#
Evasion Differential Game of Multiple Pursuers and a Single Evader with Geometric Constraints in ℓ^{2}

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{2}. The control functions of the players are subject to geometric constraints. The pursuers’ goal is to bring the state of at least one of the controlled systems to the origin of ℓ

^{2}, while the evader’s goal is to prevent this from happening in a finite interval of time. We derive a sufficient condition for evasion from any initial state and construct an evasion strategy for the evader.

## 1. Introduction

^{2}with a finite number of pursuers, when players have identical capabilities has not been studied, and no initial position has been identified from which the pursuit can be completed. The pursuit differential game of one pursuer and one evader has been studied in [34] for an infinite system of binary differential equations in the Hilbert space ℓ

^{2}. The general case of this problem was studied in [35]. Moreover, the papers [36,37,38] relate to differential games with an infinite system.

^{2}confers an advantage to the evader, enabling evasion from any finite number of pursuers.

## 2. Statement of the Problem

^{2}is the set of all sequences of real numbers

**Definition 1.**

**Definition 2.**

**Definition 3.**

## 3. The Main Result

**Theorem 1.**

**Proof.**

^{2}, where $\parallel \alpha \parallel =1$, such that the inner product $\langle {\eta}_{i}^{0},\alpha \rangle \ge 0$ for all $i=1,\cdots ,m$. We can choose the vector $\alpha $ to be an orthonormal vector to the hyperplane that passes through the points ${\eta}_{i}^{0}$, where $i=1,2,\cdots ,m$.

## 4. Conclusions

^{2}. In the construction of the evasion strategy, the fact that a finite number of points lies on a hyperplane played the key role.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Ibragimov, G.; Ruziboev, M.; Zaynabiddinov, I.; Pansera, B.A.
Evasion Differential Game of Multiple Pursuers and a Single Evader with Geometric Constraints in *ℓ*^{2}. *Games* **2023**, *14*, 52.
https://doi.org/10.3390/g14040052

**AMA Style**

Ibragimov G, Ruziboev M, Zaynabiddinov I, Pansera BA.
Evasion Differential Game of Multiple Pursuers and a Single Evader with Geometric Constraints in *ℓ*^{2}. *Games*. 2023; 14(4):52.
https://doi.org/10.3390/g14040052

**Chicago/Turabian Style**

Ibragimov, Gafurjan, Marks Ruziboev, Ibroximjon Zaynabiddinov, and Bruno Antonio Pansera.
2023. "Evasion Differential Game of Multiple Pursuers and a Single Evader with Geometric Constraints in *ℓ*^{2}" *Games* 14, no. 4: 52.
https://doi.org/10.3390/g14040052