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While for coupled hyperbolic PDEs of first order there now exist numerous PDEbackstepping designs, systems with zero speed, i.e., without convection butinvolving infinite-dimensional ODEs, which arise in many applications, fromenvironmental engineering to lasers to manufacturing, have received virtuallyno attention. In this paper, we introduce single-input boundary feedbackdesigns for a linear 1-D hyperbolic system with two counterconvecting PDEs and$n$ equations (infinite-dimensional ODEs) with zero characteristic speed. Theinclusion of zero-speed states, which we refer to as {\em atachic}, may resultin non-stabilizability of the plant. We give a verifiable condition for themodel to be stabilizable and design a full-state backstepping controller whichexponentially stabilizes the origin in the $\mathcal{L}^{2}$ sense. Inparticular, to employ the backstepping method in the presence of atachicstates, we use an invertible Volterra transformation only for the PDEs withnonzero speeds, leaving the zero-speed equations unaltered in the target systeminput-to-state stable with respect to the decoupled and stablecounterconvecting nonzero-speed equations. Simulation results are presented toillustrate the effectiveness of the proposed control design.

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