17 research outputs found
Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation
We study a damped semi-linear wave equation in a bounded domain with smooth
boundary. It is proved that any sufficiently smooth solution can be stabilised
locally by a finite-dimensional feedback control supported by a given open
subset satisfying a geometric condition. The proof is based on an investigation
of the linearised equation, for which we construct a stabilising control
satisfying the required properties. We next prove that the same control
stabilises locally the non-linear problem.Comment: 29 page
Numerical controllability of the wave equation through primal methods and Carleman estimates
This paper deals with the numerical computation of boundary null controls for
the 1D wave equation with a potential. The goal is to compute an approximation
of controls that drive the solution from a prescribed initial state to zero at
a large enough controllability time. We do not use in this work duality
arguments but explore instead a direct approach in the framework of global
Carleman estimates. More precisely, we consider the control that minimizes over
the class of admissible null controls a functional involving weighted integrals
of the state and of the control. The optimality conditions show that both the
optimal control and the associated state are expressed in terms of a new
variable, the solution of a fourth-order elliptic problem defined in the
space-time domain. We first prove that, for some specific weights determined by
the global Carleman inequalities for the wave equation, this problem is
well-posed. Then, in the framework of the finite element method, we introduce a
family of finite-dimensional approximate control problems and we prove a strong
convergence result. Numerical experiments confirm the analysis. We complete our
study with several comments
Global controllability and stabilization for the nonlinear Schrodinger equation on an interval
We prove global internal controllability in large time for the nonlinear
Schrodinger equation on a bounded interval with periodic, Dirichlet or Neumann
conditions. Our strategy combines stabilization and local controllability near
0. We use Bourgain spaces to prove this result on L2. We also get a regularity
result about the control if the data are assumed smoother
Weak observability estimates for 1-D wave equations with rough coefficients
In this paper we prove observability estimates for 1-dimensional wave
equations with non-Lipschitz coefficients. For coefficients in the Zygmund
class we prove a "classical" observability estimate, which extends the
well-known observability results in the energy space for regularity. When
the coefficients are instead log-Lipschitz or log-Zygmund, we prove
observability estimates "with loss of derivatives": in order to estimate the
total energy of the solutions, we need measurements on some higher order
Sobolev norms at the boundary. This last result represents the intermediate
step between the Lipschitz (or Zygmund) case, when observability estimates hold
in the energy space, and the H\"older one, when they fail at any finite order
(as proved in \cite{Castro-Z}) due to an infinite loss of derivatives. We also
establish a sharp relation between the modulus of continuity of the
coefficients and the loss of derivatives in the observability estimates. In
particular, we will show that under any condition which is weaker than the
log-Lipschitz one (not only H\"older, for instance), observability estimates
fail in general, while in the intermediate instance between the Lipschitz and
the log-Lipschitz ones they can hold only admitting a loss of a finite number
of derivatives. This classification has an exact counterpart when considering
also the second variation of the coefficients.Comment: submitte
Internal control of the Schrödinger equation
In this paper, we intend to present some already known results about the internal controllability of the linear and nonlinear Schrödinger equation. After presenting the basic properties of the equation, we give a self contained proof of the controllability in dimension using some propagation results. We then discuss how to obtain some similar results on a compact manifold where the zone of control satisfies the Geometric Control Condition. We also discuss some known results and open questions when this condition is not satisfied. Then, we present the links between the controllability and some resolvent estimates. Finally, we discuss the new difficulties when we consider the Nonlinear Schrödinger equation
Controllability under positivity constraints of multi-d wave equations
We consider both the internal and boundary controllability problems for wave
equations under non-negativity constraints on the controls. First, we prove the
steady state controllability property with nonnegative controls for a general
class of wave equations with time-independent coefficients. According to it,
the system can be driven from a steady state generated by a strictly positive
control to another, by means of nonnegative controls, when the time of control
is long enough. Secondly, under the added assumption of conservation and
coercivity of the energy, controllability is proved between states lying on two
distinct trajectories. Our methods are described and developed in an abstract
setting, to be applicable to a wide variety of control systems
Exact controllability for quasi-linear perturbations of KdV
We prove that the KdV equation on the circle remains exactly controllable in
arbitrary time with localized control, for sufficiently small data, also in
presence of quasi-linear perturbations, namely nonlinearities containing up to
three space derivatives, having a Hamiltonian structure at the highest orders.
We use a procedure of reduction to constant coefficients up to order zero,
classical Ingham inequality and HUM method to prove the controllability of the
linearized operator. Then we prove and apply a modified version of the
Nash-Moser implicit function theorems by H\"ormander.Comment: 39 page
Controllability of quasi-linear Hamiltonian NLS equations
We prove internal controllability in arbitrary time, for small data, for
quasi-linear Hamiltonian NLS equations on the circle. We use a procedure of
reduction to constant coefficients up to order zero and HUM method to prove the
controllability of the linearized problem. Then we apply a
Nash-Moser-H\"ormander implicit function theorem as a black box