Implicit Lyapunov control of finite dimensional Schrödinger equations

An implicit Lyapunov-based approach is proposed for generating trajectories of a finite dimensional controlled quantum system. The main difficulty comes from the fact that we consider the degenerate case where the linearized control system around the target state is not controllable. The controlled Lyapunov function is defined by an implicit equation and its existence is shown by a fix point theorem. The convergence analysis is done using LaSalle invariance principle. Closed-loop simulations illustrate the performance of such feedback laws for the open-loop control of a test case considered by chemists.