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 note, we open 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 systems in Goursat form arises. Applications include oil drilling, Saint-Venant model of shallow water waves, and Aw-Rascle-Zhang model of stop-and-go instability in congested traffic flow. In this paper 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 establish that the DeepONet-approximated gains guarantee stabilization when replacing the exact backstepping gain kernels. Taking into account anti-collocated boundary actuation and sensing, our $L^2$\emph{-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 \emph{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.

翻译：深度神经网络对非线性算子的逼近（通常称为DeepONet）已被证明能够逼近偏微分方程反步设计，其中单个Goursat型偏微分方程控制单个反馈增益函数。在耦合偏微分方程的边界控制中，耦合的Goursat型偏微分方程控制两个或多个增益核——这一偏微分方程结构迄今尚未被DeepONet所涉及。本文通过考虑一个简单的对向对流$2\times 2$耦合系统，开启了双曲型偏微分方程植物中增益核偏微分方程组近似这一课题，在该系统的控制中出现了Goursat形式的$2\times 2$核偏微分方程组。应用领域包括石油钻井、浅水波的Saint-Venant模型以及拥挤交通流中停走不稳定性的Aw-Rascle-Zhang模型。本文建立了从植物偏微分方程功能系数（共五个）到核偏微分方程解的映射的连续性，证明了DeepONet对核偏微分方程存在任意精度的逼近，并验证了基于DeepONet逼近的增益在替代精确反步增益核时能够保证镇定。考虑非同位边界驱动与传感，我们基于$L^2$全局指数镇定（GES）的近似增益核输出反馈设计隐含了控制器和观测器增益的深度学习。此外，将输出反馈律编码到DeepONet中确保了半全局实用指数稳定性（SG-PES）。DeepONet算子将控制器增益的计算速度提升了多个数量级。其理论证明的镇定能力通过仿真得到了验证。