3,925 research outputs found
On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold
In this article, we study the internal stabilization and control of the
critical nonlinear Klein-Gordon equation on 3-D compact manifolds. Under a
geometric assumption slightly stronger than the classical geometric control
condition, we prove exponential decay for some solutions bounded in the energy
space but small in a lower norm. The proof combines profile decomposition and
microlocal arguments. This profile decomposition, analogous to the one of
Bahouri-G\'erard on , is performed by taking care of possible geometric
effects. It uses some results of S. Ibrahim on the behavior of concentrating
waves on manifolds
Decay of semilinear damped wave equations:cases without geometric control condition
We consider the semilinear damped wave equation . In
this article, we obtain the first results concerning the stabilization of this
semilinear equation in cases where does not satisfy the geometric
control condition. When some of the geodesic rays are trapped, the
stabilization of the linear semigroup is semi-uniform in the sense that
for some function with when
. We provide general tools to deal with the semilinear
stabilization problem in the case where has a sufficiently fast decay
Local controllability of 1D linear and nonlinear Schr\"odinger equations with bilinear control
We consider a linear Schr\"odinger equation, on a bounded interval, with
bilinear control, that represents a quantum particle in an electric field (the
control). We prove the controllability of this system, in any positive time,
locally around the ground state. Similar results were proved for particular
models (by the first author and with J.M. Coron), in non optimal spaces, in
long time and the proof relied on the Nash-Moser implicit function theorem in
order to deal with an a priori loss of regularity. In this article, the model
is more general, the spaces are optimal, there is no restriction on the time
and the proof relies on the classical inverse mapping theorem. A hidden
regularizing effect is emphasized, showing there is actually no loss of
regularity. Then, the same strategy is applied to nonlinear Schr\"odinger
equations and nonlinear wave equations, showing that the method works for a
wide range of bilinear control systems
Coupling symmetries with Poisson structures
We study local normal forms for completely integrable systems on Poisson
manifolds in the presence of additional symmetries. The symmetries that we
consider are encoded in actions of compact Lie groups. The existence of
Weinstein's splitting theorem for the integrable system is also studied giving
some examples in which such a splitting does not exist, i.e. when the
integrable system is not, locally, a product of an integrable system on the
symplectic leaf and an integrable system on a transversal. The problem of
splitting for integrable systems with additional symmetries is also considered.Comment: 14 page
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