Gain-Only Neural Operators for PDE Backstepping

In this work we advance the recently-introduced deep learning-powered approach to PDE backstepping control by proposing a method that approximates only the control gain function -- a function of one variable -- instead of the entire kernel function of the backstepping transformation, which depends on two variables. This idea is introduced using several benchmark unstable PDEs, including hyperbolic and parabolic types, and extended to 2X2 hyperbolic systems. By employing a backstepping transformation that utilizes the exact kernel (suitable for gain scheduling) rather than an approximated one (suitable for adaptive control), we alter the quantification of the approximation error. This leads to a significant simplification in the target system, shifting the perturbation due to approximation from the domain to the boundary condition. Despite the notable differences in the Lyapunov analysis, we are able to retain stability guarantees with this simplified approximation approach. Approximating only the control gain function simplifies the operator being approximated and the training of its neural approximation, potentially reducing the neural network size. The trade-off for these simplifications is a more intricate Lyapunov analysis, involving higher Sobolev spaces for some PDEs, and certain restrictions on initial conditions arising from these spaces. It is crucial to carefully consider the specific requirements and constraints of each problem to determine the most suitable approach; indeed, recent works have demonstrated successful applications of both full-kernel and gain-only approaches in adaptive control and gain scheduling contexts.