Approximate Controllability of a Semilinear Heat Equation
We apply Rothe’s type fixed point theorem to prove the interior approximate controllability of the following semilinear heat equation: in on , 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 , and such that for all Under this condition, we prove the following statement: for all open nonempty subset of , the system is approximately controllable on . Moreover, we could exhibit a sequence of controls steering the nonlinear system from an initial state to an neighborhood of the final state at time .
In this paper, we prove the interior approximate controllability of the following semilinear heat equation: where is a bounded domain in , , is an open nonempty subset of , denotes the characteristic function of the set , the distributed control belong to , and the nonlinear function is smooth enough, and there are , and such that which implies that
Definition 1 (approximate controllability). The system (1) is said to be approximately controllable on if for every , , , there exists such that the solution of (1) corresponding to verifies see (Figure 1), where
Remark 2. It is clear that exact controllability of the system (1) implies approximate controllability, null controllability, and controllability to trajectories of the system. But it is well known (see ) 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 in 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 shall concentrate only on the study of the approximate controllability of the system (1).
Now, before proceeding with the introduction of this paper, we should mention the work of other authors and show that ours is different in relation with new perturbation and the technique used. The approximate controllability of the heat equation under nonlinear perturbations independents of and variable has been studied by several authors, particularly in [2–4], depending on conditions that impose to the nonlinear term . For instance, in [3, 4], the approximate controllability of the system (6) 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 (6) fails.
Now, we shall describe the strategy of this work.
First, we prove that the linear system is approximately controllable.
After that, we write the system (1) as follows: where is a smooth enough functions satisfying the following conditions: there are , and such that Finally, the approximate controllability of the system (11) follows from the controllability of (10), the compactness of the semigroup generated by the Laplacian operator , the condition (12) satisfied by the nonlinear term , and the following results.
Proposition 3. Let be a measure space with and . Then, and
Proof. The proof of this proposition follows from Theorem I.V.6 from  by putting and considering the relation
Theorem 4 (Rothe’s theorem [13, page 129]). Let be a Hausdorff topological vector space. Let be a closed convex subset such that the zero of is contained in the interior of .
Let be a continuous mapping with relatively compact in and .
Then, there is a point such that .
Lemma 5 (see Lemma 3.14 from [15, page 62]). Let and be two sequences of real numbers such that . Then, If and only if
2. Abstract Formulation of the Problem
In this section, we choose a Hilbert space where system (1) can be written as an abstract differential equation; to this end, we consider the following results appearing in [15, page 46], [16, page 335], and [17, page 147].
Let us consider the Hilbert space and the eigenvalues of with the Dirichlet homogeneous conditions, 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 an analytic semigroup given by
Consequently, systems (1), (10), and (11) can be written, respectively, as an abstract differential equations in : where , , , is a bounded linear operator, is defined by , for all , and . On the other hand, from condition (12), we get the following estimate.
Proposition 6. Under condition (12), the function defined by , for all , satisfies for all :
Now, since , applying Proposition 3, we obtain that
3. Interior Controllability of the Linear Equation
In this section, we shall prove the interior approximate controllability of the linear system (21). To this end, we note that, for all and , the initial value problem
where the control function belongs to , admits only one mild solution given by
Definition 7. For system (21), we define the following concept: the controllability map (for ) is given by whose adjoint operator is given by
The following lemma holds in general for a linear bounded operator between Hilbert spaces and .
Lemma 8 (see [11, 15, 18]). Equation (21) is approximately controllable on if and only if one of the following statements holds.(a).(b).(c), in .(d).(e), for all . (f)For all , we have , where
So, and the error of this approximation is given by
Remark 9. Lemma 8 implies that the family of linear operators , defined for by
is an approximate inverse for the right of the operator in the sense that
Proposition 10 (see ). If , then
Remark 11. The proof of the following theorem follows from foregoing characterization of dense range linear operators and the classical unique continuation for elliptic equations (see ), and it is similar to the one given in Theorem 4.1 in .
Theorem 12. System (21) is approximately controllable on . Moreover, a sequence of controls steering the system (21) from initial state to an neighborhood of the final state at time is given by and the error of this approximation is given by
4. Controllability of the Semilinear System
In this section, we shall prove the main result of this paper, the interior approximate controllability of the semilinear heat equation given by (1), which is equivalent to prove the approximate controllability of the system (22). To this end, for all and , the initial value problem admits only one mild solution given by
Definition 13. For the system (22), 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 14. Equation (22) is approximately controllable on if and only if .
Definition 15. The following equation shall be called the controllability equation associated with the nonlinear equation (22):
Now, we are ready to present and prove the main result of this paper, which is the interior approximate controllability of the semilinear heat equation (1)
Theorem 16. The system (22) is approximately controllable on . Moreover, a sequence of controls steering the system (22) 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. 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 fixed point . In fact, since is smooth and satisfies (12) and the semigroup given by (19) is compact (see [19, 20]), then using the result from , we obtain that the operator is compact, which implies that the operator is compact.
Moreover, In fact, putting and , we get from Proposition 6 that and from the definition of the operator , Proposition 3, and (19) we have, for , the following estimate: Now, since , applying Proposition 3, we obtain that Therefore, Consequently, Then, from condition (45), we obtain that, for a fixed , there exists big enough such that Hence, if we denote by the ball of center zero and radio , we get that . Since is compact and maps the sphere into the interior of the ball , we can apply Rothe's fixed point Theorem 4 to ensure the existence of a fixed point such that Claim. The family of fixed pint is bounded.
In fact, for the purpose of contradiction, let us assume the contrary. Then, there exists a subsequence such that On the other hand, we have that . So, Hence, which is evidently a contradiction. Then, the claim is true and there exists such that Therefore, without loss of generality, we can assume that the sequence converges to . So, if then Hence, To conclude the proof of this theorem, it is enough to prove that From Lemma 8 (d), we get that On the other hand, from Proposition 10, we get that Therefore, since converges to , we get that Consequently, So, by putting and using (38), we obtain the nice result:
5. Final Remark
Our technique is simple and can be applied to those systems involving compact semigroups like some control system governed by diffusion processes. For example, the Benjamin-Bona-Mahony equation, the strongly damped wave equations, beam equations, and so forth.
Example 17. The original Benjamin-Bona-Mahony equation is a nonlinear one in  the authors proved the approximate controllability of the linear part of this equation, which is the fundamental base for the study of the controllability of the nonlinear BBM equation. So, our next work is concerned on the controllability of nonlinear BBM equation: where and are constants, is a domain in , is an open nonempty subset of , denotes the characteristic function of the set , the distributed control , and is a nonlinear perturbation.
Example 18. We believe that this technique can be applied to prove the interior controllability of the strongly damped wave equation with Dirichlet boundary conditions: in the space , where is a bounded domain in , is an open nonempty subset of , denotes the characteristic function of the set , the distributed control , and , are positive numbers.
Example 19. Another example where this technique may be applied is partial differential equations modeling the structural damped vibrations of a string or a beam: where is a bounded domain in , is an open nonempty subset of , denotes the characteristic function of the set , the distributed control , and .
Moreover, our result can be formulated in a more general setting. Indeed, we can consider the following semilinear evolution equation in a general Hilbert space : where is an unbounded linear operator in with the following spectral decomposition: with the eigenvalues of having finite multiplicity equal to the dimension of the corresponding eigenspaces, and is a complete orthonormal set of eigenfunctions of . The operator generates a strongly continuous compact semigroup given by The control , with , is a linear and bounded operator (linear and continuous), and the function is smooth enough, and In this case, the characteristic function set is a particular operator , and the following theorem is a generalization of Theorem 16.
Theorem 20. If vectors are linearly independent in , then the system (69) is approximately controllable on .
This work has been supported by CDCHT-ULA-C-1796-12-05-AA and BCV.
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