Lyapunov control of bilinear Schrödinger equations

A Lyapunov-based approach for trajectory tracking of the Schrödinger equation is proposed. In the finite 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 closedloop system is not asymptotic. To overcome such lack of convergence, we propose, when the reference trajectory is a pure-state, a modification based on adiabatic invariance. Simulations illustrate the simplicity and also the interest for trajectory generation.