Stability and boundary stabilization of 1-D hyperbolic systems
type: Item
instance of: scholarly article
title: Stability and boundary stabilization of 1-D hyperbolic systems
zbMATH Open document ID: 1377.35001
DOI: 10.1007/978-3-319-32062-5
author: Georges Bastin
author: Jean-Michel Coron
published in: Progress in Nonlinear Differential Equations and Their Applications
publication date: 2016-04-11T00:00:00Z
review text: This very nice book concerns initial-boundary value problems for 1-D hyperbolic systems of first-order PDEs, in particular for balance laws and for conservation laws. The main topic is the boundary control problem, when the control input, the controlled output and the measured output are located at the boundary, and the goal is to identify and to characterize so-called dissipative boundary conditions and dissipative source terms, i.e. those which guarantee bounded smooth solutions converging to an equilibrium. Different criteria are derived for stability in different function space norms. Moreover, it is presented a list of mathematical models (navigable rivers, chemotaxis, networks of electrical lines, lasers, chromatography, epitemiology, road traffic, gas dynamics, shallow water equations, telegrapher equations), for which the criteria are applicable. One of the main difficulties of stability theory for hyperbolic PDEs is nicely described already in the case of linear systems of conservation laws: Necessary and sufficient for stability is the negativity of the real parts of all roots of the characteristic equation. But this criterion is not robust with respect to small perturbations of the system, in general (because of small denominator effects)! That's why the authors mainly present sufficient stability criteria by means of the Lyapunov function approach.
reviewed by: Lutz Recke
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L50
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 35Q93
Mathematics Subject Classification ID: 93D15
zbMATH DE Number: 6567166
zbMATH Keywords: boundary control problem
zbMATH Keywords: dissipative boundary conditions
zbMATH Keywords: stability criteria
zbMATH Keywords: Lyapunov function approach
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-32062-5
OpenAlex ID: W2482021336
description: scientific article; zbMATH DE number 6567166