Noether's Symmetry Theorem for Variational and Optimal Control Problems   with Time Delay
  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. 
Gastao S. F. Frederico, Delfim F. M. Torres
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