Diamond-alpha Integral Inequalities on Time Scales
Abstract
The theory of the calculus of variations was recently extended to the more general time scales setting, both for delta and nabla integrals. The primary purpose of this paper is to further extend the theory on time scales, by establishing some basic diamond-alpha dynamic integral inequalities. We prove generalized versions of H\"{o}lder, Cauchy-Schwarz, Minkowski, and Jensen's inequalities. For the particular case when alpha is equal to one or alpha is equal to zero one gets, respectively, correspondent delta and nabla inequalities. If we further restrict ourselves by fixing the time scale to the real or integer numbers, then we obtain the classical inequalities, whose role in optimal control is well known. By analogy, we trust that the diamond-alpha integral inequalities we prove here will be important in the study of control systems on times scales.
Keywords
Cite
@article{arxiv.0805.0242,
title = {Diamond-alpha Integral Inequalities on Time Scales},
author = {Rui A. C. Ferreira and Moulay Rchid Sidi Ammi and Delfim F. M. Torres},
journal= {arXiv preprint arXiv:0805.0242},
year = {2008}
}
Comments
Accepted to the 8th Portuguese Conference on Automatic Control -- CONTROLO'2008, 21 to 23 July 2008, UTAD University, Vila Real, Portugal