Semi-global weak stabilization of bilinear Schrödinger equations

2010/05/25 by Karine Beauchard, Beauchard, Karine, Vahagn Nersesyan +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1005.4558

arxiv created 2010/08/21 · arxiv updated 2010/08/24

Abstract

We consider a linear Schrödinger equation, on a bounded domain, with bilinear control, representing a quantum particle in an electric field (the control). Recently, Nersesyan proposed explicit feedback laws and proved the existence of a sequence of times (t_n)_{n ∈ ℕ} for which the values of the solution of the closed loop system converge weakly in H² to the ground state. Here, we prove the convergence of the whole solution, as t → + ∞. The proof relies on control Lyapunov functions and an adaptation of the LaSalle invariance principle to PDEs.

Related

• Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control
• Feedback stabilization for a bilinear control system under weak observability inequalities
• Simultaneous global exact controllability of an arbitrary number of 1D bilinear Schrödinger equations
• Minimal time for the bilinear control of Schrödinger equations
• Small-Time Local Controllability of the multi-input bilinear Schrödinger equation thanks to a quadratic term