Neural Operators for PDE Backstepping Control of First-Order Hyperbolic PIDE with Recycle and Delay

TL;DR

This paper extends the DeepONet framework to control hyperbolic PDEs with delays, using neural operators for gain approximation, and demonstrates stability and efficiency improvements through numerical simulations.

Contribution

It introduces a neural operator-based approach for PDE backstepping control of delayed hyperbolic PDEs, providing stability proofs and computational savings.

Findings

DeepONet accurately approximates gain functions for PDE control.

Stability of closed-loop system is maintained with neural operator approximations.

Numerical simulations show two orders of magnitude reduction in computational effort.

Abstract

The recently introduced DeepONet operator-learning framework for PDE control is extended from the results for basic hyperbolic and parabolic PDEs to an advanced hyperbolic class that involves delays on both the state and the system output or input. The PDE backstepping design produces gain functions that are outputs of a nonlinear operator, mapping functions on a spatial domain into functions on a spatial domain, and where this gain-generating operator's inputs are the PDE's coefficients. The operator is approximated with a DeepONet neural network to a degree of accuracy that is provably arbitrarily tight. Once we produce this approximation-theoretic result in infinite dimension, with it we establish stability in closed loop under feedback that employs approximate gains. In addition to supplying such results under full-state feedback, we also develop DeepONet-approximated observers and output-feedback laws and prove their own stabilizing properties under neural operator approximations. With numerical simulations we illustrate the theoretical results and quantify the numerical effort savings, which are of two orders of magnitude, thanks to replacing the numerical PDE solving with the DeepONet.