  The finite-time control problem of quantum systems is investigated in this
paper. We first define finite-time stability and present a finite-time Lyapunov
stability criterion for finite-dimensional quantum systems in coherence vector
representation. Then, for two-level quantum systems, we design a continuous
non-smooth control law with a state-dependent fractional power and prove the
uniqueness of solutions of the system dynamics with the controller via the
concept of transversality. By combining the finite-time Lyapunov stability
criterion with the homogeneity theory, the finite-time convergence of the
system to an eigenstate of its internal Hamiltonian is proved. Numerical
results on a spin-1/2 system demonstrate the effectiveness of the proposed
finite-time stabilization control scheme.
