Iterative realization of nonanticipating multivalued mappings
The method of open-loop strategies iterations was developed by \textit{A. I. Subbotin} and the author in the context of pursuit-evasion differential games [``Guaranteed optimization in control problems'' (1981; Zbl 0542.90106)]. In the reviewed work, the method is extended to positional differential games on a fixed interval with a very general class of admissible strategies. Namely, the strategies of the two players are described by uniform limits of respective sequences of step-functions \(f\) and \(g\), measurable with respect to a semi-algebra \(\mathcal L\) of subsets of a set \(E\). Semi-algebra \(\mathcal L\) is associated with a closed set of finitely additive probability measures \(P_0\). The performance criterion for the iterative construction is described by inclusion of the integrals of the functions \(f\) and \(g\) over the measures from \(P_0\) in a set \(Y\subset R\times R\). In particular, if \(Y=\{y, -y\}\) and \(P_0\) is a singleton family consisting only of the Lebesgue measure on \(E=[0,1]\), then the criterion reduces to the conventional capture criterion in a differential game with simple motion dynamics: \(\dot x = u+v\), \(|u|\leq 1\), \(|v|\leq 1\). The extension of the iterative procedure from \textit{A. I. Subbotin} and \textit{A. G. Chentsov} [loc. cit.] leads to nonanticipating counter-action strategies described by multi-valued mappings. An investigation of the extremal selectors of such mappings results in some conclusions about the existence of single-valued strategies.
scientific article; zbMATH DE number 1541588