Variational problems of Herglotz typewith time delay: DuBois--Reymond condition and Noether's first theorem
Simão P. S. Santos, Natália Martins, Delfim F. M. Torres · Discrete and Continuous Dynamical Systems · 2015
We extend the DuBois--Reymond necessary optimality conditionand Noether's first theorem to variational problems of Herglotztype with time delay. Our results provide, as corollaries,the DuBois--Reymond necessary optimality condition and the firstNoether theorem for variational problems with time delay recentlyproved in [Numer. Algebra Control Optim. 2 (2012), no. 3, 619--630].Our main result is also a generalization of the first Noether-type theoremfor the generalized variational principle of Herglotz proved in[Topol. Methods Nonlinear Anal. 20 (2002), no. 2, 261--273].