A reinterpretation of set differential equations as differential equations in a Banach space

TL;DR

This paper reinterprets set differential equations as ordinary differential equations in a Banach space using support functions, enabling broader solution analysis and overcoming limitations of the Hukuhara differential.

Contribution

It introduces a novel reformulation of set differential equations as Banach space ODEs via support functions, expanding analytical possibilities.

Findings

Existence and uniqueness results are applicable to the reformulated equations.

Support function representation allows solutions not possible with Hukuhara differential.

A simple example demonstrates the advantages of the new approach.

Abstract

Set differential equations are usually formulated in terms of the Hukuhara differential, which implies heavy restrictions for the nature of a solution. We propose to reformulate set differential equations as ordinary differential equations in a Banach space by identifying the convex and compact subsets of $\R^d$ with their support functions. Using this representation, we demonstrate how existence and uniqueness results can be applied to set differential equations. We provide a simple example, which can be treated in support function representation, but not in the Hukuhara setting.