On singular controls in the sense of the pointwise maximum principle in distributed optimization problems

It is proved that for distributed optimization problems a sufficiently typical situation is strong degeneration of the singular controls in the sense of the pointwise maximum principle, when together with the maximum principle (which is a first order necessary optimality condition in the case of spike-shaped variation) a second order necessary optimality conditions also degenerates. A derivation of constructive necessary optimality conditions for singular controls is suggested.