Backstepping Neural Operators for $2\times 2$ Hyperbolic PDEs

TL;DR

The paper addresses stabilization of 2×2 coupled linear hyperbolic PDEs using backstepping and introduces DeepONet-based neural operators to approximate the coupled Goursat-form kernel PDEs that arise in this setting. By proving continuity of the plant-to-kernel mapping and the existence of arbitrarily accurate DeepONet approximations, it shows that learned gains can guarantee stabilization (GES) when used in boundary feedback, and semi-global practical exponential stability (SG-PES) when the full output-feedback law is learned. The approach dramatically speeds up gain computation and is validated through simulations on a representative 2×2 system, with explicit stability bounds that quantify the impact of approximation error. The work thus blends proof-based Lyapunov analysis with data-driven neural operators to enable fast, reliable neural control of infinite-dimensional systems, with potential applications in oil drilling, traffic dynamics, and fluid/flow models.

Abstract

Deep neural network approximation of nonlinear operators, commonly referred to as DeepONet, has proven capable of approximating PDE backstepping designs in which a single Goursat-form PDE governs a single feedback gain function. In boundary control of coupled PDEs, coupled Goursat-form PDEs govern two or more gain kernels-a PDE structure unaddressed thus far with DeepONet. In this paper, we explore the subject of approximating systems of gain kernel PDEs for hyperbolic PDE plants by considering a simple counter-convecting $2\times 2$ coupled system in whose control a $2\times 2$ kernel PDE system in Goursat form arises. Engineering applications include oil drilling, the Saint-Venant model of shallow water waves, and the Aw-Rascle-Zhang model of stop-and-go instability in congested traffic flow. We establish the continuity of the mapping from a total of five plant PDE functional coefficients to the kernel PDE solutions, prove the existence of an arbitrarily close DeepONet approximation to the kernel PDEs, and ensure that the DeepONet-approximated gains guarantee stabilization when replacing the exact backstepping gain kernels. Taking into account anti-collocated boundary actuation and sensing, our $L^2$-Globally-exponentially stabilizing (GES) approximate gain kernel-based output feedback design implies the deep learning of both the controller's and the observer's gains. Moreover, the encoding of the output-feedback law into DeepONet ensures semi-global practical exponential stability (SG-PES). The DeepONet operator speeds up the computation of the controller gains by multiple orders of magnitude. Its theoretically proven stabilizing capability is demonstrated through simulations.

Figures (14)

• Figure 1: Learning of the kernel functions via DeepONet and through the operator described by the mapping $(\lambda,\mu,\omega,\sigma,\theta,q)\to (k_1,k_2,m_1,m_2)$. Computing multiple solutions of kernel PDEs \ref{['eq:k_1']}--\ref{['eq:k_BC_2']} in the Goursat form for different functions $\lambda(x),\ \mu(x),\ \omega(x),\ \sigma(x),\ \theta(x)$ and parameters $q$, completes the training procedure of the Neural Operator $\hat{\mathcal{K}}$.
• Figure 2: The PDE backstepping observer \ref{['eq:sys_u-obs']}--\ref{['eq:sys_BC_2-obs']} uses boundary measurement of the flux $v(0,t)$. The gains $\hat{k}_i$ and $\hat{m}_i$, $i=1,2$ are produced with the DeepONet $\hat{\mathcal{K}}$.
• Figure 3: The learning architecture of the observer-based control law in three steps.
• Figure 4: Instability of the uncontrolled plant of $u(x,t)$ and $v(x,t)$ for given coefficients $\lambda(x) = \Gamma x+1$, $\mu(x)=e^{\Gamma x}+2$, $\delta(x)=\Gamma (x+1)$, $\theta(x)=\Gamma (x + 1)$, $\omega(x)=\Gamma(\cosh(x) + 1)$, $q=\Gamma/3$, $\Gamma=2,\ 5$.
• Figure 5: The kernel of $k_1(x,\xi)$, $\hat{k}_1(x,\xi)$ and $k_1(x,\xi)-\hat{k}_1(x,\xi)$.

Theorems & Definitions (10)

• Lemma 1
• Theorem 1
• Proposition 1
• Proposition 2
• Theorem 2
• Remark 1
• Theorem 3
• Proposition 3
• Proposition 4
• Theorem 4