Neural operators in backstepping control for hyperbolic partial differential equations (PDEs) in rotary drilling systems

Designing controllers for partial differential equations (PDEs), such as backstepping controllers, requires mapping system model functions into gain functions. These mappings involve infinite-dimensional nonlinear operators, typically defined through PDEs with spatial variables. For each new coefficient of the PDE, a corresponding gain function must be determined by solving these complex equations. However, this challenge can be addressed by employing a neural operator to learn and approximate the mapping efficiently. In this context, this paper focuses on stabilizing torsional vibrations caused by bit stick–slip behavior in oilwell drilling systems, which are described by a damped wave equation. To achieve this, the second-order torsional vibration dynamics are transformed into a coupled system of $2\times 2$ first-order PDEs. Building on this formulation, we verify the continuity of the mapping from the plant PDE coefficients to the solutions of the kernel PDEs and show that a DeepONet approximation closely mirrors the exact kernel PDEs. Furthermore, we demonstrate that the DeepONet-approximated gains ensure system stabilization, effectively replacing the exact backstepping gain kernels. The stabilizing capability of the proposed approach is supported by theoretical proofs and validated through simulation results. This study extends the work by Toumiet al. (2017).