Stabilization of stochastic McKean-Vlasov equations with feedback control based on discrete-time state observation

TL;DR

This paper investigates the stabilization of stochastic McKean-Vlasov equations using feedback control based on discrete-time state observations, establishing stability results via Lyapunov functions and particle system analysis.

Contribution

It introduces a novel approach to stabilize SMVEs through feedback control derived from discrete observations, linking stability of the particle system to the original equations.

Findings

Achieved $H_{ abla}$ stability, asymptotic, and exponential stability in mean square.

Proved exponential stability of the control system is equivalent to that of the particle system.

Validated the theoretical results with a practical example.

Abstract

In this paper, we study the stability of solutions of stochastic McKean-Vlasov equations (SMVEs) via feedback control based on discrete-time state observation. By using a specific Lyapunov function, the $H_{\infty}$ stability, asymptotic stability and exponential stability in mean square for the solution of the controlled systems are obtained. Since the distribution of solution is difficult to be observed, we study the corresponding particle system which can be observed for the feedback control. We prove that the exponential stability of control system is equivalent to the the exponential stability of the corresponding particle system. Finally, an example is provided to show the effectiveness of the theory.