Neural Operator Learning for Delay Compensation Control in Backstepping of 2 × 2 Hyperbolic PDEs

To stabilize a 2 × 2 first-order hyperbolic partial differential equation system with input delay, a backstepping-based feedback controller requires kernel functions as control gains. These gains are obtained by mapping system parameters into spatial-domain functions via nonlinear operators. Due to the delay, multiple layers of mapping exist among the kernel functions, increasing the complexity of computing the control gains. Neural operators efficiently approximate these kernel functions without explicitly solving the associated equations. In this paper, we develop DeepONet-based neural operators and show that the closed-loop system remains exponentially stable with adjustable decay rates when using approximate gains. Numerical simulations illustrate the theoretical results and demonstrate the robustness of the closed-loop system with neural operators. Additionally, the simulations quantify a computational speedup of at least two orders of magnitude compared to traditional PDE solvers.