Math. Model. Nat. Phenom.
Volume 9, Number 4, 2014Optimal control
|Page(s)||6 - 19|
|Published online||20 June 2014|
Zero-Stabilization for Some Diffusive Models with State Constraints
Faculty of Mathematics, “Alexandru Ioan Cuza” University of
2 “Octav Mayer” Institute of Mathematics of the Romanian Academy, Iaşi 700506, Romania
Corresponding author. E-mail: firstname.lastname@example.org
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.
Mathematics Subject Classification: 35B35 / 35B09 / 35B51 / 92D25 / 35K57
Key words: zero-stabilizability / state constraints / comparison result / principal eigenvalue / feedback control / population dynamics / reaction-diffusion system
© EDP Sciences, 2014
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