Time-Optimal Neural Feedback Control of Nilpotent Systems as a Binary Classification Problem

Abstract

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 of the number of real roots to be found. In the second part of this paper, the polynomial systems are sampled and solved to generate a synthetic dataset for the construction of a time-optimal deep neural network (NN)—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.

Keywords

Time-optimal control / Feedback control / Supervised learning / Deep neural networks (NNs) / Nilpotent systems / Hermite quadratic forms / 93B52 / 93C05 / 68T07 / 14P10

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Funding

Air Force Office of Scientific Research(FA8655-26-1-B011)

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