Backstepping Control of a Hyperbolic PDE System with Zero Characteristic Speed States

While for coupled hyperbolic PDEs of first order there now exist numerous PDE backstepping designs, systems with zero speed, i.e., without convection but involving infinite-dimensional ODEs, which arise in many applications, from environmental engineering to lasers to manufacturing, have received virtually no attention. In this paper, we introduce single-input boundary feedback designs for a linear 1-D hyperbolic system with two counterconvecting PDEs and $n$ equations (infinite-dimensional ODEs) with zero characteristic speed. The inclusion of zero-speed states, which we refer to as {\em atachic}, may result in non-stabilizability of the plant. We give a verifiable condition for the model to be stabilizable and design a full-state backstepping controller which exponentially stabilizes the origin in the $\mathcal{L}^{2}$ sense. In particular, to employ the backstepping method in the presence of atachic states, we use an invertible Volterra transformation only for the PDEs with nonzero speeds, leaving the zero-speed equations unaltered in the target system input-to-state stable with respect to the decoupled and stable counterconvecting nonzero-speed equations. Simulation results are presented to illustrate the effectiveness of the proposed control design.