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- Karine Beauchard, Jean-Michel Coron, Mazyar Mirrahimi, Pierre Rouchon
- Systems & Control Letters
- 2007

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 andâ€¦ (More)

- Karine Beauchard
- 2013

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 SchrÃ¶dinger equation. We prove the local controllability of this nonlinear system around the ground state. Our proof uses the returnâ€¦ (More)

We consider a linear SchrÃ¶dinger 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 longâ€¦ (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 mathematicalâ€¦ (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 SchrÃ¶dinger type. We study the spectral controllability in finite time of the linearized system around the ground state. Weâ€¦ (More)

- Karine Beauchard
- 2007

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 SchrÃ¶dinger equation on a 1D bounded domain, with Dirichlet boundary conditions. We prove theâ€¦ (More)

- Karine Beauchard
- MCSS
- 2014

We study the null controllability of Kolmogorov-type equations âˆ‚t f + v âˆ‚x f âˆ’ âˆ‚2 v f = u(t, x, v)1Ï‰(x, v) in a rectangle , under an additive control supported in an open subset Ï‰ of . For Î³ = 1, with periodic-type boundary conditions, we prove that null controllability holds in any positive time, with any control support Ï‰. This improves the previousâ€¦ (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) holdsâ€¦ (More)

We study the null controllability of the parabolic equation associated with the Grushin-type operator A = âˆ‚ x+|x|âˆ‚ y , (Î³ > 0), in the rectangle Î© = (âˆ’1, 1) Ã— (0, 1), under an additive control supported in an open subset Ï‰ of Î©. We prove that the equation is null controllable in any positive time for Î³ < 1 and that there is no time for which it is nullâ€¦ (More)

- Stephane Dudret, Karine Beauchard, Fouad Ammouri, Pierre Rouchon
- 2012 American Control Conference (ACC)
- 2012

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 Cauchyâ€¦ (More)