Normality and uniqueness of Lagrange multipliers

Karla L. Cortez; Javier F. Rosenblueth

doi:10.3934/dcds.2018138

Article Contents

2018, Volume 38, Issue 6: 3169-3188. Doi: 10.3934/dcds.2018138

This issue Previous Article Sign-changing multi-bump solutions for Kirchhoff-type equations in

$\mathbb{R}^3$

IIMAS, Universidad Nacional Autónoma de México, Apartado Postal 20-126, CDMX 01000, México

Received: May 2017

Published: June 2018

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Abstract

In this paper we study, for certain problems in the calculus of variations and optimal control, two different questions related to uniqueness of multipliers appearing in first order necessary conditions. One deals with conditions under which a given multiplier associated with an extremal of a fixed function is unique, a property which, in nonlinear programming, is known to be equivalent to the strict Mangasarian-Fromovitz constraint qualification. We show that, for isoperimetric problems in the calculus of variations, a similar characterization holds, but not in optimal control where the corresponding condition is only sufficient for the uniqueness of the multiplier. The other question is related to the set of multipliers associated with all functions for which a solution to the constrained problem is given. We prove that, for both types of problems, this set is a singleton if and only if a strong normality assumption holds.

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Mathematics Subject Classification: Primary: 49K15, 49K21; Secondary: 90C30.

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