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

TL;DR

The paper addresses stabilization of a first-order PIDE with spatially varying state delay by combining PDE backstepping with a neural-operator (DeepONet) controller. A single DeepONet is trained to approximate the delay-dependent, two-branch backstepping feedback operator, and its Lipschitz continuity is established to guarantee a provable approximation quality. The control law preserves semiglobal practical stability: as the DeepONet approximation error $\epsilon$ tends to zero, the closed-loop state decays exponentially up to a small residual $\mathcal{O}(\epsilon^2)$. Numerical results show accurate approximation (loss ~ $5.89\times 10^{-4}$), significant computational speedups (at least 11x), and robustness to noisy delays, indicating real-time applicability for spatially varying delays in PDE control.

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.

Figures (9)

• Figure 1: The sketch of plug-flow tubular reactor with recycle.
• Figure 2: Integration area of $\mathcal{K}(\tau_1)-\mathcal{K}(\tau_2)$ and $\mathcal{K}_2(\tau_1)-\mathcal{K}_2(\tau_2)$.
• Figure 3: The neural operator training framework for the delay compensated controller.
• Figure 4: Closed-loop state $x(s,t)$ with initial condition $x_0=5\cos{(4\cos^{-1}(s-0.2))}$. Left: $\tau(s)=3+0.5 \cos{(5\cos^{-1}(s)}) \in \mathcal{D}_1$. Right: $\tau(s)=0.5e^{-1.6s} \in \mathcal{D}_2$. Top to bottom: states with the backstepping controller, NO-based controller, and NO-based controller for the delay with measurement noise (Gaussian noise $\mathcal{N}(0, \sigma^2)$).
• Figure 5: From top to bottom: control input $U(t)$, state $x(s,t)$, and state error between neural operator controllers and the backstepping controller. Results are shown for DeepONet ('DN'), FNO ('FN'), DeepONet with noisy delay ('DN$_\mathrm{d}$'), and FNO with noisy delay ('FN$_\mathrm{d}$'). Left and right panels correspond to $\tau \in \mathcal{D}_1$ and $\tau \in \mathcal{D}_2$, respectively.

Theorems & Definitions (22)

• Remark 1
• Remark 2
• Definition 1
• Definition 2
• Lemma 1
• Lemma 2
• Lemma 3
• proof
• Theorem 1
• Remark 3