Analysis of front interaction and control in stationary patterns of reaction-diffusion systems - art. no. 056120

We have analyzed the stability of one-dimensional patterns in one- or two-v ariable reaction-diffusion systems, by analyzing the interaction between ad jacent fronts and between fronts and the boundaries in bounded systems. We have used model reduction to a presentation that follows the front position s while using approximate expressions for front velocities, in order to stu dy various control modes in such systems. These results were corroborated b y a few numerical experiments. A stationary single front or a pattern with n fronts is typically unstable due to the interaction between fronts. The t wo simplest control modes, global control and point-sensor control (pinning ), will arrest a front in a single-variable problem since both control mode s, in fact, respond to the front position. In a two-variable system incorpo rating a localized inhibitor, in the domain of bistable kinetics, global co ntrol may stabilize a single front only in short systems while point-sensor control can arrest such a front in any system size. Neither of these contr ol modes can stabilize an n-front pattern, in either one- or two-variable s ystems, and that task calls for a distributed actuator. A single space-depe ndent actuator that is spatially qualitatively similar to the patterned set point, and which responds to the sum of deviations in front positions, may stabilize a pattern that approximates the desired state. The deviation betw een the two may be sufficiently small to render the obtained state satisfac tory. An extension of these results to diffusion-convection-reaction system s are outlined.