Representing and Computing the B-derivative of the Piecewise-Differentiable Flow of a Class of Nonsmooth Vector Fields
Representing and Computing the B-derivative of the Piecewise-Differentiable Flow of a Class of Nonsmooth Vector Fields
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DOI：
10.1115/1.4054481
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发表时间：
2022-05
影响因子：
2
通讯作者：
George Council;Shai Revzen;Samuel A. Burden
中科院分区：
文献类型：
--
作者：
George Council;Shai Revzen;Samuel A. Burden
本文研究了一类非光滑向量场产生的分段可微流的一阶近似。具体来说，我们表示并计算了由事件选择不连续的矢量场产生的分段可微流的Bouligand（或B-）导数。我们的结果非常有效：尽管期望的导数有许多“片段”，但我们提供了一种算法，该算法使用多项式时间和空间来评估其对给定切向量的作用，并通过推导b导数的表示来验证算法的正确性，该表示只需要“指数”时间和空间来构建。我们将我们的方法应用于两类说明性示例：分段常数矢量场和受单边约束的机械系统。
This paper concerns first-order approximation of the piecewise-differentiable flow generated by a class of nonsmooth vector fields. Specifically, we represent and compute the Bouligand (or B-)derivative of the piecewise-differentiable flow generated by vector fields with event-selected discontinuities. Our results are remarkably efficient: although there are factorially many "pieces" of the desired derivative, we provide an algorithm that evaluates its action on a given tangent vector using polynomial time and space, and verify the algorithm's correctness by deriving a representation for the B-derivative that requires "only" exponential time and space to construct. We apply our methods in two classes of illustrative examples: piecewise-constant vector fields and mechanical systems subject to unilateral constraints.