# American Institute of Mathematical Sciences

March  2017, 37(3): 1129-1158. doi: 10.3934/dcds.2017047

## Necessity for isoperimetric inequality constraints

 IIMAS, Universidad Nacional Autóonoma de México, Apartado Postal 20-126, México DF 01000

Received  May 2016 Revised  September 2016 Published  December 2016

Fund Project: The second author was supported by the PASPA-DGAPA program from Universidad Nacional Autónoma de México

In this paper we deal with second order necessary conditions for the problem of Lagrange in the calculus of variations posed over piecewise smooth trajectories and involving inequality and equality isoperimetric constraints. We provide a review of different approaches to derive second order necessary conditions for this problem and prove that, surprisingly, though the solution set to the problem where the conditions hold may vary, all approaches impose the same strong assumption of normality relative to the set defined by equality constraints for active indices. Based on these approaches, we also give some applications to certain optimization problems with mixed constraints.

Citation: Jorge A. Becerril, Javier F. Rosenblueth. Necessity for isoperimetric inequality constraints. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1129-1158. doi: 10.3934/dcds.2017047
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