  We consider the simplest optimal control problem with one nonregular mixed
inequality constraint, i.e. when its gradient in the control can vanish on the
zero surface. Using the Dubovitskii--Milyutin theorem on the approximate
separation of convex cones, we prove a first or der necessary condition for a
weak minimum in the form of the so-called local minimum principle, which is
formulated in terms of functions of bounded variation, integrable functions,
and Lebesgue--Stieltjes measures, and does not use functionals on the space of
measurable bounded functions. Two illustrative examples are given. The work is
based on results by Milyutin.
