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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^{2}$</tex-math></inline-formula> norm of the plant state and the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H^{1}$</tex-math></inline-formula> norm of the input delay state.