Deep Learning of Delay-Compensated Backstepping for Reaction–Diffusion PDEs

With deep neural network approximations of partial differential equation (PDE) backstepping, for each new functional coefficient of the PDE plant, the gains are obtained through a function evaluation. In this article, we expand this framework to control of cascaded PDE systems from distinct classes: a reaction–diffusion plant, which is a parabolic PDE, with input delay, which is a hyperbolic PDE. The DeepONet-approximated nonlinear operator for the control gain is a cascade/composition of the operators defined by one hyperbolic PDE of the Goursat form and one parabolic PDE on a rectangle, both of which are bilinear in their input functions and not explicitly solvable. For the DeepONet-approximated delay-compensated PDE backstepping controller, we guarantee exponential stability in the $L^{2}$ norm of the plant state and the $H^{1}$ norm of the input delay state.