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

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].

This is a preprint of a paper whose final and definite form will be published in the journal Discrete and Continuous Dynamical Systems. Series A, ISSN 1078-0947 (Print), ISSN 1553-5231 (Online). Submitted 01/May/2014; revised 14/Oct/2014; accepted for publication 20/Jan/2015

References in corpus (8)

• A Formulation of Noether's Theorem for Fractional Problems of the Calculus of Variations
• Fractional Noether's theorem in the Riesz-Caputo sense
• Noether's Theorem on Time Scales
• The Second Euler-Lagrange Equation of Variational Calculus on Time Scales
• Noether's symmetry theorem for nabla problems of the calculus of variations
• Noether's Symmetry Theorem for Variational and Optimal Control Problems with Time Delay
• Gauge Symmetries and Noether Currents in Optimal Control
• Computing ODE Symmetries as Abnormal Variational Symmetries

Cited by in corpus (3)

• Noether theorem for action-dependent Lagrangian functions: conservation laws for non-conservative systems
• A Non-Newtonian Noether's Symmetry Theorem
• A survey on sufficient optimality conditions for delayed optimal control problems