Multiple Capture in a Group Pursuit Problem with Fractional Derivatives and Phase Restrictions

Abstract

The problem of conflict interaction between a group of pursuers and an evader in a finite-dimensional Euclidean space is considered. All participants have equal opportunities. The dynamics of all players are described by a system of differential equations with fractional derivatives in the form $D^{\left(\alpha \right)}{z}_i=a{z}_i+u_i-v,u_i,v\in V,$ where $D^{\left(\alpha \right)}f$ is a Caputo derivative of order $\alpha$ of the function $f.$ Additionally, it is assumed that in the process of the game the evader does not move out of a convex polyhedral cone. The set of admissible controls V is a strictly convex compact and a is a real number. The goal of the group of pursuers is to capture of the evader by no less than m different pursuers (the instants of capture may or may not coincide). The target sets are the origin. For such a conflict-controlled process, we derive conditions on its parameters and initial state, which are sufficient for the trajectories of the players to meet at a certain instant of time for any counteractions of the evader. The method of resolving functions is used to solve the problem, which is used in differential games of pursuit by a group of pursuers of one evader.

1. Introduction

The theory of differential two-player games, originally considered by Rufus Isaacs [1], has grown to be a profound and substantial theory in which various approaches to analysis of conflict situations [2,3,4,5,6,7,8,9,10] are developed. Games involving a group of pursuers and one or several evaders [11,12,13,14] are a natural generalization of the differential two-player pursuit–evasion games. These games are of interest since they cannot be solved using two-player game theory. One of the reasons for this is that the union of sets of attainability of all pursuers and the union of all goal sets are sets that are not convex and, moreover, not connected. On the other hand, there are some applications of these games to problems concerning the motion of vehicles, avoidance of collisions of ships, etc. In this case, one of the most important directions in the development of the theory of differential pursuit–evasion games at the present time is the search for new problems to which the previously developed methods are applicable. In particular, the authors of [15,16,17,18] consider the problems of pursuit of two persons with fractional derivatives. In [19], a proof is given of the existence of the prices of the game in a differential game described by an equation with fractional derivatives. The evader–pursuit problem with phase restrictions with fractional derivatives of the order $\alpha \in (0,1)$ is addressed in [20,21]. The problem of multiple capture of an evader without phase restrictions in a differential game with fractional derivatives is treated in [22].

This paper deals with the problem of multiple capture of an evader in a differential game with fractional derivatives and phase restrictions. For such a conflict-controlled process, we derive conditions on its parameters and initial state, which are sufficient for the trajectories of the players to meet at a certain instant of time for any counteractions of the evader.

2. Formulation of the Problem

Definition 1

([23]). Let q be a natural number (positive integer) and let $f:[0,\infty )\to {\mathbb{R}}^n$ be a function, such that $f^{\left(q\right)}$ is absolutely continuous on $[0,\infty )$, $\alpha \in (q-1,q)$. A Caputo derivative of order α of the function f is a function $D^{\left(\alpha \right)}f$ of the form

$$\left(D^{\left(\alpha \right)}f\right)\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(q-\alpha )}}{\int }_0^t{\displaystyle \frac{f^{\left(q\right)}\left(s\right)}{{(t-s)}^{\alpha +1-q}}}ds,\mathrm{where}\mathrm{\Gamma }\left(\beta \right)={\int }_0^{\infty }{e}^{-s}{s}^{\beta -1}ds.$$

In the space ${\mathbb{R}}^k(k⩾2)$, we consider an $(n+1)$-player differential game which involves n pursuers $P_1,\dots ,P_n$ and an evader $E.$ The law of motion of each of pursuers $P_i$ has the form

$$\begin{array}{c}{D}^{\left(\alpha \right)}{x}_i=a{x}_i+u_i,x_i\left(0\right)=x_i^0,\dots ,x_i^{(q-1)}\left(0\right)=x_i^{q-1},u_i\in V.\end{array}$$

(1)

The law of motion of evader E has the form

$$\begin{array}{c}{D}^{\left(\alpha \right)}y=ay+v,y\left(0\right)=y^0,\dots ,y^{(q-1)}\left(0\right)=y^{q-1},v\in V.\end{array}$$

(2)

Here $i\in I=\{1,\dots ,n\},$$x_i,y,u_i,v\in {\mathbb{R}}^k,$V is a strictly convex compact ${\mathbb{R}}^k,$ q is a natural number, $\alpha \in (q-1,q),$$D^{\left(\alpha \right)}f$ is the Caputo derivative of the function f of order $\alpha ,$ and a is a real number. Assume that $x_i^{q-1}\ne y^{q-1}$ for all $i\in I.$ Additionally, it is assumed that in the process of the game evader E does not move out of the convex cone $\mathrm{\Omega }$

$$\begin{array}{c}\mathrm{\Omega }=\{y\in {\mathbb{R}}^k:(p_j,y)⩽0,j=1,\dots ,r\},\end{array}$$

where $p_1,\dots ,p_r$ are unit vectors ${\mathbb{R}}^k$ and $(a,b)$ is the scalar product of vectors a and b. If there are no phase restrictions ($\mathrm{\Omega }={\mathbb{R}}^k$), then we assume that $r=0.$

Instead of the systems (1) and (2), we consider the system

$$\begin{array}{c}{D}^{\left(\alpha \right)}{z}_i=a{z}_i+u_i-v,z_i\left(0\right)=z_i^0,\dots ,z_i^{(q-1)}\left(0\right)=z_i^{q-1},u_i,v\in V,\end{array}$$

(3)

where $z_i^l=x_i^l-y^l.$ Let $z^0=\{z_i^l,i\in I,l=0,\dots ,q-1\}$ denote the vector of initial positions.

Suppose $v:[0,\infty )\to V$ is a measurable function. Let us call the restriction of the function v on $[0,t]$ the prehistory of the function v at time t. A measurable function $v:[0,\infty )\to V$ is called admissible if $y\left(t\right)\in \mathrm{\Omega }$ for all $t⩾0.$

Definition 2.

We will say that a quasi-strategy ${\mathcal{U}}_i$ of pursuer $P_i$ is given if a map $U_i^0$ is defined which associates the measurable function $u_i\left(t\right)$ with values in $U_i$ to the initial positions $z^0,$ time t and an arbitrary prehistory of control $v_t(·)$ of evader E.

Definition 3.

In the game $G\left(n\right)$ an m-fold (m— positive integer) capture (with $m=1$ capture) occurs if there exist time $T>0$ and quasi-strategies ${\mathcal{U}}_1,\dots ,{\mathcal{U}}_n$ of pursuers $P_1,\dots ,P_n$ such that for any measurable function $v(·),$$v\left(t\right)\in V,$$t\in [0,T]$ there exist time instants ${\tau }_1,\dots ,{\tau }_m\in [0,T]$ and pairwise different indices $i_1,\dots ,i_m\in I$ for which $z_{i_l}\left({\tau }_l\right)=0$ for all $l=1,\dots ,m.$

We introduce the following notation:

$$\begin{array}{c}\lambda (h,v)=\sup \{\lambda ⩾0:-\lambda h\in V-v\},d=\underset{v\in V}{\max }\parallel v\parallel ,\\ I\left(\beta \right)=\{(i_1,\dots ,i_{\beta }):i_1,\dots ,i_{\beta }\in I\mathrm{and}\mathrm{are}\mathrm{pairwise}\mathrm{different}\}.\end{array}$$

3. Sufficient Conditions for Capture with $\mathit{a}=\mathit{0}$

Denote

$$f_i\left(t\right)=\frac{z_i^{q-1}}{\mathrm{\Gamma }\left(q\right)}+\frac{z_i^{q-2}}{t\mathrm{\Gamma }(q-1)}+\dots +\frac{z_i^1}{t^{q-2}}+\frac{z_i^0}{t^{q-1}}.$$

Lemma 1.

Let$a=0,$

$$\begin{array}{c}{\delta }_0=\underset{v\in V}{\min }\max \left\{\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }\lambda \right(\frac{z_j^{q-1}}{\mathrm{\Gamma }\left(q\right)},v),(p_1,v),\dots ,(p_r,v)\}>0.\end{array}$$

Then there exists $T>0$ such that for all $t>T$ the following inequality holds:

$$\begin{array}{c}{\delta }_t=\underset{v\in V}{\min }\max \{\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }\lambda (f_j\left(t\right),v),(p_1,v),\dots ,(p_r,v)\}>0.5{\delta }_0.\end{array}$$

Proof of Lemma 1.

It follows from ([11], Lemma 1.3.13) that the function $\lambda (z,v)$ is continuous on the set $B\times V,$ where B is an arbitrary compact subset of ${\mathbb{R}}^k$ not containing zero. Therefore, the functions

$$\begin{array}{c}{g}_{\mathrm{\Lambda }}(z_1,\dots ,z_n)=\underset{\alpha \in \mathrm{\Lambda }}{\min }\lambda (z_{\alpha },v),g(z_1,\dots ,z_n,v)=\underset{\mathrm{\Lambda }\in \mathrm{\Omega }\left(m\right)}{\max }{g}_{\mathrm{\Lambda }}(z_1,\dots ,z_n,v),\\ F(z_1,\dots ,z_n)=\max \{g(z_1,\dots ,z_n,v),(p_1,v),\dots ,(p_r,v)\},\\ F_0(z_1,\dots ,z_n)=\underset{v\in V}{\min }g(z_1,\dots ,z_n),{\delta }_t=F_0(f_1\left(t\right),\dots ,f_n\left(t\right))\end{array}$$

are continuous. Since $\underset{t\to +\infty }{\lim }{f}_j\left(t\right)=\frac{z_j^{q-1}}{\mathrm{\Gamma }\left(q\right)}$ we have $\underset{t\to +\infty }{\lim }\lambda (f_j\left(t\right),v)=\lambda (\frac{z_j^{q-1}}{\mathrm{\Gamma }\left(q\right)},v).$ Therefore, $\underset{t\to +\infty }{\lim }{\delta }_t={\delta }_0,$ whence we obtain required result. This proves the lemma. □

Lemma 2.

Let $a=0,$${\delta }_0>0,$$r=1.$ Then there exists $T_0>0$ such that for any admissible function $v(·)$ there is a set $\mathrm{\Lambda }\in I\left(m\right)$ such that for all $j\in \mathrm{\Lambda }$ the following inequality holds:

$$\begin{array}{c}\frac{1}{\mathrm{\Gamma }\left(\alpha \right)T_0^{q-1}}\underset{0}{\overset{T_0}{\int }}{(T_0-s)}^{\alpha -1}\lambda (f_j\left(T_0\right),v\left(s\right))ds⩾1.\end{array}$$

Proof of Lemma 2.

By Lemma 1, there exists $T>0$ such that ${\delta }_t>{\delta }_1=0.5{\delta }_0$ for all $t>T.$ Let $v(·)$ be an arbitrary admissible control of evader $E,$$t>T,$

$$\begin{array}{c}{y}_0\left(t\right)=\sum _{l=0}^{q-1}\frac{y^l·t^l}{\mathrm{\Gamma }(l+1)},\mu \left(t\right)=-\frac{(p_1,y_0\left(t\right))\mathrm{\Gamma }\left(\alpha \right)}{t^{q-1}},\\ T_1\left(t\right)=\{\tau \in [0,t]:(p_1,v\left(\tau \right))⩾{\delta }_1\},T_2\left(t\right)=\{\tau \in [0,t]:(p_1,v\left(\tau \right))<{\delta }_1\}.\end{array}$$

The solution of the Cauchy problem for the system (2) can be represented in the form [24] $y\left(t\right)=y_0\left(t\right)+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{0}{\overset{t}{\int }}{(t-s)}^{\alpha -1}v\left(s\right)ds.$ Since $y\left(t\right)\in \mathrm{\Omega }$ for all $t⩾0$, it follows that $(p_1,y\left(t\right))⩽0$ for all $t⩾0.$ Therefore,

$$\begin{array}{c}\mu \left(t\right)⩾\frac{1}{t^{q-1}}\underset{0}{\overset{t}{\int }}{(t-s)}^{\alpha -1}(p_1,v\left(s\right))ds⩾\frac{{\delta }_1}{t^{q-1}}\underset{T_1\left(t\right)}{\int }{(t-s)}^{\alpha -1}ds-\frac{d}{t^{q-1}}\underset{T_2\left(t\right)}{\int }{(t-s)}^{\alpha -1}ds.\end{array}$$

In addition,

$$\begin{array}{c}\frac{t^{\alpha +1-q}}{\alpha }=\frac{1}{t^{q-1}}\underset{T_1\left(t\right)}{\int }{(t-s)}^{\alpha -1}ds+\frac{1}{t^{q-1}}\underset{T_2\left(t\right)}{\int }{(t-s)}^{\alpha -1}ds.\end{array}$$

It follows from the last two relations that

$$\begin{array}{c}\frac{1}{t^{q-1}}\underset{T_2\left(t\right)}{\int }{(t-s)}^{\alpha -1}ds⩾\frac{{\delta }_1{t}^{\alpha +1-q}-\alpha \mu \left(t\right)}{\alpha (d+{\delta }_1)}.\end{array}$$

(4)

Next, we have

$$\begin{array}{c}\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }\frac{1}{t^{q-1}}\underset{0}{\overset{t}{\int }}{(t-s)}^{\alpha -1}\lambda (f_j\left(t\right),v\left(s\right))ds\\ ⩾\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\frac{1}{t^{q-1}}\underset{0}{\overset{t}{\int }}{(t-s)}^{\alpha -1}\underset{j\in \mathrm{\Lambda }}{\min }\lambda (f_j\left(t\right),v\left(s\right))ds.\end{array}$$

(5)

Since for any nonnegative numbers $a_{\mathrm{\Lambda }}(\mathrm{\Lambda }\in I\left(m\right))$ the inequality

$$\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }{a}_{\mathrm{\Lambda }}⩾\frac{1}{C_n^m}\sum _{\mathrm{\Lambda }\in I\left(m\right)}{a}_{\mathrm{\Lambda }},\mathrm{where}{C}_n^m=\frac{n!}{(n-m)!m!},$$

holds, we have the following inequality from (4) and (3):

$$\begin{array}{c}\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }\frac{1}{t^{q-1}}\underset{0}{\overset{t}{\int }}{(t-s)}^{\alpha -1}\lambda (f_j\left(t\right),v\left(s\right))ds\\ ⩾\frac{1}{C_n^m·t^{q-1}}\underset{0}{\overset{t}{\int }}{(t-s)}^{\alpha -1}\sum _{\mathrm{\Lambda }\in I\left(m\right)}\underset{j\in \mathrm{\Lambda }}{\min }\lambda (f_j\left(t\right),v\left(s\right))ds\\ ⩾\frac{1}{C_n^m·t^{q-1}}\underset{0}{\overset{t}{\int }}{(t-s)}^{\alpha -1}\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }\lambda (f_j\left(t\right),v\left(s\right))ds\\ ⩾\frac{1}{C_n^m·t^{q-1}}\underset{T_2\left(t\right)}{\int }{(t-s)}^{\alpha -1}\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }\lambda (f_j\left(t\right),v\left(s\right))ds\\ ⩾\frac{1}{C_n^m}\left[\frac{{\delta }_1{t}^{\alpha +1-q}-\alpha \mu \left(t\right)}{\alpha (d+{\delta }_1)}\right]=g\left(t\right).\end{array}$$

Since $\underset{t\to +\infty }{\lim }g\left(t\right)=+\infty ,$ there exists $T_0>T$ such that $g\left(t\right)>1$ for all $t>T_0.$ Hence, for all $t>T_0$ we have the inequality

$$\begin{array}{c}\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }\frac{1}{t^{q-1}}\underset{0}{\overset{t}{\int }}{(t-s)}^{\alpha -1}\lambda (f_j\left(t\right),v\left(s\right))ds⩾1,\end{array}$$

from which the statement of the lemma follows. This proves the lemma. □

Theorem 1.

Let $a=0,$ ${\delta }_0>0,$ $r=1.$ Then an m-fold capture occurs in the game.

Proof of Theorem 1.

Define the number

$$\begin{array}{c}\widehat{T}=\inf \{t>0:\underset{v(·)}{inf}\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }\frac{1}{t^{q-1}}\underset{0}{\overset{t}{\int }}{(t-s)}^{\alpha -1}\lambda (f_j\left(t\right),v\left(s\right))ds⩾1\}.\end{array}$$

Let $v(·)$ be an arbitrary admissible control of evader $E.$ Consider the sets

$$\begin{array}{c}{T}_i\left(v(·)\right)=\{t>0:\frac{1}{{\widehat{T}}^{q-1}}\underset{0}{\overset{t}{\int }}{(\widehat{T}-s)}^{\alpha -1}\lambda (f_i\left(\widehat{T}\right),v\left(s\right))⩾1\}.\end{array}$$

Define the quantities

$$\begin{array}{c}{t}_i\left(v(·)\right)=\left\{\begin{array}{cc}\inf \{t:t\in T_i\left(v(·)\right)\}\hfill & \mathrm{if}{T}_i\left(v(·)\right)\ne \varnothing ,\hfill \\ +\infty \hfill & \mathrm{if}{T}_i\left(v(·)\right)=\varnothing .\hfill \end{array}\right.\end{array}$$

Specify the controls of pursuers $P_i,i\in I$, assuming that

$$\begin{array}{c}{u}_i\left(t\right)=\left\{\begin{array}{cc}v\left(t\right)-\lambda (f_i\left(\widehat{T}\right),v\left(t\right))f_i\left(\widehat{T}\right),\hfill & t\in [0,t_i\left(v(·)\right)],\hfill \\ v\left(t\right),\hfill & t\in (t_i\left(v(·)\right),\widehat{T}].\hfill \end{array}\right.\end{array}$$

Then from the system (3) we obtain [24]

$$\begin{array}{c}\frac{z_i\left(\widehat{T}\right)\mathrm{\Gamma }\left(\alpha \right)}{{\widehat{T}}^{q-1}}=f_i\left(\widehat{T}\right)-\frac{1}{{\widehat{T}}^{q-1}}\underset{0}{\overset{\widehat{T}}{\int }}{(\widehat{T}-s)}^{\alpha -1}(u_i\left(s\right)-v\left(s\right))ds\\ =f_i\left(\widehat{T}\right)(1-\frac{1}{{\widehat{T}}^{q-1}}\underset{0}{\overset{t_i\left(v(·)\right)}{\int }}{(\widehat{T}-s)}^{\alpha -1}\lambda (f_i\left(\widehat{T}\right),v\left(s\right))ds).\end{array}$$

It follows from Lemma 2 that there exists $\mathrm{\Lambda }\in I\left(m\right)$ such that for all $j\in \mathrm{\Lambda }$ we have $z_j\left(\widehat{T}\right)=0.$ This proves the theorem. □

Theorem 2.

Let $a=0,$$\mathrm{\Omega }={\mathbb{R}}^k,$ ${\delta }_0=\underset{v\in V}{\min }\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }\lambda (\frac{z_j^{q-1}}{\mathrm{\Gamma }\left(q\right)},v)>0.$ Then an m-fold capture occurs in the game.

The proof of this theorem is similar to that of Theorem 1.

4. Sufficient Conditions for Capture with $\mathit{a}<\mathit{0}$

In this section we assume that $\alpha \in (1,2).$ Let us introduce the following notation. $\mathrm{Int}A$ and $\mathrm{co}A$ are, respectively, the interior and the convex hull of the set A and $E_{\rho }(z,\mu )={\displaystyle \sum _{k=0}^{\infty }}\frac{z^k}{\mathrm{\Gamma }(k{\rho }^{-1}+\mu )}$ is a generalized Mittag-Leffler function ($\rho >0,z,\mu \in {\mathbb{R}}^1$),

$$\begin{array}{c}{f}_i\left(t\right)=t^{\alpha -1}{E}_{1/\alpha }(a{t}^{\alpha },1)z_i^0+t^{\alpha }{E}_{1/\alpha }(a{t}^{\alpha },2)z_i^1,\gamma =-a\mathrm{\Gamma }(2-\alpha ),\\ \overline{E}(t,s,\alpha )={(t-s)}^{\alpha -1}{E}_{1/\alpha }(a{(t-s)}^{\alpha },\alpha ),E\left(t\right)=\underset{0}{\overset{t}{\int }}|\overline{E}(t,s,\alpha )|ds,\\ r(t,s)=\left\{\begin{array}{c}1\mathrm{if}{E}_{1/\alpha }(a{(t-s)}^{\alpha },\alpha )⩾0,\hfill \\ -1\mathrm{if}{E}_{1/\alpha }(a{(t-s)}^{\alpha },\alpha )<0,\hfill \end{array}\right.D_{\epsilon }\left(w\right)=\{z\in {\mathbb{R}}^k:\parallel z-w\parallel ⩽\epsilon \},\end{array}$$

$$\begin{array}{c}{\delta }_0^{+}=\underset{v\in V}{\min }\max \left\{\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }\lambda \right(\frac{z_j^1}{\gamma },v),(p_1,v),\dots ,(p_r,v)\},\\ {\delta }_0^{-}=\underset{v\in V}{\min }\max \left\{\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }\lambda \right(-\frac{z_j^1}{\gamma },v),(p_1,v),\dots ,(p_r,v)\},\\ {\delta }_t^{+}=\underset{v\in V}{\min }\max \left\{\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }\lambda \right(f_j\left(t\right),v),(p_1,v),\dots ,(p_r,v)\},\\ {\delta }_t^{-}=\underset{v\in V}{\min }\max \left\{\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }\lambda \right(-f_j\left(t\right),v),(p_1,v),\dots ,(p_r,v)\},\\ {\delta }_0=min\{{\delta }_0^{+},{\delta }_0^{-}\},{\delta }_t=min\{{\delta }_t^{+},{\delta }_t^{-}\},\\ y_0\left(t\right)=E_{1/\alpha }(a{t}^{\alpha },1)y^0+t{E}_{1/\alpha }(a{t}^{\alpha },2)y^1,D_{\epsilon }\left(b\right)=\{z|\parallel z-b\parallel ⩽\epsilon \}.\end{array}$$

Lemma 3.

Let $a<0,$${\delta }_0>0.$ Then there exists $T>0$ such that ${\delta }_t>{\delta }_1=0.5{\delta }_0$ for all $t>T.$

The proof of this lemma is similar to that of Lemma 1.

Lemma 4.

Let $p\in R^k,a<0.$ Then the domain of the function $\mu \left(t\right)=t^{\alpha -1}(p,y_0\left(t\right))$ is restricted to $[0,\infty ).$

Proof of Lemma 4.

For $t\to +\infty$, the following asymptotic estimates ([25], p. 12)

$$\begin{array}{c}{E}_{1/\alpha }(a{t}^{\alpha },1)=-\frac{1}{a{t}^{\alpha }\mathrm{\Gamma }(1-\alpha )}+O\left(\frac{1}{t^{2\alpha }}\right),\\ E_{1/\alpha }(a{t}^{\alpha },2)=-\frac{1}{a{t}^{\alpha }\mathrm{\Gamma }(2-\alpha )}+O\left(\frac{1}{t^{2\alpha }}\right).\end{array}$$

(6)

We represent the function $\mu \left(t\right)$ in the form

$$\begin{array}{c}\mu \left(t\right)=t^{\alpha -1}{E}_{1/\alpha }(a{t}^{\alpha },1)(p,y^0)+t^{\alpha }{E}_{1/\alpha }(a{t}^{\alpha },2)(p,y^1).\end{array}$$

It follows from (4) that the equation

$$\begin{array}{c}\mu \left(t\right)=-\frac{(p,y^0)}{at\mathrm{\Gamma }(1-\alpha )}-\frac{(p,y^1)}{a\mathrm{\Gamma }(2-\alpha )}+O\left(\frac{1}{t^{\alpha }}\right)\end{array}$$

(7)

holds for $t\to +\infty$. The continuity of the function $\mu ,$ the representation (7) and the condition $\alpha >1$ imply that the lemma holds. This proves the lemma. □

Lemma 5.

Let $r=1,a<0,{\delta }_0>0.$ Then there exists time $T_0$ such that for any admissible function $v(·)$ there is a set $\mathrm{\Lambda }\in I\left(m\right)$ such that for all $l\in \mathrm{\Lambda }$ the following inequalities hold:

$$\begin{array}{c}{T}_0^{\alpha -1}\underset{0}{\overset{T_0}{\int }}|\overline{E}(T_0,s,\alpha )|\lambda (f_l\left(T_0\right)r(T_0,s),v\left(s\right))ds⩾1.\end{array}$$

Proof of Lemma 5.

It follows from Lemma 3 that there exists time $T_1>0$ such that for all $t>T_1$ the inequality ${\delta }_t>0,5{\delta }_0$ holds. Let $T>T_1$ and let $v(·)$ be an arbitrary admissible function. Let us define the functions

$$\begin{array}{c}{h}_i(t,T,v(·))=t^{\alpha -1}\underset{0}{\overset{t}{\int }}|\overline{E}(t,s,\alpha )|\lambda (f_i\left(T\right)r(T,s),v\left(s\right))ds,t\in [0,T].\end{array}$$

Since the control $v(·)$ of evader E is admissible, the inequality $(p_1,y\left(t\right))⩽0$ holds for all $t⩾0$. By virtue of [24], the solution to the problem (2) has the form

$$\begin{array}{c}y\left(t\right)=y_0\left(t\right)+\underset{0}{\overset{t}{\int }}\overline{E}(t,s,\alpha )v\left(s\right)ds.\end{array}$$

Hence,

$$\begin{array}{c}\underset{0}{\overset{t}{\int }}\overline{E}(t,s,\alpha )(p_1,v\left(s\right))ds⩽{\mu }_0\left(t\right)=(-p_1,y_0\left(t\right)).\end{array}$$

Define the sets $({\delta }_1=0.5{\delta }_0)$

$$\begin{array}{c}{T}^{+}\left(t\right)=\{s|s\in [0,t],\overline{E}(t,s,\alpha )⩾0\},T^{-}\left(t\right)=\{s|s\in [0,t],\overline{E}(t,s,\alpha )<0\},\\ T_1^{+}\left(t\right)=\{s|s\in T^{+}\left(t\right),(p_1,v\left(s\right))⩾{\delta }_1\},T_2^{+}\left(t\right)=\{s|s\in T^{+}\left(t\right),(p_1,v\left(s\right))<{\delta }_1\}\\ T_1^{-}\left(t\right)=\{s|s\in T^{-}\left(t\right),(-p_1,v\left(s\right))⩾{\delta }_1\},T_2^{-}\left(t\right)=\{s|s\in T^{-}\left(t\right),(-p_1,v\left(s\right))<{\delta }_1\}.\end{array}$$

Then

$$\begin{array}{c}\underset{0}{\overset{t}{\int }}\overline{E}(t,s,\alpha )(p_1,v\left(s\right))ds=\underset{T^{+}\left(t\right)}{\int }\overline{E}(t,s,\alpha )(p_1,v\left(s\right))ds+\underset{T^{-}\left(t\right)}{\int }\overline{E}(t,s,\alpha )(p_1,v\left(s\right))ds\\ =\underset{T_1^{+}\left(t\right)}{\int }\overline{E}(t,s,\alpha )(p_1,v\left(s\right))ds+\underset{T_2^{+}\left(t\right)}{\int }\overline{E}(t,s,\alpha )(p_1,v\left(s\right))ds\\ +\underset{T_1^{-}\left(t\right)}{\int }(-\overline{E}(t,s,\alpha ))(-p_1,v\left(s\right))ds+\underset{T_2^{-}\left(t\right)}{\int }(-\overline{E}(t,s,\alpha ))(-p_1,v\left(s\right))ds.\end{array}$$

Since

$$\begin{array}{c}\underset{T_1^{+}\left(t\right)}{\int }\overline{E}(t,s,\alpha )(p_1,v\left(s\right))ds⩾{\delta }_1\underset{T_1^{+}\left(t\right)}{\int }\overline{E}(t,s,\alpha )ds,\\ \underset{T_2^{+}\left(t\right)}{\int }\overline{E}(t,s,\alpha )(p_1,v\left(s\right))ds⩾-d\underset{T_2^{+}\left(t\right)}{\int }\overline{E}(t,s,\alpha )ds,\\ \underset{T_1^{-}\left(t\right)}{\int }(-\overline{E}(t,s,\alpha ))(-p_1,v\left(s\right))ds⩾{\delta }_1\underset{T_1^{-}\left(t\right)}{\int }(-\overline{E}(t,s,\alpha ))ds,\\ \underset{T_2^{-}\left(t\right)}{\int }(-\overline{E}(t,s,\alpha ))(-p_1,v\left(s\right))ds⩾-d\underset{T_2^{-}\left(t\right)}{\int }(-\overline{E}(t,s,\alpha ))ds,\end{array}$$

the following inequality holds:

$$\begin{array}{c}\underset{0}{\overset{t}{\int }}\overline{E}(t,s,\alpha )(p_1,v\left(s\right))ds⩾{\delta }_1\underset{T_1^{+}\left(t\right)\cup T_1^{-}\left(t\right)}{\int }|\overline{E}(t,s,\alpha )|ds-d\underset{T_2^{+}\left(t\right)\cup T_2^{-}\left(t\right)}{\int }|\overline{E}(t,s,\alpha )|ds.\end{array}$$

This yields

$$\begin{array}{c}{\delta }_1\underset{T_1^{+}\left(t\right)\cup T_1^{-}\left(t\right)}{\int }|\overline{E}(t,s,\alpha )|ds-d\underset{T_2^{+}\left(t\right)\cup T_2^{-}\left(t\right)}{\int }|\overline{E}(t,s,\alpha )|ds⩽{\mu }_0\left(t\right),\\ \underset{T_1^{+}\left(t\right)\cup T_1^{-}\left(t\right)}{\int }|\overline{E}(t,s,\alpha )|ds+\underset{T_2^{+}\left(t\right)\cup T_2^{-}\left(t\right)}{\int }|\overline{E}(t,s,\alpha )|ds=E\left(t\right).\end{array}$$

It follows from the last two relations that for all $t\in [0,T]$ the inequality

$$\begin{array}{c}\underset{T_2^{+}\left(t\right)\cup T_2^{-}\left(t\right)}{\int }|\overline{E}(t,s,\alpha )|ds⩾\frac{{\delta }_{1}E\left(t\right)-{\mu }_0\left(t\right)}{d+{\delta }_1}\end{array}$$

(8)

holds. Further, by virtue of (8), for all $t\in [0,T]$ we have

$$\begin{array}{c}\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }{h}_j(t,T,v(·))⩾\frac{t^{\alpha -1}}{C_n^m}\underset{0}{\overset{t}{\int }}|\overline{E}(t,s,\alpha )|\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }\lambda (f_j\left(T\right)r(T,s),v\left(s\right))ds\\ ⩾\frac{t^{\alpha -1}}{C_n^m}\underset{T_2^{+}\left(t\right)\cup T_2^{-}\left(t\right)}{\int }|\overline{E}(t,s,\alpha )|\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }\lambda (f_j\left(T\right)r(T,s),v\left(s\right))ds\\ ⩾\frac{{\delta }_1}{C_n^m(d+{\delta }_1)}[{\delta }_1{t}^{\alpha -1}E\left(t\right)-t^{\alpha -1}{\mu }_0\left(t\right)].\end{array}$$

Hence, for all $T>T_1$ we have the inequality

$$\begin{array}{c}\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }{h}_j(T,T,v(·))⩾\frac{{\delta }_1}{n(d+{\delta }_1)}[{\delta }_1{T}^{\alpha -1}E\left(T\right)-T^{\alpha -1}{\mu }_0\left(T\right)].\end{array}$$

Since, according to ([26], p. 120)

$$\begin{array}{c}\underset{0}{\overset{t}{\int }}\overline{E}(t,s,\alpha )ds=t^{\alpha }{E}_{1/\alpha }(a{t}^{\alpha },\alpha +1),\end{array}$$

we have

$$\begin{array}{c}{t}^{\alpha -1}E\left(t\right)=t^{\alpha -1}\underset{0}{\overset{t}{\int }}|\overline{E}(t,s,\alpha )|ds⩾t^{\alpha -1}\underset{0}{\overset{t}{\int }}\overline{E}(t,s,\alpha )ds=t^{2\alpha -1}{E}_{1/\alpha }(a{t}^{\alpha },\alpha +1).\end{array}$$

It follows from ([25], p. 12) that for $t\to +\infty$ the following asymptotic representation holds:

$$\begin{array}{c}{E}_{1/\alpha }(a{t}^{\alpha },\alpha +1)=-\frac{1}{a{t}^{\alpha }}+O\left(\frac{1}{t^{2\alpha }}\right).\end{array}$$

By Lemma 4, there exists $c>0$ such that $|t^{\alpha -1}{\mu }_0\left(t\right)|⩽c$ for all $t⩾0.$ Therefore, for $T\to +\infty$ we have

$$\begin{array}{c}\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }{h}_j(T,T,v(·))⩾\frac{{\delta }_1}{n(d+{\delta }_1)}[-\frac{{\delta }_1{T}^{\alpha -1}}{a}-c+O\left(\frac{1}{T}\right)].\end{array}$$

Since $a<0,\alpha -1>0,$ there exists $T_0$ such that $\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }{h}_j(T_0,T_0,v(·))⩾1$ for any admissible function $v(·).$ This proves the lemma. □

Theorem 3.

Let $a<0,$$r=1,$${\delta }_0>0.$ Then an m-fold capture occurs in the game.

Proof of Theorem 3.

Define the number

$$\begin{array}{c}\widehat{T}=\inf \{t>0:\underset{v(·)}{inf}\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }{t}^{\alpha -1}\underset{0}{\overset{t}{\int }}|\overline{E}(t,s,\alpha )|\lambda (f_j\left(t\right)r(t,s),v\left(s\right))ds⩾1\}.\end{array}$$

By virtue of Lemma 5, $\widehat{T}<+\infty .$ Let $v(·)$ be an arbitrary admissible control of evader $E.$ Consider the sets

$$\begin{array}{c}{T}_i\left(v(·)\right)=\{t:{\widehat{T}}^{\alpha -1}\underset{0}{\overset{t}{\int }}|\overline{E}(\widehat{T},s,\alpha )|\lambda (f_i\left(\widehat{T}\right)r(\widehat{T},s),v\left(s\right))⩾1\}.\end{array}$$

Define the quantities

$$\begin{array}{c}{t}_i\left(v(·)\right)=\left\{\begin{array}{cc}\inf \{t:t\in T_i\left(v(·)\right)\}\hfill & \mathrm{if}{T}_i\left(v(·)\right)\ne \varnothing ,\hfill \\ +\infty \hfill & \mathrm{if}{T}_i\left(v(·)\right)=\varnothing .\hfill \end{array}\right.\end{array}$$

Specify the controls of pursuers $P_i,i\in I$, assuming that

$$\begin{array}{c}{u}_i\left(t\right)=\left\{\begin{array}{cc}v\left(t\right)-\lambda (f_i\left(\widehat{T}\right)r(\widehat{T},t),v\left(t\right))f_i\left(\widehat{T}\right)r(\widehat{T},t),\hfill & t\in [0,t_i\left(v(·)\right)],\hfill \\ v\left(t\right),\hfill & t\in (t_i\left(v(·)\right),\widehat{T}].\hfill \end{array}\right.\end{array}$$

The solution to the system (3) can be represented in the form [24]

$$\begin{array}{c}{z}_i\left(t\right)=E_{1/\alpha }(a{t}^{\alpha },1)z_i^0+t{E}_{1/\alpha }(a{t}^{\alpha },2)z_i^1+\underset{0}{\overset{t}{\int }}\overline{E}(t,s,\alpha )(u_i\left(s\right)-v\left(s\right))ds.\end{array}$$

Hence,

$$\begin{array}{c}{\widehat{T}}^{\alpha -1}{z}_i\left(\widehat{T}\right)=f_i\left(\widehat{T}\right)(1-\underset{0}{\overset{\widehat{T}}{\int }}|\overline{E}(\widehat{T},s,\alpha )|\lambda (f_i\left(\widehat{T}\right)r(\widehat{T},s),v\left(s\right))ds).\end{array}$$

It follows from Lemma 5 that there exists $\mathrm{\Lambda }\in I\left(m\right)$ such that ${\widehat{T}}^{\alpha -1}{z}_i\left(\widehat{T}\right)=0$ for all $i\in \mathrm{\Lambda }.$ This proves the theorem. □

Theorem 4.

Let $a<0,$$\mathrm{\Omega }={\mathbb{R}}^k,$${\delta }_0>0.$ Then an m-fold capture occurs in the game.

The proof of this theorem is similar to that of Theorem 3.

Lemma 6.

Let $V=D_1\left(0\right).$ Then ${\delta }_0^{+}>0$ if and only if

$$\begin{array}{c}0\in \bigcap _{\mathrm{\Lambda }\in I(n-m+1)}\mathrm{Intco}\{z_j^1,j\in \mathrm{\Lambda },p_1,\dots ,p_r\}.\end{array}$$

(9)

Proof of Lemma 6.

Assume that condition (9) is satisfied and ${\delta }_0^{+}⩽0.$ Hence, there exists $v_0\in V$ for which

$$\begin{array}{c}(p_1,v_0)⩽0,\dots ,(p_r,v_0)⩽0,\underset{\mathrm{\Lambda }\in I\left(m\right)}{\max }\underset{j\in \mathrm{\Lambda }}{\min }\lambda (z_j^1,v_0)⩽0.\end{array}$$

Therefore, for each $\mathrm{\Lambda }\in I\left(m\right)$ there is $j\in \mathrm{\Lambda }$ such that $\lambda (z_j^1,v_0)=0.$

Let ${\mathrm{\Lambda }}_1=\{1,\dots ,m\}.$ Find $i_1\in {\mathrm{\Lambda }}_1$ for which $\lambda (z_{i_1}^1,v_0)=0.$ Take ${\mathrm{\Lambda }}_2={\mathrm{\Lambda }}_1\setminus \left\{i_1\right\}\cup \{m+1\}.$ Find $i_2\in {\mathrm{\Lambda }}_2$ for which $\lambda (z_{i_2}^1,v_0)=0.$ Let us continue this process further. In the last step we take ${\mathrm{\Lambda }}_{n-m+1}={\mathrm{\Lambda }}_{n-m}\setminus \left\{i_{n-m}\right\}\cup \left\{n\right\}.$ Find $i_{n-m+1}\in {\mathrm{\Lambda }}_{n-m+1}$ for which $\lambda (z_{i_{n-m+1}}^1,v_0)=0.$ Since $V=D_1\left(0\right)$, the condition $\lambda (b,v_0)=0$ implies that $\parallel v_0\parallel =1$ and $(b,v_0)⩽0.$ This yields

$$(z_{i_1}^1,v_0)⩽0,\dots ,(z_{i_{n-m+1}}^1,v_0)⩽0.$$

Hence,

$$0\notin \mathrm{Intco}\{z_{i_1}^1,\dots ,z_{i_{n-m+1}}^1,p_1,\dots ,p_r\},$$

which contradicts (9).

Now let ${\delta }_0^{+}>0.$ Assume that condition (9) is not satisfied. Then there exists ${\mathrm{\Lambda }}^{*}\in I(n-m+1)$ for which

$$0\notin \mathrm{Intco}\{z_j^1,j\in {\mathrm{\Lambda }}^{*},p_1,\dots ,p_r\}.$$

Hence, there exists $v_0\in V,$$\parallel v_0\parallel =1$ such that $(p_1,v_0)⩽0,$$\dots ,$ $(p_r,v_0)⩽0,$ $(z_j^1,v_0)⩽0$ for all $j\in {\mathrm{\Lambda }}^{*}.$ Thus, $\lambda (z_j^1,v_0)=0$ for all $j\in {\mathrm{\Lambda }}^{*}.$ Now let $\mathrm{\Lambda }\in I\left(m\right).$ Then there exists $\alpha \in \mathrm{\Lambda }\cap {\mathrm{\Lambda }}^{*}.$ Therefore, $\underset{j\in \mathrm{\Lambda }}{\min }\lambda (z_j^1,v_0)=0.$ Hence, ${\delta }_0^{+}=0$. Thus, we have a contradiction. This proves the lemma. □

Remark 1.

If the center of the sphere V is not at the origin, Lemma 6 is not true.

Example 1

([27]). Let $k=2,$$r=1,$$n=2,$$m=1$ $b_1=(-1,-1),$$b_2=(1,-1)$ and let $p_1=(0,1),$V be a circle of radius $0.5$ with the center at point $(0,-0.5).$ Then $0\in \mathrm{Intco}\{b_1,b_2,p_1\}$, but $\delta =0.$ Indeed, taking $v_0=(0,0)$, we obtain $\lambda (b_1,v_0)=0,$ $\lambda (b_2,v_0)=0,$$(p_1,v_0)=0.$

Lemma 7.

Let $V=D_1\left(0\right).$ Then ${\delta }_0^{+}>0$ if and only if ${\delta }_0^{-}>0.$

Proof of Lemma 7.

Let ${\delta }_0^{+}>0.$ According to ([6], p. 36), this inequality is equivalent to saying that $0\in \mathrm{Intco}\{z_1^0,\cdots ,z_n^0,p_1,\cdots ,p_r\}.$ The last relation is equivalent to the condition

$$\begin{array}{c}0\in \mathrm{Intco}\{-z_1^0,\cdots ,-z_n^0,-p_1,\cdots ,-p_r\},\end{array}$$

which in turn is equivalent to the inequality ${\delta }_0^{-}>0.$ This proves the lemma. □

Theorem 5.

Let $V=D_1\left(0\right),$$r=1$ and

$$\begin{array}{c}0\in \bigcap _{\mathrm{\Lambda }\in I(n-m+1)}\mathrm{Intco}\{z_j^1,j\in \mathrm{\Lambda },p_1\}.\end{array}$$

Then an m-fold capture occurs in the game.

The validity of this statement follows from Lemmas 6 and 7 and Theorem 3.

Theorem 6.

Let $V=D_1\left(0\right),$$\mathrm{\Omega }={\mathbb{R}}^k$ and $0\in {\displaystyle \bigcap _{\mathrm{\Lambda }\in I(n-m+1)}}\mathrm{Intco}\{z_j^1,j\in \mathrm{\Lambda }\}.$ Then an m-fold capture occurs in the game.

Theorem 7.

Let $V=D_1\left(0\right)$ and suppose there exists $p_0\in {\mathbb{R}}^k$ such that $p_0\ne 0,$$D\subset D_0=\{y:(p_0,y)⩽0\}$ and

$$\begin{array}{c}0\in \bigcap _{\mathrm{\Lambda }\in I(n-m+1)}\mathrm{Intco}\{z_{\alpha }^1,\alpha \in \mathrm{\Lambda },p_0\}.\end{array}$$

(10)

Then an m-fold capture occurs in the game.

The validity of the statement of the theorem follows from Theorem 5.

Remark 2.

In the general case, condition (10) does not follow from the condition

$$\begin{array}{c}0\in \bigcap _{\mathrm{\Lambda }\in I(n-m+1)}\mathrm{Intco}\{z_{\alpha }^1,\alpha \in \mathrm{\Lambda },p_1,\dots ,p_r\}\end{array}$$

for some $p_0.$

Let $k=2,$$m=2,$$p_1=(-1,0),$$p_2=(1,0),$$z_1^1=(0,1),$$z_2^1=(0,2),$$z_3^1=(0,-1),$ $z_4^1=(0,-2).$ Then for any $\mathrm{\Lambda }\in I\left(3\right)$ the condition

$$0\in \mathrm{Intco}\{z_i^1,i\in \mathrm{\Lambda },p_1,p_2\}$$

holds, but for any $p_0\in {\mathbb{R}}^2$ one has $0\notin \mathrm{Intco}\{z_i^1,i\in \mathrm{\Lambda },p_0\}.$

Corollary 1

([22]). Let $a<0,$$V=D_1\left(0\right),$$n⩾k$ and

$$\begin{array}{c}0\in \mathrm{Intco}\{z_1^1,\cdots ,z_n^1,p_1,\cdots ,p_r\}.\end{array}$$

Then a capture occurs in the game.