631 research outputs found
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
On the cubic NLS on 3D compact domains
We prove bilinear estimates for the Schr\"odinger equation on 3D domains,
with Dirichlet boundary conditions. On non-trapping domains, they match the
case, while on bounded domains they match the generic boundary
less manifold case. As an application, we obtain global well-posedness for the
defocusing cubic NLS for data in , , with
any bounded domain with smooth boundary.Comment: 15 pages, updated references and corrected typos. To appear in
Journal of the Institute of Mathematics of Jussie
Local controllability of 1D Schr\"odinger equations with bilinear control and minimal time
We consider a linear Schr\"odinger equation, on a bounded interval, with
bilinear control.
Beauchard and Laurent proved that, under an appropriate non degeneracy
assumption, this system is controllable, locally around the ground state, in
arbitrary time. Coron proved that a positive minimal time is required for this
controllability, on a particular degenerate example.
In this article, we propose a general context for the local controllability
to hold in large time, but not in small time. The existence of a positive
minimal time is closely related to the behaviour of the second order term, in
the power series expansion of the solution
The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent
The paper deals with the genericity of domain-dependent spectral properties
of the Laplacian-Dirichlet operator. In particular we prove that, generically,
the squares of the eigenfunctions form a free family. We also show that the
spectrum is generically non-resonant. The results are obtained by applying
global perturbations of the domains and exploiting analytic perturbation
properties. The work is motivated by two applications: an existence result for
the problem of maximizing the rate of exponential decay of a damped membrane
and an approximate controllability result for the bilinear Schr\"odinger
equation
Regular propagators of bilinear quantum systems
The present analysis deals with the regularity of solutions of bilinear
control systems of the type where the state belongs to some
complex infinite dimensional Hilbert space, the (possibly unbounded) linear
operators and are skew-adjoint and the control is a real valued
function. Such systems arise, for instance, in quantum control with the
bilinear Schr\"{o}dinger equation. For the sake of the regularity analysis, we
consider a more general framework where and are generators of
contraction semi-groups.Under some hypotheses on the commutator of the
operators and , it is possible to extend the definition of solution for
controls in the set of Radon measures to obtain precise a priori energy
estimates on the solutions, leading to a natural extension of the celebrated
noncontrollability result of Ball, Marsden, and Slemrod in 1982. Complementary
material to this analysis can be found in [hal-01537743v1
On a Stability Theorem of the Optimal Control Problem For Quasi Optic Equation
In this paper, the finite difference method is applied to the optimal control problem of system governed by stationary equation of Quasi-Optic . For this aim, the finite difference scheme is constituted for considered optimal control problem. Obtained an estimation for the solution of this difference scheme, the error of the difference scheme is evaluated. Finally, the convergence according to the functional of the finite difference approximations is proved
Suboptimal feedback control of PDEs by solving HJB equations on adaptive sparse grids
International audienceAn approach to solve finite time horizon suboptimal feedback control problems for partial differential equations is proposed by solving dynamic programming equations on adaptive sparse grids. The approach is illustrated for the wave equation and an extension to equations of Schrödinger type is indicated. A semi-discrete optimal control problem is introduced and the feedback control is derived from the corresponding value function.The value function can be characterized as the solution of an evolutionary Hamilton-Jacobi Bellman (HJB) equation which is defined over a state space whose dimension is equal to the dimension of the underlying semi-discrete system. Besides a low dimensional semi-discretization it is important to solve the HJB equation efficiently to address the curse of dimensionality.We propose to apply a semi-Lagrangian scheme using spatially adaptive sparse grids. Sparse grids allow the discretization of the value functions in (higher) space dimensions since the curse of dimensionality of full grid methods arises to a much smaller extent. For additional efficiency an adaptive grid refinement procedure is explored.We present several numerical examples studying the effect the parameters characterizing the sparse grid have on the accuracy of the value function and the optimal trajectory
On the stochastic Strichartz estimates and the stochastic nonlinear Schr\"odinger equation on a compact riemannian manifold
We prove the existence and the uniqueness of a solution to the stochastic
NSLE on a two-dimensional compact riemannian manifold. Thus we generalize a
recent work by Burq, G\'erard and Tzvetkov in the deterministic setting, and a
series of papers by de Bouard and Debussche, who have examined similar
questions in the case of the flat euclidean space with random perturbation. We
prove the existence and the uniqueness of a local maximal solution to
stochastic nonlinear Schr\"odinger equations with multiplicative noise on a
compact d-dimensional riemannian manifold. Under more regularity on the noise,
we prove that the solution is global when the nonlinearity is of defocusing or
of focusing type, d=2 and the initial data belongs to the finite energy space.
Our proof is based on improved stochastic Strichartz inequalities
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