Collective dynamics, control and imaging

Stability and boundary stabilization of physical networks represented by 1-D hyperbolic balance laws

Georges Bastin

Université Catholique de Louvain


The operation of many physical networks having an engineering relevance can be represented by 1-D hyperbolic systems of balance laws. Typical example are hydraulic networks (for water supply, irrigation or navigation), road traffic networks, electrical line networks, gas transportation networks, blood flow networks etc … From an engineering perspective, for such networks, the exponential stability of the steady-states is a fundamental issue. As a matter of fact, the exponential stability closely depends on the so-called dissipativity of the boundary conditions which, in many instances, is a natural physical property of the system. If the boundary conditions are dissipative and if the smooth initial conditions are sufficiently close to the steady state, the system trajectories are guaranteed to remain smooth for all time and to exponentially converge locally to the steady state. In this talk, we shall see how robust dissipativity tests can be derived by using a Lyapunov stability approach. Surprisingly enough, it appears that, even for smooth solutions, the various function norms that can be used to measure the deviation with respect to the steady state are not may give rise to different stability tests. There are also many engineering applications where the dissipativity of the boundary conditions, and consequently the stability, is obtained by using boundary feedback control with actuators and sensors located at the boundaries. The control may be implemented with the goal of stabilization when the system is physically unstable, or simply because boundary feedback control is required to achieve an efficient regulation with disturbance attenuation. Obviously, the challenge in that case is to design the boundary control devices in order to have a good control performance with dissipative boundary conditions. In this talk, this issue will be addressed through a case-study devoted to the control of navigable rivers when the river flow is described by hyperbolic shallow water equations (Saint-Venant equations), illustrated with a real life application of the control of the Sambre and Meuse rivers in Belgium.