A Strict Lyapunov Function for Boundary Control of Hyperbolic Systems of Conservation Laws

We present a strict Lyapunov function for hyperbolic systems of conservation laws that can be diagonalized with Riemann invariants. The time derivative of this Lyapunov function can be made strictly negative definite by an appropriate choice of the boundary conditions. It is shown that the derived boundary control allows to guarantee the local convergence of the state towards a desired set point. Furthermore, the control can be implemented as a feedback of the state only measured at the boundaries. The control design method is illustrated with an hydraulic application, namely the level and flow regulation in an horizontal open channel

[1]  Axel Klar,et al.  Gas flow in pipeline networks , 2006, Networks Heterog. Media.

[2]  P. Rouchon,et al.  Active signal restoration for the telegraph equation , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[3]  G. Sallet,et al.  Exponential Stability and Transfer Functions of Processes Governed by Symmetric Hyperbolic Systems , 2002 .

[4]  Michel Rascle,et al.  Resurrection of "Second Order" Models of Traffic Flow , 2000, SIAM J. Appl. Math..

[5]  Ta-Tsien Li.  Global classical solutions for quasilinear hyperbolic systems , 1994 .

[6]  J. Coron,et al.  A Lyapunov approach to control irrigation canals modeled by saint-venant equations , 1999, 1999 European Control Conference (ECC).

[7]  J. Greenberg,et al.  The effect of boundary damping for the quasilinear wave equation , 1984 .

[8]  Li Tatsien.  Exact controllability for quasilinear hyperbolic systems and its application to unsteady flows in a network of open canals , 2004 .

[9]  B. d'Andrea-Novel,et al.  Boundary control for exact cancellation of boundary disturbances in hyperbolic systems of conservation laws , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[10]  Ole Morten Aamo,et al.  OBSERVER DESIGN USING BOUNDARY INJECTIONS FOR PIPELINE MONITORING AND LEAK DETECTION , 2006 .

[11]  C. Cheverry,et al.  Systèmes de lois de conservation et stabilité BV , 1998 .

[12]  B. d'Andrea-Novel,et al.  On boundary control design for quasilinear hyperbolic systems with entropies as Lyapunov functions , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[13]  P. Lax.  Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves , 1987 .

[14]  Jonathan de Halleux,et al.  Boundary feedback control in networks of open channels , 2003, Autom..

[15]  Georges Bastin,et al.  Boundary control design for cascades of hyperbolic 2 * 2 PDE systems via graph theory , 2004 .

[16]  Guenter Leugering,et al.  On the Modelling and Stabilization of Flows in Networks of Open Canals , 2002, SIAM J. Control. Optim..

[17]  Georges Bastin,et al.  A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws , 2004, CDC.