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 Program Systems: Theory and Applications: Year: Volume: Issue: Page: Find

 Program Systems: Theory and Applications, 2018, Volume 9, Issue 4, Pages 253–264 (Mi ps330)

Optimization Methods and Control Theory

Two-side bound of a root of an equation containing complete elliptic integrals

A. Yu. Popov, Yu. L. Sachkov

Ailamazyan Program Systems Institute of Russian Academy of Sciences

Abstract: We prove uniqueness of root of an equation arising in a problem of geometric control theory. The problem consists of description of singularity of the sub-Riemannian sphere on the Engel group near abnormal length minimizer.
During the proof, several new inequalities for complete elliptic integrals were obtained. For example, we proved that the function $K(k) E(k)$ is increasing at the segment $[0, 1)$; this fact was not noticed before in literature.
The method of investigation developed and the results obtained can be useful both for the study of elliptic integrals and for solving problems were such integrals arise (e.g. in problems of sub-Riemannian geometry). (In Russian).

Key words and phrases: asymptotics, complete elliptic integral, sub-Riemannian geometry.

 Funding Agency Grant Number Russian Science Foundation 17-11-01387

DOI: https://doi.org/10.25209/2079-3316-2018-9-4-253-264

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UDC: 517.977
02.12.2018
Accepted: 05.12.2018

Citation: A. Yu. Popov, Yu. L. Sachkov, “Two-side bound of a root of an equation containing complete elliptic integrals”, Program Systems: Theory and Applications, 9:4 (2018), 253–264

Citation in format AMSBIB
\Bibitem{PopSac18} \by A.~Yu.~Popov, Yu.~L.~Sachkov \paper Two-side bound of a root of an equation containing complete elliptic integrals \jour Program Systems: Theory and Applications \yr 2018 \vol 9 \issue 4 \pages 253--264 \mathnet{http://mi.mathnet.ru/ps330} \crossref{https://doi.org/10.25209/2079-3316-2018-9-4-253-264}