L∞ estimates on trajectories confined to a closed subset
File(s)
Accepted version
Author(s)
Bettiol, P
Frankowska, H
Vinter, RB
Type
Journal Article
Abstract
This paper concerns the validity of estimates on the distance of an arbitrary state trajectory from the set of state trajectories which lie in a given state constraint set. These so called distance estimates have wide-spread application in state constrained optimal control, including justifying the use of the Maximum Principle in normal form and establishing regularity properties of value functions. We focus on linear, L∞ distance estimates which, of all the available estimates have, so far, been the most widely used. Such estimates are known to be valid for general, closed state constraint sets, provided the functions defining the dynamic constraint are Lipschitz continuous, with respect to the time and state variables. We ask whether linear, L∞ distance estimates remain valid when the Lipschitz continuity hypothesis governing t-dependence of the data is relaxed. We show by counter-example that these distance estimates are not valid in general if the hypothesis of Lipschitz continuity is replaced by continuity. We also provide a new hypothesis, ‘absolute continuity from the left’, for the validity of linear, L∞ estimates. The new hypothesis is less restrictive than Lipschitz continuity and even allows discontinuous time dependence in certain cases. It is satisfied, in particular, by differential inclusions exhibiting non-Lipschitz t-dependence at isolated points, governed, for example, by a fractional-power modulus of continuity. The relevance of distance estimates for state constrained differential inclusions permitting fractional-power time dependence is illustrated by an example in engineering design, where we encounter an isolated, square-root type singularity, concerning the t-dependence of the data.
Date Issued
2011-09-29
Date Acceptance
2011-09-29
Citation
Journal of Differential Equations, 2011, 252 (2), pp.1912-1933
ISSN
1090-2732
Publisher
Elsevier
Start Page
1912
End Page
1933
Journal / Book Title
Journal of Differential Equations
Volume
252
Issue
2
Copyright Statement
© 2011 Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/
Subjects
Science & Technology
Physical Sciences
Mathematics
MATHEMATICS
Optimal control
Differential inclusions
State constraints
Sensitivity
CONSTRAINED OPTIMAL-CONTROL
STATE CONSTRAINTS
EXISTENCE
THEOREMS
General Mathematics
0101 Pure Mathematics
0102 Applied Mathematics
Publication Status
Published