Explicit approximate controllability of the Schrödinger equation with a polarizability term

Abstract

We consider a controlled Schrödinger equation with a dipolar and a polarizability term, used when the dipolar approximation is not valid. The control is the amplitude of the external electric field, it acts non linearly on the state. We extend in this infinite dimensional framework previous techniques used by Coron, Grigoriu, Lefter and Turinici for stabilization in finite dimension. We consider a highly oscillating control and prove the semi-global weak $H^2$ stabilization of the averaged system using a Lyapunov function introduced by Nersesyan. Then it is proved that the solutions of the Schrödinger equation and of the averaged equation stay close on every finite time horizon provided that the control is oscillating enough. Combining these two results, we get approximate controllability to the ground state for the polarizability system.

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Cited by in corpus (4)

• Global exact controllability in infinite time of Schrödinger equation: multidimensional case
• Energy Estimates for Low Regularity Bilinear Schrödinger Equations
• Approximate controllability of the Schrödinger Equation with a polarizability term in higher Sobolev norms
• Global exact controllability of a 1D Schrödinger equations with a polarizability term