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Fabre, Caroline; Puel, Jean-Pierre; Zuazua, Enrike

Approximate controllability of the semilinear heat equation. (English) Zbl 0818.93032

Proc. R. Soc. Edinb., Sect. A 125, No. 1, 31-61 (1995).
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.

Cited in 1 Review
Cited in 184 Documents

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

Keywords:

Lipschitz nonlinearity; bang-bang; approximate controllability; semilinear heat equation; boundary control; interior control
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\textit{C. Fabre} et al., Proc. R. Soc. Edinb., Sect. A, Math. 125, No. 1, 31--61 (1995; Zbl 0818.93032)
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