128 research outputs found
Chaotic dynamical systems associated with tilings of
In this chapter, we consider a class of discrete dynamical systems defined on
the homogeneous space associated with a regular tiling of , whose most
familiar example is provided by the 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
On the Benjamin-Bona-Mahony equation with a localized damping
We introduce several mechanisms to dissipate the energy in the
Benjamin-Bona-Mahony (BBM) equation. We consider either a distributed
(localized) feedback law, or a boundary feedback law. In each case, we prove
the global wellposedness of the system and the convergence towards a solution
of the BBM equation which is null on a band. If the Unique Continuation
Property holds for the BBM equation, this implies that the origin is
asymp-totically stable for the damped BBM equation
Unique continuation property and control for the Benjamin-Bona-Mahony equation on the torus
We consider the Benjamin-Bona-Mahony (BBM) equation on the one dimensional
torus T = R/(2{\pi}Z). We prove a Unique Continuation Property (UCP) for small
data in H^1(T) with nonnegative zero means. Next we extend the UCP to certain
BBM-like equations, including the equal width wave equation and the KdV-BBM
equation. Applications to the stabilization of the above equations are given.
In particular, we show that when an internal control acting on a moving
interval is applied in BBM equation, then a semiglobal exponential
stabilization can be derived in H^s(T) for any s \geq 1. Furthermore, we prove
that the BBM equation with a moving control is also locally exactly
controllable in H^s(T) for any s \geq 0 and globally exactly controllable in H
s (T) for any s \geq 1
Control and Stabilization of the Nonlinear Schroedinger Equation on Rectangles
This paper studies the local exact controllability and the local
stabilization of the semilinear Schr\"odinger equation posed on a product of
intervals (). 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. It is also proved that for the
linear Schr\"odinger equation with Dirichlet control, the exact controllability
holds in whenever the control region contains a neighborhood
of a vertex
On the reachable states for the boundary control of the heat equation
We are interested in the determination of the reachable states for the
boundary control of the one-dimensional heat equation. We consider either one
or two boundary controls. We show that reachable states associated with square
integrable controls can be extended to analytic functions onsome square of C,
and conversely, that analytic functions defined on a certain disk can be
reached by using boundary controlsthat are Gevrey functions of order 2. The
method of proof combines the flatness approach with some new Borel
interpolation theorem in some Gevrey class witha specified value of the loss in
the uniform estimates of the successive derivatives of the interpolating
function
Null controllability of one-dimensional parabolic equations by the flatness approach
We consider linear one-dimensional parabolic equations with space dependent
coefficients that are only measurable and that may be degenerate or
singular.Considering generalized Robin-Neumann boundary conditions at both
extremities, we prove the null controllability with one boundary control by
following the flatness approach, which providesexplicitly the control and the
associated trajectory as series. Both the control and the trajectory have a
Gevrey regularity in time related to the class of the coefficient in
front of .The approach applies in particular to the (possibly degenerate
or singular) heat equation with a(x)\textgreater{}0
for a.e. and , or to the heat equation with
inverse square potential with
Null controllability of the 1D heat equation using flatness
We derive in a straightforward way the null controllability of a 1-D heat
equation with boundary control. We use the so-called {\em flatness approach},
which consists in parameterizing the solution and the control by the
derivatives of a "flat output". This provides an explicit control law achieving
the exact steering to zero. We also give accurate error estimates when the
various series involved are replaced by their partial sums, which is paramount
for an actual numerical scheme. Numerical experiments demonstrate the relevance
of the approach
Controllability of the 1D Schrodinger equation by the flatness approach
We derive in a straightforward way the exact controllability of the 1-D
Schrodinger equation with a Dirichlet boundary control. We use the so-called
flatness approach, which consists in parameterizing the solution and the
control by the derivatives of a "flat output". This provides an explicit
control input achieving the exact controllability in the energy space. As an
application, we derive an explicit pair of control inputs achieving the exact
steering to zero for a simply-supported beam
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