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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
.
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
,
.
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
.
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