Stabilization of an underactuated 1+2 linear hyperbolic system with a proper control

International audienceIn this paper we consider the stabilization of an underactuated system of linear first-order hyperbolic Partial Differential Equations consisting of one rightward-convecting transport PDE and two leftward-convecting equations. The system is underactuated in the sense that only one of the leftward-convecting equations is actuated at the boundary. This is a fundamental difference compared to existing results in the literature, as it is usually assumed that (at least) one of the boundaries of the system is fully actuated. The proposed approach combines a backstepping transformation and successive state transformations which allow the reformulation of the original system as a neutral system with distributed state delays and distributed input delays. Assuming that the actuator dynamics are asymptotically stable, the stabilization of the system reduced to that of a simpler neutral system with only distributed state delays and a delayed control. Designing a predictor, it becomes possible to stabilize this simpler system. The proposed feedback law is then combined with a low pass-filter to guarantee the existence of delay-robustness margins for the closed-loop system