Abstract and Applied Analysis

Volume 2013 (2013), Article ID 729093, 11 pages

http://dx.doi.org/10.1155/2013/729093

Research Article

A Characterization of Semilinear Dense Range Operators and Applications

H. Leiva,¹ N. Merentes,² and J. Sanchez²

¹Universidad de los Andes, Facultad de Ciencias, Departamento de Matemática, Mérida 5101, Venezuela

²Universidad Central de Venezuela, Facultad de Ciencias, Departamento de Matemática, Caracas 1053, Venezuela

Received 13 October 2012; Revised 25 November 2012; Accepted 22 January 2013

Academic Editor: Valery Y. Glizer

Copyright © 2013 H. Leiva et al. 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.

Abstract.

We characterize a broad class of semilinear dense range operators given by the following formula, , where , are Hilbert spaces, , and is a suitable nonlinear operator. First, we give a necessary and sufficient condition for the linear operator to have dense range. Second, under some condition on the nonlinear term , we prove the following statement: If , then and for all there exists a sequence given by , such that . Finally, we apply this result to prove the approximate controllability of the following semilinear evolution equation: , where , are Hilbert spaces, is the infinitesimal generator of strongly continuous compact semigroup in , the control function belongs to , and is a suitable function. As a particular case we consider the controlled semilinear heat equation.

1. Introduction

It is well known from functional analysis that continuous linear surjective operators form an open set in the space of such operators; that is to say, if a surjective linear continuous operator is added to a linear continuous operator with a small enough norm, the resulting operator is still surjective; moreover, if a linear continuous surjective operator is perturbed by a nonlinear Lipschitz operator with a Lipschitz constant small enough, then the resulting operator is still surjective. This result is not true anymore for continuous linear operators that only have dense range; for instance, if a dense range continuous linear operator is perturbed by another linear operator with norm infinitely small, the resulting operator may not have dense range; in other words, the property of having dense range is not robust enough to be surjective. However, in this paper we proved the following statement: if a continuous linear operator with dense range is perturbed by a compact nonlinear operator with bounded range, then the resulting operator also has dense range. This result can have an unlimited number of applications, not only in the study of control theory for semilinear evolution equations, but it can also be used to find the approximate solution of functional equations in Hilbert spaces giving a formula for the error of this approximation, which is very important from the standpoint of numerical analysis. In addition, it is well known that approximate controllability is much more natural and common than exact controlabilidad, since most of the mechanical processes are diffusive, which implies that these systems can never be exactly controllable; and many years have passed to present a general result on semilinear operators with dense range to facilitate the study of the approximate controlabilidad for a large class of semilinear evolution equations whose dynamics are given by compact semigroups. However, in this work, by way of illustration, we only show how this result can be applied to study the approximate controllability of control systems governed by the semilinear heat equation.

Specifically, in this paper we characterize a broad class of semilinear dense range operators.

given by the following formula:

where , are Hilbert spaces, is a bounded linear operator (continuous and linear), and is a suitable nonlinear operator. First, we give a necessary and sufficient condition for the linear operator to have dense range (). Second, we prove the following statement: If and is smooth enough and is compact, then and for all there exists a sequence given by

such that

and the error of this approximation is given by

This result can be viewed as a generalization of the work done in [1–8].

We apply these results to prove the approximate controllability of the following semilinear evolution equation:

where , are Hilbert spaces, is the infinitesimal generator of strongly continuous compact semigroup in , the control function belongs to , and is a smooth enough function.

Remark 1 (see [2–4]). The function is smooth enough if

(a)the mild solutions of (5) are unique, (b)the mild solutions depends continuously on , (c)and if is a Lipschitz function, then , as a function of , is also a Lipchitz function.

As an application we consider the following example of controlled semilinear heat equation.

Example 2 (the interior controllability of the heat equation). The semilinear heat equation was studied in [8] where the authors prove the interior controllability of the following control system:

where is a bounded domain in ,  , is an open nonempty subset of , denotes the characteristic function of the set , the distributed control belongs to , and the nonlinear function is smooth enough and there are constants , with , such that

where .

We note that the interior approximate controllability of the linear heat equation,

has been study by several authors, particularly by [9], and in a general fashion in [10].

The approximate controllability of the heat equation under nonlinear perturbation independents of and variables,

has been studied by several authors, particularly in [11–13], depending on conditions imposed to the nonlinear term . For instance, in [12, 13] the approximate controllability of the system (9) is proved if is sublinear at infinity; that is,

Also, in the above reference, they mentioned that when is superlinear at the infinity, the approximate controllability of the system (9) fails.

Our result can be applied also to the semilinear Ornstein-Uhlenbeck equation, the Laguerre equation, and the Jacobi equation. Specifically, in [8], the following well-known example of reaction diffusion equations is studied.

Example 3 (see [14, 15]).     

(1)The interior controllability of the semilinear Ornstein-Uhlenbeck equation

 where , is the Gaussian measure in , is an open nonempty subset of , and the nonlinear function is smooth enough and there are constants , with , such that

 where .(2)The interior controllability of the semilinear Laguerre equation

 where , is the Gamma measure in , is an open nonempty subset of , and nonlinear function   is smooth enough and there are constant , with , such that

 where .(3)The interior controllability of the semilinear Jacobi equation

 where , , is the Jacobi measure in , is an open nonempty subset of , and the nonlinear function   is smooth enough and there are constants , with , such that

 where .

2. Dense Range Linear Operators

In this section we shall present a characterization of dense range bounded linear operators. To this end, we denote by the space of linear and bounded operators mapping to , endowed with the uniform convergence norm, and we will use the following lemma from [16] in Hilbert space.

Lemma 4. Let be the adjoint operator of . Then the following statements hold:

(i) such that

(ii).  The following lemma follows from Lemma 4 (ii).

Lemma 5 (see [1, 7, 8, 16–22]). The following statements are equivalent:

(a),(b),(c), in ,(d),(e) for all we have , where

 So, and the error of this approximation is given by the formula

Remark 6. Lemma 5 implies that the family of linear operators , defined for by

is an approximate inverse for the right of the operator , in the sense that

in the strong topology.

Proposition 7. If the , then

Proof. If , then from Lemma 4(ii) we have that

Therefore,

Then, using the Cauchy Schwartz inequality, we obtain

which is equivalents to

Consequently,

Proposition 8. If for some one has that

then

Proof. Suppose that . Then, from the following identity:

we get that

Since , we obtain that is a homeomorphism. Consequently, , which implies that .

Corollary 9. If   and , then

Moreover,

3. Dense Range Semilinear Operators

In this section we shall look for conditions under which the semilinear operator

, given by

has dense range.

Theorem 10. If , is continuous, and is compact, then , and for all there exists a sequence given by

such that

and the error of this approximation is given by

Proof. For each fixed we shall consider the following family of nonlinear operators given by

First, we shall prove that for all the operator has a fix point . In fact, since is a continuous function, the set is compact, and is a linear bounded operator, then there exists a constant such that

Therefore, the operator maps the ball of center zero and radio into itself. Hence, applying the Schauder fixed point theorem, we get that the operator has a fixed point .

Since is compact, without loss of generality, we can assume that the sequence converges to as . So, if we consider

then,

Hence,

To conclude the proof of this theorem, it is enough to prove that

From Lemma 5(d) we get that

On the other hand, from Proposition 7 we get that

Therefore, since converges to as , we get that

Consequently,

4. Controllability of Nonlinear Evolution Equations

In this section we shall apply the foregoing results to characterize the approximate controllability of the semilinear evolution equation

where , are Hilbert spaces, is the infinitesimal generator of strongly continuous compact semigroup in , the control function belongs to , and is smooth enough and there are constants such that

where and is a linear and bounded operator.

We observe that the controllability of semilinear systems has been studied by several authors, particularly interesting is the work done by [18–26].

Definition 11 (exact controllability). The system (48) is said to be exactly controllable on if for every , there exists such that the mild solution of (48) corresponding to verifies

as shown in Figure 1.

Figure 1

Definition 12 (approximate controllability). The system (48) is said to be approximately controllable on if for every , , there exists such that the solution of (48) corresponding to verifies

as shown in Figure 2.

Figure 2

Definition 13 (controllability to trajectories). The system (48) is said to be controllable to trajectories on if for every and there exists such that the mild solution of (48) corresponding to verifies:

as shown in Figure 3.

Figure 3

Definition 14 (null controllability). The system (48) is said to be null controllable on if for every there exists such that the mild solution of (48) corresponding to verifies:

as shown in Figure 4.

Figure 4

Remark 15. It is clear that exact controllability of the system (48) implies approximate controllability, null controllability, and controllability to trajectories of the system.  But, it is well known [27] that due to the diffusion effect or the compactness of the semigroup generated by , the heat equation can never be exactly controllable. We observe also that the linear case controllability to trajectories and null controllability are equivalent. Nevertheless, the approximate controllability and the null controllability are in general independent. Therefore, in this paper we will concentrated only on the study of the approximate controllability of the system (48).

Now, we shall describe the strategy of this work:

First, we characterize the approximate controllability of the auxiliary linear system

After that, we write the system (48) in the form

where is a smooth enough and bounded function.

Finally, the approximate controllability of the system (55) follows from the controllability of (54), the compactness of the semigroup generated by the operator , the uniform boundedness of the nonlinear term , and applying Schauder fixed point theorem.

Remark 16. If and , then the system is approximately controllable if and only if the system (55) is approximately controllable.

4.1. The Linear System

First, we shall characterize the approximate controllability of the linear system (54), and to this end, for all and the initial value problem

admits only one mild solution given by

Definition 17. For the system (54) we define the following concept: the controllability map (for ) is given by

whose adjoint operator is

The following lemma follows from Lemma 5.

Lemma 18. Equation (54) is approximately controllable on if and only if one of the following statements holds:

(a),(b),(c), in ,(d),(e), , (f) for all one has , where

 So, and the error of this approximation is given by the formula

Remark 19. Lemma 5 implies that the family of linear operators , defined for by

is an approximate inverse for the right of the operator , in the sense that

in the strong topology.

4.2. The Semilinear System

Now, we are ready to characterize the approximate controllability of the semilinear system (48), which is equivalent to proof of the approximate controllability of the system (55). To this end, we notice that, for all and the initial value problem

admits only one mild solution given by

Definition 20. For the system (55) we define the following concept: the nonlinear controllability map (for ) is given by

where is the nonlinear operator given by

The following lemma is trivial.

Lemma 21. Equation (55) is approximately controllable on if and only if .

Definition 22. The following equation will be called the controllability equations associated to the nonlinear equation (55)

Now, we are ready to present a result on the approximate controllability of the semilinear evolutions equation (48).

Theorem 23. If the linear system (54) is approximately controllable, then system (55) is approximately controllable on . Moreover, a sequence of controls steering the system (55) from initial state to an neighborhood of the final state at time is given by the formula

and the error of this approximation is given by

Proof. From Theorem 10, it is enough to prove that the function given by (103) is continuous and is a compact set, which follows from the compactness of the semigroup ,  the smoothness and the boundedness of the nonlinear term (see [8, 27]).

So, putting and using (65), we obtain the desired result

5. Application to the Nonlinear Heat Equation

As an application of this result we shall prove the controllability of the semilinear heat equation (6). To this end, we shall use the following strategy:

first, we prove that the auxiliary linear system

is approximately controllable.

After that, we write the system(6) as follows:

where is a smooth and bounded function.

Then to prove the controllability of the linear equation (72), we use the classical Unique Continuation Principle for Elliptic Equations (see [28]) and the following results.

Lemma 24 (see Lemma 3.14 from [16, page 62]). Let and be two sequences of real numbers such that: . Then

iff

Finally, the approximate controllability of the system (73) follows from the controllability of (72), the compactness of the semigroup generated by the Laplacean operator , and the uniform boundedness of the nonlinear term by applying Theorem 23.

5.1. Abstract Formulation of the Problem

In this part we choose a Hilbert space where system (6) can be written as an abstract differential equation; to this end, we consider the following notations.

Let us consider the Hilbert space and the eigenvalues of , each one with finite multiplicity equal to the dimension of the corresponding eigenspace. Then we have the following well-known properties.

(i) There exists a complete orthonormal set of eigenvectors of .(ii) For all we have

 where is the inner product in and

 So, is a family of complete orthogonal projections in and .(iii) generates a compact analytic semigroup given by

Consequently, systems (6), (72), and (73) can be written, respectively, as an abstract differential equations in :

where , , , is a bounded linear operator, is defined by , and .

On the other hand, the hypothesis (7) implies that

where . Therefore, is bounded and smooth enough.

5.2. The Linear Heat Equation

In this part we shall prove the interior controllability of the linear system (80). To this end, we notice that for all and the initial value problem,

admits only one mild solution given by

Definition 25. For the system (80) we define the following concept: the controllability map (for ) is given by

whose adjoint operator is given by

As a consequence of Lemma 18 and (101) one can prove the following result.

Lemma 26. Equation (80) is approximately controllable on if and only if one of the following statements holds:

(a),(b),(c), in ,(d),(e), , , (f) for all one has , where

 So, and the error of this approximation is given by

Theorem 27. The system (80) is approximately controllable on . Moreover, a sequence of controls steering the system (80) from initial state to an neighborhood of the final state at time is given by

and the error of this approximation is given by

Proof. It is enough to show that the restriction of to the space has range dense; that is, or . Consequently, takes the following form:

whose adjoint operator is given by

To this end, we observe that and . Suppose that

Then, since , this is equivalent to

On the other hand,

Hence, from Lemma 24, we obtain that

Now, putting , , we obtain that

Then, from the classical Unique Continuation Principle for Elliptic Equations (see [28]), it follows that , . So,

On the other hand, is a complete orthonormal set in , which implies that . Hence, . So, , and consequently . Hence, the system (80) is approximately controllable on , and the remainder of the proof follows from Lemma 26.

5.3. The Semilinear Heat Equation

In this part we shall prove the interior controllability of the semilinear heat equation given by (6), which is equivalent to the proof of the approximate controllability of the system (81). To this end, for all and the initial value problem,

admits only one mild solution given by

Definition 28. For the system (81) we define the following concept: the nonlinear controllability map (for ) is given by

where is the nonlinear operator given by

The following lemma is trivial.

Lemma 29. Equation (81) is approximately controllable on if and only if .

Definition 30. The following equation will be called the controllability equations associated to the nonlinear equation (81):

Now, we are ready to present a result on the interior approximate controllability of the semilinear heat equation (6).

Theorem 31. The system (81) is approximately controllable on . Moreover, a sequence of controls steering the system (81) from initial state to an neighborhood of the final state at time is given by

and the error of this approximation is given by

6. Conclusion

We believe that these results can be applied to a broad class of reaction diffusion equation like the following well-known systems of partial differential equations.

Example 32. The thermoelastic plate equation

where , , is a sufficiently regular bounded domain in , is an open nonempty subset of , denotes the characteristic function of the set , the distributed control , , , , denote the vertical deflection and the temperature of the plate, respectively, and the nonlinear terms , , are smooth enough and there are constants , with , , such that

where .

Example 33. The equation modelling the damped flexible beam:

where , , is an open nonempty subset of , , and nonlinear function is smooth enough and there are constant , with , such that

where .

Example 34. The strongly damped wave equation with Dirichlet boundary conditions:

where is a sufficiently smooth bounded domain in , , is an open nonempty subset of , , and nonlinear function is smooth enough and there are constants , with , such that

where .

Acknowledgments

This work has been supported by CDCHT-ULA-C-1796-12-05-AA and BCV.

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