- Research Article
- Open Access
Approximate Controllability of Abstract Discrete-Time Systems
Advances in Difference Equations volume 2010, Article number: 695290 (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.
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 for the state space and for the control space. We assume that and are Hilbert spaces. Moreover, throughout this paper we denote by bounded linear operators, , , bounded linear maps that represent the control action, and a map such that is continuous for each . Furthermore, , and satisfy appropriate conditions which will be specified later. We will study the controllability of control systems described by the equation
where , .
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.
For this reason, in this paper we study the approximate controllability of system (1.1). Specifically, we will compare the approximate controllability of system (1.1) with the approximate controllability of linear system
where and .
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.
Let be a Hilbert space, and let be closed subspaces of such that . Then there exists a bounded linear projection such that for each , and
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 .
Let be the evolution operator associated to the linear homogeneous equation
Furthermore, the solution of (1.2) is given by
We will abbreviate the notation by writing for this solution.
We define the bounded linear operator by
It is clear that .
The system (1.2) is said to be exactly controllable (or simply controllable) on if .
System (1.2) is said to be approximately controllable on if the space is dense in and approximately controllable in finite time if the space is dense in .
If the system (1.2) is approximately controllable on and is a finite-dimensional space, then the system (1.2) is controllable on .
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.
System (1.2) is said to be approximately controllable to the origin on if for every and approximately controllable to the origin in finite time if for every .
On the other hand, for , (1.1) has a unique solution which satisfies the equation
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 .
System (1.1) is said to be
approximately controllable on if is dense in ,
approximately controllable in finite time if is dense in ,
approximately controllable to the origin on if for every ,
approximately controllable to the origin in finite time if for every .
We next introduce some additional notations. The operators and are given by
It is clear that and are bounded linear operators. We set . Moreover, we denote by the operator defined by .
We denote by the space consisting of such that .
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).
For fixed and , we begin by defining the map by . It is clear that is a continuous map.
On the other hand, under the assumption that
we denote by the projection constructed as in Lemma 1.1 with and . We introduce the space
and we define the map by
We next study the existence of fixed points for . In the following statement, we denote .
for all . If , then has a fixed point.
It is easy to see that is a contraction map. In fact, since and are bounded linear maps, we have
which implies that is a contraction.
In what follows we always assume that satisfies the Lipschitz condition (2.10).
Under certain conditions we can modify our hypothesis .
Assume that and the space is dense in . Then .
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.
Assume that and condition (2.7) holds. Then for all .
Let be a control vector, and let be the solution of (1.2) with initial condition . In what follows, we apply our construction preceding Lemma 2.4 with the vector . Let be a fixed point of . Clearly and . We set . We now apply Lemma 1.1 to , with respect to spaces and . We set , and we define , for . It follows from this construction that , and combining the properties of and , we obtain that
for . We can also see directly that (2.12) hods for . We select a sequence such that as goes to infinity and . We denote by the solution of (2.12) when we substitute by . Hence, we can write
This expression and (2.3) show that is the solution of the equation
with initial condition . Therefore, . Since the solution of (2.3) depends continuously on , we infer that converges to as . Consequently, . Hence, from our previous considerations, we can assert that
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.
Assume that , the control system (1.2) is approximately controllable on and the space . Then the system (1.1) is approximately controllable on .
We are also in a position to establish the following result.
Assume that the following conditions hold:
the control system (1.2) is approximately controllable in finite time;
for all , the space ;
for all , .
Then system (1.1) is approximately controllable in finite time.
Proceeding as in the proof of Theorem 2.6, we can write
which shows that is dense in .
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 .
It follows from the controllability of system (1.2) that is a surjective bounded linear map. We infer that there exists a constant such that for each there exists such that and . Let , , be the solution of (1.2) corresponding to . Since and are uniformly bounded for , we can conclude that there exists a constant such that for . In the rest of this proof we apply the construction carried out in the proof of Theorem 2.6. Let be the fixed point of . From
we deduce that
which in turn implies that
which we abbreviate as
Proceeding in a similar way, we can obtain an estimate
Hence, can also be estimated as
We can choose a sequence such that and as . Therefore, we can take large enough such that . Since is the solution of (1.1) corresponding to controls , to complete the proof we only need to estimate
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.
Assume that is a space of finite dimension. Then the linear system (1.2) is controllable on if, and only if, condition (2.7) holds.
Since has finite dimension, . Assume initially that system (1.2) is controllable on . Let . Using the property of controllability, it follows from [4, Corollary .3.1] that there exists such that
We define for . This implies that
which shows that .
Conversely, assume that condition (2.7) holds; for we define . Applying (2.7), we derive the existence of and such that . The solution of (1.2) is given by
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.
Assume that is a space of finite dimension and that the linear system (1.2) is controllable on . Then there exists such that nonlinear system (1.1) is approximately controllable on when .
The assertion is an immediate consequence of Theorems 2.10 and 2.7.
Next we specialize our developments to consider systems where the associated linear system is invariant. Specifically, we will assume that and for . That is to say, we will be concerned with the nonlinear system
with linear part
In this situation, the subspaces are nondecreasing. Hence, we get the following immediate consequence.
Assume that is a space of finite dimension. If the system (2.28) is approximately controllable in finite time, then it is controllable on , for some .
Since and are closed subspaces, then there is such that .
2.2. The Projections
Next we will study a property of projections . We begin with some remarks.
Let . Since
we infer that if, and only if,
Hence, if and we define and , then .
Assume that condition (2.7) holds for and . Then
We decompose , where .
Let and . Then , where . We set and . It follows from Remark 2.13 that , and and . Therefore, using the properties of projections and established in Lemma 1.1, we get
Similarly, since , we can decompose , where . We set and . It follows from Remark 2.13 that , and and . Consequently, we have
Collecting these assertions, we get
We say that a sequence is an approximation scheme for associated to system (2.27) if are finite-dimensional subspaces of , are bounded linear projections with and , and the following conditions are fulfilled:
the subspaces and are invariant under ;
the projections are uniformly bounded with for all ;
for all , as .
We consider the control systems
in the space . We set .
If the system (2.28) is approximately controllable in finite time, then the system (2.36) is controllable on an interval for each .
We consider a fixed . It is immediate from our definition of approximation scheme that if and we consider the same values of in (2.28) and (2.36), then for all . Let . It follows from the previous remark, that if we select such that as , then
which shows that as . Hence, system (2.36) is approximately controllable in finite time. The assertion is now a consequence of Proposition 2.12.
To simplify the writing of the text, next we will assume that and . Furthermore, we take an orthonormal basis of , and is the orthogonal projection. We can establish the following property.
Assume that condition (2.7) holds in for all . Then there are constants such that
for all , .
We proceed by using mathematical induction. The assertion holds for . In fact, since and , then and
Assume now that the assertion is fulfilled for . We will prove that the assertion holds for . For , , we decompose , where . We abbreviate the notation by writing . Consequently, applying Lemma 2.14, we get
On the other hand, since is a bounded linear map on , then there exists a constant such that
and substituting these estimates in (2.40), we get that the assertion is fulfilled for .
Assume that , condition (2.7) holds in for all , and that the function in (2.35) satisfies the Lipschitz conditions
where . If
then the map defined in is a contraction.
It follows from our definition that
On the other hand, since
applying Lemma 2.16 and the definition of , we have
In view of
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.
Under the conditions of Lemma 2.17, if the system (2.28) is approximately controllable in finite time, then the system (2.35) is approximately controllable on for each .
Under the above conditions, we can apply Theorem 2.9 in the space . Consequently, there exist constants and such that for every and , there exists a sequence of controls for such that , , and , where is the solution of (2.35) corresponding to controls . Furthermore, we denote , where are the constants involved in (2.10), and we assume that
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 denote for the solution of system
and we set . It follows from (2.35) and (2.49) that
which implies that
Consequently, . Hence,
Consequently, as , which completes the proof.
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 .
We assume that is a bounded self-adjoint operator with distinct eigenvalues , , and is an orthonormal basis of consisting of eigenvectors of corresponding to eigenvalues , respectively.
We take as control space , and is given by , where is a vector such that , for all . It is clear that condition (2.7) does not hold in this case. In fact, since the space is closed, if we assume that condition (2.7) is fulfilled, then for every there is such that . In particular, for an arbitrary and and applying Remark 2.13, we obtain that . However, this means that is a finite-dimensional space, which is a contradiction. Let be given by
where are functions such that and the following Lipschitz conditions
are verified for all , , and . We assume that
We denote .
Let , and let be the orthogonal projection on . We set . Since is invariant under , we can consider the system
with , which is the restriction of system (1.2) on . It is well known that system (3.4) is exactly controllable on , for every . Furthermore,
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.
Assume that 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.
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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.