Output feedback stabilization of an ODE-heat cascade system by neural operator approximations

In this paper, we consider the output feedback stabilization of an ordinary differential equation (ODE)-heat cascade system with a variable coefficient reaction term. We design the boundary feedback controller by backstepping method, where the control design is accelerated by neural operators. For backstepping kernel functions involving spatial variables, it is difficult to obtain the analytical solutions and time-consuming to compute the numerical solutions. In order to solve this problem, we use neural operator learning framework to accelerate the generation of approximate kernel functions, and then obtain the feedback controller. Specifically, we give the continuity and boundedness of the kernel partial differential equations (PDEs) and establish the nonlinear mapping of the reaction coefficient to the kernel functions. Through DeepONet approximation of nonlinear operator, we prove the existence of kernel PDEs under DeepONet arbitrary accuracy approximation. Then we design the DeepONet-approximated observer and output feedback controller, and demonstrate the output feedback stability of the closed-loop system under DeepONet approximations. Numerical simulations verify the effectiveness of the controller and illustrate that this method is two orders of magnitude faster than PDE solvers.