Implicit Lyapunov control of multi-control Hamiltonian systems based on state distance

For closed quantum systems, if controlled systems are strongly regular and all other eigenstates are directly coupled to a target state, then such control systems can be asymptotically stabilized by means of the Lyapunov-based control. However, when the controlled systems are not strongly regular, or when there exists at least one eigenstate that is directly uncoupled to the target state, the situations will become complicated. This paper proposed a method based on an implicit Lyapunov function to overcome these two degenerate cases. Also, the proposed method is suitable for multi-control Hamiltonian systems. The convergence of closed-loop systems is analyzed by the LaSalle invariance principle. Finally, numerical simulation experiments on a 4-level system are done. The experiment results show the effectiveness of the implicit Lyapunov control method for degenerate cases and multi-control Hamiltonians.