Implicit Lyapunov control for Schrödinger equations with dipole and polarizability term

We analyze in this paper finite dimensional closed quantum systems in interaction with a laser field. The characteristic of the problem is that the interaction laser-system is modeled by a first order term (dipole coupling) and a second order term (polarisability coupling), that appear in the Hamiltonian of the Schrödinger equation. In order to determine the control, an implicit Lyapunov trajectory tracking procedure is applied when there is no direct coupling between the target state and the eigenvectors of the internal hamiltonian. The method is applied for the difficult case of degenerate systems too. The controlled Lyapunov function is defined by an implicit equation and its existence is shown by a fix point theorem. The convergence is analysed using the LaSalle invariance principle. The performance of the feedbacks is illustrated by numerical simulations.