  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.
