Imperial College London
Alternate

Emeritus ProfessorRichardVinter

Faculty of Engineering, Department of Electrical and Electronic Engineering

Emeritus Professor in Electrical and Electronic Engineering
 
 
 
//

Contact

 

+44 (0)20 7594 6287r.vinter Website

 
 
//

Assistant

 

Mrs Raluca Reynolds +44 (0)20 7594 6281

 
//

Location

 

618Electrical EngineeringSouth Kensington Campus

//

Summary

 

Publications

Citation

BibTex format

@article{Bettiol:2011:10.1016/j.jde.2011.09.007,
author = {Bettiol, P and Frankowska, H and Vinter, RB},
doi = {10.1016/j.jde.2011.09.007},
journal = {Journal of Differential Equations},
pages = {1912--1933},
title = {L∞ estimates on trajectories confined to a closed subset},
url = {http://dx.doi.org/10.1016/j.jde.2011.09.007},
volume = {252},
year = {2011}
}
Download

RIS format (EndNote, RefMan)

TY  - JOUR
AB  - 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.
AU  - Bettiol,P
AU  - Frankowska,H
AU  - Vinter,RB
DO  - 10.1016/j.jde.2011.09.007
EP  - 1933
PY  - 2011///
SN  - 1090-2732
SP  - 1912
TI  - L∞ estimates on trajectories confined to a closed subset
T2  - Journal of Differential Equations
UR  - http://dx.doi.org/10.1016/j.jde.2011.09.007
UR  - http://hdl.handle.net/10044/1/40478
VL  - 252
ER  -
Download