Noether's Theorem for Nonsmooth Extremals of Variational Problems with Time Delay

TL;DR

This paper extends Noether's symmetry theorem to nonsmooth extremals in delayed variational problems, valid for Lipschitz functions under certain conditions, including higher-order derivatives.

Contribution

It introduces a nonsmooth version of Noether's theorem for delayed variational problems, applicable to Lipschitz functions and higher-order derivatives.

Findings

Valid for Lipschitz functions with delayed Euler-Lagrange extremals

Includes higher-order derivatives in delayed variational problems

Provides a framework for nonsmooth extremals with delays

Abstract

We obtain a nonsmooth extension of Noether's symmetry theorem for variational problems with delayed arguments. The result is proved to be valid in the class of Lipschitz functions, as long as the delayed Euler-Lagrange extremals are restricted to those that satisfy the DuBois-Reymond necessary optimality condition. The important case of delayed variational problems with higher-order derivatives is considered as well.