  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.
