Zero-Stabilization for Some Diffusive Models with State Constraints

S. Aniţa^1, 2

¹ Faculty of Mathematics, “Alexandru Ioan Cuza” University of Iaşi, Iaşi 700506, Romania
² “Octav Mayer” Institute of Mathematics of the Romanian Academy, Iaşi 700506, Romania

Mathematical modelling of natural phenomena, Tome 9 (2014) no. 4, pp. 6-19.

Résumé

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.

Détail

DOI : 10.1051/mmnp/20149402

Bibliographie

[1] B. Ainseba, S. Aniţa Comm. Pure Appl. Math. 2008 491 512

[2] L.-I. Aniţa, S. Aniţa, V. Arnăutu Appl. Math. Comput. 2013 10231 10244

[3] S. Aniţa, V. Capasso Nonlin. Anal. Real World Appl. 2009 2026 2035

[4] S. Aniţa, W. Fitzgibbon, M. Langlais Discrete Cont. Dyn. Syst.- B 2009 805 822

[5] V. Barbu. Analysis and Control of Nonlinear Infinite Dimensional Systems. Academic Press, Boston, 1993.

[6] V. Barbu. Partial Differential Equations and Boundary value problems. Kluwer Academic Press, Dordrecht, 1998.

[7] V. Capasso, R.E. Wilson SIAM J. Appl. Math. 1997 327 346

[8] K. Deimling. Nonlinear Functional Analysis. Springer-Verlag, Berlin, 1985.

[9] R.A. Fisher Ann. Eugen. 1937 355 369

[10] A.V. Fursikov, O. Yu. Imanuvilov. Controllability of Evolution Equations. Lecture Notes Series 34, RIM Seoul National University, Korea, 1996.

[11] S. Genieys, V. Volpert, P. Auger Math. Modelling Nat. Phenom. 1 2006 65 82

[12] J.-M. Ghidaglia Nonlin. Anal. TMA 1986 777 790

[13] W.M. Haddad, V.S. Chellaboina, Q. Hui. Nonnegative and Compartmental Dynamical Systems. Princeton Univ. Press, Princeton, New Jersey, 2010.

[14] A. Henrot, El Haj Laamri, D. Schmitt Comm. Pure Appl. Anal. 2012 2429 2443

[15] A. Henrot, M. Pierre. Variation et Optimisation de Formes. Une Analyse Géométrique. Springer, Berlin, 2005.

[16] B. Kawohl. Rearrangements and Convexity of Level Sets in PDE. Springer Lecture Notes in Math. 1150, 1985.

[17] J. Klamka. Controllability for partial functional differential equations. Advances in Mathematical Population Dynamics, World Scientific, Ch. 7, 809-814, Houston, 1997.

[18] J. Klamka Bull. Pol. Acad. Sci., Tech. Sci. 2001 633 641

[19] G. Lebeau, L. Robbiano Comm. in PDE 1995 335 357

[20] J.-L. Lions. Controlabilité Exacte, Stabilisation et Perturbation de Systemes Distribués. RMA 8, Masson, Paris, 1988.

[21] A. Okubo. Diffusion and Ecological Problems: Mathematical Models. Springer-Verlag, Berlin, 1980.

[22] J.D. Murray. Mathematical Biology. II. Spatial Models and Biomedical Applications, 3rd edition. Springer-Verlag, New York, 2003.

[23] M.H. Protter, H.F. Weinberger. Maximum Principles in Differential Equations. Springer-Verlag, New York, 1984.

[24] J. Smoller. Shock Waves and Reaction Diffusion Equations. Springer Verlag, Berlin, 1983.