Dynamic boundary stabilization of linear and quasi-linear hyperbolic systems

Systems governed by hyperbolic partial differential equations with dynamics associated with their boundary conditions are considered in this paper. These infinite dimensional systems can be described by linear or quasi-linear hyperbolic equations. By means of Lyapunov based techniques, some sufficient conditions are derived for the exponential stability of such systems. A polytopic approach is developed for quasi-linear hyperbolic systems in order to guarantee stability in a region of attraction around an equilibrium point, given specific bounds on the parameters. The main results are illustrated on the model of an isentropic inviscid flow.

[1]  Falk M. Hante,et al.  Modeling and Analysis of Modal Switching in Networked Transport Systems , 2009 .

[2]  Luc Dugard,et al.  Contrôle de Température dans un Flux de Poiseuille , 2012 .

[3]  Luc Dugard,et al.  Restriction Model Independent Method for Non-Isentropic Outflow Valve Boundary Problem Resolution , 2012 .

[4]  Georges Bastin,et al.  Dissipative Boundary Conditions for One-Dimensional Nonlinear Hyperbolic Systems , 2008, SIAM J. Control. Optim..

[5]  J. Coron.  Control and Nonlinearity , 2007 .

[6]  Joseph J. Winkin,et al.  Robust boundary control of systems of conservation laws , 2008, Math. Control. Signals Syst..

[7]  Alireza Karimi,et al.  Robust control of polytopic systems by convex optimization , 2007, 2007 European Control Conference (ECC).

[8]  Georges Bastin,et al.  Using hyperbolic systems of balance laws for modeling, control and stability analysis of physical networks , 2009 .

[9]  Georges Bastin,et al.  A Strict Lyapunov Function for Boundary Control of Hyperbolic Systems of Conservation Laws , 2007, IEEE Transactions on Automatic Control.

[10]  GZ Georgo Angelis,et al.  System analysis, modelling and control with polytopic linear models , 2001 .

[11]  De Winterbone,et al.  Theory of Engine Manifold Design: Wave Action Methods for IC Engineers , 2001 .

[12]  Karl Henrik Johansson,et al.  Wireless ventilation control for large-scale systems: The mining industrial case: WIRELESS VENTILATION CONTROL , 2010 .

[13]  Christophe Prieur,et al.  Boundary Control of Open Channels With Numerical and Experimental Validations , 2008, IEEE Transactions on Control Systems Technology.

[14]  Alessandro D'Innocenzo,et al.  Wireless Ventilation Control for Large-Scale Systems: the Mining Industrial Case , 2009, 2009 6th IEEE Annual Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks Workshops.

[15]  Antoine Girard,et al.  Lyapunov functions for switched linear hyperbolic systems , 2012, ADHS.

[16]  F. Bruzelius.  Linear Parameter-Varying Systems - an approach to gain scheduling , 2004 .

[17]  Georges Bastin,et al.  On Lyapunov stability of linearised Saint-Venant equations for a sloping channel , 2009, Networks Heterog. Media.

[18]  Jonathan de Halleux,et al.  Boundary control of quasi-linear hyperbolic initial boundary-value problems , 2004 .

[19]  Tatsien Li,et al.  Global Propagation of Regular Nonlinear Hyperbolic Waves , 2002 .

[20]  Peter D. Lax,et al.  Hyperbolic Partial Differential Equations , 2004 .