Dynamics of a reaction-diffusion system with Brusselator kinetics under feedback control

Abstract

This paper studies the dynamics of the reaction-diffusion Brusselator model with Neumann and Dirichlet boundary conditions, under linear and nonlinear modal feedback control. The bifurcation parameters are for the Neumann problem the concentration of one of the reactants and for the Dirichlet problem the diffusion coefficient of one of the reactants. The study of the dynamics of the system is based on methods of bifurcation theory and the application of Poincar\'e maps. A direct comparison of the dynamics of the open-loop and closed-loop systems establishes that the use of feedback control significantly suppresses the rich open-loop dynamics. In addition, the superiority of the nonlinear controller over a linear controller, in attenuating the effect of bifurcations on the output of the closed-loop system, and the ability of the nonlinear controller to stabilize the system states at the spatially uniform solution provided the number of manipulated inputs is sufficiently large are shown for both the Neumann and Dirichlet problems.