heidelberg 2007 

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such that the hypersurfaces (”switching sets”) Mq := {x ∈ Rn : mq (x) = 0} are pairwise disjoint and divide Rn into open sets. Let us now introduce the following concept (see [9,3]). Definition 1. A hybrid trajectory of H is a triple X = (x, {qi }, τ ), where x(·) : [0, tf ] → Rn , {qi }i=1,...,r is a finite sequence of locations and τ is the corresponding sequence of switching times 0 = t0 < · · · < ti < · · · < tr = tf such that for each i = 0, . . . , r there exists ui (·) ∈ Ui such that:  / q∈Q Mq and xi (·) = x(·)|(ti−1 ,ti ) is an absolutely continuous – x(0) = x0 ∈ function in (ti−1 , ti ) continuously prolongable to [ti−1 , ti ], i = 1, . . . , r; – x˙ i (t) = fqi (xi (t), ui (t)) for almost all t ∈ [ti−1 , ti ], i = 1, . . . , r; – the switching condition (xi (ti ), xi+1 (ti )) ∈ Mqi holds for each i = 1, . . . , r−1. Note that the evolution equation for the continuous trajectory x(·) of a given H can be represented as follows x(t) ˙ =

r 

β(ti−1 ,ti ] (t)fqi (x(t), u(t)), a.e. on [0, tf ], i = 1, . . . , r,

(1)

i=1

where β(ti−1 ,ti ] (·) is a characteristic function of the interval (ti−1 , ti ]. Under the above assumptions for the family of vector fields {fq (x, u)}q∈Q , the right-hand side of equation (1) satisfies the Caratheodory conditions (see e.g., [4]). Next we consider x(·) as an element of the Sobolev space x(·) ∈ Wn1,∞ ([0, tf ]), which contains all absolutely continuous functions with essentially bounded derivatives. Let φ : Rn → R and g : Rn → R be continuously differentiable functions. We consider an additional target manifold given by the equation g(x) = 0 and introduce the notation Mqr (x) := {x ∈ Rn : mr (x) = 0}, where mr (x) = g(x). Given a hybrid system H we formulate the following Mayer-type hybrid optimal control problem (HOCP): minimize φ(x(tf )) over all trajectories X of H such that g(x(tf )) = 0.

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(2)

A Variant of the Hybrid Maximum Principle

We will study necessary optimality conditions for (2) with the help of the general Lagrange multiplier rule for optimization problems in Banach spaces [4,5]: Theorem 1. Let Y and Z be real Banach spaces, ψ : Y → R be a cost functional and h : Y → Z be a mapping. Let F be a convex subset of Y with a nonempty interior and y 0 be a solution of the following optimization problem minimize ψ(y) subject to h(y) = 0Z , y ∈ F,

(3)

where 0Z is the zero element of Z. Assume that ψ and h are Fr´echet differentiable in a neighborhood of y 0 , the derivative h (·) is continuous at the point y 0 and

Necessary Optimality Conditions

639

the image set h (y 0 )(Y ) is closed. Then there are a real number μ ≥ 0 and a continuous linear functional  ∈ Z ∗ (the topological dual space to Z) with (μ, ) = (0, 0Z ∗ ) such that   Ly (y 0 , μ, )(y − y 0 ) = μψ  (y 0 ) +  ◦ h (y 0 ) (y − y 0 ) ≥ 0 ∀y ∈ F, (4) where L(y, μ, ) := μψ(y) +  ◦ h(y) is the Lagrange function for (3) and ◦ is referred to as the duality pairing [5], which maps a pair from Z∗ × Z to R. Note that in the case of a regular problem (3) one can put μ = 1. We now introduce the mapping ⎛ ⎞ r · ξ(·) − x0 − β(ti−1 ,ti ] (t)fqi (ξ(t), v(t))dt ⎠. P (v(·), ξ(·)) := ⎝ 0 i=1 mqi (ξ(ti )) i=1,...,r n where (v(·), ξ(·)) ∈ Lm ∞ ([0, tf ]) × W1,∞ ([0, tf ]). The first element of P is an operator of differential equation (1) in integral form whereas the second one determines switching times ti from a sequence τ according to the following switching rule: ti = inf{t | mqi (x(t)) = 0}, where i = 1, ..., r − 1. The operator equation P (v(·), ξ(·)) = 0 determines the evolution of the hybrid control system H as a function of control v ∈ U. Note that the mapping P is specified in the same n manner as the mapping h from Theorem 1. Here Y := Lm ∞ ([0, tf ])×W1,∞ ([0, tf ]) n r and Z := W1,∞ ([0, tf ]) × R . We now rewrite the HOCP (2) in the form (3) as

minimize J(u(·), x(·)) = φ(x(tf )) subject to P (u(·), x(·)) = 0 Wn1,∞ ([0,tf ])×Rr , (u(·), x(·)) ∈ F.

(5)

n where F := {(v(·), ξ(·)) ∈ Lm ∞ ([0, tf ]) × W1,∞ ([0, tf ]) : vi (t) ∈ Uqi , i = 1, . . . , r}, and vi (·) is a restriction of the function v(·) on the time interval [ti−1 , ti ]. Note that F is a convex set. A solution of (5) is denoted by (x0 (·), u0 (·)). For (5) we introduce the Lagrange function

L((u(·), x(·)), μ, l) := μJ(u(·), x(·)) + (, a) ◦ P (u(·), x(·)), where μ ∈ R≥0 , a ∈ Rr and the continuous linear functional  belongs to the (topological) dual space of Wn1,∞ ([0, tf ]). We now are in the position to state our main result. Theorem 2. Let functions φ, fq , mq , g be continuously differentiable and the optimization problem (5) be regular. Then there exist a function p(·) from Wn1,∞ ([0, tf ]) and a vector a ∈ Rr such that ∂Hqi (x0i (t), u0i (t), p(t)) a. e. on (t0i−1 , t0i ), i = 1, ..., r, ∂x   ∂g(x0 (tf )) ∂φ(x0 (tf )) pr (tf ) = − ar + , ∂x ∂x   ∂mqi (x0 (t0i )) pi (t0i ) = pi+1 (t0i ) + ai , , i = 1, ..., r − 1, ∂x p˙ i (t) = −

(6)

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0 where p(t) = ri=1 β(t 0 0 (t)pi (t). Moreover, for every admissible control u(·) i−1 ,ti ] the following inequalities are satisfied   ∂Hqi (x0 (t), u0 (t), p(t)) , (u(t) − u0 (t)) ≤ 0 a. e. on [t0i−1 , t0i ], (7) ∂u where i = 1, ..., r and Hqi (x, u, p) := (p, fqi (x, u)) is a ”partial” Hamiltonian for the location qi ∈ Q and (·, ·) denotes the corresponding scalar product.

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Discussion

In this contribution we have proposed a version of the Hybrid Maximum Principle based on the Lagrange multiplier rule. This implies that the necessary optimality conditions for a weak minimum are obtained. However, in many practical optimal control problems weak minima coinside with strong minima. Note that the suggested proof-technique can also be applied to hybrid systems with state jumps and to some classes of hybrid systems with controlled location transitions. The Hamilton minimization conditions from Theorem 2 are presented in the form of variational inequalities. This form is closely related to the Weierstraß conditions for a strong minimum (see, e.g.,[4]) and to the gradient-based computational approach studied in [1]. Finally note that condition (7) make it possible to take into consideration some effective methods for numerical treatment of variational inequalities.

References 1. V. Azhmyakov and J. Raisch, A gradient-based approach to a class of hybrid optimal control prolems. In: Proceedings of the 2nd IFAC Conference on Analysis and Design of Hyrid Systems, Alghero, 2006, pp. 89 – 94. 2. V. Azhmyakov, S.A. Attia, D. Gromov and J. Raisch, Optimal dymanics of hybrid and switched systems, Technical Report, Fachgebiet Regelungstechnik, TU Berlin, 2007. 3. M. Garavello and B. Piccoli, Hybrid necessary priniple, SIAM J. Control Optim., 43 (2005), pp. 1867 – 1887. 4. A.D. Ioffe and V.M. Tichomirov, Theory of Extremal Problems, North Holland, Amsterdam, 1979. 5. A.J. Kurdila and M. Zabarankin, Convex Functional Analysis, Birkh¨ auser, 2005 6. B. Piccoli, Hybrid systems and optimal control. In: Proceedings of the 37th IEEE CDC, Tampa, 1998, pp. 13 – 18. 7. B. Piccoli, Necessary conditions for hybrid optimization. In: Proceedings of the 38th IEEE CDC, Phoenix, 1999, pp. 410 – 415. 8. M.S. Shaikh and P.E. Caines, On the hybrid optimal control problem: The hybrid maximum principle and dynamic programming theorem. IEEE Trans. Automat. Contr. submitted (2004). 9. H. Sussmann, A maximum principle for hybrid optimal control problems. In: Proceedings of the 38th IEEE CDC, Phoenix, 1999, pp. 425 – 430. 10. H. Sussmann, A nonsmooth hybrid maximum priniple, Stability and Stabilization of Nonlinear Systems (Ghent, 1999), LNCIS, 246, pp. 325 – 354, Springer, 1999.

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