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.

翻译：近期提出的用于偏微分方程控制的DeepONet算子学习框架，从基础双曲型和抛物型偏微分方程的应用成果，拓展至包含状态与系统输出/输入双重延迟的先进双曲型方程类别。通过反步法设计产生的增益函数是非线性算子的输出，该算子将空间域上的函数映射至另一空间域上的函数，而生成增益的算子输入为偏微分方程的系数。利用DeepONet神经网络对该算子进行逼近，可证明其逼近精度可任意收紧。在获得该无穷维逼近理论结果后，我们进一步建立了采用近似增益的反馈闭环系统稳定性。除在全状态反馈条件下获得此类结果外，我们还开发了DeepONet近似的观测器与输出反馈律，并证明其在神经算子近似下的镇定性能。通过数值仿真验证理论结果，并量化计算效率提升——由于采用DeepONet替代数值求解偏微分方程，计算量降低了两个数量级。