Boundary feedback stabilization and Riesz basis property of a 1-d first order hyperbolic linear system with L∞-coefficients

Boumediène Chentouf, Jun Min Wang

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

This paper deals with the boundary feedback stabilization problem of a wide class of linear first order hyperbolic systems with non-smooth coefficients. We propose static boundary inputs (actuators) which lead us to a closed loop system with non-smooth coefficients and non-homogeneous boundary conditions. Then, we prove the exponential stability of the closed loop system under suitable conditions on the coefficients and the feedback gains. The key idea of the proof is to combine the regularization techniques with the characteristics method. Furthermore, by the spectral analysis method, it is also shown that the closed loop system has a sequence of generalized eigenfunctions, which form a Riesz basis for the state space, and hence the spectrum-determined growth condition is deduced.

Original languageEnglish
Pages (from-to)1119-1138
Number of pages20
JournalJournal of Differential Equations
Volume246
Issue number3
DOIs
Publication statusPublished - Feb 1 2009

Fingerprint

Boundary Stabilization
Riesz Basis
Feedback Stabilization
Hyperbolic Systems
Closed loop systems
Closed-loop System
Linear systems
Stabilization
Linear Systems
First-order
Feedback
Coefficient
Nonhomogeneous Boundary Conditions
Characteristics Method
Regularization Technique
First-order System
Exponential Stability
Asymptotic stability
Growth Conditions
Spectral Analysis

Keywords

  • C-semigroup
  • First order hyperbolic linear system
  • L-coefficients
  • Regularization method
  • Riesz basis
  • Stability

ASJC Scopus subject areas

  • Analysis

Cite this

Boundary feedback stabilization and Riesz basis property of a 1-d first order hyperbolic linear system with L∞-coefficients. / Chentouf, Boumediène; Wang, Jun Min.

In: Journal of Differential Equations, Vol. 246, No. 3, 01.02.2009, p. 1119-1138.

Research output: Contribution to journalArticle

@article{e0d7646298c44892a8bd6e2810e41c02,
title = "Boundary feedback stabilization and Riesz basis property of a 1-d first order hyperbolic linear system with L∞-coefficients",
abstract = "This paper deals with the boundary feedback stabilization problem of a wide class of linear first order hyperbolic systems with non-smooth coefficients. We propose static boundary inputs (actuators) which lead us to a closed loop system with non-smooth coefficients and non-homogeneous boundary conditions. Then, we prove the exponential stability of the closed loop system under suitable conditions on the coefficients and the feedback gains. The key idea of the proof is to combine the regularization techniques with the characteristics method. Furthermore, by the spectral analysis method, it is also shown that the closed loop system has a sequence of generalized eigenfunctions, which form a Riesz basis for the state space, and hence the spectrum-determined growth condition is deduced.",
keywords = "C-semigroup, First order hyperbolic linear system, L-coefficients, Regularization method, Riesz basis, Stability",
author = "Boumedi{\`e}ne Chentouf and Wang, {Jun Min}",
year = "2009",
month = "2",
day = "1",
doi = "10.1016/j.jde.2008.08.010",
language = "English",
volume = "246",
pages = "1119--1138",
journal = "Journal of Differential Equations",
issn = "0022-0396",
publisher = "Academic Press Inc.",
number = "3",

}

TY - JOUR

T1 - Boundary feedback stabilization and Riesz basis property of a 1-d first order hyperbolic linear system with L∞-coefficients

AU - Chentouf, Boumediène

AU - Wang, Jun Min

PY - 2009/2/1

Y1 - 2009/2/1

N2 - This paper deals with the boundary feedback stabilization problem of a wide class of linear first order hyperbolic systems with non-smooth coefficients. We propose static boundary inputs (actuators) which lead us to a closed loop system with non-smooth coefficients and non-homogeneous boundary conditions. Then, we prove the exponential stability of the closed loop system under suitable conditions on the coefficients and the feedback gains. The key idea of the proof is to combine the regularization techniques with the characteristics method. Furthermore, by the spectral analysis method, it is also shown that the closed loop system has a sequence of generalized eigenfunctions, which form a Riesz basis for the state space, and hence the spectrum-determined growth condition is deduced.

AB - This paper deals with the boundary feedback stabilization problem of a wide class of linear first order hyperbolic systems with non-smooth coefficients. We propose static boundary inputs (actuators) which lead us to a closed loop system with non-smooth coefficients and non-homogeneous boundary conditions. Then, we prove the exponential stability of the closed loop system under suitable conditions on the coefficients and the feedback gains. The key idea of the proof is to combine the regularization techniques with the characteristics method. Furthermore, by the spectral analysis method, it is also shown that the closed loop system has a sequence of generalized eigenfunctions, which form a Riesz basis for the state space, and hence the spectrum-determined growth condition is deduced.

KW - C-semigroup

KW - First order hyperbolic linear system

KW - L-coefficients

KW - Regularization method

KW - Riesz basis

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=55649120663&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=55649120663&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2008.08.010

DO - 10.1016/j.jde.2008.08.010

M3 - Article

VL - 246

SP - 1119

EP - 1138

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 3

ER -