Partial Stability of Linear Hybrid Discrete–Continuous Itô Systems with Aftereffect

This paper offers several new sufficient conditions of the partial moment stability of linear hybrid stochastic systems with delay. Despite its potential applications in economics, biology and physics, this problem seems to have not been addressed before. A number of general theorems on the partial moment stability of stochastic hybrid systems are proven herein by applying a specially designed regularization method, based on the connections between Lyapunov stability and input-to-state stability, which are well known in control theory. Based on the results obtained for stochastic hybrid systems, some new conditions of the partial stability of deterministic hybrid systems are derived as well. All stability conditions are conveniently formulated in terms of the coefficients of the systems. A numerical example illustrates the feasibility of the suggested framework.