Analysis of front interaction and control in stationary patterns of reaction-diffusion systems

We have analyzed the stability of one-dimensional patterns in one- or two-variable reaction-diffusion systems, by analyzing the interaction between adjacent fronts and between fronts and the boundaries in bounded systems. We have used model reduction to a presentation that follows the front positions while using approximate expressions for front velocities, in order to study various control modes in such systems. These results were corroborated by a few numerical experiments. A stationary single front or a pattern with n fronts is typically unstable due to the interaction between fronts. The two simplest control modes, global control and point-sensor control (pinning), will arrest a front in a single-variable problem since both control modes, in fact, respond to the front position. In a two-variable system incorporating a localized inhibitor, in the domain of bistable kinetics, global control 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 control modes can stabilize an n-front pattern, in either one- or two-variable systems, and that task calls for a distributed actuator. A single space-dependent actuator that is spatially qualitatively similar to the patterned setpoint, and which responds to the sum of deviations in front positions, may stabilize a pattern that approximates the desired state. The deviation between the two may be sufficiently small to render the obtained state satisfactory. An extension of these results to diffusion-convection-reaction systems are outlined.