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Approximate controllability of the semilinear heat equation. (English) Zbl 0818.93032
Summary: This article is concerned with the study of approximate controllability for the semilinear heat equation in a bounded domain Ω when the control acts on any open and nonempty subset of Ω or on a part of the boundary. In the case of both an internal and a boundary control, the approximate controllability in L p (Ω) for 1p<+ is proved when the nonlinearity is gobally Lipschitz with a control in L . In the case of the interior control, we also prove approximate controllability in C 0 (Ω). The proof combines a variational approach to the controllability problem for linear equations and a fixed point method. We also prove that the control can be taken to be of “quasi bang-bang” form.

MSC:
93C20Control systems governed by PDE
35K60Nonlinear initial value problems for linear parabolic equations
93C10Nonlinear control systems
93B05Controllability