Robust Boundary Stabilization of Stochastic Hyperbolic PDEs

This paper proposes a backstepping boundary control design for robust stabilization of linear first-order coupled hyperbolic partial differential equations (PDEs) with Markov-jumping parameters. The PDE system consists of 4 x 4 coupled hyperbolic PDEs whose first three characteristic speeds are positive and the last one is negative. We first design a full-state feedback boundary control law for a nominal, deterministic system using the backstepping method. Then, by applying Lyapunov analysis methods, we prove that the nominal backstepping control law can stabilize the PDE system with Markov jumping parameters if the nominal parameters are sufficiently close to the stochastic ones on average. The mean-square exponential stability conditions are theoretically derived and then validated via numerical simulations.