Strong Minimizers of the Calculus of Variations on Time Scales and the Weierstrass Condition

TL;DR

This paper introduces the concept of strong local minimizers in the calculus of variations on time scales and establishes a generalized Weierstrass condition that unifies continuous and discrete cases.

Contribution

It defines strong minimizers on time scales and proves a unified Weierstrass necessary condition applicable to both continuous and discrete variational problems.

Findings

Strong local minimizers differ from weak minima on time scales.

A generalized Weierstrass condition is established for time scales.

The results unify continuous and discrete calculus of variations.

Abstract

We introduce the notion of strong local minimizer for the problems of the calculus of variations on time scales. Simple examples show that on a time scale a weak minimum is not necessarily a strong minimum. A time scale form of the Weierstrass necessary optimality condition is proved, which enables to include and generalize in the same result both continuous-time and discrete-time conditions.