Jimsher Giorgobiani

The problem of optimal performance of a power system is considered. The problem is posed in various settings within the frames of the theory of optimal storage control. Mathematical models are presented by means of the reccurent equations of dynamic programming

Informational and metastrategic extensions - metagames of lexicographic finite noncooperative game are discussed . It is proved that there always exists an equilibrium situation in the first metagames. According to an example it is shown that in the metaextensions there exists a situation which is at the same time equilibrium and optimal in Pareto's sence

In classical cooperative game theory one of the most important principle is defined by Shapley with three axioms common payoff fair distribution's Shapley value (or Shapley vector). In the last decade the field of its usage has been spread widely. At this period of time Shapley value is used in network and social systems. Naturally, the question is if it is possible to use Shapley's classical axiomatics for lexicographic cooperative games. Because of this in the article for m dimensional lexicographic cooperative game is introduced Shapley's axiomatics, as the principle of a fair distribution in the case of m dimensional payoff functions, when the criteria are strictly ranking. It has been revealed that axioms discussed by Shapley for classical games are sufficient in lexicographic cooperative games corresponding with the payoffs of distribution. Besides we are having a very interesting case: according to the proved theorem, Shapley's classical principle simultaneously transforms on the composed scalar games of a lexicographic cooperative game, nevertheless, games could not be superadditive.

Present work deals with multi-phase models of inventory control for cascade systems. Situation is similar to that of queuing theory, the inflows are used to fulfill various types of demands. Three models are introduced. The first two models are relatively easy and fit well the control of hydroenergy system. They are implemented by means of dynamic programming. The third model is general, cascade may include some types of enterprises together with their storages. In the case of homogenous productions the model can be applied for the control of cascade hydropower stations. This model is the problem of linear programming. In all cases optimality criteria is the maximum profit

The present work deals with lexicographic noncooperative (strategic) games in which the set of strategies of the players are metric compact spaces and the vector-functions of winning are continuous on the set of situations. In such games we introduce the definition of a weak nonstrict (determined by usual nonstrict lexicographic inequality) of Nash equilibrium situation in pure strategies. It has been defined the stability of such equilibrium situation and of lexicographic noncooperative game in relation to change of vectorfunctions of the winning of players, a problem of an equilibrium stable situation and availability of lexicographic noncooperative game has been studied. The conditions of their stability have been brought. The identification of the indicated conditions has been connected with those features of the task of lexicographic maximum that differs from the task of scalar maximum: the set of points of lexicographic maximum in the task of lexicographic maximum of continuous vector-function defined on metric compact is compact. And in the lexicographic noncooperative game the set of equilibrium situations may not be compact. In particular, it is certified that if in lexicographic game there is only one equilibrium situation then it is a stable situation and the relative game is stable.