Stabilization of Hybrid Stochastic McKean–Vlasov Differential Equations by Feedback Control Based on Discrete-Time State Observation

This paper addresses the stabilization problem of hybrid stochastic McKean–Vlasov differential equations via a discrete-time state observation feedback control strategy. Utilizing the coupling method and particle system approximation, Itô’s formula for Markovian switching stochastic McKean–Vlasov differential equations is established. Based on the derived formula, we construct two novel Lyapunov functionals that incorporate state processes, probability distributions, and Markovian switching signals. Using the proposed Lyapunov functionals, we further analyze three stability properties of the closed-loop system, including H∞ stability, asymptotic stability, and mean-square exponential stability. Due to the time-varying characteristics of system distributions, numerical simulation lacks fixed reference benchmarks and faces considerable difficulties. To overcome this challenge, this paper introduces a particle system approximation scheme. We further prove the exponential stability equivalence between the controlled McKean–Vlasov system and its corresponding particle system. This equivalence relation provides an effective new approach for the stability analysis of such controlled hybrid stochastic systems. Finally, an illustrative example is given to verify our theory results.