  We present detailed analysis of the convergence properties and effectiveness
of Lyapunov control design for bilinear Hamiltonian quantum systems based on
the application of LaSalle's invariance principle and stability analysis from
dynamical systems and control theory. For a certain class of Hamiltonians,
strong convergence results can be obtained for both pure and mixed state
systems. The control Hamiltonians for realistic physical systems, however,
generally do not fall in this class. It is shown that the effectiveness of
Lyapunov control design in this case is significantly diminished.

Quantum Physics