Using weighted global control for stabilizing patterned states

Abstract

A general method to control a desired patterned state in reaction-diffusion processes is presented. Weighted global control is aimed to keep weighted spatially averaged properties of state variable at preset values. It is shown that weighted global control creates a stable direction in the global space of system states and affects system dynamics globally. We apply it for a specific two-component reaction-diffusion system and show that the desired pattern is attainable for a wide range of the control parameters. (c) 1999 American Institute of Physics.

Controlling Front Position in Catalytic Diffusion−Convection−Reaction Systems

“…In the catalytic wire problem, because the source function P ( y ) is bistable within a certain domain, we replace it with a cubic source function that shows qualitatively similar behavior, and after redefining the time and length scales and the variable ( y ), we arrive at the reaction−diffusion equation above (with V = 0). The control of this problem has been extensively investigated. , …”

“…The stability analysis is based on model reduction to a model that follows the front positions using approximate solutions of the front velocity and front interaction. This approach is different from the traditional approach in chemical engineering problems of control of distributed systems, which typically employs a finite, preferably small, discretization of the underlying partial differential equations (PDEs) typically using trigonometric series and applying the Galerkin method. , Such a general approach was used for the stabilization of front-like solutions of reaction−diffusion and reaction−diffusion−convection systems, , but it does not use the specific wave property of the front-like solutions and, hence, it does not provide any insight in designing the control system.…”

Control strategies for front stabilization in a tubular reactor model

“…Thus, we restrict our discussion to the case + ( z ) = 1 and use time-dependent control only, u(z, t ) = u(t), with the technically simple sensors: a global control, in which a sensor responds to a spatially averaged property that can be measured by input and output fluxes, and point-sensor control, in which the deviation from a preset value is measured at a single point. The global control approach was tested for front stabilization in a reaction-diffusion system (Middya et al, 1994;Panfilov and Sheintuch, 1999). While it is easy to implement, it may be not sensitive enough for front stabilization.…”

“…When a second variable is added, the front can be destabilized, resulting in a rich variety of patterns (Middya et al, 1994). Several ways for stabilizing such a front were suggested (Shvartsman and Kevrekidis, 1998;Panfilov and Sheintuch, 1999).…”

Robust Control of Stationary Planar Fronts in Reaction-Diffusion Systems

“…The instabilities emerge due to the thermal effects in exothermic reactions, due to self-inhibition by a reactant and due to slow reversible modifications of the surface. The construction of a controller that enables to stabilize some inhomogeneous solutions in one-dimensional (1-D) reaction-diffusion and reactionconvection-diffusion systems is currently a subject of intensive investigation [11,12,13,14,15,16,17]. Yet, most catalytic reactors, as well as physiological systems like the heart, exhibit a behaviour that can be described by two-or even three-dimensional models.…”