Jie Qi

Jie Qi, Jing Zhang, Miroslav Krstic

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.

Jiaqi Hu, Jie Qi, Jing Zhang

Control of distributed parameter systems affected by delays is a challenging task, particularly when the delays depend on spatial variables. The idea of integrating analytical control theory with learning-based control within a unified control scheme is becoming increasingly promising and advantageous. In this paper, we address the problem of controlling an unstable first-order hyperbolic PDE with spatially-varying delays by combining PDE backstepping control strategies and deep reinforcement learning (RL). To eliminate the assumption on the delay function required for the backstepping design, we propose a soft actor-critic (SAC) architecture incorporating a DeepONet to approximate the backstepping controller. The DeepONet extracts features from the backstepping controller and feeds them into the policy network. In simulations, our algorithm outperforms the baseline SAC without prior backstepping knowledge and the analytical controller.