Approximate Controllability of Abstract Discrete-Time Systems
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Research Article Approximate Controllability of Abstract Discrete-Time Systems ´ R. Henr´ıquez1 and Claudio Cuevas2 Hernan 1 2
Departamento de Matem´atica, Universidad de Santiago (USACH) Casilla 307, Correo 2, Santiago, Chile Departamento de Matem´atica, Universidade Federal de Pernambuco, Recife, PE 50540-740, Brazil
Correspondence should be addressed to Claudio Cuevas, [email protected] Received 12 April 2010; Accepted 2 September 2010 Academic Editor: Rigoberto Medina Copyright q 2010 H. R. Henr´ıquez and C. Cuevas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Approximate controllability for semilinear abstract discrete-time systems is considered. Specifically, we consider the semilinear discrete-time system xk1 Ak xk fk, xk Bk uk , k ∈ N0 , where Ak are bounded linear operators acting on a Hilbert space X, Bk are X-valued bounded linear operators defined on a Hilbert space U, and f is a nonlinear function. Assuming appropriate conditions, we will show that the approximate controllability of the associated linear system xk1 Ak xk Bk uk implies the approximate controllability of the semilinear system.
1. Introduction In this paper we deal with the controllability problem for semilinear distributed discrete-time control systems. In order to specify the class of systems to be considered, we set X for the state space and U for the control space. We assume that X and U are Hilbert spaces. Moreover, throughout this paper we denote by Ak : X → X bounded linear operators, Bk : U → X, k ∈ N0 , bounded linear maps that represent the control action, and f : N0 × X → X a map such that fk, · is continuous for each k ∈ N0 . Furthermore, Ak , Bk , and f satisfy appropriate conditions which will be specified later. We will study the controllability of control systems described by the equation xk1 Ak xk fk, xk Bk uk ,
k ∈ N0 , x0 ∈ X,
1.1
where xk ∈ X, uk ∈ U. The study of controllability is an important topic in systems theory. In particular, the controllability of systems similar to 1.1 has been the object of several works. We only
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Advances in Difference Equations
mention here 1–11 and the references cited therein. Specially, Leiva and Uzcategui 5 have studied the exact controllability of the linear and semilinear system. However, it is well known 12–16 that most of continuous distributed systems that arise in concrete situations are not exactly controllable but only approximately controllable. A similar situation has been established in 10 in relation with the discrete wave equation and in 11 in relation with the discrete heat equation see 17–22. As mentioned in this paper, the lack of controllability is related to the fact that the spaces in which the solutions of these systems evolve are infinite dimensional. For this reason, in this paper we study the approximate controllability of system 1.1. Specifi
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