BV EXPONENTIAL STABILITY FOR NETWORKS OF SCALAR CONSERVATION LAWS USINGLINEAR OR SATURATED CONTROLS

In this paper, we investigate the BV exponential stability of general networks of scalar conservation laws with positive velocities 4 and under dissipative boundary conditions. The paper is divided in two parts, the first one focusing on linear controls while the last one deals 5 with saturated laws. For the linear case, the global exponential BV stability is proven. For the saturated case, we argue that we cannot expect 6 to have a basin of attraction larger than the region of linearity in a BV context. We rather prove an L ∞ local stability result. An explicit 7 estimate of the basin of attraction is given. The Lyapunov functional is inspired from Glimm's seminal work [13] reconsidered in [7]. 8