Analytical, Optimal, and Sparse Optimal Control of Traveling Wave Solutions to Reaction-Diffusion Systems

TL;DR

This paper explores analytical, optimal, and sparse optimal control methods for managing traveling wave solutions in reaction-diffusion systems, demonstrating their effectiveness through models like Schl"{o}gl and FitzHugh-Nagumo.

Contribution

It introduces a novel sparse optimal control approach that produces localized control signals and analyzes second order optimality conditions.

Findings

Analytical solutions for pattern control are derived.

Sparse optimal control yields highly localized control signals.

Second order optimality conditions are established for the sparse approach.

Abstract

This work deals with the position control of selected patterns in reaction-diffusion systems. Exemplarily, the Schl\"{o}gl and FitzHugh-Nagumo model are discussed using three different approaches. First, an analytical solution is proposed. Second, the standard optimal control procedure is applied. The third approach extends standard optimal control to so-called sparse optimal control that results in very localized control signals and allows the analysis of second order optimality conditions.