On question about extension of maximin problem with phase constraints

Abstract

We study the asymptotic behavior of maximin values of a payoff function, when relaxed constraints are tightened. The payoff function depends on the trajectories of controlled systems of the first and second player. An extension in the class of the Radon measures is used. The asymptotic equivalence between two types of the constraints relaxations is shown. Mathematics Subject Classification: 46N10, 49J35, 49J45, 49N99Keywords: Radon measures, extension, relaxation of constraints, maximin, asymptotic equivalenceWe study the asymptotic behavior of maximin values of a payoff function, when relaxed constraints are tightened. The payoff function depends on the trajectories of controlled systems of the first and second player. These systems can be non-linear; this is the important distinction from earlier papers [1,2], where various asymptotic effects for linear systems with impulse constraints were considered and an extension in the class of finitely additive measures was used (see also [3,4]). In the present paper a different approach is used (see [6]). We implement an extension in the class of the Radon measures. This approach is similar to the traditional one proposed in [5,7,8] and developed by N.N. Krasovskii and by his followers [9,10].