Generic controllability properties for the bilinear Schrödinger equation

Paolo Mason (1) , Mario Sigalotti (2, 3)

1 L2S - Laboratoire des signaux et systèmes
2 IECN - Institut Élie Cartan de Nancy
3 CORIDA - Robust control of infinite dimensional systems and applications

In [15] we proposed a set of sufficient conditions for the approximate controllability of a discrete-spectrum bilinear Schrödinger equation. These conditions are expressed in terms of the controlled potential and of the eigenpairs of the uncontrolled Schrödinger operator. The aim of this paper is to show that these conditions are generic with respect to the uncontrolled and the controlled potential, denoted respectively by $V$ and $W$. More precisely, we prove that the Schrödinger equation is approximately controllable generically with respect to $W$ when $V$ is fixed and also generically with respect to $V$ when $W$ is fixed and non-constant. The results are obtained by analytic perturbation arguments and through the study of asymptotic properties of eigenfunctions.

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Paolo Mason, Mario Sigalotti. Generic controllability properties for the bilinear Schrödinger equation. 2009. ⟨hal-00385279v1⟩