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

TL;DR

This paper introduces an efficient algorithm to compute the B-derivative of the flow generated by nonsmooth vector fields, enabling first-order approximations in systems with piecewise-smooth dynamics.

Contribution

It provides a novel polynomial-time algorithm for evaluating the B-derivative of piecewise-$C^r$ flows, along with a representation that can be constructed in exponential time.

Findings

Algorithm evaluates B-derivative action in polynomial time

Representation of B-derivative constructed in exponential time

Applicable to piecewise-constant and constrained mechanical systems

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.