Robust stabilization of hyperbolic PDE-ODE systems via Neural Operator-approximated gain kernels

TL;DR

This paper presents a neural operator-based control design for stabilizing coupled hyperbolic PDE-ODE systems with Markov jump parameters, significantly speeding up computations while ensuring robustness and stability.

Contribution

It introduces a DeepONet framework to learn backstepping kernels, enabling efficient and robust stabilization of stochastic PDE-ODE systems with reduced computational complexity.

Findings

Achieves over 100x speedup compared to traditional solvers.

Maintains high accuracy and robust stability under stochastic switching.

Proves stability of the neural operator-based controller using Lyapunov analysis.

Abstract

This paper investigates the mean square exponential stabilization problem for a class of coupled PDE-ODE systems with Markov jump parameters. The considered system consists of multiple coupled hyperbolic PDEs and a finite-dimensional ODE, where all system parameters evolve according to a homogeneous continuous-time Markov process. The control design is based on a backstepping approach. To address the computational complexity of solving kernel equations, a DeepONet framework is proposed to learn the mapping from system parameters to the backstepping kernels. By employing Lyapunov-based analysis, we further prove that the controller obtained from the neural operator ensures stability of the closed-loop stochastic system. Numerical simulations demonstrate that the proposed approach achieves more than two orders of magnitude speedup compared to traditional numerical solvers, while maintaining high accuracy and ensuring robust closed-loop stability under stochastic switching.