Exponential stability of distributed parameter systems governed by symmetric hyperbolic partial differential equations using Lyapunov's second method

¹ LAGEP, Bâtiment CPE, Université Claude Bernard, Lyon I, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne, France.
² Departement de Mathématiques et Informatique, Université Abdou Moumouni de Niamey, BP 10662, Niger; xu@lagep.univ-lyon1.fr

Received: 2 November 2005
Revised: 4 September 2006
Revised: 7 December 2007

Abstract

In this paper we study asymptotic behaviour of distributed parameter systems governed by partial differential equations (abbreviated to PDE). We first review some recently developed results on the stability analysis of PDE systems by Lyapunov's second method. On constructing Lyapunov functionals we prove next an asymptotic exponential stability result for a class of symmetric hyperbolic PDE systems. Then we apply the result to establish exponential stability of various chemical engineering processes and, in particular, exponential stability of heat exchangers. Through concrete examples we show how Lyapunov's second method may be extended to stability analysis of nonlinear hyperbolic PDE. Meanwhile we explain how the method is adapted to the framework of Banach spaces L^p, .