Stabilization of jump-diffusion stochastic differential equations by hysteresis switching

We address the stabilization of both classical and quantum systems modeled by jump-diffusion stochastic differential equations using a novel hysteresis switching strategy. Unlike traditional methods that depend on global Lyapunov functions or require each subsystem to stabilize the target state individually, our approach employs local Lyapunov-like conditions and state-dependent switching to achieve global asymptotic or exponential stability with finitely many switches almost surely. We rigorously establish the well-posedness of the resulting switched systems and derive sufficient conditions for stability. The framework is further extended to quantum feedback control systems governed by stochastic master equations with both diffusive and jump dynamics. Notably, our method relaxes restrictive invariance assumptions often necessary in prior work, enhancing practical applicability in experimental quantum settings. Additionally, the proposed strategy offers promising avenues for robust control under model uncertainties and perturbations, paving the way for future developments in both classical and quantum control.