Local Minimum Principle for an Optimal Control Problem with a Nonregular Mixed Constraint

We consider the simplest optimal control problem with one nonregular mixed constraint $G(x,u)\le0,$ i.e., such a that gradient $G_u(x, u)$ can vanish on surface $G = 0.$ Using Dubovitskii--Milyutin theorem approximate separation of convex cones, we prove first order necessary condition for weak minimum in form so-called local principle, which is formulated terms functions bounded variation, integrable functions, and Lebesgue--Stieltjes measures does not use functionals from $(L^\infty)^*$. Two illustrative examples are provided. The work based book by Milyutin [Maximum Principle General Problem Optimal Control, Fizmatlit, Moscow, 2001 (in Russian)].