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Finite dimensional null controllability for the semilinear heat equation. (English) Zbl 0872.93014

Summary: We study a finite-dimensional version of the null controllability problem for semilinear heat equations in bounded domains \(\Omega\) of \(\mathbb{R}^n\) with Dirichlet boundary conditions. The control acts on any open and non-empty subset of \(\Omega\). The question under consideration is the following: given an initial state, a control time \(t=T\) and a finite-dimensional subspace \(E\) of \(L^2(\Omega)\), 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 nonlinearity 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 nonlinearities 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 nonlinearity is globally Lipschitz.

MSC:

93B05 Controllability
93C10 Nonlinear systems in control theory
35K05 Heat equation
93C20 Control/observation systems governed by partial differential equations
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