Finite-time stabilization of 2*2 hyperbolic systems on tree-shaped networks

Vincent Perrollaz (1) , Lionel Rosier (2, 3)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 CAS - Centre Automatique et Systèmes
3 IECL - Institut Élie Cartan de Lorraine

We investigate the finite-time boundary stabilization of a 1-D first order quasilinear hyperbolic system of diagonal form on [0,1]. The dynamics of both boundary controls are governed by a finite-time stable ODE. The solutions of the closed-loop system issuing from small initial data in Lip([0,1]) are shown to exist for all times and to reach the null equilibrium state in finite time. When only one boundary feedback law is available, a finite-time stabilization is shown to occur roughly in a twice longer time. The above feedback strategy is then applied to the Saint-Venant system for the regulation of water flows in a network of canals.

Mots clés: network.; water management; stabilization; Finite-time stability; network; hyperbolic systems; shallow water equations

Domaines: Equations aux dérivées partielles [math.AP]; Automatique / Robotique

Vincent Perrollaz, Lionel Rosier. Finite-time stabilization of 2*2 hyperbolic systems on tree-shaped networks. SIAM Journal on Control and Optimization, 2014, 52 (1), pp.143-163. ⟨hal-00793728⟩