Backstepping neural operators for 2 × 2 hyperbolic PDEs

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 hyperbolic PDEs, coupled Goursat-form PDEs govern two or more gain kernels — a structure unaddressed thus far with DeepONet. In this contribution, we open the subject of approximating systems of gain kernel PDEs by considering a counter-convecting 2 × 2 hyperbolic system whose backstepping boundary controller and observer gains are the solutions to 2 × 2 kernel PDE systems in Goursat form. We establish the continuity of the mapping from (a total of five) functional coefficients of the plant to the kernel PDEs solutions, prove the existence of an arbitrarily close DeepONet approximation to the kernel PDEs, and ensure that the DeepONet-based approximated gains guarantee stabilization when replacing the exact backstepping gain kernel functions. Taking into account anti-collocated boundary actuation and sensing, our $L^2$-globally-exponentially stabilizing (GES) control law requires the deep learning of both the controller and the observer gains. Moreover, the encoding of the feedback law into DeepONet ensures semi-global practical exponential stability (SG-PES), as established in our result. The neural operators (NOs) speed up the computation of both controller and observer gains by multiple orders of magnitude. Its theoretically proved stabilizing capability is demonstrated through simulations.