On obtaining the stationarity conditions in an optimal control problem for a trajectory with a smooth contact with the phase boundary

We consider a class of optimal control problems with a scalar state constraint and investigate trajectory with a smooth contact with a single boundary subarc. For trajectory of this special type, we first obtain, using a special technique (two-stage variation approach), stationarity conditions without non-negativity, and then obtain the full set of stationarity conditions in the Dubovitskii-Milyutin form, including the nonnegativity of the measure density and its atoms at the junction points.