# Time-optimal neural feedback control of nilpotent systems as a binary classification problem

arXiv: https://arxiv.org/abs/2503.17581 · HTML: https://mlsanity.com/paper/2503.17581/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 36 references that are in the map.
- First posted on arXiv 21 Mar 2025.

## Builds on (most cited first, 8 of 8)

- [Adaptive Deep Learning for High-Dimensional Hamilton-Jacobi-Bellman Equations](https://mlsanity.com/paper/1907.05317/) (2019)
- [A minimax optimal control approach for robust neural ODEs](https://mlsanity.com/paper/2310.17584/) (2023)
- [Computation and Control of Unstable Steady States for Mean Field Multiagent Systems](https://mlsanity.com/paper/2406.11725/) (2024)
- [Data-driven Tensor Train Gradient Cross Approximation for Hamilton-Jacobi-Bellman Equations](https://mlsanity.com/paper/2205.05109/) (2022)
- [Solving parametric systems of polynomial equations over the reals through Hermite matrices](https://mlsanity.com/paper/2011.14136/) (2020)
- [Efficient Online Trajectory Planning for Integrator Chain Dynamics using Polynomial Elimination](https://mlsanity.com/paper/2012.07029/) (2020)
- [Time-Optimal Control for High-Order Chain-of-Integrators Systems with Full State Constraints and Arbitrary Terminal States (Extended Version)](https://mlsanity.com/paper/2311.07039/) (2023)
- [A comparison study of supervised learning techniques for the approximation of high dimensional functions and feedback control](https://mlsanity.com/paper/2402.01402/) (2024)

## Abstract (arXiv)

A computational method for the synthesis of time-optimal feedback control laws for linear nilpotent systems is proposed. The method is based on the use of the bang-bang theorem, which leads to a characterization of the time-optimal trajectory as a parameter-dependent polynomial system for the control switching sequence. A deflated Newton's method is then applied to exhaust all the real roots of the polynomial system. The root-finding procedure is informed by the Hermite quadratic form, which provides a sharp estimate on the number of real roots to be found. In the second part of the paper, the polynomial systems are sampled and solved to generate a synthetic dataset for the construction of a time-optimal deep neural network -- interpreted as a binary classifier -- via supervised learning. Numerical tests in integrators of increasing dimension assess the accuracy, robustness, and real-time-control capabilities of the approximate control law.