# HJB Optimal Feedback Control with Deep Differential Value Functions and Action Constraints

arXiv: https://arxiv.org/abs/1909.06153 · HTML: https://mlsanity.com/paper/1909.06153/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Sep 2019.

## Built on by (most cited first, 8 of 8)

- [Value Iteration in Continuous Actions, States and Time](https://mlsanity.com/paper/2105.04682/) (2021)
- [Robust Value Iteration for Continuous Control Tasks](https://mlsanity.com/paper/2105.12189/) (2021)
- [Reinforcement Learning with Non-Exponential Discounting](https://mlsanity.com/paper/2209.13413/) (2022)
- [Continuous-Time Fitted Value Iteration for Robust Policies](https://mlsanity.com/paper/2110.01954/) (2021)
- [Neural Network Optimal Feedback Control with Guaranteed Local Stability](https://mlsanity.com/paper/2205.00394/) (2022)
- [Kernel-Based Optimal Control: An Infinitesimal Generator Approach](https://mlsanity.com/paper/2412.01591/) (2024)
- [Is Bellman Equation Enough for Learning Control?](https://mlsanity.com/paper/2503.02171/) (2025)
- [Autopilot-Preserving Residual Q-Learning with HJB-Inspired Finite-Action Risk Filtering for Fixed-Wing UAV Command Supervision](https://mlsanity.com/paper/2606.01397/) (2026)

## Abstract (arXiv)

Learning optimal feedback control laws capable of executing optimal trajectories is essential for many robotic applications. Such policies can be learned using reinforcement learning or planned using optimal control. While reinforcement learning is sample inefficient, optimal control only plans an optimal trajectory from a specific starting configuration. In this paper we propose deep optimal feedback control to learn an optimal feedback policy rather than a single trajectory. By exploiting the inherent structure of the robot dynamics and strictly convex action cost, we can derive principled cost functions such that the optimal policy naturally obeys the action limits, is globally optimal and stable on the training domain given the optimal value function. The corresponding optimal value function is learned end-to-end by embedding a deep differential network in the Hamilton-Jacobi-Bellmann differential equation and minimizing the error of this equality while simultaneously decreasing the discounting from short- to far-sighted to enable the learning. Our proposed approach enables us to learn an optimal feedback control law in continuous time, that in contrast to existing approaches generates an optimal trajectory from any point in state-space without the need of replanning. The resulting approach is evaluated on non-linear systems and achieves optimal feedback control, where standard optimal control methods require frequent replanning.