Explicit approximate controllability of the Schrödinger equation with a polarizability term

Details Explicit approximate controllability of the Schrödinger equation with a polarizability term Explicit approximate controllability of the Schrödinger equation with a polarizability term

2011-10-13

arxiv.org/abs/1110.2860v1

Mathematics

Optimization and Control

Scientific paper

We consider a controlled Schr\"odinger 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 Schr\"odinger 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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