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

Tchousso, Abdoua; Besson, Thibaut; Xu, Cheng-Zhong

ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 2, pp. 403-425.

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 $L^p$, $1<p\le \infty$.

Classification : 37L15, 37L45, 93C20
Mots clés : hyperbolic symmetric systems, partial differential equations, exponential stability, strongly continuous semigroups, Lyapunov functionals, heat exchangers

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