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

Zhang, Yihuai; Auriol, Jean; Yu, Huan · 2024 · ArXiv.org

DOI: 10.48550/arxiv.2412.09019

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Summary

This paper addresses the robust stabilization of linear hyperbolic Partial Differential Equations (PDEs) subject to Markov-jumping parameter uncertainty. The research is motivated by the high computational cost associated with traditional backstepping control methods, which require solving complex kernel equations for each specific set of system parameters. While machine learning approaches like Physics-Informed Neural Networks (PINNs) and Reinforcement Learning (RL) have been applied to PDE control, they often lack theoretical stability guarantees or struggle with generalization under stochastic parameters. The authors propose using Neural Operators (NOs) to approximate the backstepping kernels, aiming to improve computational efficiency while ensuring mean-square exponential stability despite parameter uncertainties and approximation errors. The study focuses on a $2 \times 2$ heterogeneous hyperbolic PDE system where characteristic speeds, in-domain couplings, and boundary couplings are stochastic variables governed by independent Markov chains. The methodology involves two main steps. First, the authors establish the theoretical foundation by proving that the mapping from nominal system parameters to the exact backstepping kernels is locally Lipschitz continuous. This property ensures that a Neural Operator can approximate the kernel mapping with arbitrary precision. Second, they derive stability conditions for the closed-loop system using Lyapunov analysis. The control law utilizes the NO-approximated kernels to compute boundary feedback. The analysis accounts for two sources of uncertainty: the deviation of stochastic parameters from their nominal values and the approximation error introduced by the Neural Operator. The main finding is that the closed-loop system achieves mean-square exponential stability provided that the stochastic parameters remain sufficiently close to the nominal parameters on average and the Neural Operator approximation error is small enough. The authors provide explicit bounds for these uncertainties, demonstrating that the stability margin depends on the magnitude of the parameter jumps and the accuracy of the NO approximation. Unlike previous results that required specific constraints on boundary coupling coefficients for all switching modes, this approach does not impose such restrictions, offering a more general stability guarantee. The theoretical results are validated through numerical simulations applied to a freeway traffic control problem under stochastic upstream demands, demonstrating the effectiveness of the NO-based controller in regulating traffic flow. The significance of this work lies in being the first to theoretically establish the use of operator learning for the robust control of linear Markov-jumping hyperbolic PDEs. It bridges the gap between efficient data-driven approximation methods and rigorous control theory by providing stability guarantees for NO-approximated controllers. The approach extends previous Lyapunov analysis techniques to encompass Neural Operator approximations, offering a computationally efficient alternative to solving kernel equations in real-time. This has practical implications for engineering applications involving hyperbolic PDEs, such as traffic flow management, oil drilling, and gas pipeline control, where rapid adaptation to stochastic parameter changes is critical.

Key finding

The proposed neural operator-approximated backstepping controller guarantees mean-square exponential stability for linear Markov-jumping hyperbolic PDEs, provided that the stochastic parameters remain sufficiently close to nominal values on average and the neural operator approximation errors are sufficiently small.

Methodology

simulation_modeling

Provenance

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