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.
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.
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.
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 will abbreviate the notation by writing for this solution.
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 .
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 .
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 .
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.
We next study the existence of fixed points for . In the following statement, we denote .
for all . If , then has a fixed point.
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 .
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.
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.
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 .
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.
which shows that .
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.
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.
Hence, if and we define and , then .
We decompose , where .
the subspaces and are invariant under ;
the projections are uniformly bounded with for all ;
for all , as .
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 .
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.
for all , .
and substituting these estimates in (2.40), we get that the assertion is fulfilled for .
then the map defined in is a contraction.
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 .
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 .
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 denote .
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.
and and , as , then the system (2.27) is also approximately controllable in finite time.
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.
- Bittanti S, Bolzern P: Reachability and controllability of discrete-time linear periodic systems. IEEE Transactions on Automatic Control 1985,30(4):399-401. 10.1109/TAC.1985.1103961MathSciNetView ArticleMATHGoogle Scholar
- Guermah S, Djennoune S, Bettayeb M: Controllability and observability of linear discrete-time fractional-order systems. International Journal of Applied Mathematics and Computer Science 2008,18(2):213-222. 10.2478/v10006-008-0019-6MathSciNetView ArticleMATHGoogle Scholar
- Ichikawa A: Null controllability with vanishing energy for discrete-time systems. Systems & Control Letters 2008,57(1):34-38. 10.1016/j.sysconle.2007.06.008MathSciNetView ArticleMATHGoogle Scholar
- Klamka J: Controllability of Dynamical Systems, Mathematics and Its Applications (East European Series). Volume 48. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:xvi+248.MATHGoogle Scholar
- Leiva H, Uzcategui J: Exact controllability for semilinear difference equation and application. Journal of Difference Equations and Applications 2008,14(7):671-679. 10.1080/10236190701726170MathSciNetView ArticleMATHGoogle Scholar
- Negreanu M, Zuazua E: Uniform boundary controllability of a discrete 1-D wave equation. Systems & Control Letters 2003,48(3-4):261-279. 10.1016/S0167-6911(02)00271-2MathSciNetView ArticleMATHGoogle Scholar
- Phat VN: Controllability of nonlinear discrete systems without differentiability assumption. Optimization 1988,19(1):133-142. 10.1080/02331938808843327MathSciNetView ArticleMATHGoogle Scholar
- Son NK: Approximate controllability with positive controls. Acta Mathematica Vietnamica 1997,22(2):589-620.MathSciNetMATHGoogle Scholar
- Weiss G: Memoryless output feedback nullification and canonical forms, for time varying systems. International Journal of Control 2005,78(15):1174-1181. 10.1080/0020710500291986MathSciNetView ArticleMATHGoogle Scholar
- Zhang X, Zheng C, Zuazua E: Time discrete wave equations: boundary observability and control. Discrete and Continuous Dynamical Systems. Series A 2009,23(1-2):571-604.MathSciNetMATHGoogle Scholar
- Zheng C: Controllability of the time discrete heat equation. Asymptotic Analysis 2008,59(3-4):139-177.MathSciNetMATHGoogle Scholar
- Bensoussan A, Da Prato G, Delfour MC, Mitter SK: Representation and Control of Infinite-Dimensional Systems. Vol. 1, Systems & Control: Foundations & Applications. Birkhäuser, Boston, Mass, USA; 1992:xiv+315.MATHGoogle Scholar
- Bensoussan A, Da Prato G, Delfour MC, Mitter SK: Representation and Control of Infinite-Dimensional Systems. Vol. II, Systems & Control: Foundations & Applications. Birkhäuser, Boston, Mass, USA; 1993:xviii+345.View ArticleMATHGoogle Scholar
- Curtain RF, Zwart H: An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics. Volume 21. Springer, New York, NY, USA; 1995:xviii+698.View ArticleMATHGoogle Scholar
- Louis J-C, Wexler D: On exact controllability in Hilbert spaces. Journal of Differential Equations 1983,49(2):258-269. 10.1016/0022-0396(83)90014-1MathSciNetView ArticleMATHGoogle Scholar
- Henríquez HR: On non-exact controllable systems. International Journal of Control 1985,42(1):71-83. 10.1080/00207178508933347MathSciNetView ArticleMATHGoogle Scholar
- Gaĭshun IV: Controllability and stabilizability of discrete systems in a function space on a commutative semigroup. Differential Equations 2004,40(6):873-882. 10.1023/B:DIEQ.0000046864.27700.e9MathSciNetView ArticleMATHGoogle Scholar
- Klamka J: Controllability of nonlinear discrete systems. International Journal of Applied Mathematics and Computer Science 2002,12(2):173-180.MathSciNetMATHGoogle Scholar
- Komornik V: Exact Controllability and Stabilization, The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris, France; 1994:viii+156.MATHGoogle Scholar
- Sasu B, Sasu AL: Stability and stabilizability for linear systems of difference equations. Journal of Difference Equations and Applications 2004,10(12):1085-1105. 10.1080/10236190412331314178MathSciNetView ArticleMATHGoogle Scholar
- Sasu AL: Stabilizability and controllability for systems of difference equations. Journal of Difference Equations and Applications 2006,12(8):821-826. 10.1080/10236190600734218MathSciNetView ArticleMATHGoogle Scholar
- Zabczyk J: Mathematical Control Theory: An Introduction. Birkhäuser, Boston, Mass, USA; 1995.MATHGoogle Scholar
- Sukavanam N: Approximate controllability of semilinear control systems with growing nonlinearity. In Mathematical Theory of Control (Bombay, 1990), Lecture Notes in Pure and Appl. Math.. Volume 142. Dekker, New York, NY, USA; 1993:353-357.Google Scholar
This article is published under license to BioMed Central Ltd. 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.