Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup

To the memory of Roberto Conti and Yuri Borisovich
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Abstract

We study the controllability problem for a system governed by a semilinear differential inclusion in a Banach space not assuming that the semigroup generated by the linear part of inclusion is compact. Instead we suppose that the multivalued nonlinearity satisfies the regularity condition expressed in terms of the Hausdorff measure of noncompactness. It allows us to apply the topological degree theory for condensing operators and to obtain the controllability results for both upper Carathéodory and almost lower semicontinuous types of nonlinearity. As application we consider the controllability for a system governed by a perturbed wave equation.

Introduction

The investigation of controllability problems for nonlinear systems by the methods of fixed-point theory has a long history (see, for example, [2], [22] and the references therein). In recent years the corresponding parts of multivalued analysis were applied to obtain various controllability results for systems governed by semilinear differential and functional differential inclusions in infinite-dimensional Banach spaces (see [3], [4], [5], [6], [7], [11], [12], [15], [16], [17], [18], [20] and others). The attention of researchers to such systems is caused by the fact that many control processes arising in mathematical physics may be naturally presented in this form (see, e.g., [19]).

Let us mention, however, that the most of these works contain the assumption of compactness of the semigroup generated by the linear part of inclusion, as well as the supposition of the controllability of corresponding linear system, i.e., the invertibility of the linear controllability operator W. But it is known (see [23], [24]) that in infinite-dimensional case these hypotheses are in contradiction to each other. Actually, in this situation the controllability may be provided only on the subspace RangeW under additional assumption that it is invariant w.r.t. the semigroup, i.e., that the linear system can be steered to this subspace.

In the present paper we assume that the linear part of inclusion generates an arbitrary C0-semigroup. At the same time we suppose that the multivalued nonlinearity satisfies a regularity condition expressed in terms of the Hausdorff measure of noncompactness. The particular cases of this condition arise when the nonlinearity is compact or satisfies the Lipschitz condition (w.r.t. the Hausdorff metric) in the phase argument. This approach allows us to apply the technique of topological degree theory for condensing multivalued operators (see, e.g., [21], [8], [19]) and permits to obtain controllability results for both upper Carathéodory and almost lower semicontinuous types of nonlinearity.

The paper is organized as follows. In Section 2 we give the necessary preliminaries from the fields of multivalued maps, measures of noncompactness and condensing operators. In Section 3 we describe the problem and introduce main assumptions. We suppose, in particular, that the multivalued nonlinearity satisfies the upper Carathéodory conditions and define the multivalued operator Γ whose fixed points are generating solutions of the problem. We study the properties of Γ, in particular, we prove that it is condensing w.r.t. an appropriate vector-valued measure of noncompactness (Proposition 6). This allows us to present the general controllability principle in terms of the topological degree theory (Theorem 3). Some sufficient conditions producing important particular cases are given in Theorem 4, Theorem 5, Theorem 6. In Section 4 we give the outline of the corresponding results for the case when the multivalued nonlinearity is almost lower semicontinuous. In Section 5 we consider the application to the controllability problem for a system with distributed parameters governed by the wave equation under the action of controls of two types.

Section snippets

Preliminaries: Multimaps and measures of noncompactness

Let X be a metric space, Y a normed space, P(Y) denote the collection of all nonempty subsets of Y. We denote: K(Y)={DP(Y):D is compact};Kv(Y)={DK(Y):D is convex}. We recall some notions (see e.g. [9], [19] for further details).

Definition 1

A multivalued map (multimap) F:XP(Y) is:

  • (a)

    upper semicontinuous (u.s.c.) if F1(V)={xX:F(x)V} is an open subset of X for every open set VY;

  • (b)

    lower semicontinuous (l.s.c.) if F1(W) is a closed subset of X for every closed set WY.

Sometimes we will denote a multimap by

Controllability problem for upper Carathéodory nonlinearity

We will consider a nonlinear control system governed by a semilinear differential inclusion in a separable Banach space E of the form ẏ(t)Ay(t)+F(t,y(t))+Bu(t),t[0,T]Jy(0)=x0.

It will be supposed that:

(A) A:D(A)EE is a closed linear operator generating a C0-semigroup of bounded linear operators eAt.

In what follows we will denote M=suptJeAt.

We will assume that the multivalued nonlinearity F:J×EKv(E) satisfies the following conditions:

(F1) the multifunction F(,x):JKv(E) admits a

Controllability for almost lower semicontinuous nonlinearity

In this section we will consider controllability problem (3.1), (3.2), (3.3) by assuming that the nonlinear multimap F:J×EK(E) (with not necessarily convex values) satisfies the following condition of almost lower semicontinuity:

(FL)there exists a sequence of disjoint closed sets {Jn},JnJ,n=1,2, such that: (i) meas (JnJn)=0; (ii) the restriction of F on each set Jn×E is l.s.c.

It is known (see, e.g., [9], [13], [19]) that under conditions (FL), (F3) the superposition multioperator PF:C(J;E)

Example: Controllability for perturbed wave equation

We consider a vibrating string clipped at the endpoints s=0,1. We denote by z(t,s) the vertical displacement from the zero position at point s[0,1] and time t[0,T] and by z0() and z1() the initial displacement and velocity profiles. We assume that the control influence upon the vibration process can be divided into two types of actions: feedback and “absolute”.

The feedback control is characterized by m sources of external forces whose properties are depending on the state of the system.

Acknowledgements

The work of the first author is partially supported by the Russian FBR Grants 08-01-00192 and 07-01-00137. The work of the second author is partially supported by a PRIN-MIUR Grant 2007. The paper was written while the first author was guest of the University of Florence under a bilateral agreement between the University of Florence and the University of Voronezh.

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