Global non-negative controllability of the semilinear parabolic equation governed by bilinear control

Free access article

Issue		ESAIM: COCV Volume 7, 2002


Page(s)		269 - 283
DOI		10.1051/cocv:2002011

ESAIM: COCV, 2002, Vol. 7, pp. 269-283
DOI: 10.1051/cocv:2002011

Global non-negative controllability of the semilinear parabolic equation governed by bilinear control

Alexander Y. Khapalov

Department of Pure and Applied Mathematics, Washington State University, Pullman, WA 99164-3113, U.S.A. (khapala@wsu.edu).

(Received January 17, 2001. Revised May 21, September 8 and October 1, 2001. )

Abstract
We study the global approximate controllability of the one dimensional semilinear convection-diffusion-reaction equation governed in a bounded domain via the coefficient (bilinear control) in the additive reaction term. Clearly, even in the linear case, due to the maximum principle, such system is not globally or locally controllable in any reasonable linear space. It is also well known that for the superlinear terms admitting a power growth at infinity the global approximate controllability by traditional additive controls of localized support is out of question. However, we will show that a system like that can be steered in L² (0,1) from any non-negative nonzero initial state into any neighborhood of any desirable non-negative target state by at most three static ( x-dependent only) above-mentioned bilinear controls, applied subsequently in time, while only one such control is needed in the linear case.

Mathematics Subject Classification. 93, 35

Key words: Semilinear parabolic equation, global approximate controllability, bilinear control.

© EDP Sciences, SMAI 2002

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