Optimal Control Problem for Switched System with the Nonsmooth Cost Functional

Abstract

We examine the relationships between lower exhausters, quasidifferentiability (in the Demyanov and Rubinov sense), and optimal control for switching systems. Firstly, we get necessary optimality condition for the optimal control problem for switching system in terms of lower exhausters. Then, by using relationships between lower exhausters and quasidifferentiability, we obtain necessary optimality condition in the case that the minimization functional satisfies quasidifferentiability condition.

1. 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 [1–8]. 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 [8–10]. 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.

2. Some Preliminaries of Non-Smooth Analysis

Let us begin with basic constructions of the directional derivative (or its generalization) used in the sequel. Let 𝑓∶𝑋→𝑅, 𝑋⊂𝑅^𝑛 be an open set. The function 𝑓 is called Hadamard upper (lower) derivative of the function 𝑓 at the point 𝑥∈𝑋 in the direction 𝑔∈𝑋 if there exist limit such that 𝑓^↑_𝐻∶=limsup_{[𝛼,𝑔]→[+0,𝑔]}1𝛼𝑓(𝑥,𝑓𝑥+𝛼𝑔−𝑓)_↓𝐻∶=liminf_{[𝛼,𝑔]→[+0,𝑔]}1𝛼𝑓𝑥+𝛼𝑔−𝑓(𝑥), (1) where [𝛼,𝑔]→[+0,𝑔] means that 𝛼→+0 and 𝑔→𝑔.

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 𝑓 at a point 𝑥₀∈𝑋 can be defined as follows: 𝜕⁺_𝐺𝑓𝑥₀=𝑣∈𝑅^𝑛∣limsup_𝑡↓0𝑓𝑥₀+𝑡𝑔−𝑓𝑥₀𝑡≤𝑣,𝑔,∀𝑔∈𝑅^𝑛. (2) The set𝜕⁺𝑓𝑥₀=𝑣∈𝑅^𝑛∣limsup_𝑥→𝑥0𝑓(𝑥)−𝑓𝑥₀−𝑣,𝑥−𝑥₀𝑥−𝑥₀≤0 (3)is called, respectively, the upper (lower) Frechet subdifferential of the function 𝑓 at the point 𝑥₀.

As observed in [9, 10], if 𝑓 is a quasidifferentiable function then its directional derivative at a point 𝑥 is represented as 𝑓^′𝑥,𝑔=max_{𝑣∈𝜕𝑓(𝑥)}𝑣,𝑔+min_{𝑤∈(𝑥)𝜕𝑓}𝑤,𝑔, (4) where 𝜕𝑓(𝑥),𝜕𝑓(𝑥)⊂𝑅^𝑛 are convex compact sets. From the last relation, we can easily reduce that 𝑓^′𝑥,𝑔=min_{𝑤∈(𝑥)𝜕𝑓}max_{𝑣∈𝑤+𝜕𝑓(𝑥)}𝑣,𝑔=max_{𝑣∈𝜕𝑓(𝑥)}max_{𝑤∈𝑣+(𝑥)𝜕𝑓}𝑣,𝑔. (5) This means that for the function ℎ(𝑔)=𝑓^′(𝑥,𝑔) the upper and lower exhausters can be described in the following way: 𝐸^∗=𝐶=𝑤+𝜕𝑓(𝑥)∣𝑤∈(𝑥,𝐸𝜕𝑓)_∗=𝐶=𝑣+𝜕𝑓(𝑥)∣𝑣∈𝜕𝑓(𝑥). (6) 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]: 𝜕⁺_𝐹𝑓𝑥₀=𝜕⁺_𝐹𝑓^↑_𝐻𝑥₀,0_𝑛. (7)

Theorem 1. Let 𝐸_∗ be lower exhausters of the positively homogeneous function ℎ∶𝑅^𝑛→𝑅. Then, ⋂_{𝐶⊂𝐸∗}𝐶=𝜕⁺ℎ(0_𝑛), where 𝜕⁺ℎ is the Frechet upper subdifferential of the ℎ at 0_𝑛, and for the positively homogeneous function ℎ∶𝑅^𝑛→𝑅 the Frechet superdifferential at the point zero follows 𝜕⁺ℎ0_𝑛=𝑣∈𝑅^𝑛∣ℎ(𝑥)−(𝑣,𝑥)≤0,𝑥∈𝑅^𝑛. (8)

Proof. Take any 𝑣₀∈⋂_{𝐶⊂𝐸∗}𝐶. Then by using definition an lower exhausters we can write 𝑣₀(𝑥)≥ℎ(𝑥),∀𝑥∈𝑅^𝑛⟹_{𝐶⊂𝐸∗}𝐶⊂𝜕⁺ℎ0_𝑛. (9) Consider now any 𝑣₀∈𝜕⁺ℎ(0_𝑛)⇒𝑣₀(𝑥)≥ℎ(𝑥). (10) Let us consider 𝑣₀∉⋂_{𝐶⊂𝐸∗}𝐶. Then, there exists 𝐶₀∈𝐸^∗ where 𝑣₀∉𝐶₀. Then, by separation theorem, there exists 𝑥₀∈𝑅^𝑛 such that 𝑥₀,𝑣₀≤max_𝑣∈𝐶0𝑥₀,𝑣≤ℎ(𝑥). (11) It is conducts (3) and 𝑣₀∈𝐶 for every 𝐶∈𝐸^∗ and due to arbitrary. This means that 𝑣₀∈⋂_{𝐶⊂𝐸∗}𝐶. 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∶𝑅^𝑛→𝑅 be a positively homogenous function. It is not difficult to observe that every 𝑔∈𝑅^𝑛 and every 𝑡>0: ℎ0_𝑛+𝑡𝑔−ℎ0_𝑛𝑡=𝑡ℎ𝑔𝑡=ℎ𝑔. (12) Hence, the Gateaux lower subdifferential of ℎ at 0_𝑛 takes the forms 𝜕⁺_𝐺ℎ0_𝑛=𝑣∈𝑅^𝑛∣ℎ𝑔≤𝑣,𝑔,∀𝑔∈𝑅^𝑛 (13) which coincides with the representation of the Frechet upper subdifferential of the positively homogenous function (see [11, Proposition 1.9]).

3. Problem Formulation and Necessary Optimality Condition

Let investigating object be described by the differential equation 𝑥̇_𝐾(𝑡)=𝑓_𝐾𝑥_𝐾(𝑡),𝑢_𝐾(𝑡),𝑡,𝑡∈𝑡_𝑘−1,𝑡_𝑘,𝐾=1,2,…,𝑁 (14) with initial condition 𝑥₁𝑡₀=𝑥₀, (15) and the phase constraints at the end of the interval 𝐹_𝐾𝑥_𝑁𝑡_𝑁,𝑡_𝑁=0,𝐾=1,2,…,𝑁 (16) 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): 𝑥_𝐾+1𝑡_𝐾=𝑀_𝐾𝑥_𝐾𝑡_𝐾,𝑡_𝑘,𝐾=1,2,…,𝑁−1. (17) The goal of this paper is to minimize the following functional: 𝑆𝑢₁,…,𝑢_𝑁,𝑡₁,…,𝑡_𝑁=^𝑁_𝐾=1𝜑_𝐾𝑥_𝐾𝑡_𝐾+^𝑁_𝐾=1^𝑡𝐾_𝑡𝐾−1𝐿𝑥_𝐾(𝑡),𝑢_𝐾(𝑡),𝑡𝑑𝑡 (18) with the conditions (14)–(16). Namely, it is required to find the controls 𝑢₁,𝑢₂,…,𝑢_𝑁, switching points 𝑡₁,𝑡₂,…,𝑡_𝑁−1, and the end point 𝑡_𝑁 (here 𝑡₁,𝑡₂,…,𝑡_𝑁 are not fixed) with the corresponding state 𝑥₁,𝑥₂,…,𝑥_𝑁 satisfying (14)–(16) so that the functional 𝐽(𝑢₁,…,𝑢_𝑁,𝑡₁,…,𝑡_𝑁) 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).

Here 𝑓_𝐾∶ℝ×ℝ^𝑛×ℝ^𝑟→ℝ^𝑛, 𝑀_𝐾 and 𝐹_𝐾 are continuous, at least continuously partially differentiable vector-valued functions with respect to their variables, 𝐿∶ℝ^𝑛×ℝ^𝑟×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⊂ℝ^𝑟 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=(𝑡₁,𝑡₂,…,𝑡_𝑁), 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑁(𝑡)), 𝑢(𝑡)=(𝑢₁(𝑡),𝑢₂(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 point 𝑥_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(⋅).

Theorem 3 (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: 𝑥̇‌_𝐾(𝑡)=𝜕𝐻_𝐾𝑥_𝐾(𝑡),𝑢_𝐾(𝑡),𝑝_𝐾(𝑡),𝑡𝜕𝑝_𝐾,𝑡∈𝑡_𝐾−1,𝑡_𝐾; (19)(ii)Costate equation: 𝑝̇‌_𝐾(𝑡)=𝜕𝐻_𝐾𝑥_𝐾(𝑡),𝑢_𝐾(𝑡),𝑝_𝐾(𝑡),𝑡𝜕𝑥_𝐾,𝑡∈𝑡_𝐾−1,𝑡_𝐾; (20)(iii)At the switching points, 𝑡₁,𝑡₂,…,𝑡_𝑁−1, 𝑥^∗_𝐾−𝑝_𝐾𝑡_𝐾−𝑝_𝐾+1𝑡_𝐾𝜕𝑀_𝐾𝑥_𝐾𝑡_𝐾,𝑡_𝐾𝜕𝑥_𝐾=0,𝐾=1,2,…,𝑁−1; (21)(iv)Minimality condition: 𝐻_𝐾𝑥_𝐾(𝑡),𝑝_𝐾(𝑡),𝑢_𝐾(𝑡)+𝛿𝑢_𝐾(𝑡),𝑡≥𝐻_𝐾𝑥_𝐾(𝑡),𝑝_𝐾(𝑡),𝑢_𝐾(𝑡),𝑡,𝑓𝑜𝑟𝑎𝑙𝑙𝑎𝑑𝑚𝑖𝑠𝑠𝑖𝑏𝑙𝑒𝛿𝑢_𝐾,𝑡∈𝑡_𝐾−1,𝑡_𝐾; (22)(v)At the end point 𝑡_𝑁, 𝑝_𝑁𝑡_𝑁=𝑥^∗_𝑁+^𝑁_𝐾=1𝜆_𝐾𝜕𝐹_𝐾𝑥_𝑁𝑡_𝑁,𝑡_𝑁𝜕𝑥_{𝑁𝑁𝐾=1},𝜆_𝐾𝜕𝐹_𝐾𝑥_𝑁𝑡_𝑁,𝑡_𝑁𝜕𝑡_𝑁𝛿_𝐿,𝑁−1𝑁^𝑁−1_𝐾=1𝑝_𝐾+1𝑡_𝐾𝜕𝑀_𝐾𝑥_𝐾𝑡_𝐾,𝑡_𝐾𝜕𝑡_𝐾1−𝛿_𝐿,𝑁=0, (23) here 𝛿_𝐿,𝑁=1,𝐿=𝑁,0,𝐿≠𝑁,𝐿=1,2,…,𝑁 (24) is a Kronecker symbol, 𝐻_𝐾(𝑥_𝐾,𝑢_𝐾,𝑝_𝐾,𝑡)=𝐿_𝐾(𝑥_𝐾,𝑢_𝐾,𝑝_𝐾,𝑡)+𝑝^𝑇_𝐾⋅𝑓_𝐾(𝑥_𝐾,𝑢_𝐾,𝑝_𝐾,𝑡), is a Hamilton-Pontryagin function, 𝐸_∗,𝐾 is lower exhauster of the functional 𝜑_𝐾(𝑥_𝐾(𝑡_𝐾)), 𝜆_𝐾, 𝐾=1,…,𝑁 are the vectors, and 𝑝_𝑘(⋅) is defined by the conditions (ii) and (iii) in the process of the proof of the theorem, later.

Proof. Firstly, we will try to reduce optimal control problem (14)–(18) with nonsmooth cost functional to the optimal control problem with smooth minimization functional. In this way, we will use some useful theorems in [12, 13]. Let us note that smooth variational descriptions of Frechet normals theorem in [12, Theorem 1.30] and its subdifferential counterpart [12, Theorem 1.88] provide important variational descriptions of Frechet subgradients of nonsmooth functions in terms of smooth supports. To prove the theorem, take any elements from intersection of the subset of the exhauster, 𝑥^∗_𝐾∈⋂𝐶_𝐾, where 𝐶_𝐾∈𝐸_∗,𝐾, 𝐾=1,2,…,𝑁. Then by using Theorem 1, we can write that 𝑥^∗_𝐾∈𝜕⁺𝜑_𝐾(𝑥_𝐾(𝑡_𝐾)). Then, apply the variational description in [12, Theorem 1.88] to the subgradients −𝑥^∗_𝐾∈𝜕⁺(−𝜑_𝐾(𝑥_𝐾(𝑡_𝐾))). In this way, we find functions 𝑠_𝐾∶𝑋→ℝ for 𝐾=1,2,…,𝑁 satisfying the relations 𝑠_𝐾𝑥_𝐾𝑡_𝐾=𝜑_𝐾𝑥_𝐾𝑡_𝐾,𝑠_𝐾𝑥_𝐾(𝑡)≥𝜑_𝐾𝑥_𝐾(𝑡) (25) in some neighborhood of 𝑥_𝐾(𝑡_𝐾), and such that each 𝑠_𝐾(⋅) is continuously differentiable at 𝑥_𝐾(𝑡_𝐾) with ∇𝑠_𝐾(𝑥_𝐾(𝑠𝑡_𝐾))=𝑥^∗_𝐾, 𝐾=1,2,…,𝑁. It is easy to check that 𝑥_𝐾(⋅) is a local solution to the following optimization problem of type (14)–(18) but with cost continuously differentiable around 𝑥_𝐾(⋅). This means that we deduce the optimal control problem (14)–(18) with the nonsmooth cost functional to the smooth cost functional data: min𝑆𝑢₁,…,𝑢_𝑁,𝑡₁,…,𝑡_𝑁=^𝑁_𝐾=1𝑠_𝐾𝑥_𝐾𝑡_𝐾+^𝑁_𝐾=1^𝑡𝐾_𝑡𝐾−1𝐿𝑥_𝐾(𝑡),𝑢_𝐾(𝑡),𝑡𝑑𝑡, (26) taking into account that ∇𝑠_𝐾𝑥_𝐾𝑡_𝐾=𝑥^∗_𝐾,𝐾=1,2,…,𝑁. (27) We use multipliers to adjoint to constraints 𝑥̇_𝐾(𝑡)−𝑓_𝐾(𝑥_𝐾(𝑡),𝑢_𝐾(𝑡),𝑡)=0, 𝑡∈[𝑡_𝑘−1,𝑡_𝑘], 𝐾=1,2,…,𝑁 and 𝐹_𝐾(𝑥_𝑁(𝑡_𝑁),𝑡_𝑁)=0, 𝐾=1,2,…,𝑁 to 𝑆: 𝐽^′=^𝑁_𝐾=1𝑠_𝐾𝑥_𝐾𝑡_𝐾+^𝑁_𝐾=1𝜆_𝑘𝐹_𝐾𝑥_𝑁𝑡_𝑁,𝑡_𝑁+^𝑁_𝐾=1^𝑡𝐾_𝑡𝐾−1𝐿𝑥_𝐾(𝑡),𝑢_𝐾(𝑡),𝑡+𝑝^𝑇_𝐾(𝑡)×𝑓_𝐾𝑥_𝐾(𝑡),𝑢_𝐾(𝑡),𝑡−𝑥̇_𝐾(𝑡)𝑑𝑡 (28) by introducing the Lagrange multipliers 𝑝₁(𝑡),𝑝₂(𝑡),…,𝑝_𝑁(𝑡). In the following, we will find it convenient to use the function 𝐻_𝐾, called the Hamiltonian, defined as 𝐻_𝐾(𝑥_𝐾(𝑡),𝑝_𝐾(𝑡),𝑢_𝐾(𝑡),𝑡)=𝐿_𝐾(𝑥_𝐾(𝑡),𝑢_𝐾(𝑡),𝑡)+𝑝_𝐾(𝑡)𝑓_𝐾(𝑥_𝐾,𝑢_𝐾,𝑡) for 𝑡∈[𝑡_𝐾−1,𝑡_𝐾]. Using this notation, we can write the Lagrange functional as 𝐽^′=^𝑁_𝐾=1𝑠_𝐾𝑥_𝐾𝑡_𝐾+^𝑁_𝐾=1𝜆_𝑘𝐹_𝐾𝑥_𝑁𝑡_𝑁,𝑡_𝑁+^𝑁_𝐾=1^𝑡𝐾_𝑡𝐾−1𝐻_𝐾𝑥_𝐾(𝑡),𝑝_𝐾(𝑡),𝑢_𝐾(𝑡),𝑡−𝑝^𝑇_𝐾𝑥̇_𝐾(𝑡)𝑑𝑡. (29) Assume {𝑥_𝑘,𝑢_𝑘,𝜃_𝑘} is optimal control. To determine the variation 𝛿𝐽^′, we introduce the variation 𝛿𝑥_𝐾, 𝛿𝑢_𝐾, 𝛿𝑝_𝐾, and 𝛿𝑡_𝐾. From the calculus of variations, we can obtain that the first variation of 𝐽^′ as 𝛿𝐽^′=^𝑁_𝐾=1𝜕𝑠_𝐾𝑥_𝐾𝑡_𝐾𝜕𝑥_𝐾𝛿𝑥_𝐾𝑡_𝐾+^𝑁_𝐾=1𝜆_𝐾𝜕𝐹_𝐾𝑥_𝑁𝑡_𝑁,𝑡_𝑁𝜕𝑥_𝑁𝛿𝑥_𝑁𝑡_𝑁+^𝑁_𝐾=1𝜆_𝐾𝜕𝐹_𝐾𝑥_𝑁𝑡_𝑁,𝑡_𝐾𝜕𝑡_𝑁𝛿𝑡_𝑁+^𝑁_𝐾=1^𝑡𝐾_𝑡𝐾−1𝜕𝐻_𝐾𝑥_𝐾(𝑡),𝑢_𝐾(𝑡),𝑝_𝐾(𝑡),𝑡𝜕𝑥_𝐾𝛿𝑥_𝐾(𝑡)+^𝑁_𝐾=1^𝑡𝐾_𝑡𝐾−1𝜕𝐻_𝐾𝑥_𝐾(𝑡),𝑢_𝐾(𝑡),𝑝_𝐾(𝑡),𝑡𝜕𝑢_𝐾𝛿𝑢_𝐾+^𝑁_𝐾=1^𝑡𝐾_𝑡𝐾−1𝜕𝐻_𝐾𝑥_𝐾(𝑡),𝑢_𝐾(𝑡),𝑝_𝐾(𝑡),𝑡𝜕𝑝_𝐾𝛿𝑝_𝐾−^𝑁_𝐾=1^𝑡𝐾_𝑡𝐾−1𝑝_𝐾(𝑡)𝛿𝑥_𝐾(𝑡)+𝑥_𝐾(𝑡)𝛿𝑝_𝐾(𝑡)𝑑𝑡+highorderterms. (30) If we follow the steps in [3, pages 5–7] then, the first variation of the functional takes the following form:𝛿𝐽^′=^𝑁−1_𝐾=1𝜕𝑠_𝐾𝑥_𝐾𝑡_𝐾𝜕𝑥_𝐾𝛿𝑥_𝐾𝑡_𝐾+𝜕𝑠𝑥_𝑁𝑡_𝑁𝜕𝑥_𝑁𝛿𝑥_𝑁𝑡_𝑁+^𝑁_𝐾=1𝜆_𝐾𝜕𝐹_𝐾𝑥_𝑁𝑡_𝑁,𝑡_𝑁𝜕𝑥_𝑁𝛿𝑥_𝑁𝑡_𝑁+^𝑁_𝐾=1𝜆_𝐾𝜕𝐹_𝐾𝑥_𝑁𝑡_𝑁,𝑡_𝑁𝜕𝑡_𝑁𝛿𝑡_𝑁+^𝑁_𝐾=1^𝑡𝐾_𝑡𝐾−1𝜕𝐻_𝐾𝑢_𝐾,𝑥_𝐾,𝑝_𝐾,𝑡𝜕𝑢_𝐾𝛿𝑢_𝐾+^𝑁_𝐾=1^𝑡𝐾_𝑡𝐾−1𝜕𝐻_𝐾𝑢_𝐾(𝑡),𝑥_𝐾(𝑡),𝑝_𝐾(𝑡),𝑡𝜕𝑝_𝐾𝛿𝑝_𝐾−^𝑁−1_𝐾=1𝑝_𝐾𝑡_𝐾𝛿𝑥_𝐾𝑡_𝐾−𝑝_𝑁𝑡_𝑁𝛿𝑥_𝑁𝑡_𝑁−^𝑁−1_𝐾=1𝑝_𝐾+1𝑡_𝐾𝜕𝑀_𝐾𝑥_𝐾𝑡_𝐾,𝑡_𝐾𝜕𝑥_𝐾𝛿𝑥_𝐾𝑡_𝐾−^𝑁−1_𝐾=1𝑝_𝐾+1𝑡_𝐾𝜕𝑀_𝐾𝑥_𝐾𝑡_𝐾,𝑡_𝐾𝜕𝑡_𝐾𝛿𝑡_𝐾−^𝑁_𝑘=1𝑝_𝐾(𝑡)𝛿𝑥_𝐾𝑡_𝐾=^𝑁−1_𝐾=1𝜕𝜑_𝐾𝑥_𝐾𝑡_𝐾𝜕𝑥_𝐾−𝑝_𝐾𝑡_𝐾−𝑝_𝐾+1𝑡_𝐾𝜕𝑀_𝐾𝑥_𝐾𝑡_𝐾,𝑡_𝐾𝜕𝑥_𝐾𝛿𝑥_𝐾𝑡_𝐾+𝜕𝜑_𝑁𝑥_𝑁𝑡_𝑁𝜕𝑥_𝑁+^𝑁_𝐾=1𝜆_𝐾𝜕𝐹_𝐾𝑥_𝑁𝑡_𝑁,𝑡_𝑁𝜕𝑥_𝑁−𝑝_𝑁𝑡_𝑁𝛿𝑥_𝑁𝑡_𝑁+^𝑁_𝐿=1^𝑁_𝐾=1𝜆_𝐾𝜕𝐹_𝐾𝑥_𝑁𝑡_𝑁,𝑡_𝑁𝜕𝑡_𝑁𝛿_𝐿,𝑁−1𝑁^𝑁−1_𝐾=1𝑝_𝐾+1𝑡_𝐾𝜕𝑀_𝐾𝑥_𝐾𝑡_𝐾,𝑡_𝐾𝜕𝑡_𝐾×1−𝛿_𝐿,𝑁𝛿𝑡_𝐿+^𝑁_𝐾=1^𝑡𝐾_𝑡𝐾−1𝜕𝐻_𝐾𝑥_𝐾(𝑡),𝑝_𝐾(𝑡),𝑢_𝐾(𝑡),𝑡𝜕𝑥_𝐾−𝑝̇_𝐾(𝑡)𝛿𝑥_𝐾+^𝑁_𝐾=1^𝑡𝐾_𝑡𝐾−1𝜕𝐻_𝐾𝑥_𝐾(𝑡),𝑝_𝐾(𝑡),𝑢_𝐾(𝑡),𝑡𝜕𝑢_𝐾𝛿𝑢_𝐾+^𝑁_𝐾=1^𝑡𝐾_𝑡𝐾−1𝜕𝐻_𝐾𝑥_𝐾(𝑡),𝑝_𝐾(𝑡),𝑢_𝐾(𝑡),𝑡𝜕𝑝_𝐾−𝑥̇_𝐾(𝑡)𝛿𝑝_𝐾. (31)The latter sum is known because 𝜕𝐻_𝐾𝑥_𝐾(𝑡),𝑢_𝐾(𝑡),𝑝_𝐾(𝑡),𝑡𝜕𝑝_𝐾=𝑥̇_𝐾(𝑡), (32) and it is easy to check that ^𝑁_𝐾=1𝜆_𝐾𝜕𝐹_𝐾𝑥_𝑁𝑡_𝑁,𝑡_𝑁𝜕𝑡_𝑁𝛿𝑡_𝑁−^𝑁−1_𝐾=1𝑝_𝐾+1𝑡_𝐾𝜕𝑀_𝐾𝑥_𝐾𝑡_𝐾,𝑡_𝐾𝜕𝑡_𝐾𝛿𝑡_𝐾=^𝑁_𝐿=1^𝑁_𝐾=1𝜆_𝐾𝜕𝐹_𝐾𝑥_𝑁𝑡_𝑁,𝑡_𝑁𝜕𝑡_𝑁𝛿_𝐿,𝑁−1𝑁^𝑁−1_𝐾=1𝑝_𝐾+1𝑡_𝐾𝜕𝑀_𝐾𝑥_𝐾𝑡_𝐾,𝑡_𝐾𝜕𝑡_𝐾×1−𝛿_𝐿,𝑁𝛿𝑡_𝐿. (33) If the state equations (14) are satisfied, 𝑝̇_𝑘 is selected so that coefficient of 𝛿𝑥_𝑘 and 𝛿𝑡_𝑁 is identically zero. Thus, we have 𝛿𝑆^′=^𝑁−1_𝐾=1^𝑡𝐾_𝑡𝐾−1𝜕𝐻_𝐾𝑥_𝐾(𝑡),𝑝_𝐾(𝑡),𝑢_𝐾(𝑡),𝑡𝜕𝑢_𝐾𝛿𝑢_𝐾+highorderterms. (34) The integrand is the first-order approximation to the change in 𝐻_𝐾 caused by 𝜕𝐻_𝐾𝑥_𝐾(𝑡),𝑝_𝐾(𝑡),𝑢_𝐾(𝑡),𝑡𝜕𝑢_𝐾𝛿𝑢_𝐾^𝑇𝛿𝑢_𝐾(𝑡)=𝐻_𝐾𝑥_𝐾(𝑡),𝑝_𝐾(𝑡),𝑢_𝐾(𝑡)+𝛿𝑢_𝐾(𝑡),𝑡−𝐻_𝐾𝑥_𝐾(𝑡),𝑝_𝐾(𝑡)𝑢_𝐾(𝑡),𝑡. (35) Therefore, 𝛿𝑆^′=^𝑁−1_𝐾=1^𝑡𝐾_𝑡𝐾−1𝐻_𝐾𝑥_𝐾(𝑡),𝑝_𝐾(𝑡),𝑢_𝐾(𝑡)+𝛿𝑢_𝐾(𝑡),𝑡−𝐻_𝐾𝑥_𝐾(𝑡),𝑝_𝐾(𝑡),𝑢_𝐾(𝑡),𝑡𝛿𝑢_𝐾(𝑡)+highorderterms. (36) If 𝑢_𝐾+𝛿𝑢_𝐾 is in a sufficiently small neighborhood of 𝑢_𝐾 then the high-order terms are small and the integral in last equation dominates the expression of 𝛿𝑆^′. Thus, for 𝑢_𝐾 to be a minimizing control it is necessary that ^𝑁−1_𝐾=1^𝑡𝐾_𝑡𝐾−1𝐻_𝐾𝑥_𝐾,𝑝_𝐾,𝑢_𝐾+𝛿𝑢_𝐾,𝑡−𝐻_𝐾𝑥_𝐾,𝑝_𝐾,𝑢_𝐾,𝑡𝛿𝑢_𝐾≥0 (37) for all admissible 𝛿𝑢_𝐾. We assert that in order for the last inequality to be satisfied for all admissible 𝛿𝑢_𝐾 in the specified neighborhood, it is necessary that 𝐻_𝐾(𝑥_𝐾,𝑝_𝐾,𝑢_𝐾+𝛿𝑢_𝐾,𝑡)≥𝐻_𝐾(𝑥_𝐾,𝑝_𝐾,𝑢_𝐾,𝑡) for all admissible 𝛿𝑢_𝐾 and for all 𝑡∈[𝑡_𝐾−1,𝑡_𝐾]. To show this, consider the control Δ𝑢_𝐾=𝑢_𝐾(𝑡),𝑡∈𝑡_𝐾−1,𝑡_𝐾,𝑢_𝐾(𝑡)+𝛿𝑢_𝐾(𝑡),𝑡∈𝑡_𝐾−1,𝑡_𝐾, (38) where 𝑡∈[𝑡_𝐾−1,𝑡_𝐾] is an arbitrarily small, but nonzero time interval and 𝛿𝑢_𝐾 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, 𝛿𝐽^′=0. Setting to zero, the coefficients of the independent increments 𝛿𝑥_𝑁(𝑡_𝑁), 𝛿𝑥_𝐾(𝑡_𝐾)𝛿𝑥_𝐾, 𝛿𝑢_𝐾 and 𝛿𝑝_𝐾, and taking into account that ∇𝑠_𝐾𝑥_𝐾𝑡_𝐾=𝑥^∗_𝐾,𝐾=1,2,…,𝑁, (39) yield the necessary optimality conditions (i)–(v) in Theorem 3.
This completes the proof of the theorem.

Theorem 4 (Necessary optimality conditions for switching optimal control system in terms of Quasidiffereniability). Let the minimization functional 𝜑_𝐾(⋅) be positively homogenous, quasidifferentiable at a point 𝑥_𝐾(⋅), and let (𝑢_𝐾(⋅),𝑥_𝐾(⋅),𝜃) be an optimal solution to the control problem (14)–(18). Then, there exist vector functions 𝑝_𝐾(𝑡), 𝐾=1,…,𝑁, and there exist convex compact and bounded set 𝑀(𝜑_𝐾(⋅)), in which for any elements 𝑥^∗_𝐾∈𝑀(𝜑_𝐾(⋅)), the necessary optimality conditions (i)–(v) in Theorem 3 are satisfied.

Proof. Let minimization functional 𝜑_𝐾(⋅) be positively homogenous and quasidifferentiable at a point 𝑥_𝐾(𝑡_𝐾). Then, there exist totally bounded lower exhausters 𝐸_∗,𝐾 for the 𝜑_𝐾(⋅) [9, Theorem 4]. Let us make the substitution 𝑀(𝜑_𝐾(⋅))=𝐸_∗,𝐾; take any element 𝑥^∗_𝐾∈𝑀(𝜑_𝐾(⋅)), then 𝑥^∗_𝐾∈𝐸_∗,𝐾 also, and if we follow the proof description and result in Theorem 3 in the current paper, we can prove Theorem 4. If we use the relationship between the Gateaux upper subdifferential and Dini upper derivative [9, Lemma 3.6], substitute ℎ_𝐾(𝑔)=𝜑⁺_𝐾,𝐻(𝑥_𝐾(𝑡_𝐾),𝑔), then we can write the following corollary (here 𝜑⁺_𝐾,𝐻(𝑥_𝐾(𝑡_𝐾), 𝑔 is the Hadamard upper derivative of the minimizing functional 𝜑_𝐾(⋅) in the direction 𝑔).

Corollary 5. Let the minimization functional 𝜑_𝐾(⋅) be positively homogenous, and let the Dini upper differentiable at a point 𝑥_𝐾(𝑡_𝐾) and (𝑢_𝐾(⋅),𝑥_𝐾(⋅),𝜃) be an optimal solution to the control problem (14)–(18). Then for any elements 𝑥^∗_𝐾∈𝜕⁺_𝐺ℎ_𝐾(0_𝑛), there exist vector functions 𝑝_𝐾(𝑡), 𝐾=1,…,𝑁 in which the necessary optimality conditions (i)–(v) in the Theorem 3 hold.

Proof. Let us take any element 𝑥^∗_𝐾∈𝜕⁺_𝐺ℎ_𝐾(0_𝑛). Then by using the lemma in [9, Lemma 3.8] we can write 𝑥^∗_𝐾∈𝜕⁺_𝐹ℎ_𝐾(0_𝑛). Next, if we use the lemma in [9, Lemma 3.2], then we can put 𝑥^∗_𝐾∈𝜕⁺_𝐹𝜑_𝐾(𝑥_𝐾(𝑡_𝐾)). At least, if we follow Theorem 1 (relationship between upper Frechet subdifferential and exhausters) and Theorem 3 (necessary optimality condition in terms of exhausters) in the current paper, we can prove the result of Corollary 5.