The Hamiltonian Inclusion for Nonconvex Velocity Sets
File(s)
Published version
Author(s)
Vinter, RB
Type
Journal Article
Abstract
Since Clarke's 1973 proof of the Hamiltonian inclusion for optimal control problems with convex velocity sets, there has been speculation (and, more recently, speculation relating to a stronger, partially convexified version of the Hamiltonian inclusion) as to whether these necessary conditions are valid in the absence of the convexity hypothesis. The issue was in part resolved by Clarke himself when, in 2005, he showed that $L^{\infty}$ local minimizers satisfy the Hamiltonian inclusion. In this paper it is shown, by counterexample, that the Hamiltonian inclusion (and so also the stronger partially convexified Hamiltonian inclusion) are not in general valid for nonconvex velocity sets when the local minimizer in question is merely a $W^{1,1}$ local minimizer, not an $L^{\infty}$ local minimizer. The counterexample demonstrates that the need to consider $L^{\infty}$ local minimizers, not $W^{1,1}$ local minimizers, in the proof of the Hamiltonian inclusion for nonconvex velocity sets is fundamental, not just a technical restriction imposed by currently available proof techniques. The paper also establishes the validity of the partially convexified Hamiltonian inclusion for $W^{1,1}$ local minimizers under a normality assumption, thereby correcting earlier assertions in the literature.
Date Issued
2014-04-10
Date Acceptance
2013-12-09
Citation
SIAM Journal on Control and Optimization, 2014, 52 (2), pp.1237-1250
ISSN
1095-7138
Publisher
Society for Industrial and Applied Mathematics
Start Page
1237
End Page
1250
Journal / Book Title
SIAM Journal on Control and Optimization
Volume
52
Issue
2
Copyright Statement
© 2014 Society for Industrial and Applied Mathematics
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Grant Number
EP/G066477/1
Subjects
Science & Technology
Technology
Physical Sciences
Automation & Control Systems
Mathematics, Applied
Mathematics
AUTOMATION & CONTROL SYSTEMS
MATHEMATICS, APPLIED
optimal control
differential inclusions
Hamiltonian inclusion
EULER-LAGRANGE
DIFFERENTIAL-INCLUSIONS
VARIATIONAL-PROBLEMS
Industrial Engineering & Automation
0102 Applied Mathematics
0906 Electrical And Electronic Engineering
0913 Mechanical Engineering
Publication Status
Published