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

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  • 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].
    Mathematics Subject Classification: Primary: 49K15, 49S05; Secondary: 34H05.

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