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

~Received 31 December 1997! 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 Poincaremaps. 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.@S1063-651X~99!05201-0# Reaction-diffusion ~RD! systems arise frequently in the study of chemical and biological phenomena and are natu- rally modeled by parabolic partial differential equations ~PDEs!. The dynamics of RD systems has been the subject of intense research activity over the past decades. The reason is that RD systems exhibit very rich dynamic behavior includ- ing periodic and quasiperiodic solutions and chaos ~see, for example, @1-4# for some results and reference lists!. A RD system whose dynamics has been studied exten- sively is the Brusselator reaction scheme in one- and two- dimensional domains. In @5-7# extensive bifurcation studies of the Brusselator model showed that the system exhibits very rich dynamic behavior for different regions in the pa- rameter space. In @8# the dynamics of the Brusselator model with Dirichlet boundary conditions was studied using the length of the domain as the bifurcation parameter and evi- dence of chaotic behavior was presented. Theoretical justifi- cation of the existence of aperiodic solutions based on bifur- cation theory was presented in @9#, while secondary and homoclinic bifurcations were analyzed in @10-12# for the Brusselator reaction scheme for both one- and two- dimensional domains. Even though the analysis of complex dynamics of RD systems has been a research subject for more than 30 years, the use of feedback control to supress complex dynamics of RD systems and the study of the dynamics of RD systems under feedback control have been addressed only recently. In @13# a linear adaptive control strategy was applied to the Gray-Scott model in order to control the formation of pat- terns in a one-dimensional domain, while in @14# an experi- mental application of linear modal feedback control for sup- pressing chaotic temporal fluctuations of spatiotemporal thermal patterns on a catalytic wafer was reported. In @8# a feedback controller based on the singular value decomposi- tion of the spatial differential operator was used to control complex dynamics of the Brusselator model and a detailed bifurcation analysis of the closed-loop system was per- formed. Finally, the effect of linear modal feedback control on the open-loop dynamics of the FitzHugh-Nagumo model was studied in @15#. In this paper we present a study of the dynamics of the RD Brusselator model with Neumann and Dirichlet bound- ary conditions, under nonlinear and linear 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 re- actants. The main objective of the study, which is based on methods of bifurcation theory and the application of Poin- caremaps, is to determine to what extent the rich dynamic behavior exhibited by the open-loop system is suppressed by the use of feedback control. The paper is structured as fol- lows. Initially, the process model is presented and an analy- sis of the open-loop system is performed to determine values of the bifurcation parameter for which Hopf bifurcations oc- cur. Then the nonlinear modal feedback controller is synthe- sized and the closed-loop system is analyzed. Finally, the projections of the Poincaremaps on suitably chosen planes of the open-loop and closed-loop systems, for various values of the bifurcation parameter, as well as the closed-loop out- put responses and spatial profiles of the process states under nonlinear and linear control, are presented and compared.