Evgenii S. Polovinkin

Moscow Institute of Physics and Technology | MIPT · Department of Higher Mathematics

We consider the distance function (DF), given by the caliber (the Minkowski gauge function) of a convex body, from a point to strictly, strongly, and weakly convex sets in an arbitrary Hilbert space. Some properties of the caliber of a strongly convex set and the conditions for obtaining a strict, strong, or weak convexity of Lebesgue sets for the...

The property of the continuous dependence of the trajectories of the differential inclusion with a measurable pseudo-Lipschitz unbounded right-hand side on the initial approximations in Euclidean space is proved. This property is an important part of the direct method of obtaining the necessary optimality conditions in the extremal problems with co...

Развивается прямой вариационный метод Понтрягина для получения необходимых условий в экстремальной задаче Майера на фиксированном отрезке, ограничение на траектории в которой задается дифференциальным включением с неограниченной, вообще говоря, правой частью. Полученные необходимые условия оптимальности содержат сопряженное дифференциальное включен...

We develop Pontryagin’s direct variational method, which allows us to obtain necessary conditions in the Mayer extremal problem on a fixed interval under constraints on the trajectories given by a differential inclusion with generally unbounded right-hand side. The established necessary optimality conditions contain the Euler—Lagrange differential...

We study the properties of a parameterized sequence of countably additive vector measures with densities defined on a compact space T with a nonnegative nonatomic Radon measure μ and taking values in a separable Banach space. Each vector measure of this sequence depends continuously on a parameter belonging to a metric space. We assume that a count...

Based on the properties of solutions of a differential inclusion with unbounded measurable-pseudo-Lipschitz right-hand side, the necessary conditions in the Mayer extremal problem on set of trajectories of the unbounded differential inclusion are obtained. DOI: 10.1134/S0081543813080099

It is shown that for some classes of functions all epiderivatives and subdifferentials of the Clarke, Michel—Penot, and other types coincide. Several rules of calculation of epiderivatives and subdifferentials for the difference of two convex functions are obtained. Some examples are considered.

A differential inclusion with values in a reflexive Banach space such that its righthand side is at each time a convex closed cone is considered. The form of the weak polar cone of the cone of strongly bounded solutions to the Cauchy problem for this inclusion is found. A solution is called strongly bounded if it is an absolutely continuous functio...

Based on the properties of solutions of a differential inclusion with unbounded measurable-pseudo-Lipschitz right-hand side, the necessary conditions in time optimum problem on set of solutions of the unbounded differential inclusion are obtained. DOI: 10.1134/S0081543813080099

We study the properties of the trajectories of a differential inclusion with unbounded measurable–pseudo-Lipschitz right-hand side that takes values in a separable Banach space and consider the problem of minimizing a functional over the set of trajectories of such a differential inclusion on an interval. We obtain necessary optimality conditions i...

An example of multivalued convex-valued Lipschitz mapping from ℝn into ℝm such that, at any point, the support function of this mapping has no mixed derivatives in the sense of Gâteaux with respect to the initial and conjugate variables is constructed.

We obtain existence theorems and Filippov-Ważewski type relaxation theorems for differential inclusions in Banach spaces with measurable-pseudo-Lipschitz right-hand side. For the solution sets of these differential inclusions, we also describe some properties that extend classical theorems on continuous dependence and on differentiation of solution...

A general form of the polar cone is obtained for the solution set of an arbitrary differential inclusion such that the graph of its right-hand side is a convex closed cone and the solutions take values in a reflexive Banach space.

We consider the possibility of generalizing the averaging theorem from the case of sets from n-dimensional Euclidean space to the case of sets from Banach spaces. The result is a cornerstone for constructing the theory of the Riemann integral for non-convex-valued multivalued mappings and for proving the convexity of this multivalued integral. We o...

A survey of the author’s results is presented, and the research is continued of the known differential geometry problem on the construction of bodies of constant width containing an arbitrary given bounded set of the same diameter as the width of the bodies. The problem is considered for sets from reflexive Banach spaces in which the unit ball is a...

A numerical algorithm for the construction of stable Krasovskii bridges, Pontryagin alternating sets, and also of piecewise program strategies solving two-person linear differential (pursuit or evasion) games on a fixed time interval is developed on the basis of a general theory. The aim of the first player (the pursuer) is to hit a prescribed targ...

Properties of strongly convex sets (that is, of sets that can be represented as intersections of balls of radius fixed for each particular set) are investigated. A connection between strongly convex sets and strongly convex functions is established. The concept of a strongly convex R-hull of a set (the minimal strongly convex set containing the giv...

The mathematical model of all the activities of a state are presented. The state is not understood in a narrow sense as a part of the society's political sphere. A mathematical model of a more modern unitary state, which is called a nome is discussed. The sets of the first level are related to the second nome level. A graph is called connected if a...

Considered is the problem of searching the new classes of generating sets, in particular of algorithms for constructing constant-width bodies containing a given set. The formula is derived, which can be used for constructing nontrivial bodies of constant width but only for non-central-symmetric sets. However, with employing the lemma based on the d...

The authors solve the competition problem for two enterprises. The problem reduces to an antagonistic dynamic game on a fixed time interval with a convex compact objective set. The problem is solved in the interests of one enterprise. The set of initial positions is constructed, for which an enterprise necessarily attains the objective at a given f...

For subsets of a Banach space the notions of a generating set M and an M-strongly convex set are introduced. The latter can be represented as the intersection of sets of the form M+x, which are translates of the generating set M. A generating set must satisfy a condition that ensures a special support principle, as shown in the paper. Using this su...

For a long time the detailed investigation of optimization problems for differential inclusions encountered great principal difficulties for lack of suitable properties of solutions of differential inclusions. F. Clark avoided those difficulties mainly by using his approximated method and Ekeland’s Theorem (see F. Clark [5]). But we have to remark...

A class of differential games is delineated, in which the main pursuit operator /1,2/ is computed analytically. The support function of set is written out in explicit form. It is proved that for this class of games the optimal pursuit time coincides with the maximin pursuit time /1–5/ introduced by Kelendzheridze. A sufficient condition for the com...

Lebesgue integral of a set-valued function is a conception of growing importance for the theories of the optimal control and of the differential games (see for example [1]). For solving concrete problems it will be often useful to reduce a Lebegue set-integral to a riemannien one. In this article we shall find sufficient and necessary conditions of...

We shall prove that, in the case of a stable terminal set, the maximal operator T → t 1 t 2 of a differential pursuit game coincides with the sufficient operator F → t 1 t 2 ; this implies the optimization of the time of the alternative integral. We establish sufficient conditions for the optimization of the pursuit time when the terminal set is st...