State Feedback Regulation of ODE-Parabolic Cascade Systems via Neural Operators

In this paper, we consider the neural operator (NO)-based state feedback regulation for the ordinary differential equation (ODE)-parabolic partial integro differential equation (PIDE) cascade systems with spatially varying in-domain coefficients, in which both the body equations and the uncontrolled end are subject to disturbances. The feedback regulator is constructed via the backstepping method, and the design procedure is significantly accelerated by NOs. DeepONet, a representative NO designed for learning nonlinear operators, has shown considerable promise in approximating backstepping-based controllers for PDEs. Our approach demonstrates that DeepONet generates the kernel functions with a loss on the order of 10^-3, nearly two orders of magnitude faster than conventional PDE solvers. By integrating DeepONet-approximated kernels into the feedback regulator, Lyapunov-based analysis rigorously confirms that the system output exponentially tracks the reference trajectory.