Geometrically, the affine connection is the main ingredient that underlies the covariant derivative, the parallel transport, the auto-parallel curves, the torsion tensor field, and the curvature tensor field on a finite-dimensional differentiable manifold. In this paper, we come up with a new idea of controllability and observability of states by using auto-parallel curves, and the minimum time problem controlled by the affine connection. The main contributions refer to the following: (i) auto-parallel curves controlled by a connection, (ii) reachability and controllability on the tangent bundle of a manifold, (iii) examples of equiaffine connections, (iv) minimum time problem controlled by a connection, (v) connectivity by stochastic perturbations of auto-parallel curves, and (vi) computing the optimal time and the optimal striking time. The connections with bounded pull-backs result in bang–bang optimal controls. Some significative examples on bi-dimensional manifolds clarify the intention of our paper and suggest possible applications. At the end, an example of minimum striking time with simulation results is presented.

中文翻译：

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仿射连接控制的最短时间问题

在几何上，仿射连接是有限维可微流形上协变导数、平行传输、自平行曲线、扭转张量场和曲率张量场的主要成分。在本文中，我们通过使用自平行曲线和仿射连接控制的最小时间问题提出了状态的可控性和可观察性的新思想。主要贡献如下：（i）由连接控制的自动平行曲线，（ii）流形切丛上的可达性和可控性，（iii）等仿射连接的例子，（iv）由以下控制的最小时间问题连接，（v）通过自动平行曲线的随机扰动进行连接，以及（vi）计算最佳时间和最佳打击时间。与有界回拉的联系导致 bang-bang 最优控制。一些关于二维流形的重要例子阐明了我们论文的意图并提出了可能的应用。最后，给出了一个带有模拟结果的最小打击时间的例子。