Appl Math Optim 56:105–130 (2007)
2007 Springer Science+Business Media, Inc.
Pointwise Stabilization of a Hybrid System
and Optimal Location of Actuator
and Abdelkader Sa¨ıdi
Department of Mathematics, Faculty of Sciences of Monastir,
5019 Monastir, Tunisia
Institut de Recherche Math´ematique Avanc´ee, Universit´e Louis Pasteur,
7 rue Ren´e Descartes, F-67084 Strasbourg, France
Abstract. We consider a pointwise stabilization problem for a model arising in
the control of noise. We prove that we have exponential stability for the low frequen-
cies but not for the high frequencies. Thus, we give an explicit polynomial decay
estimation at high frequencies that is valid for regular initial data while clarifying
that the behavior of the constant which intervenes in this estimation there, functions
as the frequency of cut. We propose a numerical approximation of the model and
study numerically the best location of the actuator at low frequencies.
Key Words. Pointwise stabilization, Observability inequality, Coupled partial
differential equations, Optimal location.
AMS Classiﬁcation. 35A05, 35B40, 35B37, 93B07.
Let = (0, 1) × (0, 1) ⊂ R
, ∂ =
(0, y), y ∈ (0, 1)
. We consider the wave equation coupled to the string equation with
The research by Ka¨ıs Ammari was supported by the RIP program of the Oberwolfach Institut and by
MRST under Grant 02/UR/15-01, and that by Abdelkader Sa¨ıdi was supported by the RIP program of the