Operator Learning for Robust Stabilization of Linear Markov-Jumping Hyperbolic PDEs

TL;DR

This paper develops a neural operator-based control method for robustly stabilizing Markov-jumping hyperbolic PDEs, demonstrating stability under uncertainty and approximation errors with applications to traffic control.

Contribution

It introduces a neural operator approach combined with backstepping for stabilizing hyperbolic PDEs with Markov-jumping parameters, enhancing computational efficiency and robustness.

Findings

Achieves mean-square exponential stability under parameter uncertainty.

Validates the approach with numerical simulations in traffic control.

Provides stability conditions accounting for neural operator approximation errors.

Abstract

This paper addresses the problem of robust stabilization for linear hyperbolic Partial Differential Equations (PDEs) with Markov-jumping parameter uncertainty. We consider a 2 x 2 heterogeneous hyperbolic PDE and propose a control law using operator learning and the backstepping method. Specifically, the backstepping kernels used to construct the control law are approximated with neural operators (NO) in order to improve computational efficiency. The key challenge lies in deriving the stability conditions with respect to the Markov-jumping parameter uncertainty and NO approximation errors. The mean-square exponential stability of the stochastic system is achieved through Lyapunov analysis, indicating that the system can be stabilized if the random parameters are sufficiently close to the nominal parameters on average, and NO approximation errors are small enough. The theoretical results are applied to freeway traffic control under stochastic upstream demands and then validated through numerical simulations.