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arXiv · 2603.17933

Minimum-Energy Control For Control-Affine Systems

Cyprien Tamekue·Zongxi Yu·ShiNung Ching
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Abstract

Fixed-horizon minimum-energy control for control-affine systems rarely admits closed-form formulas beyond the linear setting. In this article, under the assumption of unconstrained controls, we derive a fixed-horizon minimum-energy control for a broad class of control-affine systems as a fixed point of a Lagrange multiplier synthesis map. Compared with our earlier general steering synthesis, the present construction is based on a generally non-symmetric Gramian-like matrix and solves the associated $L^2$-energy problem. More specifically, under a comparison estimate criterion and a self-mapping assumption, we prove that the synthesis map has a unique fixed point in the feasible coercive classes, which can be computed via a standard Picard iteration. As a demonstration of concept, we use uniform complete controllability results for linear time-varying systems to derive a bracket-generating condition ensuring complete controllability for time-dependent planar control-affine systems with scalar input, and our minimum energy framework applies under standing assumptions. Special treatment for the unicycle kinematic model is also provided, and numerical examples illustrate the approach's effectiveness.

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Cyprien Tamekue, Zongxi Yu, ShiNung Ching. 2026-03-18. Minimum-Energy Control For Control-Affine Systems. https://arxiv.org/abs/2603.17933

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