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Approximate controllability and its well-posedness for the semilinear reaction-diffusion equation with internal lumped controls

Published online by Cambridge University Press:  15 August 2002

Alexander Khapalov
Alexander Khapalov*
Affiliation:
Department of Pure and Applied Mathematics, Washington State University, Pullman, WA 99164-3113, USA; [email protected].
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Abstract

We consider the one dimensional semilinear reaction-diffusion equation, governed in Ω = (0,1) by controls, supported on any subinterval of (0, 1), which are the functions of time only. Using an asymptotic approach that we have previously introduced in [9], we show that such a system is approximately controllable at any time in both L2(0,1)( and C0[0,1], provided the nonlinear term f = f(x,t, u) grows at infinity no faster than certain power of log |u|. The latter depends on the regularity and structure of f (x, t, u) in x and t and the choice of the space for controllability. We also show that our results are well-posed in terms of the “actual steering” of the system at hand, even in the case when it admits non-unique solutions.

Keywords

The semilinear reaction-diffusion equationapproximate controllabilityinternal lumped control multiple solutions.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 83 - 98
Copyright
© EDP Sciences, SMAI, 1999

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