Backstepping Controllers Based on Neural Operators for Coupled PDE-ODE Systems

This letter explores the stabilization of a wide class of coupled dynamical systems, consisting of partial and ordinary differential equations with spatially varying coefficients. The main challenge in controlling these systems is the presence of variable spatial coefficients, which prevents the use of conventional control design methods. To address this challenge, we introduce an innovative approach employing deep neural networks, specifically DeepONet. A key advantage of this method is that it removes the need for pre-solving differential equations. The ability of DeepONet to approximate nonlinear operators is utilized to design a backstepping control strategy that ensures exponential stability of the system. The efficiency of the proposed method in stabilizing complex coupled systems is demonstrated through simulation results.