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

2021/02/21 by George Council, Council, George, Shai Revzen +3
Chemical Engineering · Computer Science · Engineering · Mathematics · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Electrical engineering #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Rheology and Fluid Dynamics Studies #Systems and Control (eess.SY) #cs.SY #eess.SY #electronic engineering #information engineering #math.DS

paper · pdf · doi:10.48550/arxiv.2102.10702

openalex publication_date 2021/02/21 · arxiv created 2021/03/08 · arxiv updated 2021/03/09 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/30

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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