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\(\varepsilon \)-Capture in Nonlinear Differential Games Described by System of Order Two

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Abstract

A differential game described by a nonlinear system of differential equations of order two is considered in a finite Euclidean space. The value set of the pursuer control is a finite set. The value set of the evader control is a compact set. The purpose of the pursuer is a translation of the system in a finite time to any given neighborhood of zero. The pursuer uses a piecewise open-loop strategies constructed only by using information on the state coordinates and the velocity in the partition points of a time interval. Sufficient conditions of the capture problem solvability in the piecewise open-loop strategies class are obtained. These conditions are imposed on the system vectogram at zero and have a geometric nature. Also, it is proved that the capture time tends to zero with the approach the initial position to zero. It happens independently of the evader’s actions. The solution of the problem is based on a positive basis notion.

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  1. Udmurt State University, ul. Universitetskaya, 1, Izhevsk, Russia, 426034

    Kirill Shchelchkov

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  1. Kirill Shchelchkov
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Correspondence to Kirill Shchelchkov.

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This research was funded by the Ministry of Science and Higher Education of the Russian Federation in the framework of state assignment no. 075-00232-20-01, project 0827-2020-0010 “Development of the theory and methods of control and stabilization of dynamical systems" and under grant 20-01-00293 from the Russian Foundation for Basic Research. The research was performed using computing resources of the collective use center of IMM UB RAS “Supercomputer center of IMM UB RAS."

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Shchelchkov, K. \(\varepsilon \)-Capture in Nonlinear Differential Games Described by System of Order Two. Dyn Games Appl 12, 662–676 (2022). https://doi.org/10.1007/s13235-021-00393-0

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