Neural Operators for PDE Backstepping Control of First-Order Hyperbolic PIDE with Recycle and Delay
This work addresses control of advanced PDEs with delays for applications in engineering, representing an incremental extension of existing operator-learning methods to a more complex class.
The authors tackled the problem of controlling a class of hyperbolic partial integro-differential equations with delays by extending the DeepONet operator-learning framework to approximate gain functions, achieving stability with provably tight accuracy and reducing numerical effort by two orders of magnitude in simulations.
The recently introduced DeepONet operator-learning framework for PDE control is extended from the results for basic hyperbolic and parabolic PDEs to an advanced hyperbolic class that involves delays on both the state and the system output or input. The PDE backstepping design produces gain functions that are outputs of a nonlinear operator, mapping functions on a spatial domain into functions on a spatial domain, and where this gain-generating operator's inputs are the PDE's coefficients. The operator is approximated with a DeepONet neural network to a degree of accuracy that is provably arbitrarily tight. Once we produce this approximation-theoretic result in infinite dimension, with it we establish stability in closed loop under feedback that employs approximate gains. In addition to supplying such results under full-state feedback, we also develop DeepONet-approximated observers and output-feedback laws and prove their own stabilizing properties under neural operator approximations. With numerical simulations we illustrate the theoretical results and quantify the numerical effort savings, which are of two orders of magnitude, thanks to replacing the numerical PDE solving with the DeepONet.