Optimal Control Problem for Switched System with the Nonsmooth Cost Functional

and Applied Analysis 3 Here f K : R × Rn × Rr → R, M K and F K are continuous, at least continuously partially differentiable vectorvalued functions with respect to their variables, L : Rn ×Rr × R → R are continuous and have continuous partial derivative with respect to their variables, φ k (⋅) has Frechet upper subdifferentiable (superdifferentiable) at a point x K (t K ) and positively homogeneous functional, and u K (t) : R → U K ⊂ Rr are controls. The sets U K are assumed to be nonempty and open. Here (16) is switching conditions. If we denote this as follows: θ = (t 1 , t 2 , . . . , t N ), x(t) = (x 1 (t), x 2 (t), . . . , x N (t)), u(t) = (u 1 (t), u 2 (t), . . . , u N (t)), then it is convenient to say that the aim of this paper is to find the triple (x(t), u(t), θ) which solves problem (14)–(18). This triple will be called optimal control for the problem (14)–(18). At first we assume that φ k (⋅) is the Hadamar upper differentiable at the pointx K (t K ) in the direction of zero.Then,φ k (⋅) is upper semicontinuous, and it has an exhaustive family of lower concave approximations of φ k (⋅). Theorem3 (Necessary optimality condition in terms of lower exhauster). Let (u K (⋅), x K (⋅), θ) be an optimal solution to the control problem (14)–(18). Then, for every element x∗ K from intersection of the subsets C K of the lower exhauster E ∗,K of the functional φ K (x K (t K )), that is, x∗ K ∈ ⋂ CK∈E∗,K C K , K = 1, 2, . . . , N, there exist vector functions p K (t), K = 1, . . . , N for which the following necessary optimality condition holds: (i) State equation: ?̇? K (t) = ∂H K (x K (t) , u K (t) , p K (t) , t) ∂p K , t ∈ [t K−1 , t K ] ;


Introduction
A switched system is a particular kind of hybrid system that consists of several subsystems and a switching law specifying the active subsystem at each time instant.There are some articles which are dedicated to switching system [1][2][3][4][5][6][7][8].Examples of switched systems can be found in chemical processes, automotive systems, and electrical circuit systems, and so forth.
Regarding the necessary optimality conditions for switching system in the smooth cost functional, it can be found in [1,4,6].The more information connection between quasidifferential, exhausters and Hadamard differential are in [8][9][10].Concerning the necessary optimality conditions for discrete switching system is in [5], and switching system with Frechet subdifferentiable cost functional is in [3].This paper addresses the role exhausters and quasi-differentiability in the switching control problem.This paper is also extension of the results in the paper [5] (additional conditions are switching points unknown, and minimization functional is nonsmooth) in the case of first optimality condition.The rest of this paper is organized as follows.Section 2 contains some preliminaries, definitions, and theorems.Section 3 contains problem formulations and necessary optimality conditions for switching optimal control problem in the terms of exhausters.Then, the main theorem in Section 3 is extended to the case in which minimizing function is quasidifferentiable.

Some Preliminaries of Non-Smooth Analysis
Let us begin with basic constructions of the directional derivative (or its generalization) used in the sequel.Let  :  → ,  ⊂   be an open set.The function  is called Hadamard upper (lower) derivative of the function  at the point  ∈  in the direction  ∈  if there exist limit such that where [, ] → [+0, ] means that  → +0 and  → .Note that limits in (1) always exist, but there are not necessary finite.This derivative is positively homogeneous functions of direction.The Gateaux upper (lower) subdifferential of the function  at a point  0 ∈  can be defined as follows: 2

Abstract and Applied Analysis
The set is called, respectively, the upper (lower) Frechet subdifferential of the function  at the point  0 .
As observed in [9,10], if  is a quasidifferentiable function then its directional derivative at a point  is represented as where (), () ⊂   are convex compact sets.From the last relation, we can easily reduce that This means that for the function ℎ() =   (, ) the upper and lower exhausters can be described in the following way: It is clear that the Frechet upper subdifferential can be expressed with the Hadamard upper derivative in the following way; see [9, Lemma 3.2]: Theorem 1.Let  * be lower exhausters of the positively homogeneous function ℎ :   → .Then, ⋂ ⊂ *  = ∂+ ℎ(0  ), where ∂+ ℎ is the Frechet upper subdifferential of the ℎ at 0  , and for the positively homogeneous function ℎ :   →  the Frechet superdifferential at the point zero follows Proof.Take any V 0 ∈ ⋂ ⊂ * .Then by using definition an lower exhausters we can write Consider now any Let us consider V 0 ∉ ⋂ ⊂ * .Then, there exists  0 ∈  * where V 0 ∉  0 .Then, by separation theorem, there exists  0 ∈   such that It is conducts (3) and V 0 ∈  for every  ∈  * and due to arbitrary.This means that V 0 ∈ ⋂ ⊂ * .The proof of the theorem is complete.

Lemma 2. The Frechet upper and Gateaux lower subdifferentials of a positively homogeneous function at zero coincide.
Proof.Let ℎ :   →  be a positively homogenous function.It is not difficult to observe that every  ∈   and every  > 0: Hence, the Gateaux lower subdifferential of ℎ at 0  takes the forms which coincides with the representation of the Frechet upper subdifferential of the positively homogenous function (see [11,Proposition 1.9]).

Problem Formulation and Necessary Optimality Condition
Let investigating object be described by the differential equation with initial condition and the phase constraints at the end of the interval and switching conditions on switching points (the conditions which determine that at the switching points the phase trajectories must be connected to each other by some relations): The goal of this paper is to minimize the following functional: with the conditions ( 14)-( 16).Namely, it is required to find the controls  1 ,  2 , . . .,   , switching points  1 ,  2 , . . .,  −1 , and the end point   (here  1 ,  2 , . . .,   are not fixed) with the corresponding state  1 ,  2 , . . .,   satisfying ( 14)-( 16) so that the functional ( 1 , . . .,   ,  1 , . . .,   ) in ( 18) is minimized.We will derive necessary conditions for the nonsmooth version of these problems (by using the Frechet superdifferential and exhausters, quasidifferentiable in the Demyanov and Rubinov sense).
From the calculus of variations, we can obtain that the first variation of   as If we follow the steps in [3, pages 5-7] then, the first variation of the functional takes the following form: If the state equations ( 14) are satisfied, ṗ  is selected so that coefficient of   and   is identically zero.Thus, we have The integrand is the first-order approximation to the change in   caused by where  ∈ [ −1 ,   ] is an arbitrarily small, but nonzero time interval and   are admissible control variations.After this, if we consider proof description of the maximum principle in [4], we can come to the last inequality.
According to the fundamental theorem of the calculus of the variation, at the extremal point the first variation of the functional must be zero, that is,   = 0. Setting to zero, the coefficients of the independent increments   (  ),   (  )  ,   and   , and taking into account that yield the necessary optimality conditions (i)-(v) in Theorem 3.This completes the proof of the theorem.