Variational problems of Herglotz type with time delay: DuBois-Reymond condition and Noether's first theorem

TL;DR

This paper extends classical optimality conditions and symmetry theorems to Herglotz-type variational problems with time delay, broadening their applicability in control and optimization theory.

Contribution

It generalizes the DuBois-Reymond condition and Noether's theorem to include time delay in Herglotz variational problems, building on recent and prior results.

Findings

Extended DuBois-Reymond condition with time delay

Generalized Noether's first theorem for delayed problems

Unified framework for variational problems with delay

Abstract

We extend the DuBois-Reymond necessary optimality condition and Noether's first theorem to variational problems of Herglotz type with time delay. Our results provide, as corollaries, the DuBois-Reymond necessary optimality condition and the first Noether theorem for variational problems with time delay recently proved in [Numer. Algebra Control Optim. 2 (2012), no. 3, 619-630]. Our main result is also a generalization of the first Noether-type theorem for the generalized variational principle of Herglotz proved in [Topol. Methods Nonlinear Anal. 20 (2002), no. 2, 261-273].