  We consider a class of optimal control problems with a state constraint and
investigate a trajectory with a single boundary interval (subarc). Following
R.V. Gamkrelidze, we differentiate the state constraint along the boundary
subarc, thus reducing the original problem to a problem with mixed
control-state constraints, and show that this way allows one to obtain the full
system of stationarity conditions in the form of A.Ya. Dubovitskii and A.A.
Milyutin, including the sign definiteness of the measure (state constraint
multiplier), i.e., the nonnegativity of its density and atoms at junction
points. The stationarity conditions are obtained by a two-stage variation
approach, proposed in this paper. At the first stage, we consider only those
variations, which do not affect the boundary interval, and obtain optimality
conditions in the form of Gamkrelidze. At the second stage, the variations are
concentrated on the boundary interval, thus making possible to specify the
stationarity conditions and obtain the sign of density and atoms of the
measure.
