Feedback Stabilization for a Reaction-Diffusion System with Nonlocal Reaction Term

We consider a two-component reaction-diffusion system with a nonlocal reaction term. A necessary condition and a sufficient condition for the internal stabilizability to zero of one of the two components of the solution while preserving the nonnegativity of both components have been established in [...

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Bibliographic Details
Published in:Numerical functional analysis and optimization Vol. 32; no. 4; pp. 351 - 369
Main Authors: Aniţa, Sebastian, Arnăutu, Viorel, Dodea, Smaranda
Format: Journal Article
Language:English
Published: Philadelphia, PA Taylor & Francis Group 07.03.2011
Taylor & Francis
Subjects:
Approximation
Control systems
Control theory
Dynamics
Eigenvalues
Exact sciences and technology
Feedback
Feedback control
Feedback stabilization
Mathematical models
Mathematics
Nonlinear algebraic and transcendental equations
Nonlocal reaction term
Numerical analysis
Numerical analysis. Scientific computation
Numerical iterative algorithm
Numerical linear algebra
Prey-predator system
Primary 92D25, 35K57, 93D15
Principal eigenvalue
Reaction-diffusion system
Sciences and techniques of general use
Secondary 35P15, 35B40
Stabilization
Numerical iterative algorithm
Necessary and sufficient condition
Stabilization
Eigenvalue
Iterative method
Predator prey system
Feedback control
Prey-predator system
Non linear equation
Secondary 35P15
Feedback
Nonlocal reaction term
Feedback stabilization
Transcendental equation
Approximate method
Eigenvector
Numerical linear algebra
Principal eigenvalue
Reaction diffusion equation
35K57
Primary 92D25
Algorithm
Numerical analysis
Reaction-diffusion system
93D15
Algebraic equation
Population dynamics
35B40
Self adjoint operator
ISSN:0163-0563, 1532-2467
Online Access:Get full text
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Summary:We consider a two-component reaction-diffusion system with a nonlocal reaction term. A necessary condition and a sufficient condition for the internal stabilizability to zero of one of the two components of the solution while preserving the nonnegativity of both components have been established in [ 6 ]. In case of stabilizability, a feedback stabilizing control of harvesting type has been indicated. The rate of stabilization (for the indicated feedback control) is given by the principal eigenvalue of a certain non-selfadjoint operator. A large principal eigenvalue leads to a fast stabilization. The first main goal of this article is to approximate this principal eigenvalue. This is done in two steps. First, we investigate the large-time behavior of the solution to a logistic population dynamics with migration, and next we derive as a consequence a method to approximate the principal eigenvalue. The other main goal is to derive a conceptual iterative algorithm to improve the position of the support of the control in order to get a faster stabilization. Our results apply to prey-predator systems.
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 23
ISSN:0163-0563
1532-2467
DOI:10.1080/01630563.2010.542266