Lionel Rosier
Chaotic dynamical systems associated with tilings of $\R^N$.
Mon, 02/08/2010 - 16:01 — SomNambulAuthors: Lionel Rosier
Subjects: Analysis of PDEs
AbstractIn this chapter, we consider a class of discrete dynamical systems defined on
the homogeneous space associated with a regular tiling of $\R^N$, whose most
familiar example is provided by the $N-$dimensional torus $\T ^N$. It is proved
that any dynamical system in this class is chaotic in the sense of Devaney, and
that it admits at least one positive Lyapunov exponent. Next, a
chaos-synchronization mechanism is introduced and used for masking information
in a communication setup.Control and Stabilization of the Nonlinear Schroedinger Equation on Rectangles.
Mon, 02/08/2010 - 16:01 — SomNambulAuthors: Lionel Rosier, Bing-Yu Zhang
Subjects: Analysis of PDEs
AbstractThis paper studies the local exact controllability and the local
stabilization of the semilinear Schr\"odinger equation posed on a product of
$n$ intervals ($n\ge 1$). Both internal and boundary controls are considered,
and the results are given with periodic (resp. Dirichlet or Neumann) boundary
conditions. In the case of internal control, we obtain local controllability
results which are sharp as far as the localization of the control region and
the smoothness of the state space are concerned.Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line.
Mon, 02/08/2010 - 16:01 — SomNambulAuthors: Lionel Rosier, Ademir Pazoto
Subjects: Analysis of PDEs
AbstractStudied here is the large-time behavior of solutions of the Korteweg-de Vries
equation posed on the right half-line under the effect of a localized damping.
Assuming as in \cite{linares-pazoto} that the damping is active on a set
$(a_0,+\infty)$ with $a_0>0$, we establish the exponential decay of the
solutions in the weighted spaces $L^2((x+1)^mdx)$ for $m\in \N ^*$ and
$L^2(e^{2bx}dx)$ for $b>0$ by a Lyapunov approach. The decay of the spatial
derivatives of the solution is also derived.Null controllability of a parabolic system with a cubic coupling term.
Tue, 02/02/2010 - 16:01 — SomNambulAuthors: Jean-Michel Coron, Sergio Guerrero, Lionel Rosier
Subjects: Optimization and Control
AbstractWe consider a system of two parabolic equations with a forcing term present
in one equation and a cubic coupling term in the other one. We prove that the
system is locally null controllable.
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