Abstract
We consider the simplest optimal control problem with one nonregular mixed constraint $G(x,u)\le0,$ i.e., such a constraint that the gradient $G_u(x, u)$ can vanish on the surface $G = 0.$ Using the Dubovitskii--Milyutin theorem on the approximate separation of convex cones, we prove a first order 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 from $(L^\infty)^*$. Two illustrative examples are provided. The work is based on the book by Milyutin [Maximum Principle in the General Problem of Optimal Control, Fizmatlit, Moscow, 2001 (in Russian)].
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