Mathematics of Control, Signals, and Systems

, Volume 25, Issue 3, pp 407–432 | Cite as

Explicit approximate controllability of the Schrödinger equation with a polarizability term

Original Article

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 nonlinearly 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 with explicit controls. Numerical simulations are presented to illustrate those theoretical results.

Keywords

Approximate controllability Schrödinger equation Polarizability Oscillating controls Averaging Feedback stabilization LaSalle invariance principle 

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Copyright information

© Springer-Verlag London 2012

Authors and Affiliations

  1. 1.CMLA UMR 8536, ENS CachanCachanFrance
  2. 2.CMLS UMR 7640, Ecole PolytechniquePalaiseauFrance

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