Boundary Consensus of Networked Hyperbolic Systems of Conservation Laws

Concerning the consensus problem of a networked hyperbolic system of two linear conservation laws, this article proposes a boundary consensus protocol and establishes a theoretical framework for consensus analysis under both fixed and switching communication topologies. First, we present a consensus analysis under fixed topologies by utilizing the Lyapunov approach, where both undirected graph and digraph cases are considered. Under the assumption that the undirected graph is connected or the digraph is balanced and rooted, we derive sufficient conditions w.r.t. the boundary control matrices and Laplacian eigenvalues for ensuring the asymptotic consensus. Second, we further consider the consensus problem under switching topologies, in which the undirected graph (or balanced digraph) is relaxed to be jointly connected (or jointly rooted). By using the Lyapunov approach combined with the related space decomposition technique, we derive sufficient conditions w.r.t. the boundary control gain based on a priori knowledge of possible Laplacian matrices for ensuring the asymptotic consensus. Finally, we provide numerical examples and an application to the consensus control of a multilane road traffic flow system described by the Aw-Rascle equations, to demonstrate the effectiveness of our theoretical results.