  The convergence of closed quantum systems in the degenerate cases to the
desired target state by using the quantum Lyapunov control based on the average
value of an imaginary mechanical quantity is studied. On the basis of the
existing methods which can only ensure the single-control Hamiltonian systems
converge toward a set, we design the control laws to make the multi-control
Hamiltonian systems converge to the desired target state. The convergence of
the control system is proved, and the convergence to the desired target state
is analyzed. How to make these conditions of convergence to the target state to
be satisfied is proved or analyzed. Finally, numerical simulations for a three
level system in the degenrate case transfering form an initial eigenstate to a
target superposition state are studied to verify the effectiveness of the
proposed control method.
