Representing and computing the b-derivative of the piecewise-differentiable flow of a class of nonsmooth vector fields

George Council, Shai Revzen, Samuel A Burden

Journal of Computational and Nonlinear Dynamics, vol. 17, no. 9, pp. 091004, 2022

Abstract

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 a vector field with event-selected discontinuities. Our results are remarkably efficient: although there are factorially many ?pieces? of the derivative, we provide an algorithm that evaluates its action on a 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.

BibTeX

@article{council2022representing,
  author = {Council, George and Revzen, Shai and Burden, Samuel A},
  title = {Representing and computing the b-derivative of the piecewise-differentiable flow of a class of nonsmooth vector fields},
  journal = {Journal of Computational and Nonlinear Dynamics},
  year = {2022},
  volume = {17},
  number = {9},
  pages = {091004},
  doi = {10.1115/1.4054481},
  publisher = {American Society of Mechanical Engineers},
  cvtopic = {dsctrl}
}