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 Izv. IMI UdGU, 2018, Volume 52, Pages 75–85 (Mi iimi362)

The evasion problem in a nonlinear differential game with discrete control

A. Ya. Narmanova, K. A. Shchelchkovb

a National University of Uzbekistan, ul. Universitetskaya, 4, Tashkent, 100174, Uzbekistan
b Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia

Abstract: A two-agent differential game is considered. The game is described by the following system of differential equations: $\dot x = f(x, v) + g(x, u),$ where $x \in \mathbb R^k$, $u \in U$, $v \in V$. The evader's admissible control set is a finite subset of phase space. The pursuer's admissible control set is a compact subset of phase space. The pursuer's purpose is to avoid an encounter, that is, to ensure a system position no closer than some neighborhood of zero. Sufficient conditions for avoidance of an encounter in the class of piecewise open-loop strategies on infinite and any finite-time intervals are obtained. The conditions are superimposed on the velocity vectogram at the zero point of phase space. When the game is considered on an infinite time interval, the conditions provide the evader with some advantage. The properties of a positive basis play a major role in proving the theorems.

Keywords: differential game, nonlinear system, avoidance of an encounter, discrete control.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-51-41005_Óçá_ò16-01-00346_à The work of the first author was supported by the grant of MRU-10/17. The work of the second author was supported by the Russian Foundation for Basic Research (project no. 18–51–41005, 16–01–00346).

DOI: https://doi.org/10.20537/2226-3594-2018-52-06

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Bibliographic databases:

UDC: 517.977
MSC: 49N70, 49N75

Citation: A. Ya. Narmanov, K. A. Shchelchkov, “The evasion problem in a nonlinear differential game with discrete control”, Izv. IMI UdGU, 52 (2018), 75–85

Citation in format AMSBIB
\Bibitem{NarShc18} \by A.~Ya.~Narmanov, K.~A.~Shchelchkov \paper The evasion problem in a nonlinear differential game with discrete control \jour Izv. IMI UdGU \yr 2018 \vol 52 \pages 75--85 \mathnet{http://mi.mathnet.ru/iimi362} \crossref{https://doi.org/10.20537/2226-3594-2018-52-06} \elib{https://elibrary.ru/item.asp?id=36508457}