Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-23T02:04:29.819Z Has data issue: false hasContentIssue false
  • English
  • Français

Zero-Stabilization for Some Diffusive Modelswith State Constraints

Published online by Cambridge University Press:  20 June 2014

S. Aniţa
S. Aniţa*
Affiliation:
Faculty of Mathematics, “Alexandru Ioan Cuza” University of Iaşi, Iaşi 700506, Romania “Octav Mayer” Institute of Mathematics of the Romanian Academy, Iaşi 700506, Romania
*
⋆Corresponding author. E-mail: [email protected]
Get access

Abstract

We discuss the zero-controllability and the zero-stabilizability for the nonnegative solutions to some Fisher-like models with nonlocal terms describing the dynamics of biological populations with diffusion, logistic term and migration. A necessary and sufficient condition for the nonnegative zero-stabilizabiity for a linear integro-partial differential equation is obtained in terms of the sign of the principal eigenvalue to a certain non-selfadjoint operator. For a related semilinear problem a necessary condition and a sufficient condition for the local nonnegative zero-stabilizability are also derived in terms of the magnitude of the above mentioned principal eigenvalue. The rate of stabilization corresponding to a simple feedback stabilizing control is dictated by the principal eigenvalue. A large principal eigenvalue leads to a fast stabilization to zero. A necessary condition and a sufficient condition for the stabilization to zero of the predator population in a predator-prey system is also investigated. Finally, a method to approximate the above mentioned principal eigenvalues is indicated.

Keywords

zero-stabilizabilitystate constraintscomparison resultprincipal eigenvaluefeedback controlpopulation dynamicsreaction-diffusion system
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 4: Optimal control , 2014 , pp. 6 - 19
Copyright
© EDP Sciences, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ainseba, B., Aniţa, S.. Internal nonnegative stabilization for some parabolic equations. Comm. Pure Appl. Math., 7 (2008), no. 3, 491–512. Google Scholar
Aniţa, L.-I., Aniţa, S., Arnăutu, V.. Internal null stabilization for some diffusive models in population dynamics. Appl. Math. Comput., 219 (2013), no. 20, 10231–10244. Google Scholar
Aniţa, S., Capasso, V.. A stabilization strategy for a reaction-diffusion system modelling a class of spatially structured epidemic systems (think globally, act locally). Nonlin. Anal. Real World Appl. 10 (2009), no. 4, 2026–2035. CrossRefGoogle Scholar
Aniţa, S., Fitzgibbon, W., Langlais, M.. Global existence and internal stabilization for a class of predator-prey systems posed on non coincident spatial domains. Discrete Cont. Dyn. Syst.- B, 11 (2009), no. 4, 805–822 CrossRefGoogle Scholar
V. Barbu. Analysis and Control of Nonlinear Infinite Dimensional Systems. Academic Press, Boston, 1993.
V. Barbu. Partial Differential Equations and Boundary value problems. Kluwer Academic Press, Dordrecht, 1998.
Capasso, V., Wilson, R.E.. Analysis of a reaction-diffusion system modelling man-environment-man epidemics. SIAM J. Appl. Math. 57 (1997), no. 2, 327–346. Google Scholar
K. Deimling. Nonlinear Functional Analysis. Springer-Verlag, Berlin, 1985.
Fisher, R.A.. The wave of advance of advantageous genes. Ann. Eugen. 7 (1937), 355–369. CrossRefGoogle Scholar
A.V. Fursikov, O. Yu. Imanuvilov. Controllability of Evolution Equations. Lecture Notes Series 34, RIM Seoul National University, Korea, 1996.
Genieys, S., Volpert, V., Auger, P.. Pattern and waves for a model in population dynamics with nonlocal consumption of resources. Math. Modelling Nat. Phenom. 1, (2006), no. 1, 65–82. Google Scholar
Ghidaglia, J.-M.. Some backward uniqueness results. Nonlin. Anal. TMA, 10 (1986), no. 8, 777–790. CrossRefGoogle Scholar
W.M. Haddad, V.S. Chellaboina, Q. Hui. Nonnegative and Compartmental Dynamical Systems. Princeton Univ. Press, Princeton, New Jersey, 2010.
Henrot, A., Haj Laamri, El, Schmitt, D.. On some spectral problems arising in dynamic populations. Comm. Pure Appl. Anal. 11 (2012), no. 6, 2429–2443. CrossRefGoogle Scholar
A. Henrot, M. Pierre. Variation et Optimisation de Formes. Une Analyse Géométrique. Springer, Berlin, 2005.
B. Kawohl. Rearrangements and Convexity of Level Sets in PDE. Springer Lecture Notes in Math. 1150, 1985.
J. Klamka. Controllability for partial functional differential equations. Advances in Mathematical Population Dynamics, World Scientific, Ch. 7, 809-814, Houston, 1997.
Klamka, J.. Reachability and controllability of positive continuous-time linear systems with time variable coefficients. Bull. Pol. Acad. Sci., Tech. Sci., 49 (2001), no. 4, 633–641. Google Scholar
Lebeau, G., Robbiano, L.. Contrôle exact de l’équation de la chaleur. Comm. in PDE, 20 (1-2) (1995), 335–357. CrossRefGoogle Scholar
J.-L. Lions. Controlabilité Exacte, Stabilisation et Perturbation de Systemes Distribués. RMA 8, Masson, Paris, 1988.
A. Okubo. Diffusion and Ecological Problems: Mathematical Models. Springer-Verlag, Berlin, 1980.
J.D. Murray. Mathematical Biology. II. Spatial Models and Biomedical Applications, 3rd edition. Springer-Verlag, New York, 2003.
M.H. Protter, H.F. Weinberger. Maximum Principles in Differential Equations. Springer-Verlag, New York, 1984.
J. Smoller. Shock Waves and Reaction Diffusion Equations. Springer Verlag, Berlin, 1983.