Dynamic boundary stabilization of linear and quasi-linear hyperbolic systems

Abstract

Systems governed by hyperbolic partial differential equations with dynamics associated with their boundary conditions are considered in this paper. These infinite dimensional systems can be described by linear or quasi-linear hyperbolic equations. By means of Lyapunov based techniques, some sufficient conditions are derived for the exponential stability of such systems. A polytopic approach is developed for quasilinear hyperbolic systems in order to guarantee stability in a region of attraction around an equilibrium point, given specific bounds on the parameters. The main results are illustrated on the model of an isentropic inviscid flow.

PDE Observer Design for Counter-Current Heat Flows in a Heat-Exchanger

“…In this framework, our contribution is to synthesize a boundary observer for a system of linear, hyperbolic balance laws with a rightward and a leftward transport equation. For this, we get the inspiration from Castillo et al (2012) for a type of infinite-dimensional Luenberger-type observer, which results in a (relatively) simple observer architecture, e.g. in comparison with the backstepping approaches.…”

“…The issue of boundary control of such class of systems is also present in the literature. In Castillo et al (2012) the problem of the stability of a boundary control applied to a system of one-dimensional rightward conservation transport PDE is considered. Diagne et al (2012) give a method for selecting the dissipative boundary conditions that ensure the stability of one-dimensional linear hyperbolic systems of balance laws, using Lyapunov exponential stability techniques.…”

Modeling and control of flow with dynamical boundary actions

“…Moreover, a strict Lyapunov function approach is proposed in [13] for the boundary control with integral actions of hyperbolic systems of conservation law that can be diagonalized by means of Riemann invariants. The stability problem of linear and quasi-linear hyperbolic systems in the presence of dynamic behavior at the boundary conditions is addressed valter@ieee.org in [5]. Reference [6] addresses the problem of boundary observer design for one-dimensional first order linear and quasi-linear strict hyperbolic systems with n rightward convecting transport.…”

Lyapunov stability of a singularly perturbed system of two conservation laws

“…This kind of systems is interesting for analysis as far as it describes numerous important phenomenon in many domains, for example, a two-dimensional motion of an incompressible viscous fluid problem in Nefedov [1988]. The model of gas transport in a constant section tube from the work of Castillo et al [2012] provides the principal motivation for this paper. This model contains two time scales for propagation speed, precisely the propagation speed of gas is much slower than the sound speed, which can be considered as a singular perturbation problem.…”