Abstract In this paper, we consider the signature-to-path reconstruction problem from the control-theoretic perspective. Namely, we design an optimal control problem whose solution leads to the minimal-length path that generates a given signature. In order to do that, we minimize a cost functional consisting of two competing terms, i.e., a weighted final-time cost combined with the $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -norm squared of the controls. Moreover, we can show that, by taking the limit to infinity of the parameter that tunes the final-time cost, the problem $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> -converges to the problem of finding a sub-Riemannian geodesic connecting two signatures. Finally, we provide an alternative reformulation of the latter problem, which is particularly suitable for the numerical implementation.

Marco Rauscher

Alessandro Scagliotti

Felipe Pagginelli Patricio

Mathematics of Control Signals and Systems

控制、信号和系统数学（MCSS）是一本致力于数学控制和系统理论的国际期刊，包括信号处理的系统理论方面。它的独特之处在于它专注于数学系统理论；它专注于具有输入和/或输出的系统的数学理论，以及通常由确定性或随机常微分或偏微分方程、微分代数方程或差分方程描述的动力学。潜在主题包括但不限于可控性、可观察性、和实现理论，非线性系统的稳定性理论，系统辨识，开关、混合、网络和随机系统的数学方面，以及最优控制和其他控制器设计技术的系统理论方面。如果它们包含重要的理论贡献，则欢迎面向应用的论文。

The journal "Mathematics of Control, Signals, and Systems" is an international publication dedicated to mathematical control and systems theory, including aspects of system theory in signal processing. Its uniqueness lies in its focus on mathematical systems theory, studying the mathematical theory of systems with inputs and/or outputs, typically described by deterministic or stochastic ordinary differential equations, partial differential equations, differential algebraic equations, or difference equations. Potential topics include, but are not limited to, controllability, observability, and realization theory, stability theory of nonlinear systems, system identification, mathematical aspects of switched, hybrid, networked, and stochastic systems, as well as systems theory aspects of optimal control and other controller design techniques. Papers that contain significant theoretical contributions are also welcome, even if they are application-oriented.

《Mathematics of Control, Signals, and Systems》是一本国际期刊，专注于数学控制和系统理论，涵盖信号处理的系统理论方面。该期刊的独特之处在于其专注于数学系统理论，研究具有输入和/或输出的系统的数学理论，通常由确定性或随机的常微分方程、偏微分方程、微分代数方程或差分方程描述的动力学。潜在的研究主题包括但不限于可控性、可观察性和实现理论，非线性系统的稳定性理论，系统辨识，开关、混合、网络和随机系统的数学方面，以及最优控制和其他控制器设计技术的系统理论。如果论文包含重要的理论贡献，面向应用的研究也受到欢迎。

Category	Quartile
计算机科学	4
计算机科学, 自动化与控制系统	4
计算机科学, 工程电子与电气	4
计算机科学, 数学跨学科应用	4

Shortest-path recovery from signature with an optimal control approach

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Category	Quartile
MATHEMATICS, INTERDISCIPLINARY APPLICATIONS	2