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

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.