Optimal Orbit Stabilization by Approximating Lagrangian Submanifolds with Neural Networks

In this work, we propose a novel approach to periodic orbit stabilization using the periodic Linear Quadratic Regulator (LQR) framework. While conventional methods rely on transverse coordinates for control design and controller implementation, our approach transforms the optimal feedback back to the original coordinates, enabling seamless application. By leveraging the symplectic geometric structure of the control problem, we approximate the value function, which defines the Lagrangian submanifold representing the optimal feedback in a coordinate independent way, using a neural network. This eliminates the need for explicit online transverse coordinate computation. The method is demonstrated on two nonlinear systems in two and five dimensions.