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On considère l’équation des ondes sur un rectangle avec un feedback de type Dirichlet. On se place dans le cas où la condition de contrôle géométrique n’est pas satisfaite (BLR Condition), ce qui implique qu’on n’a pas stabilité exponentielle dans l’espace d’énérgie. On prouve qu’on peut trouver un sous espace de l’espace d’énergie tel qu’on a stabilité exponentielle. De plus, on montre un résultat de décroissance polynomiale pour toute donnée initiale régulière.
We consider here the Interior Penalty Discontinuous Galerkin (IPDG) discretization of the wave equation. We show how to derive the optimal penalization parameter involved in this method in the case of regular meshes. Moreover, we provide necessary stability conditions of the global scheme when IPDG is coupled with the classical Leap–Frog scheme for the time discretization. Numerical experiments illustrate the fact that these conditions are also sufficient.
We consider abstract second order evolution equations with unbounded
feedback with delay. Existence results are obtained under some
realistic assumptions. Sufficient and explicit conditions are derived that guarantee the exponential or polynomial stability. Some new examples that enter into our abstract framework are presented.
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 . We obtain results that are radically different from those known in the case of the oscillator. We consider periodic functions : typically is equal to on , equal to on and is -periodic. We study the boundary case and next the locally distributed case, and we give optimal results of stability. In both cases,...
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 .
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....
The energy in a square membrane Ω subject to constant viscous damping
on a subset decays exponentially in time
as soon as ω
satisfies a geometrical condition known as the “Bardos-Lebeau-Rauch” condition. The rate
of this decay satisfies
(see Lebeau [Math.
Phys. Stud.19 (1996) 73–109]). Here denotes the spectral abscissa of the
damped wave equation operator and is a number called
the geometrical quantity of ω and defined as follows.
A ray in Ω is the trajectory generated by the
free motion...
We introduce and study the linear symmetric systems associated with the modified Cherednik operators. We prove the well-posedness results for the Cauchy problem for these systems. Eventually we describe the finite propagation speed property of it.
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