Neural Lyapunov and Optimal Control

Optimal control (OC) is an effective approach to controlling complex dynamical systems. However, traditional approaches to parameterising and learning controllers in optimal control have been ad-hoc, collecting data and fitting it to neural networks. However, this can lead to learnt controllers ignoring constraints like optimality and time variability. We introduce a unified framework that simultaneously solves control problems while learning corresponding Lyapunov or value functions. Our method formulates OC-like mathematical programs based on the Hamilton-Jacobi-Bellman (HJB) equation. We leverage the HJB optimality constraint and its relaxation to learn time-varying value and Lyapunov functions, implicitly ensuring the inclusion of constraints. We show the effectiveness of our approach on linear and nonlinear control-affine problems. Additionally, we demonstrate significant reductions in planning horizons (up to a factor of 25) when incorporating the learnt functions into Model Predictive Controllers.