Karine Beauchard

Implicit Lyapunov control of finite dimensional Schrödinger equations

An implicit Lyapunov-based approach is proposed for generating trajectories of a finite dimensional controlled quantum system. The main difficulty comes from the fact that we consider the degenerate case where the linearized control system around the target state is not controllable. The controlled Lyapunov function is defined by an implicit equation and its existence is shown by a fix point theorem. The convergence analysis is done using LaSalle invariance principle. Closed-loop simulations illustrate the performance of such feedback laws for the open-loop control of a test case considered by chemists.

Practical Stabilization of a Quantum Particle in a One-Dimensional Infinite Square Potential Well

We consider a nonrelativistic charged particle in a one-dimensional infinite square potential well. This quantum system is subjected to a control, which is a uniform (in space) time-depending electric field. It is represented by a complex probability amplitude solution of a Schrodinger equation on a one-dimensional bounded domain, with Dirichlet boundary conditions. We prove the almost global practical stabilization of the eigenstates by explicit feedback laws.

Null controllability of Grushin-type operators in dimension two

We study the null controllability of the parabolic equation associated with the Grushintype operator

DEGENERATE PARABOLIC OPERATORS OF KOLMOGOROV TYPE WITH A GEOMETRIC CONTROL CONDITION

A = \partial_x^2 + |x|^{2\gamma}\partial_y^2 (\gamma > 0)

Stabilization for an ensemble of half-spin systems

in the rectangle

Local Controllability of a One-Dimensional Beam Equation

\Omega = (-1, 1) \times (0, 1)

Null controllability of Kolmogorov-type equations

, under an additive control supported in an open subset

Stability and asymptotic observers of binary distillation processes described by nonlinear convection/diffusion models

\omega

Inverse source problem and null controllability for multidimensional parabolic operators of Grushin type

of

Minimal time for the bilinear control of Schrödinger equations

\Omega