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arXiv · 2102.10702

Representing and computing the B-derivative of an $EC^r$ vector field's $PC^r$ flow

George Council·Shai Revzen·Samuel A. Burden
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Abstract

This paper concerns the 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-$C^r$ flow generated by an event-selected $C^r$ vector field. 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.

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George Council, Shai Revzen, Samuel A. Burden. 2021-02-21. Representing and computing the B-derivative of an $EC^r$ vector field's $PC^r$ flow. https://arxiv.org/abs/2102.10702

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