Using hyperbolic systems of balance laws for modeling, control and stability analysis of physical networks

The operation of many physical networks having an engineering relevance may be represented by hyperbolic systems of balance laws in one space dimension. Among the potential applications we mention for instance hydraulic networks (for irrigation or navigation), electric line networks, road traffic networks or gas pipeline networks. In this talk, our main concern is to discuss the exponential stability of the classical solutions for such physical network systems. The analysis relies on Lyapunov functions. Essentially, we shall see that the time derivative of the Lyapunov function can be made negative if the boundary conditions are dissipative. There is therefore an underlying control problem which is the problem of designing the control laws at the network junctions in order to make the corresponding boundary conditions dissipative.