Backstepping stabilization of 2×2 linear hyperbolic PDEs coupled with potentially unstable actuator and load dynamics

Abstract

We consider the problem of full-state feedback stabilization of a (possibly unstable) system of hyperbolic partial differential equations (PDEs). Unlike previous works, boundary couplings to linear ordinary differential equations (ODEs) at both boundaries are considered and actuation is available through one of these ODE dynamics. This structure can arise when considering linear (or linearized) systems of balance laws with finite-dimensional actuator and load dynamics. The feedback law proposed in this paper is constructed using an invertible transform based on the (infinite-dimensional) backstepping method.