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Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities. (English) Zbl 0959.93025

This work provides a new proof of the approximate controllability for the semilinear heat equation, when the nonlinear term depends on the state and its spatial gradient in a global Lipschitz way and the control is acting on a nonempty open subset of the domain. Denoting by \(R\) the range of the solutions at the final time, it is shown by means of the classical fixed point approach that \(R\) is dense in \(L^2(\Omega)\). A first proof of this property was obtained by the reviewer and the author [J. Optimization Theory Appl. 101, No. 2, 307-328 (1999; Zbl 0952.49003)] by using the penalization of an optimal control problem. Indeed, we proved a stronger result: the density of \(R\) in the Sobolev space \(H^s_0(\Omega)\) for every \(s \in [0,1)\).

MSC:

93C20 Control/observation systems governed by partial differential equations
93B05 Controllability
93C10 Nonlinear systems in control theory

Citations:

Zbl 0952.49003
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