Abstract
It is shown how an entropy-based Lyapunov function can be used for the stability analysis of equilibria in networks of scalar conservation laws. The analysis gives a sufficient stability condition which is weaker than the condition which was previously known in the literature. Various extensions and generalisations are briefly discussed. The approach is illustrated with an application to ramp-metering control of road traffic networks.
Highlights
Conservation laws are first-order partial differential equations that are commonly used to express the fundamental balance laws that occur in many physical systems and engineering problems when small friction or dissipation effects are neglected (e.g.[4])
We have presented an entropy-based Lyapunov function which is used for the stability analysis of systems of conservation laws
We have shown that this analysis gives a sufficient stability condition which is weaker than the condition which was previously known in the literature
Summary
Conservation laws are first-order partial differential equations that are commonly used to express the fundamental balance laws that occur in many physical systems and engineering problems when small friction or dissipation effects are neglected (e.g.[4]). Physical networks described by systems of 2x2 conservation laws have been recently considered in the literature. We may mention for instance Saint-Venant equations for hydraulic networks (e.g.[11],[7]), isothermal Euler equations for gas pipeline networks (e.g.[1]), or Aw-Rascle equations for road traffic networks (e.g.[10], [8]). Our concern is to analyse the stability (in the sense of Lyapunov) of the steady-states of such networks. For the sake of simplicity, we shall restrict ourselves to networks of scalar conservation laws. Typical examples include LWR models for road traffic networks [9, Chapter 6]), Eulerian flow models for air traffic networks [3]) and fluid models for switched-packets networks The case of networks of 2x2 conservation
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