An optimisation approach for stability analysis and controller synthesis of linear hyperbolic systems
Lamare, Pierre-Olivier1 ; Girard, Antoine2 ; Prieur, Christophe3
1 BIOCORE project-team, Inria Sophia Antipolis – Méditerranée, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis cedex, France.
2 Laboratoire des signaux et systèmes (L2S), CNRS, Centrale Supélec, Université Paris-Sud, Université Paris-Saclay, 3, rue Joliot-Curie, 91192 Gif-sur-Yvette cedex, France.
3 Department of Automatic Control, Gipsa-lab, 11 rue des Mathématiques, BP 46, 38402 Saint Martin d’Hères cedex, France.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1236-1263.

In this paper, we consider the problems of stability analysis and control synthesis for first-order hyperbolic linear Partial Differential Equations (PDEs) over a bounded interval with spatially varying coefficients. We propose Linear Matrix Inequalities (LMI) conditions for the stability and for the design of boundary and distributed control for the system. These conditions involve an infinite number of LMI to solve. Hence, we show how to overapproximate these constraints using polytopic embeddings to reduce the problem to a finite number of LMI. We show the effectiveness of the overapproximation with several examples and with the Saint-Venant equations with friction.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2016038
Classification : 49J20, 37N35, 93B52
Mots-clés : Hyperbolic PDE, Lyapunov method, LMI
Lamare, Pierre-Olivier 1 ; Girard, Antoine 2 ; Prieur, Christophe 3

1 BIOCORE project-team, Inria Sophia Antipolis – Méditerranée, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis cedex, France.
2 Laboratoire des signaux et systèmes (L2S), CNRS, Centrale Supélec, Université Paris-Sud, Université Paris-Saclay, 3, rue Joliot-Curie, 91192 Gif-sur-Yvette cedex, France.
3 Department of Automatic Control, Gipsa-lab, 11 rue des Mathématiques, BP 46, 38402 Saint Martin d’Hères cedex, France. 
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Lamare, Pierre-Olivier; Girard, Antoine; Prieur, Christophe. An optimisation approach for stability analysis and controller synthesis of linear hyperbolic systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1236-1263. doi : 10.1051/cocv/2016038. https://www.numdam.org/articles/10.1051/cocv/2016038/

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