Optimization of nonlocal time-delayed feedback controllers

TL;DR

This paper develops an optimal control framework for nonlocal time-delayed feedback controllers in the Schl"ogl model, aiming to design kernels that produce desired spatio-temporal patterns with improved accuracy.

Contribution

It introduces a nonlinear optimal control approach to identify optimal kernels for Pyragas type controllers, with theoretical analysis and numerical validation.

Findings

Optimal kernels minimize the difference between the controlled solution and desired pattern.

The framework handles time-periodic target patterns effectively.

Numerical examples demonstrate the effectiveness of the proposed method.

Abstract

A class of Pyragas type nonlocal feedback controllers with time-delay is investigated for the Schl\"ogl model. The main goal is to find an optimal kernel in the controller such that the associated solution of the controlled equation is as close as possible to a desired spatio-temporal pattern. An optimal kernel is the solution to a nonlinear optimal control problem with the kernel taken as control function. The well-posedness of the optimal control problem and necessary optimality conditions are discussed for different types of kernels. Special emphasis is laid on time-periodic functions as desired patterns. Here, the cross correlation between the state and the desired pattern is invoked to set up an associated objective functional that is to be minimized. Numerical examples are presented for the 1D Schl\"ogl model and a class of simple step functions for the kernel.