#### Title

Zero-stabilization for some diffusive models with state constraints

#### Document Type

Article

#### Publication Date

1-1-2014

#### 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. © EDP Sciences, 2014.

#### Recommended Citation

Hritonenko, N., Marinoschi, G., Swierniak, A., Aniţa, S., & Aniţa, S. (2014). Zero-stabilization for some diffusive models with state constraints. Retrieved from https://digitalcommons.pvamu.edu/mathematics-facpubs/21