Noether's Symmetry Theorem for Variational and Optimal Control Problems with Time Delay

TL;DR

This paper extends Noether's symmetry theorem to variational and optimal control problems with time delays, providing both Lagrangian and Hamiltonian formulations to address delayed systems.

Contribution

It introduces a generalized version of Noether's theorem applicable to delayed variational and control problems, including new necessary conditions and symmetry results.

Findings

Extended DuBois-Reymond condition for delayed systems

Proved Lagrangian and Hamiltonian Noether's theorems with delays

Applicable to calculus of variations and optimal control with time delays

Abstract

We extend the DuBois-Reymond necessary optimality condition and Noether's symmetry theorem to the time delay variational setting. Both Lagrangian and Hamiltonian versions of Noether's theorem are proved, covering problems of the calculus of variations and optimal control with delays.