An optimisation approach for stability analysis and controller synthesis of linear hyperbolic systems

Pierre-Olivier Lamare (1) , Antoine Girard (2) , Christophe Prieur (3)

1 BIOCORE - Biological control of artificial ecosystems
2 L2S - Laboratoire des signaux et systèmes
3 GIPSA-SYSCO - GIPSA - Systèmes non linéaires et complexité

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.

Mots clés: 37N35; 93B52; 1991 Mathematics Subject Classification 49J20

Domaines: Optimisation et contrôle [math.OC]; Automatique / Robotique; Systèmes et contrôle [cs.SY]

Pierre-Olivier Lamare, Antoine Girard, Christophe Prieur. An optimisation approach for stability analysis and controller synthesis of linear hyperbolic systems. ESAIM: Control, Optimisation and Calculus of Variations, 2016, 22 (4), pp.1236-1263. ⟨10.1051/cocv/2016038⟩. ⟨hal-01331730⟩