# On the stability analysis of deep neural network representations of an optimal state-feedback

arXiv: https://arxiv.org/abs/1812.02532 · HTML: https://mlsanity.com/paper/1812.02532/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Dec 2018.

## Built on by (most cited first, 10 of 10)

- [Neural representation of a time optimal, constant acceleration rendezvous](https://mlsanity.com/paper/2203.15490/) (2022)
- [Guidance and Control Networks with Periodic Activation Functions](https://mlsanity.com/paper/2405.18084/) (2024)
- [Selected Trends in Artificial Intelligence for Space Applications](https://mlsanity.com/paper/2212.06662/) (2022)
- [Optimality Principles in Spacecraft Neural Guidance and Control](https://mlsanity.com/paper/2305.13078/) (2023)
- [Neural Network Optimal Feedback Control with Guaranteed Local Stability](https://mlsanity.com/paper/2205.00394/) (2022)
- [Neural network optimal feedback control with enhanced closed loop stability](https://mlsanity.com/paper/2109.07466/) (2021)
- [Certifying Guidance & Control Networks: Uncertainty Propagation to an Event Manifold](https://mlsanity.com/paper/2410.03729/) (2024)
- [High-order expansion of Neural Ordinary Differential Equations flows](https://mlsanity.com/paper/2504.08769/) (2025)
- [Guidance and Control Neural Network Acceleration using Memristors](https://mlsanity.com/paper/2509.02369/) (2025)
- [Pretrained Approximators for Low-Thrust Trajectory Cost and Reachability](https://mlsanity.com/paper/2605.26790/) (2026)

## Abstract (arXiv)

Recent work have shown how the optimal state-feedback, obtained as the solution to the Hamilton-Jacobi-Bellman equations, can be approximated for several nonlinear, deterministic systems by deep neural networks. When imitation (supervised) learning is used to train the neural network on optimal state-action pairs, for instance as derived by applying Pontryagin's theory of optimal processes, the resulting model is referred here as the guidance and control network. In this work, we analyze the stability of nonlinear and deterministic systems controlled by such networks. We then propose a method utilising differential algebraic techniques and high-order Taylor maps to gain information on the stability of the neurocontrolled state trajectories. We exemplify the proposed methods in the case of the two-dimensional dynamics of a quadcopter controlled to reach the origin and we study how different architectures of the guidance and control network affect the stability of the target equilibrium point and the stability margins to time delay. Moreover, we show how to study the robustness to initial conditions of a nominal trajectory, using a Taylor representation of the neurocontrolled neighbouring trajectories.