Adv Syst Sci Appl 2025; 1:86–98
Published online at https://ijassa.ipu.ru.

Multiple Capture of Coordinated Evaders in the Linear Group
Pursuit Problem with a Simple Matrix and Phase Restrictions

Nikolay Petrov

Udmurt State University, Izhevsk, Russia

Abstract: In finite-dimensional Euclidean space, an analysis is made of the problem of pursuit
of two evaders by a group of pursuers, which is described by a system of the form

żij = αzij + ui − v, ui, v ∈ V.

It is assumed that the evaders use the same control and do not move out of a convex polyhedral
set. The pursuers use counterstrategies based on information about the initial positions and the
prehistory of the evaders’ control. The set of admissible controls V is a sphere of unit radius with
its center at the origin, and the goal sets are the origin. The goal of the group of pursuers is the
capture of at least one evader by a given number of pursuers. In terms of the initial positions and
parameters of the game, a sufficient condition for capture is obtained. The method of resolving
functions, which is used as a basis for analysis, provides sufficient conditions for solvability of
the problem of pursuit in some guaranteed time.

Keywords: differential games, pursuer, evader, capture, multiple capture, conflict-controlled
processes

1. INTRODUCTION

Mathematical control theory and dynamical game theory provide a fundamental framework
for investigating controlled processes of different nature. The schools of N.N. Krasovskii and
L.S. Pontryagin played a key role in establishing these theories and developing a number of
classical methods in this area.

The methods for investigating dynamical games include those aimed at constructing
optimal strategies [9] and methods that ensure a guaranteed result [19]. The main condition
in this case is to achieve the goal in hand and to accomplish the task under specific
conditions. To obtain the guaranteed result, use is often made of the first method of L.S.
Pontryagin [19] and the method of resolving functions [3,6,7]. These methods give sufficient
conditions to complete the game in finite time from given initial positions. Two-player
differential game theory was developed further by studying the problem of pursuit of one
evader by a group of pursuers and the problem of evasion of one evader from a group of
pursuers [4, 5, 8, 10, 22]. B.N. Pshenichny obtained [20] necessary and sufficient conditions
for capture of the evader. N.L. Grigorenko introduced the notion of multiple capture. For the
problem with simple motions and equal opportunities he presented necessary and sufficient
conditions [7] for multiple capture of the evader. In [14], sufficient conditions were obtained
for multiple capture of the evader in L.S. Pontryagin’s example with equal opportunities for
all participants. The problem of multiple capture of an evader in the presence of defenders
was discussed in [2].

∗Corresponding author: kma3@list.ru



MULTIPLE CAPTURE OF COORDINATED EVADERS... 87

A natural generalization of these group pursuit problems is the situation of conflict
interaction of a group of pursuers and a group of evaders. The goal of the group of pursuers
is to capture a given number of evaders, and the goal of the group of evaders is the opposite
one [1, 12, 21].

The authors of [23] addressed the problem of pursuit of a group of evaders by a group of
pursuers under the condition that all evaders use the same control. Sufficient conditions for
capture of at least one evader were obtained. In what follows, we will call the pursuit problem
in which all evaders use the same control the problem of pursuing coordinated evaders. The
study presented in [23] was developed, in particular, in [11, 16, 17], where sufficient, and in
some cases necessary, conditions for capture of at least one evader were obtained under the
condition that all participants have equal opportunities and all evaders use the same control.

The problem of pursuing two coordinated evaders in which the goal of the group of
pursuers was the capture of at least one evader by two pursuers was treated in [15, 18].

This paper addresses the linear problem of pursuit of two coordinated evaders by a group
of pursuers in a differential game with a simple matrix and phase restrictions on the states of
the evaders. The goal of the group of pursuers is the capture of at least one evader by a given
number of pursuers.

2. FORMULATION OF THE PROBLEM

In the space Rk (k ⩾ 2) we consider a differential game Γ(n+ 2) involving n+ 2 players: n
pursuers P1, . . . , Pn and two evaders E1, E2.

The law of motion of each of pursuers Pi has the form

ẋi = αxi + ui, xi(0) = x0
i , ui ∈ V. (2.1)

The law of motion of each of evaders Ej has the form

ẏj = αyj + v, yj(0) = y0j , v ∈ V. (2.2)

Here i ∈ I = {1, . . . , n}, j ∈ {1, 2}, xi, yj , ui, V =
{
v | v ∈ Rk, ∥v∥ ⩽ 1

}
, α ∈ R1,

α ⩽ 0. In addition, x0
i ̸= y0j for all i ∈ I, j = 1, 2.

Additionally, it is assumed that each evader Ej, j ∈ {1, 2} does not move out of a convex
polyhedral set

Ω = {y ∈ Rk | (pl, y) ⩽ βl, l = 1, . . . , r},

where p1, . . . , pr are the unit vectors Rk, β1, . . . , βr are the real numbers, and (u, v) is the
scalar product. Assume that Ω = Rk for r = 0.

Introduce new variables zij = xi − yj. Then instead of the systems (2.1) and (2.2) we
obtain the system

żij = αzij + ui − v, zij(0) = z0ij = x0
i − y0j . (2.3)

The measurable function v : [0,∞) → Rk is called admissible if v(t) ∈ V, y1(t) ∈ Ω, y2(t) ∈
Ω for all t ⩾ 0. Let us call the restriction of the function v to the interval [0, t] the prehistory
vt(·) of the function v(·) at time t.

The actions of the evaders can be interpreted as follows: there is a center that chooses the
same control v(t) for evaders E1 and E2.
Definition 2.1:
We will say that a quasi-strategy Ui of pursuer Pi is given if a map Ui(t, z

0, vt(·)) is defined
which associates a measurable function ui(t) = Ui(t, z

0, vt(·)) with values in V to the initial
position z0 = (z0ij), time t and an arbitrary prehistory vt(·) of admissible control v(·) of
evaders Ej .

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88 N. PETROV

Definition 2.2:
A q-fold capture (if q = 1, then it is a capture) occurs in the game Γ(n+ 2) if there exist time
T0 = T (z0), quasi-strategies U1, . . . ,Un of pursuers P1, . . . , Pn such that for any admissible
measurable function v(·) there are numbers i1, . . . , iq ∈ I, l ∈ {1, 2}, and time instants
τi1 , . . . , τiq ∈ [0, T0] such that zisl(τis) = 0 for all s = 1, . . . , q.

Remark 2.1:
The capture condition implies that q pursuers perform a capture of one evader and that the
time instants may or may not coincide. Such a situation can arise if the system consists of q
blocks, and in order to put it out of operation, one needs to damage all blocks.

3. AUXILIARY RESULTS

Definition 3.1 (see [13]):
The vectors a1, a2, . . . , as form a positve basis in Rk if for any x ∈ Rk there exist nonnegative
real numbers α1, α2, . . . , αs such that

x = α1a1 + α2a2 + . . .+ αsas.

We introduce the following notation. IntX and coX are the interior and the convex hull
of the set X ⊂ Rk, respectively, and |J | is the number of elements of the finite set J ,

λ(h, v) = sup{λ ⩾ 0 | − λh ∈ V − v},
Ωl(K) = {(i1, . . . , il) | i1, . . . , il ∈ K and are pairwise different},

where K is a finite set of natural numbers and l is a natural number.

Theorem 3.1 (see [13]):
Let a1, a2, . . . , am ∈ Rk. The following statements are equivalent.

1. The vectors a1, a2, . . . , am form a positive basis in Rk.
2. 0 ∈ Intco {a1, . . . , am}.
3. For any p ∈ Rk, p ̸= 0, there exists a number l ∈ {1, . . . ,m} for which (al, p) > 0.

Lemma 3.1:
Let a1, . . . , am, c1, p1 ∈ Rk such that, for each J0 ⊂ J = {1, . . . ,m}, |J0| = q − 1, one has

0 ∈ Intco{as, s ∈ J \ J0, c1, p1}.

Then for any b1, . . . , bq ∈ Rk there is µ̂ > 0 such that for all µ > µ̂ the following inequality
holds:

δ(µ) = min
v∈V

max
{

max
Λ∈Ω0

q(J1)
min
i∈Λ

λ(ωi, v), (p1, v)
}
> 0,

where

J1 = J ∪ {m+ 1, . . . ,m+ q}, Ω0
q(J1) = Ωq(J) ∪ {(m+ 1, . . . ,m+ q)},

ωi =

{
ai if i ∈ J,
bi−m + µc1 if i = m+ 1, . . . ,m+ q.

Proof
Assume that the statement of the lemma is not valid. Then there exist b01, . . . , b

0
q ∈ Rk for

which for any µ̂ > 0 there is µ > µ̂ such that δ(µ) = 0. It follows from the condition δ(µ) = 0

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



MULTIPLE CAPTURE OF COORDINATED EVADERS... 89

that there exists vµ ∈ V such that (p1, vµ) ⩽ 0 and

max
Λ∈Ω0

q(J1)
min
i∈Λ

λ(ωi, vµ) = 0. (3.4)

We show that it follows from condition (3.4) that there exist a set J(µ) ⊂ J0, |J(µ)| = q − 1,
and a number s(µ) ∈ {m+ 1, . . . ,m+ q} for which the following inequalities hold:

(wi, vµ) ⩽ 0 for all i ∈
(
J0 \ J(µ)

)
∪{s(µ)}. (3.5)

It follows from condition (3.4) that for any Λ ∈ Ω0
q(J1) there is a number iΛ ∈ Λ such that

λ(ωiΛ , vµ) = 0. By virtue of the properties of the function λ [7] we find that ∥vµ∥ = 1.
Therefore, the condition λ(ωiΛ , vµ) = 0 is equivalent to the condition (ωiΛ , vµ) ⩽ 0.

Let Λ1 = {1, . . . , q}. Then there is a number i1 ∈ Λ1 such that (wi1 , vµ) ⩽ 0. Take Λ2 =
Λ1 \ {i1} ∪ {q + 1}. Then there is a number i2 ∈ Λ2 such that (ωi2 , vµ) ⩽ 0. Continuing
this process further, we find that for the set Λm−q+1 = Λm−q \ {im−q} ∪ {m} there is a
number im−q+1 ∈ Λm−q+1 for which (ωim−q+1 , vµ) ⩽ 0. In addition, for the set Λ = {(m+
1, . . . ,m+ q)} there is a number s(µ) such that (ωs(µ), vµ) ⩽ 0. Consider a set J(µ) =
J \ {i1, . . . , im−q+1}. Then |J(µ)| = q − 1 and (ωs, vµ) ⩽ 0 for all s ∈ J \ J(µ). This proves
(3.5).

We now show that there exists a set J0 ⊂ J, |J0| = q − 1, such that

0 /∈ Intco{as, s ∈ J \ J0, c1, p1}. (3.6)

Let µ̂ = 1. Then there is µ1 > µ̂ for which there are a vector v1 ∈ V, ∥v1∥ = 1, a set
J(µ1) ⊂ J, |J(µ1)| = q − 1, and an index s(µ1) ∈ {m+ 1, . . . ,m+ q} such that

(ai, v1) ⩽ 0 for all i ∈ J \ J(µ1),

(b0s(µ1)−m + µ1c1, v1) ⩽ 0, (p1, v1) ⩽ 0.

Take µ̂ = µ1 + 1. Then there is µ2 > µ̂ for which there are a vector v2 ∈ V, ∥v2∥ = 1, a set
J(µ2) ⊂ J, |J(µ2)| = q − 1, and an index s(µ2) ∈ {m+ 1, . . . ,m+ q} such that

(ai, v2) ⩽ 0 for all i ∈ J \ J(µ2),

(b0s(µ2)−m + µ2c1, v1) ⩽ 0, (p1, v1) ⩽ 0.

Continuing this process further, we find that there exist a sequence of real numbers {µl}∞l=1,
lim

l→+∞
µl = +∞, a sequence of vectors {vl}∞l=1, ∥vl∥ = 1, a sequence of sets {J(µl)}∞l=1,

J(µl) ⊂ J , |J(µl)| = q − 1, and a sequence of natural numbers {s(µl)}∞l=1, s(µl) ∈ {m+
1, . . . ,m+ q} for which the following inequalities hold:

(ai, vl) ⩽ 0 for all i ∈ J \ J(µl), (b0s(µl)−m + µlc1, v1) ⩽ 0, (p1, v1) ⩽ 0. (3.7)

It follows from (3.7) that there exist a subsequence {vlp}∞p=1, ∥vlp∥ = 1, a set J0 ⊂ J,
|J0| = q − 1, and a natural number s ∈ {m+ 1, . . . ,m+ q} for which for all p = 1, 2, . . .
the following inequalities hold:

(ai, vlp) ⩽ 0 for all i ∈ J \ J0, (b0s−m + µlpc1, vlp) ⩽ 0, (p1, vlp) ⩽ 0. (3.8)

From the sequence {vlp} one can choose a subsequence converging to v0, ∥v0∥ = 1. Assume
that the sequence {vlp} itself converges. Passing in (3.8) to the limit as p → +∞, we find that

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90 N. PETROV

the following inequalities hold:

(ai, v0) ⩽ 0 for all i ∈ J \ J0, (c1, v0) ⩽ 0, (p1, v0) ⩽ 0.

By Theorem 3.1, this implies that

0 /∈ Intco{ai, i ∈ J \ J0, c1, p1}.
This proves (3.6). We have obtained a contradiction with the condition of the lemma. The
lemma is proved.

Lemma 3.2:
Let r = 1, α = 0, a1, . . . , am, c1, p1 ∈ Rk such that for any J0 ⊂ J = {1, . . . ,m}, |J0| =
q − 1, one has

0 ∈ Intco{al, l ∈ J \ J0, c1, p1}.

Then for any b1, . . . , bq ∈ Rk there is µ̂ > 0 such that for each µ > µ̂ there is a time instant
T (µ) > 0 such that for any admissible control v(·) of evaders E1 and E2 there is a set
Λ∗ ∈ Ω0

q(J1) such that for all l ∈ Λ∗ the following inequalities hold:∫ T (µ)

0

λ(ωl, v(s)) ds ⩾ 1,

where

J1 = J ∪ {m+ 1, . . . ,m+ q}, Ω0
q(J1) = Ωq(J) ∪ {(m+ 1, . . . ,m+ q)},

ωi =

{
ai if i ∈ J,
bi−m + µc1 if i = m+ 1, . . . ,m+ q.

Proof
Take arbitrary vectors b1, . . . , bq ∈ Rk. By Lemma 3.1, there exists µ̂ > 0 such that for all
µ > µ̂ the inequality δ(µ) > 0 is satisfied, where

δ(µ) = min
v∈V

max
{

max
Λ∈Ω0

q(J1)
min
i∈Λ

λ(ωi, v), (p1, v)
}
.

Prove that the statement of the lemma is valid for all µ > µ̂. Let v(·) be an arbitrary admissible
control of the evaders. For each t > 0, µ > µ̂ we define the sets

T1(t, µ) = {τ | τ ∈ [0, t], (p1, v(τ)) ⩾ δ(µ)}, T2(t, µ) = {τ | τ ∈ [0, t], (p1, v(τ)) < δ(µ)}.

Since v(·) is the admissible control of the evaders, the following inequalities hold for all
t ⩾ 0:

(p1, y1(t)) ⩽ β1, (p1, y2(t)) ⩽ β1.

Therefore, for all t ⩾ 0 the following inequality holds:∫ t

0

(p1, v(s)) ds ⩽ β0, where β0 = min{β1 − (p1, y
0
1), β1 − (p1, y

0
2)}.

Then one has

β0 ⩾
∫ t

0

(p1, v(s)) ds ⩾ δ(µ)

∫
T1(t,µ)

ds−
∫
T2(t,µ)

ds,

t =

∫
T1(t,µ)

ds+

∫
T2(t,µ)

ds.

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



MULTIPLE CAPTURE OF COORDINATED EVADERS... 91

This implies that the following inequality holds:∫
T2(t,µ)

ds ⩾
δ(µ)t− β0

1 + δ(µ)
.

Next, we have

max
Λ∈Ω0

1(J1)
min
i∈Λ

∫ t

0

λ(ωi, v(s)) ds ⩾ max
Λ∈Ω0

1(J1)

∫ t

0

min
i∈Λ

λ(ωi, v(s)) ds.

For any nonnegative numbers γΛ
(
Λ ∈ Ω0

1(J1)
)

the following inequality holds:

max
Λ∈Ω0

1(J1)
γΛ ⩾

1

M

∑
Λ∈Ω0

1(J1)

γΛ, where M =
m!

q!(m− q)!
+ 1.

Therefore,

max
Λ∈Ω0

1(J1)

∫ t

0

min
i∈Λ

λ(ωi, v(s)) ds. ⩾
1

M

∫ t

0

∑
Λ∈Ω0

q(J1)

min
i∈Λ

λ(ωi, v(s)) ds ⩾

⩾
1

M

∫ t

0

max
Λ∈Ω0

1(J1)
min
i∈Λ

λ(ωi, v(s)) ds ⩾
1

M

∫
T2(t,µ)

max
Λ∈Ω0

1(J1)
min
i∈Λ

λ(ωi, v(s)) ds ⩾

⩾
δ(µ)

M

∫
T2(t,µ)

ds ⩾
δ(µ)

M

(δ(µ)t− β0

1 + δ(µ)

)
.

Hence,

max
Λ∈Ω0

1(J1)
min
i∈Λ

∫ t

0

λ(ωi, v(s)) ds ⩾
δ(µ)

M

(δ(µ)t− β0

1 + δ(µ)

)
.

The required inequality follows from the last inequality. This proves the lemma.

Lemma 3.3:
Let r = 1, α < 0, β1 = 0, a1, . . . , am, c1, p1 ∈ Rk such that for any J0 ⊂ J = {1, . . . ,m},
|J0| = q − 1, one has

0 ∈ Intco{al, l ∈ J \ J0, c1, p1}.

Then for any b1, . . . , bq ∈ Rk there is µ̂ > 0 such that for each µ > µ̂ there is a time
instant T (µ) > 0 such that for any admissible control of evaders E1 and E2 there is a set
Λ∗ ∈ Ω0

q(J1) such that for all l ∈ Λ∗ the following inequalities hold:∫ T (µ)

0

e−αsλ(ωl, v(s)) ds ⩾ 1,

where

J1 = J ∪ {m+ 1, . . . ,m+ q}, Ω0
q(J1) = Ωq(J) ∪ {(m+ 1, . . . ,m+ q)},

ωi =

{
ai if i ∈ J,
bi−m + µc1 if i = m+ 1, . . . ,m+ q.

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



92 N. PETROV

Proof
Let v(·) be the admissible control of the evaders. Then for all t ⩾ 0 the following inequalities
hold: (p1, y1(t)) ⩽ 0, (p1, y2(t)) ⩽ 0. Therefore, for all t ⩾ 0 the following inequality is
valid: ∫ t

0

e−as(p1, v(s)) ds ⩽ β0, where β0 = min{−(p1, y
0
1),−(p2, y

0
2)}.

Further reasoning is similar to that used in the proof of Lemma 3.2.

4. SUFFICIENT CONDITIONS FOR CAPTURE WITH α = 0

Theorem 4.1:
Let r = 1 and suppose that there exists a set I0 ⊂ I, |I0| = n− q, such that for any J0 ⊂ I0,
|J0| = q − 1, one has

0 ∈ Intco{x0
l − y01, x

0
l − y02, l ∈ I0 \ J0, p1}. (4.9)

Then a q-fold capture occurs in the game Γ(n+ 2).

Proof
Assume that I0 = {1, . . . , n− q}. Denote c = y01 − y02. Since the equation

x0
l − y02 = x0

l − y01 + c

holds for all l ∈ I , it follows from (4.9) that for any J0 ⊂ I0, |J0| = q − 1, we have

0 ∈ Intco{z0l1, l ∈ I0 \ J0, c, p1}.

It follows from Lemma 3.1 that there exists µ > 0 such that

δ(µ) = min
v∈V

max
{

max
Λ∈Ω0

q(I)
min
i∈Λ

λ(ωi, v), (p1, v)
}
> 0,

where Ω0
q(I) = Ωq(I0) ∪ {(n− q + 1, . . . , n)},

ωi =

{
z0i1 if i ∈ I0,
z0i2 + µc if i = n− q + 1, . . . , n.

It follows from Lemma 3.2 that

T0 = min{t ⩾ 0 | inf
v(·)

max
Λ∈Ω0

q(I)
min
i∈Λ

∫ t

0

λ(ωi, v(s)) ds ⩾ 1}

is finite. Let v(·) be the admissible control of the evaders. Define the functions

hi(t) = 1−
∫ t

0

λ(ωi, v(s)) ds.

Let pursuer Pi, i ∈ I , construct a control as follows. If the inequality hi(t) ⩾ 0 holds at time
t, then we assume

ui(t) = v(t)− λ(ωi, v(t))ωi.

If τ is the first time instant for which hi(τ) = 0, then we assume that λ(ωi, v(t)) = 0 for all
t ⩾ τ.

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MULTIPLE CAPTURE OF COORDINATED EVADERS... 93

From the definition of controls and the system (2.3) it follows that for all t ⩾ 0 the
following equations hold:

zi1(t) = z0i1hi(t), i ∈ I0, zi2(t) = z0i2hi(t)− µc(1− hi(t)), i ∈ I \ I0. (4.10)

It follows from Lemma 3.2 that there exists Λ∗ ∈ Ω0
q(T ) for which hi(T0) = 0 for all i ∈ Λ∗.

Two cases are possible.
1. Λ∗ ∈ Ωq(I0). In this case, zi1(T0) = 0 for all i ∈ Λ∗. Hence, a q-fold capture of evader

E1 occurs.
2. Λ∗ = {(n− q + 1, . . . , n)}. Then zi2(T0) = −µc for all i ∈ I \ I0. We show that for

any J0 ⊂ I , |J0| = q − 1, we have

0 ∈ Intco{zi2(T0), i ∈ I \ J0, p1}. (4.11)

Assume that this is not the case. Then there exists J0 ⊂ I , |J0| = q − 1, for which

0 /∈ Intco{zi2(T0), i ∈ I \ J0, p1}.

Hence, there exists a vector v0 ∈ V, ∥v0∥ = 1 such that

(p1, v0) ⩽ 0, (zi2(T0), v0) ⩽ 0 for all i ∈ I \ J0.

It follows from the condition |J0| = q − 1 that there exists a number l ∈ I \ I0 such that
l ∈ Λ∗. Then from the equation zl2(T0) = −µc we find that (c, v0) ⩾ 0.

Since the equation zi1(T0) = zi2(T0)− c holds for all i ∈ I0, the inequality (zi1(T0), v0) ⩽
0 holds for all i ∈ I0 \ J0. Therefore, it follows from (4.10) that (z0i1, v0) ⩽ 0 for all i ∈
I0 \ J0. In addition, from the equation (i ∈ I0)

zi2(T0) = zi1(T0) + c = z0i1hi(T0) + z0i2 − z0i1

it follows that
z0i2 = zi2(T0) + z0i1(1− hi(T0))

and hence (z0i2, v0) ⩽ 0 for all i ∈ I0 \ J0. Thus, the following inequalities hold:

(p1, v0) ⩽ 0, (z0i1, v0) ⩽ 0, (z0i2, v0) ⩽ 0 for all i ∈ I0 \ J0.

Let J2 ⊂ I0 \ J0 be a set such that for the set J1 = (I0 ∩ J0) ∪ J2 the equation |J1| = q − 1
is satisfied. Then, by Theorem 3.1,

0 /∈ Intco{z0i1, z0i2, i ∈ I0 \ J1, p1},

which contradicts the condition of the theorem. Thus, (4.11) is proved. Taking T0 as the initial
time instant and using Theorem [14], we find that a q-fold capture of evader E2 occurs in the
game Γ(n+ 2). This proves the theorem.

Theorem 4.2:
Let D = Rk and suppose that there exists a set I0 ⊂ I, |I0| = n− q, such that for any
J0 ⊂ I0, |J0| = q − 1, we have

Intco{x0
l , l ∈ I0 \ J0} ∩ co{y01, y02} ≠ ∅. (4.12)

Then a q-fold capture occurs in the game Γ(n+ 2).

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



94 N. PETROV

Proof
We prove that it follows from condition (4.12) that

0 ∈ Intco{x0
l − y01, x

0
l − y02, l ∈ I0 \ J0}. (4.13)

Assume that there exists J0 ⊂ I0, |J0| = q − 1, for which (4.12) is satisfied, but

0 /∈ Intco{x0
l − y01, x

0
l − y02, l ∈ I0 \ J0}.

Hence, {0} and co{x0
l − y01, x

0
l − y02, l ∈ I0 \ J0} are separable. Therefore, there exists v0 ∈

V, ∥v0∥ = 1, such that

(x0
l − y01, v0) ⩽ 0, (x0

l − y02, v0) ⩽ 0 for all l ∈ I0 \ J0.

Therefore,
(x0

l , v0) ⩽ γ ⩽ (y0j , v0) for all l ∈ I0 \ J0, j ∈ {1, 2},

where γ = min{(y01, v0), (y02, v0)}. Hence, the sets co{x0
l , l ∈ I0 \ J0} and co{y01, y02} are

separable. Therefore,

Intco{x0
l , l ∈ I0 \ J0} ∩ co{y01, y02} = ∅,

which contradicts (4.12). Thus, (4.13) is proved. Further reasoning is similar to that used in
the proof of Theorem 4.1. This proves the theorem.

Theorem 4.3:
Let there exist a vector p ∈ Rk, p ̸= 0, a number γ ∈ R1 and a set I0 ⊂ I , |I0| = n− q, such
that

1) D ⊂ {x ∈ Rk | (p, x) ⩽ γ};
2) for any set J0 ⊂ I0, |J0| = q − 1, we have

0 ∈ Intco{x0
l − y01, x

0
l − y02, l ∈ I0 \ J0, p}.

Then a q-fold capture occurs in the game Γ(n+ 2).
The validity of this theorem follows from Theorem 4.1.
Example 1. Let q = 2, α = 0, k = 2, D = R2, x0

1 = (0, 0), x0
2 = (1, 1), x0

3 = (−1, 0),
x0
4 = (−1, 1), x0

5 = (1,−1), x0
6 = (0, 1), y01 = (0,−2), y02 = (1, 2). Taking I0 = {2, 3, 4, 5},

we find that the conditions of Theorem 3 are satisfied and hence a two-fold capture occurs in
the game Γ(8).

Example 2. Let q = 2, α = 0, k = 2, D = R2, p1 = (0,−1), β1 = 3, x0
1 = (−1, 1),

x0
2 = (1, 1), x0

3 = (−1, 2), x0
4 = (1, 2), x0

5 = (0, 1), x0
6 = (0, 4), y01 = (0, 0), y02 = (0,−1).

Taking I0 = {2, 3, 4, 5}, we find that the conditions of Theorem 4.2 are satisfied and hence a
two-fold capture occurs in the game Γ(8). Taking I0 = {1, 2, 3, 4}, we find that the conditions
of Theorem 4.1 are satisfied and hence a two-fold capture occurs in the game Γ(8).

5. SUFFICIENT CONDITIONS FOR CAPTURE WITH α < 0

Theorem 5.1:
Let r = 1, α < 0, β1 = 0 and suppose that there exists a set I0 ⊂ I, |I0| = n− q, such that
for any J0 ⊂ I0, |J0| = q − 1, one has

0 ∈ Intco{x0
l − y01, x

0
l − y02, l ∈ I0 \ J0, p1}. (5.14)

Then a q-fold capture occurs in the game Γ(n+ 2).

Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)



MULTIPLE CAPTURE OF COORDINATED EVADERS... 95

Proof
Assume that I0 = {1, . . . , n− q}. Denote c = y01 − y02. Since the equation

x0
l − y02 = x0

l − y01 + c

holds for all l ∈ I , it follows from (5.14) that for any J0 ⊂ I0, |J0| = q − 1, we have

0 ∈ Intco{z0l1, l ∈ I0 \ J0, c, p1}.

It follows from Lemma 3.1 that there exists µ > 0 such that

δ(µ) = min
v∈V

max
{

max
Λ∈Ω0

q(I)
min
i∈Λ

λ(ωi, v), (p1, v)
}
> 0,

where Ω0
q(I) = Ωq(I0) ∪ {(n− q + 1, . . . , n)},

ωi =

{
z0i1 if i ∈ I0,
z0i2 + µc if i = n− q + 1, . . . , n.

It follows from Lemma 3.3 that

T0 = min{t ⩾ 0 | inf
v(·)

max
Λ∈Ω0

q(I)
min
i∈Λ

∫ t

0

e−αsλ(ωi, v(s)) ds ⩾ 1}

is finite. Let v(·) be the admissible control of the evaders. Define the functions

hi(t) = 1−
∫ t

0

e−αsλ(ωi, v(s)) ds.

Let pursuer Pi, i ∈ I , construct a control as follows. If the inequality hi(t) ⩾ 0 holds at time
t, then we assume

ui(t) = v(t)− λ(ωi, v(t))ωi.

If τ is the first time instant for which hi(τ) = 0, then we assume that λ(ωi, v(t)) = 0 for all
t ⩾ τ.

It follows from (2.3) that the solution to the Cauchy problem has the form

zij(t) = eαt ·
(
z0ij +

∫ t

0

e−αs(ui(s)− v(s)) ds
)
. (5.15)

From the definition of the controls of the pursuers and (5.15) we find that for all t ⩾ 0 the
following equations hold:

zi1(t)e
−αt = z0i1hi(t), i ∈ I0, zi2(t)e

−αt = z0i2hi(t)− µc(1− hi(t)), i ∈ I \ I0. (5.16)

It follows from Lemma 2 that there exists Λ∗ ∈ Ω0
q(T ) for which hi(T0) = 0 for all i ∈ Λ∗.

Two cases are possible.
1. Λ∗ ∈ Ωq(I0). In this case we find that zi1(T0) = 0 for all i ∈ Λ∗. Hence, a q-fold capture

of evader E1 occurs.
2. Λ∗ = {(n− q + 1, . . . , n)}. Then zi2(T0) = −µeαT0c for all i ∈ I \ I0. We show that

for any J0 ⊂ I , |J0| = q − 1,

0 ∈ Intco{zi2(T0), i ∈ I \ J0, p1}. (5.17)

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96 N. PETROV

Assume that this is not the case. Then there exists J0 ⊂ I , |J0| = q − 1, for which

0 /∈ Intco{zi2(T0), i ∈ I \ J0, p1}.

Consequently, there exists a vector v0 ∈ V, ∥v0∥ = 1, such that

(p1, v0) ⩽ 0, (zi2(T0), v0) ⩽ 0 for all i ∈ I \ J0.

Since |J0| = q − 1, there exists a number l ∈ I \ I0 such that l ∈ Λ∗. Then it follows from
the equation zl2(T0) = −µeαT0c that (c, v0) ⩾ 0.

Since the equation zi1(T0) = zi2(T0)− ceαT0 holds for all i ∈ I0, the inequality
(zi1(T0), v0) ⩽ 0 holds for all i ∈ I0 \ J0. Therefore, it follows from (5.16) that (z0i1, v0) ⩽ 0
for all i ∈ I0 \ J0. Also, from the equation (i ∈ I0)

zi2(T0) = zi1(T0) + eαT0c = z0i1hi(T0) + eαT0(z0i2 − z0i1)

it follows that
z0i2 = e−αT0zi2(T0) + z0i1(1− hi(T0))

and hence (z0i2, v0) ⩽ 0 for all i ∈ I0 \ J0. Thus, the following inequalities hold:

(p1, v0) ⩽ 0, (z0i1, v0) ⩽ 0, (z0i2, v0) ⩽ 0 for all i ∈ I0 \ J0.

Let J2 ⊂ I0 \ J0 be a set such that for the set J1 = (I0 ∩ J0) ∪ J2 the equation |J1| = q − 1
is satisfied. Then, by Theorem 3.1,

0 /∈ Intco{z0i1, z0i2, i ∈ I0 \ J1, p1},

which contradicts the condition of the theorem. Thus, (5.17) is proved. Taking T0 as the initial
time instant and using Theorem [14], we find that a q-fold capture of evader E2 occurs in the
game Γ(n+ 2). This proves the theorem.

Theorem 5.2:
Let D = Rk, α < 0 and suppose that there exists a set I0 ⊂ I, |I0| = n− q, such that for any
J0 ⊂ I0, |J0| = q − 1,

Intco{x0
l , l ∈ I0 \ J0} ∩ co{y01, y02} ≠ ∅. (5.18)

Then a q-fold capture occurs in the game Γ(n+ 2).
This theorem is proved along the same lines as Theorem 4.1.

Theorem 5.3:
Let there exist a vector p ∈ Rk, ∥p∥ = 1, and a set I0 ⊂ I , |I0| = n− q, such that

1) D ⊂ {x ∈ Rk | (p, x) ⩽ 0};
2) for any set J0 ⊂ I0, |J0| = q − 1, one has

0 ∈ Intco{x0
l − y01, x

0
l − y02, l ∈ I0 \ J0, p}.

Then a q-fold capture occurs in the game Γ(n+ 2).
The validity of this theorem follows from Theorem 5.1.

ACKNOWLEDGEMENTS

This research was funded by the Ministry of Science and Higher Education of the Russian
Federation in the framework of state assignment, project FEWS-2024-0009.

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MULTIPLE CAPTURE OF COORDINATED EVADERS... 97

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Copyright © 2025 ASSA. Adv Syst Sci Appl (2025)


	Introduction
	Formulation of the problem
	Auxiliary results
	Sufficient conditions for capture with = 0
	Sufficient conditions for capture with < 0