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

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.

💡 Research Summary

This paper tackles the robust stabilization of linear hyperbolic partial differential equations (PDEs) whose coefficients jump according to a finite‑state Markov process. The authors focus on a 2 × 2 heterogeneous hyperbolic system, a canonical model for many transport phenomena (e.g., traffic flow, oil drilling, gas pipelines). The state variables (u(x,t)) and (v(x,t)) propagate with time‑varying characteristic speeds (\lambda(t)>0) and (\mu(t)>0). In‑domain couplings (\sigma^{\pm}(t)) and boundary couplings (\phi(t),\varrho(t)) are also stochastic, each modeled as an independent Markov chain with a finite set of modes and known transition rates.

Classical backstepping design for such systems requires solving a set of kernel PDEs to obtain Volterra‑type integral transformations. The resulting boundary feedback law involves four kernels (K_{uu},K_{uv},K_{vu},K_{vv}) defined on the triangular domain ({0\le\xi\le x\le1}). While the backstepping transformation guarantees exponential stability for the nominal (constant‑parameter) system, solving the kernel equations is computationally intensive and hampers real‑time implementation, especially when parameters change.

To overcome this bottleneck, the authors propose to learn the mapping from system parameters to the backstepping kernels with a neural operator (NO), such as DeepONet or Fourier Neural Operator. Lemma 1 establishes that the kernel operator (K:U\to (C^{1}(T))^{4}) is locally Lipschitz on any bounded parameter set (U). Consequently, universal approximation results for neural operators guarantee the existence of an approximator (\widehat K) that can achieve an arbitrary uniform error (\epsilon>0) (Lemma 2). The approximated control law (U_{\text{NO}}(t)) is obtained by substituting (\widehat K) into the original backstepping formula.

Lemma 3 proves well‑posedness of the closed‑loop stochastic system under the NO‑approximated kernels: for any square‑integrable initial condition and any realization of the Markov parameters, a unique solution exists and its second‑moment remains bounded. The core stability analysis builds a Lyapunov functional (V(t)=\mathbb{E}\big