Backstepping Stabilization of Stochastic Hyperbolic PDE-ODE Systems

This paper establishes a constructive boundary control framework for stabilizing coupled hyperbolic PDE-ODE systems subject to stochastic parameter variations. The system under consideration exhibits mixed-directional dynamics, with three positive and one negative characteristic speeds in its PDE subsystem. Building upon nominal backstepping designs for deterministic systems, we develop a parameter-adaptive control law that maintains stabilization capability when system parameters undergo random switching. Through a novel composite Lyapunov analysis, we derive explicit stability conditions relating the controller’s robustness to both the magnitude of parameter variations and their transition statistics. The theoretical results demonstrate that mean-square exponential stability can be preserved when the stochastic parameters remain within probability-dependent neighborhoods of their nominal values. A distinctive feature of our approach lies in the unified treatment of PDE-PDE and PDE-ODE coupling terms under stochastic perturbations, which has not been addressed in existing Markov-jumping PDE literature. The control design is validated through numerical experiments, showing consistent stabilization performance across different parameter switching scenarios.