Stationarity and Regularity of Infinite Collections of Sets

This article investigates extremality, stationarity, and regularity properties of infinite collections of sets in Banach spaces. Our approach strongly relies on the machinery developed for finite collections. When dealing with an infinite collection of sets, we examine the behavior of its finite subcollections. This allows us to establish certain primal-dual relationships between the stationarity/regularity properties some of which can be interpreted as extensions of the Extremal principle. Stationarity criteria developed in the article are applied to proving intersection rules for Fréchet normals to infinite intersections of sets in Asplund spaces. © 2012 Springer Science+Business Media, LLC

New Necessary Conditions for Optimal Control Problems in Discontinuous Dynamic Systems

In the paper we derive new necessary optimality conditions for optimal control of differential equations systems with discontinuous right hand side. The main attention is paid to a situation when an optimal trajectory slides on the discontinuity surface. The new conditions, derived in the paper, are essential and do not follow from any known necessary conditions for such systems

Optimality conditions (In Pontryagin form)

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