Lyapunov control of bilinear Schrodinger equations

Mazyar Mirrahimi (1) , Pierre Rouchon (2) , Gabriel Turinici (3)

1 SISYPHE - SIgnals and SYstems in PHysiology & Engineering
2 CAS - Centre Automatique et Systèmes
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

A Lyapunov-based approach for trajectory tracking of the SchrÄodinger equation is proposed. In the ¯nite dimensional case, convergence is precisely analyzed. Connection between the controllability of the linear tangent approximation around the reference trajectory and asymptotic tracking is studied. When the linear tangent approximation is controllable, such a feedback ensures almost global asymptotic convergence. When the linear tangent system is not controllable, the stability of the closed- loop system is not asymptotic. To overcome such lack of convergence, we propose, when the reference trajectory is a pure-state, a modi¯cation based on adiabatic invariance. Simulations illustrate the simplicity and also the interest for trajectory generation.

Domaines: Analyse numérique [math.NA]; Physique Quantique [quant-ph]

Mazyar Mirrahimi, Pierre Rouchon, Gabriel Turinici. Lyapunov control of bilinear Schrodinger equations. Automatica, 2005, 41, pp.1987-1994. ⟨10.1016/j.automatica.2005.05.018⟩. ⟨hal-00798328⟩