Karine Beauchard

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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(More)
We consider a non relativistic charged particle in a 1-D box of potential. This quantum system is subject to a control, which is a uniform electric field. It is represented by a complex probability amplitude solution of a Schrödinger equation. We prove the local controllability of this nonlinear system around the ground state. Our proof uses the return(More)
We consider a linear Schrödinger equation, on a bounded interval, with bilinear control, that represents a quantum particle in an electric eld (the control). We prove the exact controllability of this system, in any positive time, locally around the ground state. Similar results were proved for particular models [14, 15, 17], in non optimal spaces, in long(More)
We study the controllability of the Bloch equation, for an ensemble of non interacting half-spins, in a static magnetic field, with dispersion in the Larmor frequency. This system may be seen as a prototype for infinite dimensional bilinear systems with continuous spectrum, whose controllability is not well understood. We provide several mathematical(More)
We consider a quantum particle in an infinite square potential well of Rn, n = 2, 3, subjected to a control which is a uniform (in space) electric field. Under the dipolar moment approximation, the wave function solves a PDE of Schrödinger type. We study the spectral controllability in finite time of the linearized system around the ground state. We(More)
We consider a non relativistic charged particle in a 1D 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 Schrödinger equation on a 1D bounded domain, with Dirichlet boundary conditions. We prove the(More)
We study the null controllability of Kolmogorov-type equations ∂t f + v ∂x f − ∂2 v f = u(t, x, v)1ω(x, v) in a rectangle , under an additive control supported in an open subset ω of . For γ = 1, with periodic-type boundary conditions, we prove that null controllability holds in any positive time, with any control support ω. This improves the previous(More)
We consider a one dimensional Bose-Einstein condensate in a in nite square-well (box) potential. This is a nonlinear control system in which the state is the wave function of the Bose Einstein condensate and the control is the length of the box. We prove that local exact controllability around the ground state (associated with a xed length of the box) holds(More)
We study the null controllability of the parabolic equation associated with the Grushin-type operator A = ∂ x+|x|∂ y , (γ > 0), in the rectangle Ω = (−1, 1) × (0, 1), under an additive control supported in an open subset ω of Ω. We prove that the equation is null controllable in any positive time for γ < 1 and that there is no time for which it is null(More)
Distillation column monitoring requires shortcut nonlinear dynamic models. On the basis of a classical wave-model and time-scale reduction techniques, we derive a one-dimensional partial differential equation describing the composition dynamics where convection and diffusion terms depend non-linearly on the internal compositions and the inputs. The Cauchy(More)