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

Free access article

Issue		ESAIM: COCV Volume 4, 1999


Page(s)		83 - 98
DOI		10.1051/cocv:1999104

DOI: 10.1051/cocv:1999104

ESAIM: COCV, April 1999, Vol. 4, p. 83-98

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

Contrôlabilité approchée et son caractère bien posé pour l'équation de la chaleur semi-linéaire gouvernée par des contrôles internes constants par morceaux

Alexander Khapalov
Department of Pure and Applied Mathematics, Washington State University, Pullman, WA 99164-3113, USA; (khapala@delta.math.wsu.edu)

Received April 22, 1997. Revised June 24, 1998.

Abstract: We consider the one dimensional semilinear reaction-diffusion equation, governed in $\Omega = (0,1)$ 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 $\log \mid u \mid$ . 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.

Résumé: On étudie l'équation de la chaleur semi-linéaire sur l'intervalle (0,1) avec des contrôles internes sur un sous-intervalle et qui ne sont que des fonctions du temps. Utilisant une approche asymptotique que nous avons précédemment introduite dans [9], on montre la contrôlabilité approchée pour tout temps à la fois dans L² (0,1) et dans C₀[0, 1] si le terme non linéaire f = f(x,t, u) ne croit pas plus vite qu'une certaine puissance de $\log \mid u \mid$ . Celle-ci dépend de la régularitée et de la structure de f (x, t, u) par rapport à x et t et du choix de l'espace pour la contrôlabilité. On montre aussi que nos résultats permettent de piloter le système considéré, même dans le cas où l'on n'a pas unicité des solutions.

Keywords and phrases: The semilinear reaction-diffusion equation, approximate controllability, internal lumped control, multiple solutions.

AMS Subject Classification: 93 35

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