Stabilization of the wave equation by on-off and positive-negative feedbacks

Free access article

Issue		ESAIM: COCV Volume 7, 2002


Page(s)		335 - 377
DOI		10.1051/cocv:2002015

ESAIM: COCV, 2002, Vol. 7, pp. 335-377
DOI: 10.1051/cocv:2002015

Stabilization of the wave equation by on-off and positive-negative feedbacks

Patrick Martinez and Judith Vancostenoble

M.I.P. Université Paul Sabatier Toulouse III, 118 route de Narbonne, 31062 Toulouse Cedex 4, France; martinez@mip.ups-tlse.fr. vancoste@mip.ups-tlse.fr.

(Received June 25, 2001. Revised November 27, 2001.)

Abstract
Motivated by several works on the stabilization of the oscillator by on-off feedbacks, we study the related problem for the one-dimensional wave equation, damped by an on-off feedback a(t)u_t. We obtain results that are radically different from those known in the case of the oscillator. We consider periodic functions a: typically a is equal to 1 on (0,T), equal to 0 on (T, qT) and is qT-periodic. We study the boundary case and next the locally distributed case, and we give optimal results of stability. In both cases, we prove that there are explicit exceptional values of T for which the energy of some solutions remains constant with time. If T is different from those exceptional values, the energy of all solutions decays exponentially to zero. This number of exceptional values is countable in the boundary case and finite in the distributed case. When the feedback is acting on the boundary, we also study the case of postive-negative feedbacks: a(t) = a₀ >0 on (0,T), and a(t) = -b₀ <0 on (T,qT), and we give the necessary and sufficient condition under which the energy (that is no more nonincreasing with time) goes to zero or goes to infinity. The proofs of these results are based on congruence properties and on a theorem of Weyl in the boundary case, and on new observability inequalities for the undamped wave equation, weakening the usual "optimal time condition" in the locally distributed case. These new inequalities provide also new exact controllability results.

Mathematics Subject Classification. 35L05, 35B35, 35B40, 11A07.

Key words: Damped wave equation, asymptotic behavior, on-off feedback, congruences, observability inequalities.

© EDP Sciences, SMAI 2002

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