Regular propagators of bilinear quantum systems

Nabile Boussaid (1) , Marco Caponigro (2) , Thomas Chambrion (3, 4)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 M2N - Modélisation mathématique et numérique
3 EDP - Equations aux dérivées partielles
4 CORIDA - Robust control of infinite dimensional systems and applications

This present analysis deals with the regularity of the solutions of bilinear control systems of the type $x'=(A+u(t)B)x$ where the state $x$ belongs to some complex infinite dimensional Hilbert space, the (possibly unbounded) linear operators $A$ and $B$ are skew-adjoint and the control $u$ is a real valued function. Such systems arise for instance in quantum control with the bilinear Schr\"{o}dinger equation. Under some hypotheses on the commutator of the operators $A$ and $B$, it is possible to define the solution of the system for controls in the set of Radon measures and to obtain precise a priori energy estimates of the solutions. This leads to a natural extension of a celebrated non-controllability result of Ball, Marsden and Slemrod. Moreover, upper bounds on the error made by replacing the original systems by its finite dimensional Galerkin approximation are obtain from the energy estimates. This allows to use geometric (finite dimensional) techniques to obtain approximate controllability results for the original infinite dimensional problem.

Mots clés: bilinear control; Shrodinger equation; bounded variation

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

Nabile Boussaid, Marco Caponigro, Thomas Chambrion. Regular propagators of bilinear quantum systems. 2014. ⟨hal-01016299v1⟩