Gain-Only Neural Operator Approximators of PDE Backstepping Controllers

For the recently introduced deep learning-powered approach to PDE backstepping control, we present an advancement applicable across all the results developed thus far: approximating the control gain function only (a function of one variable), rather than the entire kernel function of the backstepping transformation (a function of two variables). We introduce this idea on a couple benchmark (unstable) PDEs, hyperbolic and parabolic. We alter the approach of quantifying the effect of the approximation error by replacing a backstepping transformation that employs the approximated kernel (suitable for adaptive control) by a transformation that employs the exact kernel (suitable for gain scheduling). A major simplification in the target system arises, with the perturbation due to the approximation shifting from the domain to the boundary condition. This results in a significant difference in the Lyapunov analysis, which nevertheless results in a guarantee of the stability being retained with the simplified approximation approach. The approach of approximating only the control gain function simplifies the operator being approximated and the training of its neural approximation, with an expected reduction in the neural network size. The price for the savings in approximation is paid through a somewhat more intricate Lyapunov analysis, in higher Sobolev spaces for some PDEs, as well as some restrictions on initial conditions that result from higher Sobolev spaces. It is essential to carefully consider the specific requirements and constraints of each problem to determine the most appropriate approach; indeed, recent works have demonstrated the successful application of both full-kernel and gain-only approaches in both adaptive control and gain scheduling contexts.