Non-smooth Lyapunov function-based global stabilization for quantum filters

This paper addresses the global stabilization problem for a class of quantum filters via non-smooth Lyapunov functions. Due to the intrinsic symmetric topology of filter state space, the smooth controls synthesized via the classical Lyapunov stochastic stability theory fail to obtain the global stabilizability because of the existence of the so-called antipodal eigenstates. As such, for the first time, we introduce a non-smooth 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 continuous control in saturation form are constructed for the quantum filters, with consideration of the sliding motion of filter state. The non-smooth property enables these controls to deal with the symmetric topology of filter state space and to solve the problem of global stabilization for quantum filters. The eigenstate-transferring is obtained as a special result, distinguishing these non-smooth Lyapunov-based controls from classical control approaches for quantum filters. The effectiveness of the non-smooth Lyapunov-based controls is illustrated through the control design for the Spin- 1 / 2 systems. Simulation results are presented and discussed to show the effectiveness of the controls.