Necessary and Sufficient Conditions on the Exponential Stability of Positive Hyperbolic Systems

Abstract

In this paper, a strict linear Lyapunov function is developed in order to investigate the exponential stability of a linear hyperbolic partial differential equation with positive boundary conditions. Based on the method of characteristics, some properties of the positive solutions are derived for the hyperbolic initial boundary value problems. The dissipative boundary condition in terms of linear inequalities is proven to be not only sufficient but also necessary under an extra assumption on the velocities of the hyperbolic systems. An application to control of the freeway traffic modeled by the Aw–Rascle traffic flow equation illustrates and motivates the theoretical results. The boundary control strategies are designed by integrating the on-ramp metering with the mainline speed limit. Finally, the proposed feedback laws are tested under simulation, first in the free-flow case and then in the congestion mode, which show adequate performance to stabilize the local freeway traffic.