Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

DeepONet of dynamic event-triggered backstepping boundary control for reaction-diffusion PDEs

Hongpeng Yuan, Ji Wang, Mamadou Diagne

TL;DR

This paper advances boundary control for a reaction-diffusion PDE by integrating backstepping with DeepONet to learn the backstepping kernels and associated gains, enabling a neural-operator-based, event-triggered control scheme. By approximating the kernel operator, the authors design a dynamic event-triggering mechanism that updates the boundary actuation without inducing Zeno behavior, while preserving exponential stability in the $L^2$ sense. Theoretical results establish a positive dwell time, compute a Lyapunov-based decay rate, and show convergence of the state and triggering variable under small approximation error $\iota$, with a comprehensive numerical demonstration. Overall, the work demonstrates computational efficiency and robustness to parameter variation in infinite-dimensional ETC design for reaction-diffusion systems, through an explicit NO-augmented backstepping framework.

Abstract

We present an event-triggered boundary control scheme for a class of reaction-diffusion PDEs using operator learning and backstepping method. Our first-of-its-kind contribution aims at learning the backstepping kernels, which inherently induces the learning of the gains in the event trigger and the control law. The kernel functions in constructing the control law are approximated with neural operators (NOs) to improve the computational efficiency. Then, a dynamic event-triggering mechanism is designed, based on the plant and the continuous-in-time control law using kernels given by NOs,to determine the updating times of the actuation signal. In the resulting event-based closed-loop system, a strictly positive lower bound of the minimal dwell time is found, which is independent of initial conditions. As a result, the absence of a Zeno behavior is guaranteed. Besides, exponential convergence to zero of the L_2 norm of the reaction-diffusion PDE state and the dynamic variable in the event-triggering mechanism is proved via Lyapunov analysis. The effectiveness of the proposed method is illustrated by numerical simulation.

DeepONet of dynamic event-triggered backstepping boundary control for reaction-diffusion PDEs

TL;DR

This paper advances boundary control for a reaction-diffusion PDE by integrating backstepping with DeepONet to learn the backstepping kernels and associated gains, enabling a neural-operator-based, event-triggered control scheme. By approximating the kernel operator, the authors design a dynamic event-triggering mechanism that updates the boundary actuation without inducing Zeno behavior, while preserving exponential stability in the sense. Theoretical results establish a positive dwell time, compute a Lyapunov-based decay rate, and show convergence of the state and triggering variable under small approximation error , with a comprehensive numerical demonstration. Overall, the work demonstrates computational efficiency and robustness to parameter variation in infinite-dimensional ETC design for reaction-diffusion systems, through an explicit NO-augmented backstepping framework.

Abstract

We present an event-triggered boundary control scheme for a class of reaction-diffusion PDEs using operator learning and backstepping method. Our first-of-its-kind contribution aims at learning the backstepping kernels, which inherently induces the learning of the gains in the event trigger and the control law. The kernel functions in constructing the control law are approximated with neural operators (NOs) to improve the computational efficiency. Then, a dynamic event-triggering mechanism is designed, based on the plant and the continuous-in-time control law using kernels given by NOs,to determine the updating times of the actuation signal. In the resulting event-based closed-loop system, a strictly positive lower bound of the minimal dwell time is found, which is independent of initial conditions. As a result, the absence of a Zeno behavior is guaranteed. Besides, exponential convergence to zero of the L_2 norm of the reaction-diffusion PDE state and the dynamic variable in the event-triggering mechanism is proved via Lyapunov analysis. The effectiveness of the proposed method is illustrated by numerical simulation.

Paper Structure

This paper contains 11 sections, 5 theorems, 67 equations, 4 figures.

Table of Contents

  1. INTRODUCTION
  2. Problem Formulation and Continuous-in-time Control Design
  3. Problem statement
  4. Continuous-in-time Control Design
  5. NO-approximated Continuous-in-time Control Design
  6. Event-Triggered NO-approximated Boundary Control Design
  7. Main Result
  8. Proof of property $1$ of Theorem $\ref{['theo1']}$
  9. Proof of property $2$ of Theorem $\ref{['theo1']}$
  10. Numerical Simulations
  11. Conclusion

Key Result

Theorem 1

(DeepONet universal approximation theorem) Let $X \subset \mathbb{R}^{d_x}, Y \subset \mathbb{R}^{d_y}$ be compact sets of vectors $x \in X$ and $y \in Y, d_x, d_y \in \mathbb{N}$. Let $\mathcal{U}: X \rightarrow$$\mathbb{U} \subset \mathbb{R}^{d_u}$ and $\mathcal{V}: Y \rightarrow \mathbb{V} \subse for all functions $\mathbf{u} \in \mathcal{U}$ and all values $y \in Y$ of $\mathcal{G}(\mathbf{u})

Figures (4)

  • Figure 1: Block diagram of the closed-loop system.
  • Figure 2: Results of the kernel $k(x, y)$, learned kernel $\hat{k}(x,y)$, and the kernel error $k(x,y)-\hat{k}(x,y)$.
  • Figure 3: Results for open-loop system.
  • Figure 4: Results under the event-triggered NO-approximated control input $U_d$.

Theorems & Definitions (8)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Lemma 2
  • proof