On Certain Control Problems for a Class of Singularly Perturbed Parabolic Equations

Adelaida B. Vasil'eva
Department of Physics,
Moscow State University,
Moscow, 119899 Russia
E-mail: abvas@mathabv.phys.msu.su

Leonid Kalachev
Leonid V. Kalachev Department of Mathematical Sciences,
University of Montana,
Missoula, MT 59812, USA
E-mail: kalachev@mso.umt.edu

and

Alexander A. Plotnikov
Department of Physics,
Moscow State University,
Moscow, 119899 Russia

Abstract

We consider singularly perturbed parabolic equations that have alternating boundary layer type solutions: two boundary layer type solutions (upper and lower) may exist, and at certain instants of time the switching between these two solutions is observed. The problem formulations for these parabolic equations contain parameters that can be chosen to regulate the duration of periods corresponding to upper solution and to lower solution stages. We present a biological model for which it is possible to choose the values of control parameters in such a way that only one of the mentioned above boundary layer type solutions persists.

Keywords: singular perturbations, parabolic equations, boundary function method, Dirichlet boundary conditions, control

AMS Subject Classification:34E10, 35B05, 35B25