# On the Relation Between Two Approaches to Necessary Optimality Conditions in Problems with State Constraints

## TL;DR

This paper explores two methods for deriving necessary optimality conditions in control problems with state constraints, demonstrating how differentiating along boundary subarcs yields comprehensive stationarity conditions including measure sign definiteness.

## Contribution

It introduces a two-stage variation approach to derive full stationarity conditions, linking Gamkrelidze's and Dubovitskii-Milyutin's methods for problems with boundary state constraints.

## Key findings

- Full stationarity conditions include measure sign definiteness.
- Differentiating along boundary subarc simplifies the problem.
- Two-stage variation approach clarifies the relation between methods.

## Abstract

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.