Backstepping Control of a Hyperbolic PDE System with Zero Characteristic Speed

In this paper, we study the single-input boundary feedback stabilization of 3 × 3 linear hyperbolic partial differential equations (PDEs) with two counterconvecting PDEs and the third one with zero characteristic speed. We design a full-state backstepping controller which exponentially stabilizes the origin in the L2 sense. The zero transport velocity makes the previous backstepping designs inapplicable (their application would result in a controller with infinite gains). To employ backstepping in the presence of zero speed, we use an invertible Volterra transformation only for the PDEs with nonzero speeds, leaving the state of the zero-speed PDE unaltered in the target system, but making the target zero-speed PDE input-to-state stable with respect to the decoupled and stable counterconvecting nonzero-speed PDEs. In addition to achieving stabilization, we produce an explicit bound on the rate of convergence of the target system by a method of successive approximations and the use of Laplace transform. Simulation results are presented to illustrate the effectiveness of the proposed control design.