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

TL;DR

This work extends PDE backstepping control to first-order hyperbolic PIDEs with state and measurement delays by learning the backstepping kernels and observer gains with DeepONet. By offline-training three neural operators for the state-feedback kernels $K,L,J$ and two for the observer gains $Q_1,Q_2$, the method provides provable approximation accuracy in infinite dimensions and ensures exponential stability of the closed-loop systems under neural gains. Theoretical results establish Lipschitz continuity of the kernel operators and stability bounds for state-feedback, observer, and output-feedback, confirming a neural-operator version of the separation principle. Numerical experiments demonstrate large computational speedups (up to tens of times faster than PDE solvers) while maintaining accurate control performance, with kernel and observer gains reaching orders of magnitude $10^{-3}$ and $10^{-2}$ respectively. These findings enable real-time, data-driven backstepping control for complex delayed PDEs and point to future work on delay-adaptive control and higher-dimensional extensions.

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.

Figures (10)

• Figure 1: The neural operator learning framework for backstepping delay compensation control.
• Figure 2: The DeepONet structure for kernel $K$.
• Figure 3: (a) The loss of the neural control kernel for $K$. (b) The loss of the neural control kernels for $L$ and $J$. (c) The loss of neural observer gains.
• Figure 4: The first row shows the kernel functions $K(s,q)$, the learned kernel functions $\hat{K}(s,q)$ and the errors $K(s,q)-\hat{K}(s,q)$. The second row shows the kernel functions $L(s)$, the learned kernel functions $\hat{L}(\phi)$ and the errors $L(\phi)-\hat{L}(\phi)$. The last row shows the kernel functions $J(\sigma)$, the learned kernel functions $\hat{J}(\sigma)$ and the errors $J(\sigma)-\hat{J}(\sigma)$.
• Figure 5: The first row shows the analyzed control gains $K(0,q)$, $L(hr)$, $J(\eta r)$, and the learned control gains $\hat{K}(0,q)$, $\hat{L}(hr)$, $\hat{J}(hr)$. The last row shows the errors $K_1(0,q)- \hat{K}_1(0,q)$, $L(hr)- \hat{L}(hr)$, $J(\eta r)- \hat{J}(\eta r)$.

Theorems & Definitions (16)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1
• Definition 2
• Lemma 1
• proof
• Lemma 2
• Theorem 4
• Theorem 5