Hight-order Noether's Theorem for Nonsmooth Extremals of Isoperimetric Variational Problems with Time Delay

TL;DR

This paper extends Noether's theorem to nonsmooth, higher-order isoperimetric variational problems with time delay, applicable to Lipschitz functions and delayed optimal control, broadening the theoretical framework.

Contribution

It provides a higher-order, nonsmooth version of Noether's theorem for delayed variational problems, including isoperimetric and optimal control cases.

Findings

Established a higher-order Noether's theorem for Lipschitz functions with delays.

Extended the theorem to include delayed isoperimetric optimal control problems.

Validated the theorem under the delayed higher-order DuBois-Reymond condition.

Abstract

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