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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 $L\sp p(\Omega)$ for $1\le p< +\infty$ is proved when the nonlinearity is gobally Lipschitz with a control in $L\sp \infty$. In the case of the interior control, we also prove approximate controllability in $C\sb 0(\Omega)$. 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.

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