arXiv subjects

Dmitrii Serkov

Publications and source records attributed to Dmitrii Serkov.

3 recordsLinked to original sources

arXiv2016

On approximation of joint fixed points

For a given partially ordered set (poset) and a given family of mappings of the poset into itself, we study the problem of the description of joint fixed points of this family. Well-known Tarski's theorem gives the structure of the set of joint fixed points of isotone automorphisms on a complete lattice. This theorem has several generalizations (see., e.g., Markowsky, Ronse) that weaken demands on the order structure and upgrade in an appropriate manner the assertion on the structure of the set of joint fixed points. However, there is a lack of the statements similar to Kantorovich or Kleene theorems, describing the set of joint fixed points in terms of convergent sequences of the operator degrees. The paper provids conditions on the poset and on the family; these conditions ensure that the iterative sequences of elements of this family approximate the set of joint fixed points. The result obtained develops in a constructive direction the mentioned theorems on joint fixed points.

math.LO↗

arXiv2015

On fixed points of multivalued mappings

The paper discusses the conditions for the existence of fixed points of multivalued mappings that are not based on the linear structure of the set. The descriptions for the sets of fixed points for mappings with closed graph in compact Hausdorff spaces and in metrizable spaces, as well as for continuous functions in spaces with convergence (Frechet topology) and in spaces with Scott topology are provided. Applications to the problem of the equilibrium are given: the sets of saddle points and of Nash equilibria for compact Hausdorff and metrizable spaces of strategies of players are described.

math.GN↗

arXiv2014

On non-improvability of full-memory strategies in problems of optimization of the guaranteed result

The paper addresses the problem of optimization of a guaranteed (worst case) result for a control system driven by a controlling side in presence of a dynamical disturbance. The disturbances as functions of time are subject to functional constraints belonging to a given family of constraints. The latter family is known to the controlling side that does not observe the disturbance and uses full-memory strategies to form the control actions. The study is focused on the case where disturbance varies in open-loop disturbances chosen in advance and the case where the disturbances are restricted to a $L_2$--compact set fixed in advance but unknown to the controlling side. In these cases it is shown that the optimal guaranteed result is non-improvable in the sense that it coincides with that obtained in the class of quasi-strategies -- nonantisipatory transformations of disturbances into controls. An $\varepsilon$--optimal full-memory strategy is constructed. An illustrative nonlinear example is given.

math.OC↗