  This paper presents a detailed Lyapunov-based theory to control and stabilize
continuously-measured quantum systems, which are driven by Stochastic
Schrodinger Equation (SSE). Initially, equivalent classes of states of a
quantum system are defined and their properties are presented. With the help of
equivalence classes of states, we are able to consider global phase invariance
of quantum states in our mathematical analysis. As the second mathematical
modelling tool, the conventional Ito formula is further extended to
non-differentiable complex functions. Based on this extended Ito formula, a
detailed stochastic stability theory is developed to stabilize the SSE. Main
results of this proposed theory are sufficient conditions for stochastic
stability and asymptotic stochastic stability of the SSE. Based on the main
results, a solid mathematical framework is provided for controlling and
analyzing quantum system under continuous measurement, which is the first step
towards implementing weak continuous feedback control for quantum computing
purposes.
