Necessary conditions for a minimum in variational problems with delay in the presence of degeneracies

TL;DR

This paper investigates necessary conditions for a minimum in variational problems with delay, especially when the Weierstrass condition degenerates, by analyzing variations and deriving inequalities and equalities for local minima.

Contribution

It introduces a novel approach using right and left variations to derive necessary conditions for minima under degeneracy of the Weierstrass condition in delayed variational problems.

Findings

Derived inequalities and equalities for minima under degeneracy

Established necessary conditions for strong and weak local minima

Provided an example demonstrating the applicability of the results

Abstract

This article explores minimum of an extremal in the variational problem with delay under the degeneracy of the Weierstrass condition. Here for study the minimality of extremal, variations of the Weierstrass type are used in two forms: in the form of variations on the right with respect to the given point, and in the form of variations on the left with respect to the same point. Further, using these variations, formulas for the increments of the functional are obtained. The exploring of the minimality of the extremal with the help of these formulas is conducted under the assumption that the Weierstrass condition degenerates. As a result, considering different forms of degenerations (degeneracy of the Weieristrass condition at a single point and at points of a certain interval), we obtain the necessary conditions of the inequality type and the equality type for a strong and weak local minimum. A specific example is given to demonstrate the effectiveness of the results obtained in this article.