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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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