Optimal control of parabolic equation with an infinite number of variables for non‐standard functional and time delay

A distributed control problem for the parabolic operator with an infinite number of variables and time delay is considered. The performance index has an integral form. Constraints on controls are imposed. To obtain optimality conditions for the Neumann problem, the generalization of the Dubovitskii–Milyutin theorem given by Walczak in WALCZAK, S. Folia Mathematics, 1, 187–196 and WALCZAK, S. J. Optim. Theory Appl., 42, 561–582 was applied.