Pursuit Differential Game with Slow Pursuers on the 1-Skeleton Graph of the Icosahedron

Abstract

A differential game of m, $3\le m\le 6$, pursuers and one evader is studied on an icosahedron in ${\mathbb{R}}^3$. All the players move only along the 1-skeleton graph of the icosahedron when the maximal speeds of the pursuers are less than the speed of the evader. Pursuit is said to be completed if the state of a pursuer coincides with the state of evader at some time. We give a sufficient condition of the completion of pursuit in the game.

1. Introduction

The main constraints on the controls for differential games are integral constraint and geometric constraint. In the case of geometric constraints on the controls of players, players choose the values of control functions from given convex subsets of ${\mathbb{R}}^d$ for some $d\ge 1$. Differential games of many players is one of the important chapters of differential games. For the multi-player differential games with geometric constraints, interesting results were obtained by [1,2,3,4,5,6,7,8,9,10,11]. A detailed survey is given in the paper [12] for such differential games.

In [10], a game of m pursuers and one evader is studied. All the players have the same dynamic capability. The game is completed if the states of k pursuers simultaneously coincide with the state of the evader. If fewer than k pursuers reach the evader, then all of these pursuers are destroyed by the pursuer. In the case of a discrete time game, the necessary and sufficient condition of game completion is obtained in terms of k-hull.

Additionally, the paper [2] studies a differential game of a group of rigidly coordinated evaders and many pursuers with equal capabilities. Simultaneous multiple capture occurs in the game if a certain number of pursuers catches the evaders at the same time. The main result of the paper is the sufficient and necessary conditions for simultaneous multiple capture of evaders, where pursuers use piecewise-program counter strategies.

The differential games of many players, where the control resources of players are bounded, are an interesting and difficult area of differential games. In such games, players need to optimize the expenditure of the resource as well (see, for example, refs. [13,14,15]). In [14], the simple motion differential game of many pursuers and many evaders was studied, and it was proved that if the total energy of the evaders is greater than or equal to that of the pursuers, then evasion is possible from any initial position of the players.

There are many papers devoted to differential games with state constraints (see, for example, refs. [16,17]). An interesting class of differential games with state constraint is differential games on graphs (see, for example, refs. [18,19,20,21,22]). If the graph is connected and finite, then only one fast pursuer can catch the evader. Therefore, the case of many slow pursuers is of importance, and the problems for such games are minimizing the number of pursuers to complete the game and for the construction of pursuit and evasion strategies.

Azamov et al. [22] studied differential games of many pursuers and one evader on the edge graphs of regular polyhedrons in ${\mathbb{R}}^3$. The speeds of all players do not exceed 1. It was established that the minimum number of pursuers to complete the game for the tetrahedron, octahedron and cube is two, and that for the dodecahedron and icosahedron is three.

Later on Azamov et al. [21] showed that on the edge graphs of the d-dimensional regular simplex, cocube, and cube, and it was shown that the minimum numbers of pursuers to complete the game for these polyhedrons are 2, 2, and [d/2] + 1, respectively. In the work [19], it was established that the minimum number of pursuers to complete the game for the regular 24-gone and 120-gone in ${\mathbb{R}}^4$ is equal to 3.

The paper [23] is devoted to the differential game of many slow pursuers and one evader on the edge graphs of the n-dimensional cocube. The problem of finding the optimal number of pursuers to complete the game was solved in that paper. It should be noted that the $\Pi$-strategy (see, for example, ref. [24]) played a key role in constructing the strategies of players.

It should be noted that there are other types of dynamic games on abstract graphs, such as multimove games, where players move from one vertex of the graph to an adjacent one by jumping (see, for example, refs. [25,26,27,28,29,30] and search games (see, for example, ref. [31]).

The present paper is devoted to studying a pursuit game of m, $3\le m\le 6$, slow pursuers and one evader on the 1-skeleton graph of the regular icosahedron K in the Euclidean space ${\mathbb{R}}^3$ (Figure 1).

We recall that a regular icosahedron is one of the five Platonic solids, and it is a convex polyhedron with 30 edges, 12 vertices, and 20 faces. We construct strategies for the slow pursuers to complete the game.

2. Statement of Problem

In the present paper, a differential game of m pursuers $x_1$, $x_2$, …, $x_m$ and one evader y is considered. Players move according to the following equations:

$$\begin{array}{cc}{\dot{x}}_i=u_i,\hfill & x_i\left(0\right)=x_{i0},i=1,2\dots m,\hfill \\ \dot{y}=v,\hfill & y\left(0\right)=y_0,\hfill \end{array}$$

(1)

where $x_i,y,x_{i0},y_0\in {\mathbb{R}}^3$, $x_{i0}\ne y_0$, $i=1,2,\dots ,m$; $u_1$, $u_2$, …, $u_m$ and v are the control parameters of pursuers $x_1$, $x_2$, …, $x_m$ and evader, respectively. All the pursuers and the evader move along the edge graphs of icosahedron K. The maximal speed of evader y is $\sigma$, and the maximal speeds of the pursuers $x_1$, $x_2$, …, $x_m$ are ${\rho }_1$, ${\rho }_2$, …, ${\rho }_m$, respectively, i.e., $|u_i|\le {\rho }_i$, $i=1,2,\dots ,m$, and $\left|v\right|\le \sigma$.

Let $B\left(r\right)$ denote the ball of radius r and centered at the origin of ${\mathbb{R}}^3$.

Definition 1.

We call the function $u_i(·)$, $u_i:[0,\infty )\to B\left({\rho }_i\right)$ an admissible control of the pursuer $x_i$, $i\in \{1,2,\dots ,m\}$, if the solution $x_i(·)$ of the initial value problem

$${\dot{x}}_i=u_i,x_i\left(0\right)=x_{i0},$$

satisfies the condition $x_i\left(t\right)\in K$, $t\ge 0$.

Definition 2.

We call the function $v(·)$, $v:[0,\infty )\to B\left(\sigma \right)$ an admissible control of the evader y, if for the solution $y(·)$ of the initial value problem

$$\dot{y}=v,y\left(0\right)=y_0,$$

we have $y\left(t\right)\in K$, $t\ge 0$.

We consider a pursuit differential game where the pursuers apply strategies and the evader uses an arbitrary admissible control. We give a definition for the strategies of pursuers.

Definition 3.

We call the functions $(t,x_1,x_2,\dots ,x_m,y,v)\to U_i(t,x_1,x_2,\dots ,x_m,y,v)$, $i=1,2,\dots ,$ m, strategies of pursuers $x_i$, $i=1,2,\dots ,m$, if, for $u_i=U_i(t,x_1,x_2,\dots ,x_m,y,v)$, $i=1,2,\dots ,m$, and for any admissible control $v=v\left(t\right)$ of the evader, the initial value problem (1) has a unique solution $x_1\left(t\right),x_2\left(t\right),\dots ,x_m\left(t\right)$, $y\left(t\right)\in K$, $t\ge 0$.

This definition shows that to construct the strategy of the pursuers, the information about the states of players $x_1\left(t\right)$, $x_2\left(t\right)$, $x_3\left(t\right)$, $y\left(t\right)$, and velocity of evader $v\left(t\right)$ at current time t is needed.

Definition 4.

If, for some number $T>0$ and any initial states of players $x_{10},x_{20},\dots ,x_{m0},y_0\in K$, there exist strategies of pursuers such that $x_i\left(\tau \right)=y\left(\tau \right)$ at some $0<\tau \le T$ and $i\in \{1,2,\dots ,m\}$, then we say that pursuit is completed in the game in K for the time T.

The pursuers try to complete the game as early as possible. The evader tries to maintain the inequality $x_i\left(t\right)\ne y\left(t\right)$ as long as possible.

Problem 1.

The problem is to find a sufficient condition on ${\rho }_1,{\rho }_2,\dots ,{\rho }_m$ and σ for the completion of pursuit in the game in K.

Note that in the case where ${\rho }_1=1$, ${\rho }_2>0$, ${\rho }_3>0$, and $\sigma =1$ the problem was studied in [20] and it was shown that pursuit can be completed in the game on icosahedron K.

3. Main Result

Let us consider a regular icosahedron $A{A}_1{A}_2{A}_3{A}_4{A}_5{\overline{A}}_1{\overline{A}}_2{\overline{A}}_3{\overline{A}}_4{\overline{A}}_5\overline{A}$ with the edges of length 1, where $\overline{A},{\overline{A}}_1,{\overline{A}}_2,{\overline{A}}_3,{\overline{A}}_4$, and ${\overline{A}}_5$ are antipodal of vertices $A,A_1,A_2,A_3,A_4$, and $A_5$, respectively (Figure 1). We assume that the length of the edges of the icosahedron is equal to 1.

3.1. The Case of Three Pursuers $P_1,P_2,P_3$

In this subsection, we consider a differential game of three pursuers $P_1,P_2,P_3$ and one evader on the icosahedron. We prove the following statement.

Theorem 1.

If the maximal speeds of two pursuers belong to $[1/2,1)$ and the speed of the third pursuer is in the interval $[2/3,1)$, then pursuit can be completed in the game on icosahedron K.

Without loss of generality, we assume that $1/2\le {\rho }_1<1$, $1/2\le {\rho }_2<1$, and $2/3\le {\rho }_3<1$. It is sufficient to show that pursuit can be completed when ${\rho }_1={\rho }_2=1/2$, ${\rho }_3=2/3$.

Proof. 

We use the graph of icosahedron on the plane illustrated in Figure 2. We denote $A_{ij}\left(k\right)=A_i+k\overrightarrow{A_i{A}_j}$. That is, $A_{ij}\left(k\right)$ is the point of the edge $A_i{A}_j$ whose distance from the point $A_i$ is k, $0\le k\le 1$.

We want to place first the pursuer $P_3$ of speed 2/3 on the edge $A_1{A}_5$ to guard this edge from the evader. To this end, we define the point $F\left(M\right)$ associated with any point M of icosahedron K for which $A_{1}M\le 1$ and $M\notin A_1{A}_5$, that is, with the point on the edges $A_{1}A$, $A_1{A}_2$, $A_1{\overline{A}}_3$, $A_1{\overline{A}}_4$ (highlighted in blue) by the equation $A_{1}F\left(M\right)=\frac{2}{3}{A}_{1}M$ (Figure 2). Clearly, if the evader is at the point M and the pursuer $P_3$ is at the point $F\left(M\right)$, then they can reach the point $A_1$ at the same time. Note that if pursuer $P_3$ of speed 2/3 is at the point $F\left(M\right)$ and evader E of speed 1 is at the point M, then pursuer $P_3$ can reach the vertex $A_1$ not later than evader E.

Similarly, we define the point $F\left(N\right)$ associated with any point N, for which $A_{5}N\le 1$ and $N\notin A_1{A}_5$, that is, with any point on the edges $A_{5}A$, $A_5{A}_4$, $A_5{\overline{A}}_2$, $A_5{\overline{A}}_3$ (highlighted in pink) by the equation $A_{5}F\left(N\right)=\frac{1}{3}{A}_{5}N$. Clearly, if evader is at the point N and pursuer $P_3$ is at the point $F\left(N\right)$, then they can reach the point $A_5$ at the same time. Note that if pursuer $P_3$ of speed 2/3 is at the point $F\left(N\right)$ and evader E of speed 1 is at the point N, then pursuer $P_3$ can reach the vertex $A_5$ not later than evader E.

If $A_{5}M\ge 1$ and $A_{1}M\ge 1$, that is, if M is on an edge highlighted in red, then $F\left(M\right)$ is defined by the equation $F\left(M\right)=A_{51}(1/3)$.

Next, we construct strategies for the pursuers $P_1$, $P_2$, and $P_3$ and show that pursuit can be completed in a finite time in the game. We divide this process into two stages. In Stage 1, we bring the pursuers to some points associated with the position of the evader. Then, in Stage 2, we prove that pursuit is completed.

Stage 1. Pursuer $P_3$ comes to the point $A_1$ and then moves along $A_1{A}_5$ until $P_3\left(t_1\right)=F\left(E\left(t_1\right)\right)$ at some time $t_1$. Starting from the time $t_1$ to some unspecified time $t_2$ to be defined later, pursuer $P_3$ maintains the equation $P_3\left(t\right)=F\left(E\left(t\right)\right)$.

Next, to describe a strategy for the pursuer $P_2$, we define the point $G\left(Q\right)\in \overline{A}{\overline{A}}_3$ associated with each point Q for which $\overline{A}Q\le 1$ and $Q\notin \overline{A}{\overline{A}}_3$, that is, with each point on the edges $\overline{A}{\overline{A}}_1$, $\overline{A}{\overline{A}}_2$, $\overline{A}{\overline{A}}_4$, $\overline{A}{\overline{A}}_5$ (highlighted in cyan) by the equation $\overline{A}G\left(Q\right)=\frac{1}{2}\overline{A}Q$. Clearly, the evader of speed 1 at the point Q and pursuer $P_2$ of speed 1/2 at the point $G\left(Q\right)$ can reach the point $\overline{A}$ at the same time.

We also define the point $G\left(Q\right)\in \overline{A}{\overline{A}}_3$ associated with each point Q for which ${\overline{A}}_{3}Q\le 1$ and $Q\notin \overline{A}{\overline{A}}_3$, that is, with each point on the edges ${\overline{A}}_3{A}_1$, ${\overline{A}}_3{A}_5$, ${\overline{A}}_3{\overline{A}}_2$, ${\overline{A}}_3{\overline{A}}_4$ (highlighted in violet) by the equation ${\overline{A}}_{3}G\left(Q\right)=\frac{1}{2}{\overline{A}}_{3}Q$ (Figure 3). Clearly, the evader of speed 1 is at the point Q and pursuer $P_2$ of speed 1/2 at the point $G\left(Q\right)$ can reach the point ${\overline{A}}_3$ at the same time. If $\overline{A}Q>1$ and ${\overline{A}}_{3}Q>1$, then we define $G\left(Q\right)$ as the mid point of the edge $\overline{A}{\overline{A}}_3$. To indicate that point $G\left(Q\right)$ is the point of the edge $\overline{A}{\overline{A}}_3$, we use the symbol $G_{\overline{A}{\overline{A}}_3}\left(Q\right)$.

We let the pursuer $P_2$ first come to the point $\overline{A}$ and move toward ${\overline{A}}_3$ until $P_2\left(t_1^{\prime }\right)=G\left(E\left(t_1^{\prime }\right)\right)$ at some time $t_1^{\prime }$. Pursuer $P_2$ further moves on the point $G\left(E\right(t\left)\right)$ keeping it until the time $t_2$, which will be defined later.

Next, to describe a strategy for the pursuer $P_1$, we define point $H\left(L\right)$ on the triangle ${\overline{A}}_1\overline{A}{\overline{A}}_5$ associated with each point L on the icosahedron K, except for the edge $A_1{A}_5$ (Figure 4). First, we define $H\left(L\right)\in {\overline{A}}_1{\overline{A}}_5$ for $L\notin {\overline{A}}_1{\overline{A}}_5$. We let

(1)

$H\left(A_2\right)={\overline{A}}_5$, $H\left(A_4\right)={\overline{A}}_1$,

(2)

$H\left(L\right)={\overline{A}}_1$ if $L\in {\overline{A}}_1{A}_4$, and $H\left(L\right)={\overline{A}}_5$ if $L\in {\overline{A}}_5{A}_2$,

(3)

$H\left(L\right)=A_{\overline{1}\overline{5}}(1/2)$ if $L\in A{A}_1\cup A{A}_3\cup A{A}_5$ (recall $A_{\overline{1}\overline{5}}(1/2)$ is midpoint of ${\overline{A}}_1{\overline{A}}_5$). This set is highlighted in yellow in Figure 4.

(4)

If $L\in A{A}_4$ (Figure 4), then $H\left(L\right)\in {\overline{A}}_1{\overline{A}}_5$ is defined by the equation ${\overline{A}}_{1}H\left(L\right)=\frac{1}{2}{A}_{4}L$. Similarly, if L is on one of the edges $A_{4}A$, $A_{2}A$, $A_2{A}_3$, ${\overline{A}}_5{A}_3$, ${\overline{A}}_1{A}_3$, $A_4{A}_3$ whose vertex is $X\in \{A_2,{\overline{A}}_5,{\overline{A}}_1,A_4\}$, then $H\left(L\right)\in {\overline{A}}_1{\overline{A}}_5$ is defined by the equation $H\left(X\right)H\left(L\right)=\frac{1}{2}XL$.

In a similar fashion, we define $H\left(L\right)\in \overline{A}{\overline{A}}_1$ and $H\left(L\right)\in \overline{A}{\overline{A}}_5$ if point L is on the edges highlighted in cyan and blue.

Note that if the pursuer $P_1$ of speed 1/2 catches the point $H\left(E\right)$ for some position of evader E at some time, then $P_1$ can further move on the point $H\left(E\right)$. Therefore, pursuer $P_1$ first tries to catch the point $H\left(E\right)$ (Figure 5).

To this end, pursuer $P_1$ first comes to the vertex $\overline{A}$ and starting from the time $max\{t_1,t_1^{\prime }\}$ moves along the sides of the triangle $\overline{A}{\overline{A}}_5{\overline{A}}_1$ in the direction $\overline{A}\to {\overline{A}}_5\to {\overline{A}}_1\to \overline{A}$. Since the speed of the point $H\left(E\right)$ does not exceed 1/2, and the evader cannot pass through the points $A_1,A_5,\overline{A},{\overline{A}}_3$, therefore we have $P_1\left(t_2\right)=H\left(E\left(t_2\right)\right)$ at some $t_2$.

Stage 2. Starting from the time $t_2$, pursuers chase the evader as follows. Pursuer $P_1$ maintains the equation $P_1\left(t\right)=H\left(E\left(t\right)\right)$, $t\ge t_2$. Clearly, if the evader comes to a point of the triangle $\overline{A}{\overline{A}}_5{\overline{A}}_1$, then the evader is captured by the pursuer $P_1$. In other words, pursuer $P_1$ can guard the triangle $\overline{A}{\overline{A}}_5{\overline{A}}_1$.

Pursuer $P_3$ continues to apply their strategy to guard the edge $A_1{A}_5$, that is $P_3\left(t\right)=F\left(E\left(t\right)\right)$, $t\ge t_2$.

Pursuer $P_2$, starting from time $t_2$, moves to the point ${\overline{A}}_3$ and comes to the point ${\overline{A}}_3$ at some time $t_2^{\prime }$. If at this time the evader is on one of the edges ${\overline{A}}_3{A}_1$ and ${\overline{A}}_3{A}_5$, then E is trapped by pursuers $P_2$ and $P_3$ and the evader will be captured. If E is not on the edges ${\overline{A}}_3{A}_1$ and ${\overline{A}}_3{A}_5$, then E is on an edge whose one endpoint is either A or $A_2$ or $A_3$ or $A_4$ or ${\overline{A}}_2$ or ${\overline{A}}_4$.

If E is on an edge with an endpoint ${\overline{A}}_2$ or $A_4$ at the time $t_2^{\prime }$, then pursuer $P_2$ moves along the path ${\overline{A}}_3\to {\overline{A}}_2\to A_4$ and then moves by the edge $A_{4}A$ or $A_4{A}_3$ where E is, forcing the evader to come to one of the points, A and $A_3$. Evader E may also be trapped by $P_2$ and one of the pursuers $P_1$ and $P_3$ if E stays on an edge with endpoint at ${\overline{A}}_2$ or $A_4$ highlighted in blue (Figure 6) when $P_2$ reaches the green endpoint of the edge. In the latter case, the evader will be captured. Therefore, we assume that E comes to one of the points A and $A_3$.

As E reaches one of the points A and $A_3$ at some time $t_3$, pursuer $P_2$ returns back and moves along the sides of hexagon $A_2{A}_3{A}_4{\overline{A}}_2{\overline{A}}_3{\overline{A}}_4$ clockwise and comes to the point $A_2$.

Recall that when E reaches one of the points A and $A_3$ at time $t_3$, by the strategy of pursuer $P_3$, they will be at the point $A_{51}(1/3)$. Starting from time $t_3$, pursuer $P_3$ applies the following strategy. If evader E moves along the path $A\to A_4\to {\overline{A}}_2$ or $A_3\to A_4\to {\overline{A}}_2$ toward the vertex ${\overline{A}}_2$ after time $t_3$, then pursuer $P_3$ moves along the path $A_{51}(1/3)\to A_5\to {\overline{A}}_2$ maintaining the equation $\frac{2}{3}d(A,E)=d(A_{51}(1/3),P_3)$ or $\frac{2}{3}d(A_3,E)=d(A_{51}(1/3),P_3)$, respectively, where $d(A_3,E)$ denotes the distance of evader E from $A_3$ along the path $A_3\to A_4\to {\overline{A}}_2$. If $F\left(E\left(t\right)\right)\in A_1{A}_{51}(1/3)$, then $P_3\left(t\right)=F\left(E\left(t\right)\right)$. This strategy of pursuer $P_3$ allows them to guard the edges $A_1{A}_5$ and $A_5{\overline{A}}_2$ from the evader for $t\ge t_3$ (Figure 7).

If E is on an edge with an endpoint ${\overline{A}}_4$ or $A_2$ or A but not on the edge with the endpoint $A_4$ at time $t_2^{\prime }$, then pursuer $P_2$ moves along the path ${\overline{A}}_3\to {\overline{A}}_4\to A_2$. We assume again that E is not trapped by two pursuers before $P_2$ reaches $A_2$.

What is the behavior of pursuer $P_2$ after reaching point $A_2$? To describe further the strategy of pursuer $P_2$, let us define the projection $E^{\prime }$ of $E\in S=A{A}_1\cup A{A}_2\cup A{A}_3\cup A{A}_4\cup A{A}_5$ on the edge $A_2{A}_3$ by the following equations: (a) $A_2{E}^{\prime }=\frac{1}{2}{A}_{2}E$ if $E\in A{A}_2$, (b) $A_3{E}^{\prime }=\frac{1}{2}{A}_{3}E$ if $E\in A{A}_3$, (c) $E^{\prime }=A_{23}(1/2)$, that is, $E^{\prime }$ is the midpoint of $A_2{A}_3$, if $E\in A{A}_1\cup A{A}_5$, and (d) $A_3{E}^{\prime }=\frac{1}{2}{A}_{4}E$ if $E\in A{A}_4$.

As the pursuer reaches the point $A_2$, they move along the edge $A_2{A}_3$ until pursuer $P_2$ coincides with the projection $E^{\prime }$ of $E\in S$ on the edge $A_2{A}_3$ at some time $t_4$. When pursuer $P_2$ coincides with $E^{\prime }$, they further move, maintaining the equation $P_2\left(t\right)=E^{\prime }\left(t\right)$ provided $E\in S$. This is possible since the speed of $E^{\prime }$ does not exceed 1/2.

This strategy of the pursuer $P_2$ does not allow the evader to pass through the points $A_2$ and $A_3$ for $t\ge t_4$, that is, $P_2$ guards the edge $A_2{A}_3$ from the evader. If the evader reaches one of these points, it will be captured by $P_2$.

The evader can leave the set S only from the point $A_4$ at some time since the points $A_1,A_5$ are guarded by $P_3$, and $A_2,A_3$ are guarded by $P_2$. When the evader comes to the point $A_4$, according to the strategy above, pursuer $P_2$ will be at point $A_3$ at that time. We let the pursuer $P_2$ further stay at $A_3$ until E enters S through point $A_4$. As E enters S through the point $A_4$ we let again $P_2\left(t\right)=E^{\prime }\left(t\right)$.

Starting from the time $T=max\{t_3,t_4\}$ pursuer $P_1$ moves toward point ${\overline{A}}_1$. As $P_1$ reaches the point ${\overline{A}}_1$, $P_1$ moves toward the point $A_4$ along the edge ${\overline{A}}_1{A}_4$. When $P_1$ reaches point $A_4$, the evader E is either on the edge $A_4{\overline{A}}_2$, on $A_4{A}_5$ or on $A_4{A}_3$, or on an edge with the vertex A.

If $E\in A_4{\overline{A}}_2$ or $E\in A_4{A}_5$, then E is trapped by pursuers $P_1$ and $P_3$ and soon will be captured. If $E\in A_4{A}_3$, then E is trapped by pursuers $P_1$ and $P_2$ and will be captured. If the evader E is on an edge with the vertex A, pursuer $P_1$ moves along the edge $A_{4}A$ then toward the evader, and the evader will be captured. The proof of the theorem is complete. □

3.2. The Case of More than Three Pursuers

In this subsection, we consider a differential game of m, $m\ge 4$ pursuers and one evader. For the case of four pursuers $P_1,P_2,P_3,P_4$ ($m=4$) and one evader on the icosahedron K, we prove the following statement.

Theorem 2.

If the maximal speeds of two pursuers belong to $[1/2,1)$ and the maximal speeds of the other two pursuers are positive, then the pursuit can be completed in the game on icosahedron K.

Proof. 

Without loss of generality, we assume that $1/2\le {\rho }_1<1$, $1/2\le {\rho }_2<1$, and ${\rho }_3,{\rho }_4>0$. It is sufficient to show that the pursuit can be completed when ${\rho }_1={\rho }_2=1/2$ and ${\rho }_3,{\rho }_4>0$.

We construct strategies for the pursuers $P_1$, $P_2$, $P_3$, and $P_4$ and show that pursuit can be completed in a finite time in the game. We divide this process into two stages. In Stage 1, we bring the pursuers to some specific points. Then, in the Stage 2, we prove that pursuit is completed.

Stage 1. We use the definition of the point $G\left(Q\right)$ defined in the proof of Theorem 1. We let the pursuer $P_1$ come to the vertex $A_2$, then move along $A_2{A}_3$ toward $A_3$ until the state of $P_1\left(t\right)$ coincides with the point $G_{A_2{A}_3}\left(E\left(t\right)\right)$, and further move on this point keeping it. Then, clearly, if the evader reaches one of the points $A_2$ or $A_3$ at some time, it is captured at that time (Figure 8). Note that when the pursuer $P_1$ moves along the edge $A_2{A}_3$, they can collide with the evader E as well. Then, clearly, the evader is captured. Therefore, we exclude such a collision.

We let the pursuer $P_2$ come to the vertex $A_1$, then move along $A_1{\overline{A}}_3$ toward the point ${\overline{A}}_3$ until the state of $P_2\left(t\right)$ coincides with the point $G_{A_1{\overline{A}}_3}\left(E\left(t\right)\right)$, and further move on this point, keeping it. Let the pursuers $P_3$ and $P_4$ come to the points ${\overline{A}}_2$ and $A_4$, respectively, and stay there until the first time $t=t_1$ when all the equations $P_1\left(t\right)=G_{A_2{A}_3}\left(E\left(t\right)\right)$, $P_2\left(t\right)=G_{A_1{\overline{A}}_3}\left(E\left(t\right)\right)$, $P_3\left(t\right)={\overline{A}}_2$, and $P_4\left(t\right)=A_4$ are satisfied.

Stage 2. We show that pursuit can be completed. Indeed, if the evader is on one of the edges $A_1{A}_2$, $A_3{A}_4$, $A_4{\overline{A}}_2$, or ${\overline{A}}_2\overline{A}$ (highlighted in pink) at time $t_1$, then the evader is trapped by two pursuers, and so the evader will be captured. For example, if $E\left(t_1\right)\in A_1{A}_2$, then starting from time $t_1$, the pursuer $P_2$ of speed ${\rho }_2$ comes to the point $A_1$ and then moves toward the point $A_2$ along $A_1{A}_2$ to catch the evader.

If at time $t_1$ the evader is on an edge with an endpoint at A or $A_5$ (highlighted in blue), then we let the pursuer $P_3$ of speed ${\rho }_3$ come to the point $A_5$ at some time $t_2$, (that is, $P_3\left(t_2\right)=A_5$) and pursuers $P_1$ and $P_2$ further control the edges $A_2{A}_3$ and $A_1{\overline{A}}_3$, respectively, and $P_4\left(t\right)=A_4$. If the evader is on one of the edges $A_5{A}_1$, $A_5{\overline{A}}_3$, $A_5{\overline{A}}_2$, $A_5{A}_4$ at the time $t_2$, then the evader is trapped by two pursuers, and so the evader will be captured.

If the evader is not on any of these edges, then $P_3$ of speed ${\rho }_3$ starting from time $t_2$ moves toward point A, and we let $P_3\left(t_3\right)=A$ at some time $t_3$. Then at time $t_3$, the evader is trapped by two pursuers, and so the evader will be captured.

If the evader is on an edge with the vertex $\overline{A}$ or ${\overline{A}}_1$ or ${\overline{A}}_4$ or ${\overline{A}}_5$ (highlighted in green) at the time $t_1$, then we let the pursuer $P_4$ of speed ${\rho }_4$ come to the vertex ${\overline{A}}_1$ at some time $t_2^{\prime }$. The evader is trapped by two pursuers at time $t_2^{\prime }$ if it is on the edge ${\overline{A}}_1{\overline{A}}_2$ or ${\overline{A}}_1{A}_3$. If the evader is not on these edges, we let the pursuer $P_3$ of speed ${\rho }_3$ come to the vertex $\overline{A}$ at some time $t_3^{\prime }$. If the evader is not trapped by two pursuers at the time $t_3^{\prime }$, then pursuer $P_4$ moves along the path ${\overline{A}}_1\to {\overline{A}}_5\to {\overline{A}}_4$, and $P_3$ stays at the vertex $\overline{A}$ for $t\ge t_3^{\prime }$. Then, clearly, the evader is trapped by two pursuers when pursuer $P_4$ reaches ${\overline{A}}_5$ or ${\overline{A}}_4$, and so it will be captured. The proof of the theorem is complete. □

Next, for the case of five pursuers $P_1,P_2,P_3,P_4,P_5$ ($m=5$) and one evader on the icosahedron K, the following statement is true.

Theorem 3.

If the maximal speed of one pursuer belong to $[1/2,1)$ and the maximal speeds of the other three pursuers are positive, then the pursuit can be completed in the game on icosahedron K.

The proof of this theorem is very similar to the proof of the previous theorem. Therefore, we just give the main idea of the proof. Let ${\rho }_1=1/2$ and ${\rho }_2,{\rho }_3,{\rho }_4,{\rho }_5>0$. First, the pursuer $P_5$ catches the point $G_{A_2{A}_3}\left(E\left(t\right)\right)$ and further moves on this point, and pursuers $P_1,P_2,P_3,P_4$ come to the points $A_1$, ${\overline{A}}_3$, ${\overline{A}}_2$, $A_4$, respectively. Then, we let pursuer $P_3$ move along the path $A_2\to A_5\to A$ until the evader is trapped by two pursuers if the evader is on an edge with an endpoint at A or $A_5$. If the evader is on an edge with an endpoint at $\overline{A}$ or ${\overline{A}}_1$ or ${\overline{A}}_4$ or ${\overline{A}}_5$, then $P_4$ moves from $A_4$ to ${\overline{A}}_1$, then $P_3$ moves from ${\overline{A}}_2$ to $\overline{A}$, and then $P_4$ moves along the path ${\overline{A}}_1\to {\overline{A}}_5\to {\overline{A}}_4$ until the evader is trapped by two pursuers. For the case of six pursuers $P_1,P_2,P_3,P_4,P_5,P_6$ ($m=6$) and one evader on the icosahedron K, the following statement is true.

Theorem 4.

If the maximal speeds of pursuers are positive, then pursuit can be completed in the game on icosahedron K.

We let the pursuers $P_1,P_2,P_3,P_4,P_5,P_6$ move to the points $A_1$, ${\overline{A}}_3$, ${\overline{A}}_2$, $A_4$, $A_3$, and $A_2$, respectively. The rest of the reasoning is as above.

4. Conclusions and Discussion

We studied a differential game of three slow pursuers and one evader of speed 1 on 1-skeleton of an icosahedron. Previous research [20] shows that three pursuers of speeds ${\rho }_1=1$, ${\rho }_2>0$, ${\rho }_3>0$ can catch one evader of speed 1, but two pursuers of speed 1 cannot catch one evader of speed 1 on icosahedron K. Therefore, in the present paper, we studied the case of slow pursuers. We obtained a sufficient condition for speeds of pursuers to complete the game when the number of pursuers is $m=3,4,5,6$. We also constructed strategies for the pursuers to complete the game. According to Theorem 4, six pursuers with positive speeds can catch one evader of speed 1.

If the speeds of the pursuers are not in the intervals considered in Theorems 1–4, then differential game has not been studied yet. For the further investigation, we give some open problems for the differential game on the graph edge of an icosahedron. Can three pursuers catch one evader of speed 1 if the speeds of the pursuers are (1) ${\rho }_1<1/2$, ${\rho }_2=1/2$, ${\rho }_3=2/3$, and (2) ${\rho }_1=1/2$, ${\rho }_2=1/2$, ${\rho }_3<2/3$?

Additionally, the case where there are obstacles along the edges is very interesting. In particular, such an obstacle can be considered a pursuer with the speed equal to 0. Such a pursuer cannot move, but if the state of the evader coincides with that of this pursuer, the pursuit is completed.