  A Lyapunov-based method is presented for stabilizing and controlling of
closed quantum systems. The proposed method is constructed upon a novel quantum
Lyapunov function of the system state trajectory tracking error. A
positive-definite operator in the Lyapunov function provides additional degrees
of freedom for the designer. The stabilization process is analyzed regarding
two distinct cases for this operator in terms of its vanishing or non-vanishing
commutation with the Hamiltonian operator of the undriven quantum system. To
cope with the global phase invariance of quantum states as a result of the
quantum projective measurement postulate, equivalence classes of quantum states
are defined and used in the proposed Lyapunov-based analysis and design.
Results show significant improvement in both the set of stabilizable quantum
systems and their invariant sets of state trajectories generated by designed
control signals. The proposed method can potentially be applied for
high-fidelity quantum control purposes in quantum computing frameworks.
