Finite dimensional null controllability for the semilinear heat equation

Abstract

We study a finite dimensional version of the null controllability problem for semilinear heat equations in bounded domains Ω of R n with Dirichlet boundary conditions. The control acts on any open an non-empty subset of Ω. The question under consideration is the following: given an initial state, a control time t = T and a finite dimensional subspace E of L 2(Ω) , is there a control such that the orthogonal projection over E of the solution at time t = T vanishes? Under rather natural growth conditions on the non-linearity we show that this can be done provided the initial data is sufficiently small. The method of proof combines the Implicit Function Theorem with a constructive method to solve this finite controllability problem in the linear case. We then consider non-linearities with the “good sign”. Using the decay properties of solutions and the fact that the problem is solvable for small data, we show that the problem is solvable for large data too. When analyzing the linear heat equation we will prove that with one sole control one can obtain simultaneously approximate controllability and exact reachability of a finite number of constraints. The same result holds when the non-linearity is globally Lipschitz.