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.

翻译：非线性算子的深度神经网络逼近（通常称为DeepONet）已被证明能够逼近偏微分方程（PDE）的反步设计，其中单个Goursat形式的PDE控制单个反馈增益函数。在耦合PDE的边界控制中，耦合的Goursat形式PDE控制两个或更多增益核——这是一种迄今为止DeepONet尚未涉及的PDE结构。本文通过考虑一个简单的反向对流$2\times 2$耦合系统，探讨了为双曲型PDE对象逼近增益核PDE系统的问题，在该系统的控制中出现了Goursat形式的$2\times 2$核PDE系统。工程应用包括石油钻井、浅水波的Saint-Venant模型以及拥堵交通流中起停不稳定性的Aw-Rascle-Zhang模型。我们建立了从总共五个对象PDE函数系数到核PDE解的映射的连续性，证明了存在一个可以任意逼近核PDE的DeepONet近似，并确保当替换精确的反步增益核时，DeepONet逼近的增益能够保证稳定性。考虑到反配置的边界驱动与传感，我们基于$L^2$全局指数稳定（GES）的近似增益核输出反馈设计意味着对控制器增益和观测器增益的深度学习。此外，将输出反馈律编码到DeepONet中确保了半全局实际指数稳定性（SG-PES）。DeepONet算子将控制器增益的计算速度提高了数个数量级。其理论证明的稳定能力通过仿真得到了验证。