Constants of motion for Isoperimetric Variational Problems with Time Delay

In the present work, we obtain the constants of motion for isoperimetric variational problems with time delay. We consider a constrained optimization problem where the Lagrangian function defining the functional depends on time delayed arguments. We prove the isoperimetric Euler–Lagrange and DuBois–Reymond type optimality conditions and, in order to investigate the constants of motion for this problem, we obtain a nonsmooth extension of Noether’s symmetry theorem for isoperimetric variational problems with delayed arguments.