Non-smooth lyapunov function-based global stabilization for 2-dimensional quantum filters

Abstract

This paper addresses the global stabilization problem for 2-dimensional quantum filters via non-smooth Lyapunov functions. Due to the symmetric topology of filter state space, the smooth controls synthesized via smooth Lyapunov stochastic stability theory fail to obtain the global stabilizability. As such, for the first time, we introduce a nonsmooth Lyapunov-like theory for generic stochastic nonlinear systems, which includes a continuous Lyapunov-like theorem and a discontinuous Lyapunov-like theorem for stability in probability. Applying the non-smooth Lyapunov-like theory, switching control and saturation-form control are constructed for 2-dimensional quantum filters with the consideration of sliding motion. The non-smooth property enables these controls to deal with the symmetric topology of filter state space and to globally asymptotically render the filter state to the final desired state almost surely. The effectiveness of the proposed controls is illustrated through the control design for the Spin-1/2 system. Simulation results are presented and discussed. ©2010 IEEE.