Event-Selected Vector Field Discontinuities Yield Piecewise-Differentiable Flows

S A Burden, S S Sastry, D E Koditschek, Shai Revzen

SIAM Journal of Applied Dynamical Systems, vol. 15, no. 2, pp. 1227–1267, 2016

Abstract

We study a class of discontinuous vector fields that arise in biomechanics and neuroscience. Under the conditions that (i) the vector field’s discontinuities are locally confined to a finite number of smooth submanifolds and (ii) the vector field is transverse to these surfaces in an appropriate sense, we show that the vector field yields a well-defined flow that is Lipschitz continuous and piecewise-differentiable. This implies that although the flow is not classically differentiable, nevertheless it admits a first–order approximation (known as a Bouligand derivative). We exploit this first-order approximation to infer existence of piecewise-differentiable impact maps and assess structural stability of the flow

BibTeX

@article{Burden-2015-multi,
  author = {Burden, S A and Sastry, S S and Koditschek, D E and Revzen, Shai},
  title = {Event-Selected Vector Field Discontinuities Yield Piecewise-Differentiable Flows},
  journal = {SIAM Journal of Applied Dynamical Systems},
  year = {2016},
  volume = {15},
  number = {2},
  pages = {1227--1267},
  doi = {10.1137/15M1016588},
  cvtopic = {dsctrl}
}