Neural Operators for Robust State feedback Regulation of Hyperbolic PDEs

The recently introduced neural operator (NO) has been employed as a gain approximator in the backstepping stabilization control of hyperbolic and parabolic PDE systems. Due to the global approximation ability of the DeepONet, the NO generates the approximate spatial gain function with arbitrary accuracy. The closed-loop system stability can be ensured by the backstepping controller involving the approximate gain with sufficiently small error. In this paper, we extend the NO theory to a more complex field of robust regulation control for a class of hyperbolic PDE systems. The NO is trained offline on a dataset containing a sufficient number of prior solutions of the kernel equation, so as to generate feedback gain for the robust regulator. Based on the internal model principle, the regulator is inherently robust to the parameter uncertainty and the error in approximate gain. Therefore, the tracking error can still converge to 0 if the extended regulator equation is solvable. We provide a series of theory proofs of the regulation problem and a numerical test under the approximate gain. It is shown that NO is almost three orders of magnitude faster than PDE solver in generating kernel function, and the loss remains on the order of 10 -4 . This provides an opportunity to use the NO methodology for accelerated gain scheduling regulation for PDEs with time-varying parameters.