Multiple Capture of a Given Number of Evaders in a Problem with Fractional Derivatives and a Simple Matrix
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ltiple Capture of a Given Number of Evaders in a Problem with Fractional Derivatives and a Simple Matrix N. N. Petrov1,∗ and A. Ya. Narmanov2,∗∗ Received May 6, 2019; revised June 19, 2019; accepted June 24, 2019
Abstract—A problem of pursuing a group of evaders by a group of pursuers with equal capabilities of all the participants is considered in a finite-dimensional Euclidean space. The system is described by the equation D(α) zij = azij + ui − vj , ui , vj ∈ V, where D(α) f is the Caputo fractional derivative of order α of the function f , the set of admissible controls V is strictly convex and compact, and a is a real number. The aim of the group of pursuers is to capture at least q evaders; each evader must be captured by at least r different pursuers, and the capture moments may be different. The terminal set is the origin. Assuming that the evaders use program strategies and each pursuer captures at most one evader, we obtain sufficient conditions for the solvability of the pursuit problem in terms of the initial positions. Using the method of resolving functions as a basic research tool, we derive sufficient conditions for the solvability of the approach problem with one evader at some guaranteed instant. Hall’s theorem on a system of distinct representatives is used in the proof of the main theorem. Keywords: differential game, group pursuit, multiple capture, pursuer, evader, fractional derivative.
DOI: 10.1134/S0081543820040136 INTRODUCTION An important direction in the modern theory of differential games is associated with the development of solution methods for game problems of pursuit and evasion with several objects [1–4]. In this area, not only are the classical solution methods deepened, but also new problems are sought to which the existing methods are applicable. In particular, in [5,6], problems of pursuing two objects described by equations with fractional derivatives were considered and sufficient conditions of a capture were obtained. Recently, Gomoyunov [7] proved the existence of the value of a nonlinear differential game with fractional derivatives. We consider a linear problem of pursuing a group of evaders by a group of pursuers provided that all the participants have equal capabilities. The problem of simple pursuit of a single evader by a group of pursuers was considered by Pshenichnyi [8], who obtained necessary and sufficient 1 2
Udmurt State University, Izhevsk, 426034 Russia National University of Uzbekistan, Tashkent, 700174 Uzbekistan e-mail: ∗ [email protected], ∗∗ [email protected]
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conditions of a capture. A multiple capture of an evader in a simple group pursuit problem was studied by Grigorenko in [9]. The problem of capturing a given number of evaders in a simple pursuit problem under the conditions that the set of admissible controls is a unit ball centered at zero, the terminal sets are the origins, the evaders use program strategies, and each pursuer captures at most one evader, is presented in [10], where necessary and sufficient conditions for the solvabil
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