Robust Stabilization of Linear Markov-Jumping Hyperbolic PDEs with Boundary Input Delay

Abstract

This paper studies the robust stabilization of 2 $\times$ 2 linear hyperbolic partial differential equations (PDEs) with Markov-jumping parameters and boundary input delay. The main challenge arises from the simultaneous presence of stochastic parameter variations and input delay, which complicates both the stability analysis and controller design. To address this issue, a nominal delay-compensating backstepping controller is first designed for a fixed nominal system. Applying the nominal transformation to the stochastic system yields a target system with additional perturbation terms induced by parameter mismatch. A mode-independent Lyapunov functional is then constructed to establish a pathwise exponential estimate, which directly implies mean-square exponential stability under an explicit small-mismatch condition. The proposed analysis provides a direct robustness certificate for nominal delay compensation without using mode-dependent Lyapunov functionals. Finally, we present simulation results and discuss how the conservative small-mismatch condition should be interpreted for the numerical example.