# A characterization of the Filippov convention

## TL;DR

This paper provides a necessary and sufficient condition for an interpolation scheme of piecewise-continuous vector fields to match the Filippov convention, clarifying when the Filippov approach can be uniquely characterized.

## Contribution

It establishes a precise criterion for when an interpolation scheme aligns with the Filippov convention, enhancing understanding of discontinuous vector field analysis.

## Key findings

- Characterizes when interpolation schemes match Filippov's convention
- Provides a necessary and sufficient condition for equivalence
- Clarifies the applicability of Filippov's method in discontinuous systems

## Abstract

The Filippov convention is widely used in the literature to define vector fields on a discontinuity set of piecewise-continuous vector fields. The aim of this paper is to give a sufficient and necessary condition for an interpolation scheme of piecewise-continuous vector fields to coincide with the Filippov convention. That is, we show that a map from a space of piecewise-continuous vector fields with two components to the space of vector fields coincides with the Filippov convention where the latter can be applied, if it is sufficiently well-behaved as a generalization of continuous vector fields.