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

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Abstract
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.
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