On the Construction of a Nonanticipating Selection of a Multivalued Mapping
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the Construction of a Nonanticipating Selection of a Multivalued Mapping D. A. Serkov1,2,∗ and A. G. Chentsov1,2,∗∗ Received May 15, 2019; revised June 19, 2019; accepted June 24, 2019
Abstract—We study the properties of multivalued mappings of general form with respect to the possibility of finding their nonanticipating selections. The property of nonanticipation is formulated for an arbitrary domain by specifying some family of “test” subsets. Sufficient conditions for the existence of a nonanticipating selection of a nonanticipating multivalued mapping are proposed: the values of the mapping must be nonempty compact sets, and the family of “test” subsets must be totally ordered with respect to inclusion. We illustrate the results by showing their applicability to the pursuit–evasion differential game in the form of P. Varaya and J. Lin. Keywords: quasistrategy, nonanticipation, selection, topology.
DOI: 10.1134/S008154382004015X INTRODUCTION The property of nonanticipation plays an important role in the theory of differential games in connection with the construction of idealized resolving strategies. In early papers, idealized strategies (quasi-strategies) were defined in the form of operators on functional spaces of controls or trajectories with the property of physical realizability or nonanticipation (see [1–4] et al.). On the other hand, nonanticipating multivalued mappings and, as a consequence, multivalued quasi-strategies naturally arise in some constructions leading to fixed points of program absorption operators (see [5]). The question on a selection of a multivalued mapping with the nonanticipation property preserved was considered in [6] for mappings on spaces of generalized controls. The study [6] essentially employed the specific features of measure controls and was carried out in connection with the question posed by N. N. Krasovskii to one of the authors. It is known that, in approach–evasion differential games, the formalization in the class of quasi-strategies is equivalent (see [7]) to the positional formalization, i.e., to the formalization in the class of feedback procedures (see [8, 9]). The basic result concerning the structure of differential games is Krasovskii and Subbotin’s theorem of the alternative. It was used to establish the existence of a saddle point for typical performance indices. It is important to note that the mentioned positional formalization can be implemented in terms of step-by-step motions with time discretization, which is natural for engineering applications. In this connection, control procedures with a guide (a model) proposed 1
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia 2 Ural Federal University, Yekaterinburg, 620002 Russia e-mail: ∗ [email protected], ∗∗ [email protected]
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by Krasovskii and Subbotin (see [9, Sect. 57]) played an essential role. It turned out that such procedures can employ quasi-strategies (see [10]). Procedures constructed with the u
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