Implicit Lyapunov control of closed quantum systems

In this paper, we investigate the state convergence problem for closed quantum systems under degenerate cases. An implicit Lyapunov-based control strategy is proposed for the convergence analysis of finite dimensional bilinear Schrodinger equations. The degenerate cases that the systems do not satisfy the strong regular condition and the condition ¿¿_i|H₁|¿_j¿ ¿ 0, i, j ¿ k for eigenstates ¿_i, ¿_j of H₀ different from target state ¿_k, are considered. First the Lyapunov function is defined by the implicit function and the existence is guaranteed by a fixed point theorem. Then the convergence analysis is investigated by the LaSalle invariance principle. Finally, an example is provided to show the effectiveness of proposed results.