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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 \(\Omega\) when the control acts on any open and nonempty subset of \(\Omega\) or on a part of the boundary. In the case of both an internal and a boundary control, the approximate controllability in \(L^ p(\Omega)\) for \(1\leq p< +\infty\) is proved when the nonlinearity is gobally Lipschitz with a control in \(L^ \infty\). In the case of the interior control, we also prove approximate controllability in \(C_ 0(\Omega)\). 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:
93C20 Control/observation systems governed by partial differential equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
93C10 Nonlinear systems in control theory
93B05 Controllability
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