A reinterpretation of set differential equations as differential equations in a Banach space
File(s)
Author(s)
Rasmussen, M
Rieger, J
Webster, KN
Type
Journal Article
Abstract
Set differential equations are usually formulated in terms of the
Hukuhara differential. As a consequence, the theory of set differential
equations is perceived as an independent subject, in which all results
are proved within the framework of the Hukuhara calculus.
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 rep-
resentation, standard existence and uniqueness theorems for ordinary
differential equations can be applied to set differential equations. We
provide a geometric interpretation of the main result, and we demon-
strate that our approach overcomes the heavy restrictions the use of
the Hukuhara differential implies for the nature of a solution.
Hukuhara differential. As a consequence, the theory of set differential
equations is perceived as an independent subject, in which all results
are proved within the framework of the Hukuhara calculus.
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 rep-
resentation, standard existence and uniqueness theorems for ordinary
differential equations can be applied to set differential equations. We
provide a geometric interpretation of the main result, and we demon-
strate that our approach overcomes the heavy restrictions the use of
the Hukuhara differential implies for the nature of a solution.
Date Issued
2018-04-01
Date Acceptance
2016-06-22
Citation
Proceedings of the Royal Society of Edinburgh Section A-Mathematics, 2018, 148 (2), pp.429-446
ISSN
1473-7124
Publisher
Cambridge University Press (CUP)
Start Page
429
End Page
446
Journal / Book Title
Proceedings of the Royal Society of Edinburgh Section A-Mathematics
Volume
148
Issue
2
Copyright Statement
© Royal Society of Edinburgh 2018
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Commission of the European Communities
Grant Number
EP/I004165/1
EP/L00187X/1
PIEF-GA-2013-624526
Subjects
Science & Technology
Physical Sciences
Mathematics, Applied
Mathematics
set differential equations
ordinary differential equations in Banach spaces
existence and uniqueness
math.CA
math.CA
58D25, 34G20, 26E25
0101 Pure Mathematics
0102 Applied Mathematics
Publication Status
Published
Date Publish Online
2017-11-20