  The condition of a quantum Lyapunov-based control which can be well used in a
closed quantum system is that the method can make the system convergent but not
just stable. In the convergence study of the quantum Lyapunov control, two
situations are classified: non-degenerate cases and degenerate cases. In this
paper, for these two situations, respectively, the target state is divided into
four categories: eigenstate, the mixed state which commutes with the internal
Hamiltonian, the superposition state, and the mixed state which does not
commute with the internal Hamiltonian state. For these four categories, the
quantum Lyapunov control methods for the closed quantum systems are summarized
and analyzed. Especially, the convergence of the control system to the
different target states is reviewed, and how to make the convergence conditions
be satisfied is summarized and analyzed.
