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Approximate Controllability of Abstract Discrete-Time Systems
Advances in Difference Equations volume 2010, Article number: 695290 (2010)
Abstract
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.
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 
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 [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). 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.
Lemma 1.1.
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 
.
Definition 2.1.
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.
Definition 2.2.
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 
.
Definition 2.3.
System (1.1) is said to be
-
(a)
approximately controllable on
if 
is dense in 
, -
(b)
approximately controllable in finite time if
is dense in 
, -
(c)
approximately controllable to the origin on
if 
for every 
, -
(d)
approximately controllable to the origin in finite time if
for every 
.
if 
is dense in 
,
is dense in 
,
if 
for every 
,
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 [23] 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 
.
Lemma 2.4.
Assume that
for all 
. If 
, then 
has a fixed point.
Proof.
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 
.
Lemma 2.5.
Assume that 
and the space 
is dense in 
. Then 
.
Proof.
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.
Theorem 2.6.
Assume that 
and condition (2.7) holds. Then 
for all 
.
Proof.
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.
Theorem 2.7.
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.
Theorem 2.8.
Assume that the following conditions hold:
-
(a)
the control system (1.2) is approximately controllable in finite time;
-
(b)
for all
, the space 
; -
(c)
for all
, 
.
, the space 
;
, 
.Then system (1.1) is approximately controllable in finite time.
Proof.
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.
Theorem 2.9.
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 
.
Proof.
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.
Theorem 2.10.
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.
Proof.
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.
Corollary 2.11.
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 
.
Proof.
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.
Proposition 2.12.
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 
.
Proof.
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.
Remark 2.13.
Let 
. Since
we infer that 
if, and only if,
Hence, if 
and we define 
and 
, then 
.
Lemma 2.14.
Assume that condition (2.7) holds for 
and 
. Then
where 
.
Proof.
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:
-
(i)
the subspaces
and 
are invariant under 
; -
(ii)
the projections
are uniformly bounded with 
for all 
; -
(iii)
for all
, 
as 
.
and 
are invariant under 
;
are uniformly bounded with 
for all 
;
, 
as 
.We consider the control systems
in the space 
. We set 
.
Theorem 2.15.
If the system (2.28) is approximately controllable in finite time, then the system (2.36) is controllable on an interval 
for each 
.
Proof.
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.
Lemma 2.16.
Assume that condition (2.7) holds in 
for all 
. Then there are constants 
such that
for all 
, 
.
Proof.
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 
.
Lemma 2.17.
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.
Proof.
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.
Corollary 2.18.
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 
.
Remark 2.19.
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.
Theorem 2.20.
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.
Proof.
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.
3. Application
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.
Proposition 3.1.
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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Acknowledgments
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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Henríquez, H.R., Cuevas, C. Approximate Controllability of Abstract Discrete-Time Systems. Adv Differ Equ 2010, 695290 (2010). https://doi.org/10.1155/2010/695290
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DOI: https://doi.org/10.1155/2010/695290