Approximate Controllability of Abstract Discrete-Time Systems
© H. R. Henríquez and C. Cuevas. 2010
Received: 12 April 2010
Accepted: 2 September 2010
Published: 15 September 2010
Approximate controllability for semilinear abstract discrete-time systems is considered. Specifically, we consider the semilinear discrete-time system , , where are bounded linear operators acting on a Hilbert space , are -valued bounded linear operators defined on a Hilbert space , and is a nonlinear function. Assuming appropriate conditions, we will show that the approximate controllability of the associated linear system implies the approximate controllability of the semilinear system.
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 mention here [1–11] and the references cited therein. Specially, Leiva and Uzcategui  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  in relation with the discrete wave equation and in  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.
Throughout this paper, for Hilbert spaces , , we denote by the Banach space of bounded linear operators from into , and we abbreviate this notation by for . Moreover, for a linear operator we denote by the range space of .
The following property of Hilbert spaces is essential for our treatment of controllability.
In the next section we study the controllability of systems of type (1.1) when the state space is a Hilbert space and, in Section 3, we will apply our results to study the controllability of a typical system.
2. Approximate Controllability
Throughout this section, we assume that and are Hilbert spaces endowed with an inner product denoted generically by . In this case, for , and are also Hilbert spaces. The inner product in is given by for , , and similarly for .
We introduce the reachability set of system (1.2) as the set consisting of the values . Clearly, system (1.2) is approximately controllable on if and only if is dense in for every . A weaker property of controllability is established in the following definition.
Proceeding as in Definitions 2.1 and 2.2, we next consider the approximate controllability for system (1.1). Let be the solution of (1.1) with initial condition and control function . We introduce the reachability set of system (1.1) as the set consisting of the values .
Next we will show that a modification of an argument of Sukavanam  can be applied to compare the approximate controllability of systems (1.1) and (1.2).
Let . There exist sequences in and in such that as . Let be the orthogonal projection on . Therefore, as . Since and , we can assert that the sequence converges to some element and the sequence converges to some element . Consequently, , which completes the proof.
Related to this result, it is worthwhile to point out that if has a continuous left inverse for each , then the space is closed. Moreover, if and the range of is a closed subspace, which occurs, for instance, when is a finite dimensional space, then has a continuous left inverse.
which completes the proof.
Now we are able to establish the following criteria for the approximate controllability of system (1.1). The next property is an immediate consequence of Theorem 2.6.
We are also in a position to establish the following result.
the control system (1.2) is approximately controllable in finite time;
Then system (1.1) is approximately controllable in finite time.
Similar results for approximate controllability to the origin can be established. On the other hand, with appropriate hypotheses we can estimate the controls involved in the strategies of controllability and approximate controllability. This property allows us to compare the controllability in spaces of infinite dimension with the controllability in spaces of finite dimension.
Assume that the control system (1.2) is controllable on , condition (2.7) holds, each operator has a continuous left inverse , for , and . Then there exists constants such that for every and there exists a control sequence , , with and , where , , is the solution of (1.1) corresponding to .
and the assertion is consequence of (2.22).
2.1. The Finite-Dimensional Case
Certainly condition (2.7) considered in our previous results is strong. However, the following property holds.
which completes the proof.
We will apply Theorem 2.10 to reduce the study of controllability of system (1.1) to the controllability of systems with finite-dimensional state space.
The assertion is an immediate consequence of Theorems 2.10 and 2.7.
collecting the above estimate, we get the assertion.
Using now Theorem 2.15 and Lemma 2.17 we can emphasize the assertion of Corollary 2.11.
Finally, we are in a position to establish the following result of controllability.
Assume that there exists an approximation scheme and the system (2.28) is approximately controllable in finite time. If, in addition, , , and and as , then the system (2.27) is also approximately controllable in finite time.
Let and . It follows from Corollary 2.18 that system (2.35) is approximately controllable on . Since as , for , we chose such that . It follows from Remark 2.19 that there exists a sequence of controls for such that , and , where is the solution of (2.35) corresponding to controls .
We complete this paper with an application of the results established in Section 2.
In this application we are concerned with a general class of systems that satisfy the conditions considered previously. Specifically, we consider a control system of type (1.1) with state space of infinite dimension and operators and for .
Let , , be the constants introduced in Lemma 2.16, and let be the constants introduced in Remark 2.19. At this point it is worth to note that the constants for and depend on and for and while and depend on and , respectively, for . We can establish the following immediate consequence of Theorem 2.20.
The authors are grateful to the referees for providing nice comments and suggestions. H. R. Henríquez was supported in part by CONICYT under Grant FONDECYT no. 1090009. C. Cuevas was partially supported by CNPq/Brazil.
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