  Quantum Lyapunov control was developed in order to transform a quantum system
from arbitrary initial states to a target state. The idea is to find control
fields that steer the Lyapunov function to zero as $t\rightarrow \infty$,
meanwhile the quantum system is driven to the target state. In order to shorten
the time required to reach the target state, we propose two designs to optimize
Lyapunov control in this paper. The first design makes the Lyapunov function
decrease as fast as possible with a constraint on the total power of control
fields, and the second design has the same purpose but with a constraint on
each control field. Examples of a three-level system demonstrate that the
evolution time for Lyapunov control can be significantly shortened, especially
when high control fidelity is required. Besides, this optimal Lyapunov-based
quantum control is robust against uncertainties in the free Hamiltonian and
decoherence in the system compared to conventional Lyapunov control.
