Controlling the position of traveling waves in reaction-diffusion systems

TL;DR

This paper introduces a method to control the position of traveling waves in reaction-diffusion systems over time, using integral equations derived perturbatively, applicable to various protocols and systems.

Contribution

The authors develop a general analytical framework for controlling traveling wave positions in reaction-diffusion systems, including explicit control functions and solutions to integral equations.

Findings

Control functions closely match numerically optimized controls

Analytical expressions valid for arbitrary protocols

Control expressed using wave profile and velocity

Abstract

We present a method to control the position as a function of time of one-dimensional traveling wave solutions to reaction-diffusion systems according to a pre-specified protocol of motion. Given this protocol, the control function is found as the solution of a perturbatively derived integral equation. Two cases are considered. First, we derive an analytical expression for the space ($x$) and time ($t$) dependent control function $f\left(x,t\right)$ that is valid for arbitrary protocols and many reaction-diffusion systems. These results are close to numerically computed optimal controls. Second, for stationary control of traveling waves in one-component systems, the integral equation reduces to a Fredholm integral equation of the first kind. In both cases, the control can be expressed in terms of the uncontrolled wave profile and its propagation velocity, rendering detailed knowledge of the reaction kinetics unnecessary.