# Optimal Control Problem for Switched System with the Nonsmooth Cost Functional

Authors: Shahlar Meherrem; Rufan Akbarov
Journal: Abstract and Applied Analysis
 (2013)

---

## 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.


---

## Body

## 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. Letf:X→R, X⊂Rn 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
(1)fH↑:=limsup[α,g]→[+0,g]1α[f(x+αg)-f(x)],(fH↓:=liminf[α,g]→[+0,g]1α[f(x+αg)-f(x)]),
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:
(2)∂G+f(x0)={v∈Rn∣limsupt↓0f(x0+tg)-f(x0)t≤(v,g),∀g∈Rnf(x0+tg)-f(x0)t}.
The set(3)∂^+f(x0)={v∈Rn∣limsupx→x0f(x)-f(x0)-〈v,x-x0〉∥x-x0∥≤0}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
(4)f′(x,g)=maxv∈∂_f(x)(v,g)+minw∈∂¯f(x)(w,g),
where ∂_f(x),∂¯f(x)⊂Rn are convex compact sets. From the last relation, we can easily reduce that
(5)f′(x,g)=minw∈∂¯f(x)maxv∈w+∂_f(x)(v,g)=maxv∈∂_f(x)maxw∈v+∂¯f(x)(v,g).
This means that for the function h(g)=f′(x,g) the upper and lower exhausters can be described in the following way:
(6)E*={C=w+∂_f(x)∣w∈∂¯f(x)},E*={C=v+∂¯f(x)∣v∈∂_f(x)}.
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]:
(7)∂F+f(x0)=∂F+fH↑(x0,0n).Theorem 1.
LetE* be lower exhausters of the positively homogeneous function h:Rn→R. Then, ⋂C⊂E*C=∂^+h(0n), where ∂^+h is the Frechet upper subdifferential of the h at 0n, and for the positively homogeneous function h:Rn→R the Frechet superdifferential at the point zero follows
(8)∂^+h(0n)={v∈Rn∣h(x)-(v,x)≤0,x∈Rn}.Proof.
Take anyv0∈⋂C⊂E*C. Then by using definition an lower exhausters we can write
(9)v0(x)≥h(x),∀x∈Rn⟹⋂C⊂E*C⊂∂^+h(0n).
Consider now any v0∈∂^+h(0n)⇒(10)v0(x)≥h(x).
Let us consider v0∉⋂C⊂E*C. Then, there exists C0∈E* where v0∉C0. Then, by separation theorem, there exists x0∈Rn such that
(11)(x0,v0)≤maxv∈C0(x0,v)≤h(x).
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.
Leth:Rn→R be a positively homogenous function. It is not difficult to observe that every g∈Rn and every t>0:
(12)h(0n+tg)-h(0n)t=th(g)t=h(g).
Hence, the Gateaux lower subdifferential of h at 0n takes the forms
(13)∂G+h(0n)={v∈Rn∣h(g)≤(v,g),∀g∈Rn}
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(14)x˙K(t)=fK(xK(t),uK(t),t),t∈[tk-1,tk],K=1,2,…,N
with initial condition
(15)x1(t0)=x0,
and the phase constraints at the end of the interval
(16)FK(xN(tN),tN)=0,K=1,2,…,N
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):
(17)xK+1(tK)=MK(xK(tK),tk),K=1,2,…,N-1.
The goal of this paper is to minimize the following functional:
(18)S(u1,…,uN,t1,…,tN)=∑K=1NφK(xK(tK))+∑K=1N∫tK-1tKL(xK(t),uK(t),t)dt
with the conditions (14)–(16). Namely, it is required to find the controls u1,u2,…,uN, switching points t1,t2,…,tN-1, and the end point tN (here t1,t2,…,tN are not fixed) with the corresponding state x1,x2,…,xN satisfying (14)–(16) so that the functional J(u1,…,uN,t1,…,tN) 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).HerefK:ℝ×ℝn×ℝr→ℝn, MK and FK are continuous, at least continuously partially differentiable vector-valued functions with respect to their variables, L:ℝn×ℝr×ℝ→ℝ are continuous and have continuous partial derivative with respect to their variables, φk(·) has Frechet upper subdifferentiable (superdifferentiable) at a point x¯K(tK) and positively homogeneous functional, and uK(t):ℝ→UK⊂ℝr are controls. The sets UK are assumed to be nonempty and open. Here (16) is switching conditions. If we denote this as follows: θ=(t1,t2,…,tN), x(t)=(x1(t),x2(t),…,xN(t)), u(t)=(u1(t),u2(t),…,uN(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 φk(·) is the Hadamar upper differentiable at the point x¯K(tK) in the direction of zero. Then, φk(·) is upper semicontinuous, and it has an exhaustive family of lower concave approximations of φk(·).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 xK* from intersection of the subsets CK of the lower exhauster E*,K of the functional φK(x¯K(t¯K)), that is, xK*∈⋂CK∈E*,KCK, 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:(19)x-˙K(t)=∂HK(x-K(t),u-K(t),p-K(t),t)∂pK,t∈[t-K-1,t-K];(ii)
Costate equation:(20)p-˙K(t)=∂HK(x-K(t),u-K(t),p-K(t),t)∂xK,t∈[t-K-1,t-K];(iii)
At the switching points,t-1,t-2,…,t-N-1,
(21)xK*-p-K(t-K)-p-K+1(t-K)∂MK(x-K(t-K),t-K)∂xK=0,K=1,2,…,N-1;(iv)
Minimality condition:(22)HK(x-K(t),p-K(t),u-K(t)+δuK(t),t)≥HK(x-K(t),p-K(t),u-K(t),t),foralladmissibleδuK,t∈[t-K-1,t-K];(v)
At the end pointt-N,
(23)p-N(t-N)=xN*+∑K=1NλK∂FK(x-N(t-N),t-N)∂xN,(∑K=1NλK∂FK(x-N(t-N),t-N)∂tN)δL,N-1N(∑K=1N-1p-K+1(t-K)∂MK(x-K(t-K),t-K)∂tK)(1-δL,N)=0,
here
(24)δL,N={1,L=N,0,L≠N,L=1,2,…,N
is a Kronecker symbol, HK(xK,uK,pK,t)=LK(xK,uK,pK,t)+pKT·fK(xK,uK,pK,t), is a Hamilton-Pontryagin function, E*,K is lower exhauster of the functional φK(xK(tK)), λK, K=1,…,N are the vectors, and pk(·) 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, xK*∈⋂CK, where CK∈E*,K, K=1,2,…,N. Then by using Theorem 1, we can write that xK*∈∂^+φK(x-K(tK)). Then, apply the variational description in [12, Theorem 1.88] to the subgradients -xK*∈∂^+(-φK(x-K(t-K))). In this way, we find functions sK:X→ℝ for K=1,2,…,N satisfying the relations
(25)sK(x¯K(tK))=φK(x¯K(tK)),sK(xK(t))≥φK(xK(t))
in some neighborhood of x¯K(tK), and such that each sK(·) is continuously differentiable at x¯K(t-K) with ∇sK(x¯K(st-K))=xK*, 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:
(26)minS(u1,…,uN,t1,…,tN)=∑K=1NsK(xK(tK))+∑K=1N∫tK-1tKL(xK(t),uK(t),t)dt,
taking into account that
(27)∇sK(x-K(t-K))=xK*,K=1,2,…,N.
We use multipliers to adjoint to constraints x˙K(t)-fK(xK(t),uK(t),t)=0, t∈[tk-1,tk], K=1,2,…,N and FK(xN(tN),tN)=0, K=1,2,…,N to S:
(28)J′=∑K=1NsK(xK(tK))+∑K=1NλkFK(xN(tN),tN)+∑K=1N∫tK-1tK(L(xK(t),uK(t),t)+pKT(t)×(fK(xK(t),uK(t),t)-x˙K(t))pKT(t))dt
by introducing the Lagrange multipliers p1(t),p2(t),…,pN(t). In the following, we will find it convenient to use the function HK, called the Hamiltonian, defined as HK(xK(t),pK(t),uK(t),t)=LK(xK(t),uK(t),t)+pK(t)fK(xK,uK,t) for t∈[tK-1,tK]. Using this notation, we can write the Lagrange functional as
(29)J′=∑K=1NsK(xK(tK))+∑K=1NλkFK(xN(tN),tN)+∑K=1N∫tK-1tK(HK(xK(t),pK(t),uK(t),t)-pKTx˙K(t))dt.
Assume {x-k,u-k,θ-k} is optimal control. To determine the variation δJ′, we introduce the variation δxK, δuK, δpK, and δtK. From the calculus of variations, we can obtain that the first variation of J′ as
(30)δJ′=∑K=1N∂sK(x-K(t-K))∂xKδxK(tK)+∑K=1NλK∂FK(x-N(t-N),t-N)∂xNδxN(tN)+∑K=1NλK∂FK(x-N(t-N),t-K)∂tNδtN+∑K=1N∫tK-1tK∂HK(x-K(t),u-K(t),p-K(t),t)∂xKδxK(t)+∑K=1N∫tK-1tK∂HK(x-K(t),u-K(t),p-K(t),t)∂uKδuK+∑K=1N∫tK-1tK∂HK(x-K(t),u-K(t),p-K(t),t)∂pKδpK-∑K=1N∫tK-1tK(p-K(t)δxK(t)+x-K(t)δpK(t))dt+highorderterms.
If we follow the steps in [3, pages 5–7] then, the first variation of the functional takes the following form:(31)δJ′=∑K=1N-1∂sK(x-K(t-K))∂xKδxK(tK)+∂s(x-N(t-N))∂xNδxN(tN)+∑K=1NλK∂FK(x-N(t-N),t-N)∂xNδxN(tN)+∑K=1NλK∂FK(x-N(t-N),t-N)∂tNδtN+∑K=1N∫tK-1tK∂HK(u-K,x-K,p-K,t)∂uKδuK+∑K=1N∫tK-1tK∂HK(u-K(t),x-K(t),p-K(t),t)∂p-KδpK-∑K=1N-1p-K(tK)δxK(tK)-p-N(tN)δxN(tN)-∑K=1N-1pK+1(t-K)∂MK(x-K(t-K),t-K)∂xKδxK(t-K)-∑K=1N-1p-K+1(t-K)∂MK(x-K(t-K),t-K)∂tKδtK-∑k=1Np-K(t)δxK(tK)=∑K=1N-1(∂φK(x-K(t-K))∂xK-p-K(t-K)-p-K+1(t-K)∂MK(x-K(t-K),t-K)∂xK)δxK(t-K)+(∂φN(x-N(t-N))∂xN+∑K=1NλK∂FK(x-N(t-N),t-N)∂x-N-p-N(tN)∑K=1NλK∂FK(x-N(t-N),t-N)∂x-N)δxN(tN)+∑L=1N[(∑K=1NλK∂FK(x-N(t-N),t-N)∂t-N)δL,N-1N(∑K=1N-1p-K+1(t-K)∂MK(x-K(t-K),t-K)∂tK)×(1-δL,N)(∑K=1NλK∂FK(x-N(t-N),t-N)∂t-N)]δtL+∑K=1N∫t-K-1t-K(∂HK(x-K(t),p-K(t),u-K(t),t)∂xK-p-˙K(t))δxK+∑K=1N∫t-K-1t-K∂HK(x-K(t),p-K(t),u-K(t),t)∂uKδuK+∑K=1N∫tK-1tK(∂HK(x-K(t),p-K(t),u-K(t),t)∂pK-x-˙K(t))δpK.The latter sum is known because
(32)∂HK(x-K(t),u-K(t),p-K(t),t)∂pK=x-˙K(t),
and it is easy to check that
(33)∑K=1NλK∂FK(x-N(t-N),t-N)∂tNδtN-∑K=1N-1p-K+1(t-K)∂MK(x-K(t-K),t-K)∂tKδtK=∑L=1N[(∑K=1NλK∂FK(x-N(t-N),t-N)∂tN)δL,N-1N(∑K=1N-1p-K+1(t-K)∂MK(x-K(t-K),t-K)∂tK)×(1-δL,N)(∑K=1NλK∂FK(x-N(t-N),t-N)∂tN)]δtL.
If the state equations (14) are satisfied, p˙k is selected so that coefficient of δxk and δtN is identically zero. Thus, we have
(34)δS′=∑K=1N-1∫tK-1tK∂HK(x-K(t),p-K(t),u-K(t),t)∂uKδuK+highorderterms.
The integrand is the first-order approximation to the change in HK caused by
(35)[∂HK(x-K(t),p-K(t),u-K(t),t)∂uKδuK]TδuK(t)=HK(x-K(t),p-K(t),u-K(t)+δuK(t),t)-HK(x-K(t),p-K(t)u-K(t),t).
Therefore,
(36)δS′=∑K=1N-1∫tK-1tK[HK(x-K(t),p-K(t),u-K(t)+δuK(t),t)-HK(x-K(t),p-K(t),u-K(t),t)]δuK(t)+highorderterms.
If u-K+δuK 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
(37)∑K=1N-1∫tK-1tK[HK(x-K,p-K,u-K+δuK,t)-HK(x-K,p-K,u-K,t)]δuK≥0
for all admissible δuK. We assert that in order for the last inequality to be satisfied for all admissible δuK in the specified neighborhood, it is necessary that HK(x-K,p-K,u-K+δuK,t)≥HK(x-K,p-K,u-K,t) for all admissible δuK and for all t∈[tK-1,tK]. To show this, consider the control
(38)ΔuK={u-K(t),t∈[tK-1,tK],u-K(t)+δuK(t),t∈[tK-1,tK],
where t∈[tK-1,tK] is an arbitrarily small, but nonzero time interval and δuK 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 δxN(tN), δxK(tK)δxK, δuK and δpK, and taking into account that
(39)∇sK(x¯K(tK))=xK*,K=1,2,…,N,
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 functionalφK(·) 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 pK(t), K=1,…,N, and there exist convex compact and bounded set M(φK(·)), in which for any elements xK*∈M(φK(·)), the necessary optimality conditions (i)–(v) in Theorem 3 are satisfied.Proof.
Let minimization functionalφK(·) be positively homogenous and quasidifferentiable at a point x¯K(t¯K). Then, there exist totally bounded lower exhausters E*,K for the φK(·) [9, Theorem 4]. Let us make the substitution M(φK(·))=E*,K; take any element xK*∈M(φK(·)), then xK*∈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 hK(g)=φK,H+(x¯K(t¯K),g), then we can write the following corollary (here φK,H+(x¯K(t¯K), g is the Hadamard upper derivative of the minimizing functional φK(·) in the direction g).Corollary 5.
Let the minimization functionalφK(·) be positively homogenous, and let the Dini upper differentiable at a point xK(t¯K) and (u¯K(·),x¯K(·),θ¯) be an optimal solution to the control problem (14)–(18). Then for any elements xK*∈∂G+hK(0n), there exist vector functions pK(t), K=1,…,N in which the necessary optimality conditions (i)–(v) in the Theorem 3 hold.Proof.
Let us take any elementxK*∈∂G+hK(0n). Then by using the lemma in [9, Lemma 3.8] we can write xK*∈∂F+hK(0n). Next, if we use the lemma in [9, Lemma 3.2], then we can put xK*∈∂F+φK(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.



---

*Source: 681862-2013-10-24.xml*