# Control of traveling localized spots

## TL;DR

This paper develops an analytical open-loop control method to guide stable traveling localized spots in reaction-diffusion systems along desired paths without altering their shape, confirmed by numerical optimal control solutions.

## Contribution

It provides an explicit analytical control strategy based on Goldstone modes for guiding spots, bypassing complex reaction kinetics details and enhancing computational efficiency.

## Key findings

- Control signal expressed analytically in terms of Goldstone modes.
- Numerical confirmation of the control's optimality.
- Analytical expressions serve as effective initial guesses for optimal control.

## Abstract

Traveling localized spots represent an important class of self-organized two-dimensional patterns in reaction-diffusion systems. We study open-loop control intended to guide a stable spot along a desired trajectory with desired velocity. Simultaneously, the spot's concentration profile does not change under control. For a given protocol of motion, we first express the control signal analytically in terms of the Goldstone modes and the propagation velocity of the uncontrolled spot. Thus, detailed information about the underlying nonlinear reaction kinetics is unnecessary. Then, we confirm the optimality of this solution by demonstrating numerically its equivalence to the solution of a regularized, optimal control problem. To solve the latter, the analytical expressions for the control are excellent initial guesses speeding-up substantially the otherwise time-consuming calculations.