Neural Operator Feedback for a First-Order PIDE with Spatially-Varying State Delay

TL;DR

This paper develops a neural operator approach to approximate a complex two-branch feedback law for a PDE with spatially-varying delay, demonstrating its stability and effectiveness through numerical experiments.

Contribution

It introduces a neural operator method for a novel two-branch feedback law in PDE control with spatially-varying delay, proving stability.

Findings

Neural operator accurately approximates the two-branch feedback law.

The proposed method ensures semiglobal practical stability.

Numerical results confirm effective training and stabilization.

Abstract

A transport PDE with a spatial integral and recirculation with constant delay has been a benchmark for neural operator approximations of PDE backstepping controllers. Introducing a spatially-varying delay into the model gives rise to a gain operator defined through integral equations which the operator's input -- the varying delay function -- enters in previously unencountered manners, including in the limits of integration and as the inverse of the `delayED time' function. This, in turn, introduces novel mathematical challenges in estimating the operator's Lipschitz constant. The backstepping kernel function having two branches endows the feedback law with a two-branch structure, where only one of the two feedback branches depends on both of the kernel branches. For this rich feedback structure, we propose a neural operator approximation of such a two-branch feedback law and prove the approximator to be semiglobally practically stabilizing. With numerical results we illustrate the training of the neural operator and its stabilizing capability.