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.

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