State-Feedback Stabilization of \({2 \times 2}\) Hyperbolic Systems with Distributed Actuation

Abstract. This paper addresses the stabilization of a [Formula: see text] system of heterodirectional linear first-order hyperbolic partial differential equations using distributed in-domain actuation. By applying a backstepping Volterra transformation, the PDE system is reformulated as an integral difference equation. We then propose a controller with an auto-regressive structure, formulated as a distributed delayed feedback involving both the system state and the control input. The controller gains are computed by solving a set of Fredholm integral equations, whose solvability is ensured via an approximate controllability condition. The proposed methodology is further extended to a mixed actuation setting, in which the same control input also acts at one boundary of the domain. Numerical simulations are provided to illustrate and validate the theoretical results.