Optimal time delays in a class of reaction-diffusion equations

TL;DR

This paper develops an optimization framework for selecting time delays and weights in reaction-diffusion equations to best approximate a desired state, supported by theoretical proofs and numerical examples.

Contribution

It introduces a novel method for optimizing multiple time delays in reaction-diffusion equations with proven differentiability and optimality conditions.

Findings

Optimization of time delays improves approximation accuracy.

Theoretical proof of differentiability of the solution mapping.

Numerical examples demonstrate practical applicability.

Abstract

A class of semilinear parabolic reaction diffusion equations with multiple time delays is considered. These time delays and corresponding weights are to be optimized such that the associated solution of the delay equation is the best approximation of a desired state function. The differentiability of the mapping is proved that associates the solution of the delay equation to the vector of weights and delays. Based on an adjoint calculus, first-order necessary optimality conditions are derived. Numerical test examples show the applicability of the concept of optimizing time delays.