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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| Published in: | Numerical functional analysis and optimization Vol. 32; no. 4; pp. 351 - 369 |
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| Language: | English |
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Philadelphia, PA
Taylor & Francis Group
07.03.2011
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| ISSN: | 0163-0563, 1532-2467 |
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| Abstract | 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. |
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| AbstractList | 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. 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. |
| Author | Dodea, Smaranda Arnăutu, Viorel Aniţa, Sebastian |
| Author_xml | – sequence: 1 givenname: Sebastian surname: Aniţa fullname: Aniţa, Sebastian email: sanita@uaic.ro organization: Institute of Mathematics "Octav Mayer," – sequence: 2 givenname: Viorel surname: Arnăutu fullname: Arnăutu, Viorel organization: Faculty of Mathematics , University "Al.I. Cuza," – sequence: 3 givenname: Smaranda surname: Dodea fullname: Dodea, Smaranda organization: Faculty of Mathematics , University "Al.I. Cuza," |
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| Keywords | 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 |
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| SubjectTerms | 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 |
| Title | Feedback Stabilization for a Reaction-Diffusion System with Nonlocal Reaction Term |
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