On pursuit and evasion game problems with Gro¨\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{\text {o}}$$\end{document}nwall-type constraints

We study a fixed duration pursuit-evasion differential game problem of one pursuer and one evader with Grönwall-type constraints (recently introduced in the work of Samatov et al. (Ural Math J 6:95–107, 2020b)) imposed on all players’ control functions. The players’ dynamics are governed by a generalized dynamic equation. The payoff is the greatest lower bound of the distances between the evader and the pursuers when the game is terminated. The pursuers’ goal, which contradicts that of the evader, is to minimize the payoff. We obtained sufficient conditions for completion of pursuit and evasion as well. To this end, players’ attainability domain and optimal strategies are constructed.

where a(t) = 1, and are given positive numbers, and k is a given non-negative number.They constructed optimal strategies for the players and obtained the optimal pursuit time of the game.The problems considered in Samatov et al. (2020a); Samatov et al. (2020b) brought forth interesting research questions such as: for an arbitrary scalar function a(t), can we find conditions for completion of pursuit in the game described by (1) with the Grönwall-type constraints (2-3)?what conditions can guarantee evasion in the game described by (1) with the constraints (2-3)?
Summarizing, the main objective of this research is to address the research questions stated above.That is, finding conditions for completion of pursuit and also for evasion.To this end, we will construct the players' attainability domain and optimal Grönwall-type strategies.

Problem formulation
Let the dynamics of the pursuer P and evader E be governed by the equations (1) (with i = j = 1) , where x, y, x 0 , y 0 , u, v ∈ ℝ n , and also let the function a(⋅) be a positive scalar function on the interval [0, ∞) .The duration of the game, denoted , is fixed.The payoff function is the infimum of the distances between the evader and the pursuers at : The pursuer's goal is to minimize the payoff, and the evader's goal is to maximize it.
Definition 1 Samatov et al. (2020b) 2) and (3) are called the admissible controls of the pursuer and evader respectively.
Given the players admissible controls u(⋅) and v(⋅) , the corresponding paths x(t), y(t), at any time t > 0 of the players determined by u(⋅), v(⋅), for any initial positions x 0 , y 0 , (respectively) are given by ( 2) Lemma 1 Let (t), t ≥ 0 be a measurable function, and k be non-negative real numbers.Then whenever Proof Let the assumptions of the lemma and ( 8 Definition 2 A continuous function U(x 0 , y 0 , t, v), such that the system has a unique solution for an admissible control v(t) of the evader is called a strategy of the pursuer.The strategy U is said to be admissible if each control generated by this strategy is admissible.
Definition 3 A continuous function V(x 0 , y 0 , t, x, y), is called a strategy of the evader if the following initial valued problem 1 3 has a unique solution (x(t), y), t ≥ 0 .The strategy V is said to be admissible if each control generated by this strategy is admissible.

Definition 4
The strategy U = U(x 0 , y 0 , t, v(t)) guarantees the completion of pursuit at time if, for any admissible control of the evader v(t), t ≥ 0 , we have x( ) = y( ) at some time ∈ [0, ] , where (x(⋅), y(⋅)) is the solution of the initial value problem

Definition 5
The strategy V(x 0 , y 0 , t) guarantees evasion in the game ( 1)-( 3) with an initial positions x 0 , y 0 , if, for any admissible control of the pursuer u(t), t ≥ 0 , the relation x(t) ≠ y(t) holds for all t ≥ 0.

Main results
3.1 The extended 5 Gr −strategy The following is an extension of the Π Gr − strategy constructed in Samatov et al. (2020a) for the simple motion differential game of one-pursuer-one-evader where v(⋅) ∈ ℝ n is an admissible control of the evader.It can be verified that the strategy (13) satisfies The following lemma is crucial in establishing the admissibility of the strategy (13).

Lemma 2 The relation holds for all positive real-valued function a(s)
.
The relation in the lemma 2 follows by integrating both sides of ( 17) from s = 0 to s = t .From ( 14), ( 15) and the lemma 2, we obtain the admissibility of (13) as follows.
That is, ‖U

Attainability domains of players
The attainability domain of the pursuer P at any given time from the initial state x 0 is the closed balls B(x 0 , r P (0, )) , where Based on the classical method of showing players attainability domain (see, for example Ibragimov and Salimi 2009), we must establish the following i.‖x( ) − x 0 ‖ ≤ r P (0, ); ii. given any point x in B(x 0 , r P (0, ) there exists an admissible control of the pursuer P that guarantees x() = x .We show (i) using (5) as follows.
Hence x() = x .Moreover, the admissibility of (20) follows easily from (19).That is Using similar argument, it can be shown that the attainability domain of the evader E at time from the initial state y 0 is the closed ball B(y 0 , r E (0, )) , where

Conditions for completion of pursuit
To state the conditions, we introduce the following notations.Consider the game problem (1-3) and let: and the half space X be defined as follows Theorem 1 If ≥ 0 and y( ) ∈ X , then the Π Gr -strategy (13) guarantees the completion of pursuit in the game (1)-(3) for the pursuer.
Proof Let the assumptions of the theorem hold and the strategy (13) be defined for all t in the interval [0, ] .For t in ( , ], we set where is the time instant at which x( ) = y( ) .Indeed, the strategy (24) is admissible in the time interval ( , ] since Now let x 0 ≠ y 0 .By ( 5) and ( 6) we have y(t) − x(t) = 0 f (t) where Since f (0) = ‖y 0 − x 0 ‖ > 0 , then the conclusion of the Theorem 1 follows if we can establish f ( ) ≤ 0 .That is, f ( ) ≤ 0 implies the existence of ∈ [0, ] such that f ( ) = 0 .To this end, we further introduce the function Observe that where K ∶= � ∫ 0 a(s)e k ∫ s 0 a(r)dr ds
Since we already have That is, x( ) = y( ) .◻

Conditions for evasion
Theorem 2 If  < 0 for all , then evasion is possible in the game (1-3).
Proof Let the hypothesis hold.Consider the evader's strategy Indeed, the strategy (32) is admissible since from lemma 2 we have Let u(⋅) ∈ ℝ n be any admissible control of the pursuer, we show evasion using lemma 1 as follows.

Concluding remarks and suggestions for further research
We have studied a fixed duration pursuit-evasion differential game problem with the Grönwall-type constraints on players control functions.By virtue of the constraints on the players control functions, we constructed the players attainability domains.For the pursuit problem, we constructed the admissible Π Gr strategy which is an extension of the well- known P−strategy, and proved under mild conditions on a certain half-space that the Π Gr strategy can guarantee completion for pursuit.For the evasion problem, we proved that if the total energy resources of the pursuer is less than that of the evader, then evasion is guaranteed through out the game.The problem studied in this paper with multiple players, and also estimating the game value for the game (1) with the Grönwall-type constraints (2)-( 3) are open problems for further research.
Author contributions All authors contributed to the study conception and design.Material preparation, data collection and analysis were performed by JR, MF and AJB.The first draft of the manuscript was written by BAP.All authors commented on previous versions of the manuscript.All authors read and approved the final manuscript.
Funding Open access funding provided by Università degli Studi Mediterranea di Reggio Calabria within the CRUI-CARE Agreement.This work was partially supported by a grant from the Simons Foundation.permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.To view a copy of this licence, visit http:// creat iveco mmons.org/ licen ses/ by/4.0/.

Conflict of interest
follows by integrating both sides of (9) from s = 0 to s = t .◻ According to the lemma 1, if u(⋅) and v(⋅) are admissible controls then we must have Denote by B(O, r) the ball of radius r centered at the origin O.

≥
‖x(t) − y(t)‖ = ‖x 0 − y 0 − � ‖x 0 − y 0 ‖ +  � t 0 a(s)e k ∫ s 0 a(r)dr ds −  � t 0 a(s)e k ∫ s 0 a(r)dr ds ≥ ‖x 0 − y 0 ‖ −  � t 0 a(s)e k ∫ s 0 a(r)dr ds > 0 The authors have no relevant financial or non-financial interests to disclose.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.If material is not included in the article's Creative Commons licence and your intended use is not