Research on stabilization of coupled hyperbolic PDEs has been dominated by the focus on pairs of counter-convecting ("heterodirectional") transport PDEs with distributed local coupling and with controls at one or both boundaries. A recent extension allows stabilization using only one control for a system containing an arbitrary number of coupled transport PDEs that convect at different speeds against the direction of the PDE whose boundary is actuated. In this paper we present a solution to the fully general case, in which the number of PDEs in either direction is arbitrary, and where actuation is applied on only one boundary (to all the PDEs that convect downstream from that boundary). To solve this general problem, we solve, as a special case, the problem of control of coupled "homodirectional" hyperbolic linear PDEs, where multiple transport PDEs convect in the same direction with arbitrary local coupling.Our approach is based on PDE backstepping and yields solutions to stabilization, by both full-state and observer-based output feedback, trajectory planning, and trajectory tracking problems.L. Hu is with Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005,

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We solve the problem of stabilizing a linear ODE having a system of a linearly coupled hyperbolic PDEs in the actuating and sensing paths. The system is exponentially stabilized by mapping it to a target system with a cascade structure using a Volterra transformation.Key words: Stabilization, Distributed Parameter Systems IntroductionThe interest for coupled Ordinary Differential EquationsPartial Differential Equations (ODE-PDE) systems has first emerged when considering delays in the actuating and sensing paths of ODE. Delays can be seen as first-order hyperbolic PDEs. There are many approaches to deal with input or measurement delays, usually divided into two categories: memoryless controllers, which extend standard control techniques without explicitly accounting for the delay in the control design [16,25,9]; and prediction-based controllers aiming at explicitly compensating the delay [20,4,2].The use of Lyapunov and backstepping methods enabled dealing with more involved PDEs in the actuating and sensing paths. In [13], an output feedback control law is derived for an ODE having a heat equation in the actuating and sensing paths. The coupled PDE-ODE system is stabilized using an observer-controller structure relying on a backstepping approach. The same approach has been used to deal with ODEs coupled (rather than cascaded) with parabolic PDEs The first application of the backstepping approach to deal with hyperbolic PDE-ODE couplings is [14] where actuator and sensor delays are explicitly compensated. While this problem had already been tackled by, e.g., the Smith predictor [20], the reformulation of the delay as a linear ODE enabled numerous related problem to be tackled, most notably non-constant and uncertain delays [2,4]. In [22], the problem of stabilizing a multi-input ODE with distinct delays is tackled using a backstepping approach. In [6], an observer is designed for an ODE having a homodirectional 1 hyperbolic PDE in the sensing path, relying on a Lyapunov analysis requiring to solve Linear Matrix Inequalities (LMI). As will appear, the systems of [22] are particular cases of the system considered here, although the control design approaches are different.Here, we solve the problem of stabilizing an ODE with a system of first-order linear hyperbolic PDEs in the sensing 1 i.e. where all the states transport in the same direction