Equivariant Pyragas control of discrete waves

TL;DR

This paper develops a method to stabilize discrete wave patterns with symmetries using equivariant Pyragas control, extending control techniques to complex spatio-temporal systems with broad applicability.

Contribution

It provides sufficient conditions for stabilizing discrete waves with symmetries using an adapted Pyragas control method, including systems far from bifurcation points.

Findings

Derived stability conditions for discrete waves

Extended Floquet theory to symmetric systems

Reduced eigenvalue problems to zero-finding tasks

Abstract

Equivariant Pyragas control is a delayed feedback method that aims to stabilize spatio-temporal patterns in systems with symmetries. In this article, we apply equivariant Pyragas control to discrete waves, which are periodic solutions that have a finite number of spatio-temporal symmetries. We prove sufficient conditions under which a discrete wave can be stabilized via equivariant Pyragas control. The result isapplicable to a broad class of discrete waves, including discrete waves that are far away from a bifurcation point. Key ingredients of the proof are an adaptation of Floquet theory to systems with symmetries, and the use of characteristic matrix functions to reduce the infinite dimensional eigenvalue problem to a one dimensional zero finding problem.