Neural ODEs for Optimal Control: Learning Continuous Trajectories and Feedback Policies via Adaptive Discretisation

In this article, a novel method to compute continuous optimal control trajectories, and policies, using Neural Ordinary Differential Equations is proposed. The proposed method introduces a novel framework that allows solving open-loop optimal control problems with arbitrary state and/or control constraints by using a penalty method and bounded activation functions, while also learning the initial control value. Computational cost is significantly reduced by computing the state and control variables in a single ODE solver pass. Leveraging adaptive ODE solvers, our framework dynamically adjusts time steps during integration, eliminating manual grid refinement and ensuring high accuracy with optimal computational efficiency, as if computed on an infinitely fine grid. Furthermore, the continuous-time control trajectories, given by ODEs, allow for flexible evaluation of state and control variables at any desired time points, within and beyond the solving time horizon, without the need to re-solve the problems.Building on this, a second framework that enables feedback control is proposed, allowing the system to adapt to disturbances and uncertainties through a Neural ODE-based policy that dynamically maps states to control actions rather than relying on fixed open-loop trajectories. Simulation results of the proposed frameworks solving six optimal control problems with different types of constraints and complexity are presented. The results demonstrate that the proposed frameworks achieve comparable or superior performance to the multi-shooting indirect method, both in solving the problems and in predicting beyond the original time horizon—without requiring re-solving the problem or manual tuning of the discretisation.