%0 Journal Article
%T Approximate Controllability of a Semilinear Heat Equation
%A Hugo Leiva
%A N. Merentes
%A J. Sanchez
%J International Journal of Partial Differential Equations
%D 2013
%R 10.1155/2013/424309
%X 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 . 1. Introduction 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 Figure 1 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 [1]) 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¨C4], 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
%U http://www.hindawi.com/journals/ijpde/2013/424309/