On two approaches to necessary conditions for an extended weak minimum in optimal control problems with state constraints

摘要

We consider a class of optimal control problems with a scalar state constraint. For a trajectory with a single boundary subarc, we first obtain, using a special technique (two-stage var iation approach), optimality conditions in the form of Gamkrelidze, and then obtain the full set of optimality conditions in the Dubovitskii-Milyutin form, including the nonnegativity of the measure density and its atoms at the junction points.