Indirect stabilization of locally coupled wave-type systems

¹ Present position : Délégation CNRS at MAPMO, UMR 6628, Current position : LMAM, Université Paul Verlaine-Metz and CNRS (UMR 7122) Metz Cedex 1, France
alabau@univ-metz.fr
² Université Pierre et Marie Curie Paris 6, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
³ CNRS, UMR 7598 LJLL, 75005 Paris, France
leautaud@ann.jussieu.fr
⁴ Laboratoire POEMS, INRIA Paris-Rocquencourt/ENSTA, CNRS UMR 2706, 78153 Le Chesnay, France

Received: 3 March 2010
Revised: 2 November 2010

Abstract

We study in an abstract setting the indirect stabilization of systems of two wave-like equations coupled by a localized zero order term. Only one of the two equations is directly damped. The main novelty in this paper is that the coupling operator is not assumed to be coercive in the underlying space. We show that the energy of smooth solutions of these systems decays polynomially at infinity, whereas it is known that exponential stability does not hold (see [F. Alabau, P. Cannarsa and V. Komornik, J. Evol. Equ. 2 (2002) 127–150]). We give applications of our result to locally or boundary damped wave or plate systems. In any space dimension, we prove polynomial stability under geometric conditions on both the coupling and the damping regions. In one space dimension, the result holds for arbitrary non-empty open damping and coupling regions, and in particular when these two regions have an empty intersection. Hence, indirect polynomial stability holds even though the feedback is active in a region in which the coupling vanishes and vice versa.