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\(D^{(\alpha)}z_{ij}=az_{ij}+u_{i}-v_{j},\ \ u_{i},v_{j}\in V,\)where \(D^{(\alpha)}f\) is the Caputo fractional derivative of order \(\alpha\) 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.

中文翻译：

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具有分数导数和简单矩阵的问题中给定数量的规避值的多重捕获

在有限维欧几里德空间中考虑了一个问题，即由一群追求者在所有参与者之间具有同等能力来追求逃避者的问题。系统由等式\（D ^ {（\\ alpha）} z_ {ij} = az_ {ij} + u_ {i} -v_ {j}，\ \ u_ {i}，v_ {j} \ in来描述V，\）其中\（D ^ {（\\ alpha}} f \）是 函数 \（f \）的可阶\（\ alpha \）的Caputo分数导数 ，这是一组允许的控件 \（V \）是严格凸且紧的，  \（a \）是实数。追赶者小组的目的是至少捕获 \（q \）个逃避者；每个逃避者必须至少被\（r \）捕获  不同的追随者，捕获的时刻可能会有所不同。终端机是原点。假设逃避者使用程序策略并且每个追踪者最多捕获一个逃避者，我们就初始位置而言具有足够的条件来解决追踪问题。使用解决功能的方法作为基础研究工具，我们得出了在某个保证的时刻使用一个躲避者解决逼近问题的充分条件。在主要定理的证明中使用了霍尔在不同代表系统上的定理。