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6URAL MATHEMATICAL JOURNAL, Vol. 10, No. 1, 2024, pp. 112-122
DOI: 10.15826/umj.2024.1.010
A TWO-FOLD CAPTURE OF COORDINATED EVADERS IN THE PROBLEM OF A SIMPLE PURSUIT ON TIME SCALES1
Elena S. Mozhegova^, Nikolai N. Petrov^
Udmurt State University, Universitetskaya Str., 1, Izhevsk, 426034, Russian Federation
t [email protected] tt [email protected]
Abstract: In finite-dimensional Euclidean space, we study the problem of a simple pursuit of two evaders by a group of pursuers in a given time scale. 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 on the initial positions and on the prehistory of the control of evaders. The set of admissible controls of each of the participants 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 two pursuers. In terms of the initial positions and parameters of the game, a sufficient condition for capture is obtained. The study is based on the method of resolving functions, which makes it possible to obtain sufficient conditions for solvability of the pursuit problem in some guaranteed time.
Keywords: Differential game, Group pursuit, Evader, Pursuer, Time scale.
1. Introduction
The modern theory of differential pursuit-evasion games involves the development of methods for solving problems of conflict interaction of groups of pursuers and evaders [3, 6, 7, 10]. In particular, it is concerned with searching for new classes of problems which can be analyzed using the previously developed methods, for example, the method of resolving functions. It was pointed out in [1, 9] that some results obtained separately for the theories of differential and difference equations may be considered from a unified point of view if one admits the possibility of specifying dynamical systems on arbitrary closed subsets R1 called time scales. Time scales find applications in constructing various mathematical models [2, 4]. A nonantagonistic game of N persons in a time scale was considered in [11]. Sufficient conditions for the capture of one evader in the problem of a simple group pursuit in a given time scale were obtained in [15].
Ref. [14] addressed the problem of a simple pursuit of a group of rigidly coordinated evaders by a group of pursuers in a given time scale, where sufficient conditions for the capture of at least one evader were obtained. The problem of a multiple capture of a given number of evaders in time scales, under the condition that the evaders use programmed strategies, each pursuer catches no more than one evader and the motions of the players are simple was treated in [13].
Ref. [17] dealt with the problem of a simple pursuit of rigidly coordinated evaders in a given time scale, under the condition that the evaders do not move out of a convex polyhedral set. The goal of the pursuers was either the capture of one evader by two pursuers or the capture of two evaders. Sufficient conditions for capture were obtained.
In this paper we consider, in a given time scale, the problem of a simple pursuit of two evaders by a group of pursuers who use the same control, under the condition that the evaders do not move
1This work was supported by the Ministry of Science and Higher Education of the Russian Federation in the framework of state assignment, project FEWS-2024-0009.
out of a convex polyhedral set. The goal of the pursuers is the capture of at least one evader by two different pursuers. Sufficient conditions for capture are obtained.
2. Auxiliary definitions and facts
In this section we will outline the basic facts from time scale theory. All results presented below can be found, for example, in [5, 8].
Definition 1. A nonempty closed subset T c R1 such that sup t = is called a time scale.
teT
Definition 2. Let T be a time scale. A function a : T — R1 of the form
a(t) = inf {s € T | s > t}
is called a translation function.
Definition 3. A function f : T — R1 is called A-differentiable at point t € T if there exists a number y € R1 such that for any e > 0 there exists a neighborhood W of point t such that the inequality
If (a(t)) - f(s) - y(a(t) - s)| < e|a(t) - s|
holds for all s € T n W.
In this case, the number y is called the A-derivative of the function f at point t. The Aderivative of the function f at point t will be denoted by f A(t) = y.
Definition 4. A function f : T — Rn, f (t) = (f1(t),..., fn(t)) is called A-differentiable at point t € T if all functions f1,...,fn are A-differentiable at point t.
Let T be a time scale, E c T. Denote
R(E) = {t € E I a(t) > t}.
Then the set R(E) is no more than countable.
Definition 5. The set E c T is called A-measurable if the set
E = E U |J (t,a(t))
teR(E)
is measurable in the sense of Lebesgue.
Definition 6. A function f : T — R1 is called A-measurable on a A-measurable set E if a function f of the form
f (t), t € E,
(t) =
is measurable on the set E.
f(t) I f (ti), t € (ti,a(ti)), tt € R(E)
Definition 7. A function f : E ^ R1, E c T is called A-integrable on a A-measurable set E if the function f is integrable in the sense of Lebesgue on the set E. If f is A-integrable on the set E, then we define JE f (s)As, assuming
IE
where ^ is the Lebesgue measure.
Í f (s)As = / fd/j,,
JE JE
3. Formulation of the problem
Let a time scale T, t0 € T be given.
In the space Rk (k ^ 2) we consider the differential game r(n, 2) of n + 2 persons: n pursuers Pi,..., Pn and two evaders Ei, E2 with laws of motion of the form
xf = Ui, Xj(i0) = x0, Ui € V, (3.1)
yf = v, yj(to) = y0, v € V. (3.2)
Here xi,yj ,x0,yo,Ui ,v € Rk, i € I = {1,...,n}, j € J = {1, 2}, V = {v € Rk | ||vM < 1}. We assume that x0 = yj0 for all i € I, j € J. Additionally, we assume that in the process of the game evaders E1 and E2 do not move out of a convex set D with a nonempty interior of the form
D = {y € Rk | (pi,y) < l = 1,...,r},
where pi,... ,pr are unit vectors Rk, ..., are real numbers, and (a, b) is a scalar product. We also assume that D = Rk with r = 0.
Introduce new variables Zj = xi — yj. Then instead of the systems (3.1) and (3.2) we obtain the system
zf = Ui — v, Zij(t0) = z0j = x0 — y0, Ui, v € V. (3.3)
We will say the A-measurable function v : T ^ Rk is A-admissible if v(t) € V, yj(t) € D for all t € T, j € J. Here yj(t) is a solution to the Cauchy problem (3.2) with a given function v(-).
We will say that the prehistory vt(-) of the function v at time t € T is a restriction of the function v to [t0, t) n T. Let
z0 = {z0j | i € I, j € J}
denote the vector of initial positions.
The actions of the evaders can be interpreted as follows: there is a center which for all evaders E1 and E2 chooses the same control v(t).
Definition 8. We will say that a quasi-strategy Ui of pursuer Pi is given if a map Up is defined which associates the A-measurable function Ui(t) = Ui(z0, t, vt(-)) with values in V to the initial positions z0, time t € T and an arbitrary prehistory of the control vt(-) of evaders Ei and E2.
Definition 9. A two-fold capture occurs in the game r(n, 2) if there exist time T0 = T(z0) and quasi-strategies Ui,..., Un of pursuers P^ ..., Pn such that, for any measurable function v(-), v(t) € V, y(t) € D, t € [t0,T0] n T, there are numbers l,m € I, (m = l), j € {1,2} and times t1, t2 € [t0,T0] n T such that zj(t1) = 0, zmj(t2) = 0.
4. Sufficient conditions for capture
Definition 10 [12]. The vectors ai,a2,... ,am form a positive basis in Rk if for any x € Rk there exist nonnegative real numbers a1, a2,..., am such that
x = aiai + a2a2 + ... + am am.
Let Int X, co X denote, respectively, the interior and the convex hull of the set X c Rk.
Theorem 1 [12]. The vectors a1, a2,..., am form a positive basis in Rk if and only if
0 € Intco {a1,..., am}.
Lemma 1. Let m ^ 3, a1,..., am, b1, b2, p1,... ,pr € Rk be such that
1) for each q € J0 = {1,..., m - 2}
0 € Intco{a» - b1, aj - 62, i € J0 \ {q},p1,...,pr},
2) am-1 - 62 = t1(&1 - 62), am - 62 = t2(61 - 62) for some t1 < 0, t2 < 0. Then for each l € J = {1,..., m} the following inclusion holds:
0 € Intco{a» - 62, i € J \ {l},P1,... ,Pr}. (4.1)
Proof. If m = 3, then it follows from condition 1) of the lemma that
0 € Intco{p1,...,pr}.
Therefore, the condition (4.1) is satisfied automatically. Let m ^ 4. Assume that there exists q € J for which
0 / Intco{a» - 62, i € J \ {q},p1,...,pr}.
Then, by the separability theorem, there exists a unit vector x € Rk such that
(a» - 62, x) ^ 0 for all i € J \ {q}, (pj, x) ^ 0, for all j = 1,..., r. (4.2)
It follows from condition 2) of the lemma that (61 - 62,x) ^ 0. Then
(a» - 61,x) = (ai - 62,x) + (62 - 61,x) ^ 0 for all i € J \ {q}. (4.3)
Inequalities (4.2) and (4.3) contradict condition 1) of the lemma. This proves the lemma. □
Let us introduce the following notation:
A(h,v)=sup{A ^ 0 | - Ah € V - v},
K(t) = [ As, Q(J) = {(i1 ,i2)|i1,i2 € J, i1 = i2},
J to
where J is a finite set of natural numbers.
Lemma 2. Let m ^ 4, a1,..., am-2, c, p1 € Rk be such that for each q € J0 = {1,..., m - 2} the vectors {a»,i € J0 \ {q},c,p1} form a positive basis Rk. Then for any 61,62 € Rk there exists p0 > 0 such that for all p > p0 the following inequality holds:
¿(p)=minmax{ max min A(w,-,v), (pi,v)j> 0,
vev Aen°(J) »eA
where J = {1,..., m}, Q0(J) = Q(J0) U {(m - 1, m)},
M, i € J0,
w» = ^ 61 + pc, i = m - 1, 62 + pc, i = m.
Proof. Assume that the statement of the lemma is false. Then there exist 61,62 € Rk such that for any p° > 0 there is p > po for which ¿(p) = 0. It follows from the definition of ¿(p) that there exists vp € V such that (p1,vp) ^ 0 and for all A € Q°(J)
min A(w^vp) = 0, with ||vp|| = 1.
isA
From the last condition it follows that there exist J(p) C J°, | J(p)| = m — 3 and j(p) € {m — 1, m} such that X(wi ,vp) = 0 for all i € J (p) U {j(p)}.
Let p° = 1. Then there are p1 > p°, v1 € V, J(p1) for which
(p1,v1) ^ 0, (wi,v1) ^ 0 for all i € J(p1) U {j(p1)}, and For p0 = p1 + 1 there are p2 > p0, v2 € V, J(p2) for which
(P1,V2) ^ 0, (wj, V2) ^ 0 for all i € J(P2) U {j(p2)}, and Continuing this process further, we find that there exist sequences {ps}^=1,
lim ps =
s—
{vs}, {J(ps)}, {j(ps)} for which
(p1,Vs) ^ 0, (wj ,Vs) ^ 0 for all i € J (ps) U {j(ps)}, and
= 1.
= 1.
Ilv.,11 = 1.
Consequently, there exists a subsequence {psi}, lim psi = for which there are a subsequence
i—
{vsi}, a set J0, J0 c J0, | J01 = m - 3, and an index j € {m - 1, m} such that for all j the following inequalities hold:
) ^ 0, ) ^ 0 for all i € J° U {j}, and
= 1.
From the sequence one can single out a subsequence {v/} converging to vo, with Therefore, we have
= 1.
(Pi,vi) < 0, (wî,vi) < 0 for all i € J°,
w:
Pl
+ c,vi K 0
Passing in the last inequalities to the limit as l ^ we obtain
(P1,v°) < 0, (wi,v°) < 0 for all i € J°, (c,v°) < 0.
Therefore, by virtue of Theorem 1 the set of vectors {wi,i € J°,c,p1} does not form the positive basis Rk, which contradicts the condition of the lemma. This proves the lemma. □
v
.
j
Lemma 3. Let a1,..., am,p1 € Rk be such that
ö = min max{ max min A(a,-, v), (p1, v)j > 0,
vev Aen°(J) je A
where Jo = {1,..., m - 2}, Q°(J) = Q(J°) U {m - 1, m}.
Then there exists T° > t°, T° € T such that for any admissible control v(-) of evaders there is A = (a,ß) € Q°(J) such that
T° T°
J A(aa,v(s))As ^ 1, J A(aß,v(s))As ^ 1.
^a )
t° t°
Proof. Let v(-) be an admissible control of evaders. From [5] it follows that the functions A(aj, v(t)) are A-measurable and A-integrable. For each t € T we define the sets
T(t) = {t € T | (pi,v(t)) ^ T2(t) = {t € T | (pi,v(t)) < 5}.
Since (yj(t),p1) ^ for all t € T, j = 1, 2, the following inequality holds:
t
J(pi,v(s))As ^ ^q = min{^i - (pi,y°),^i - (pi,y0)}.
t°
Therefore,
t
^o ^y(pi,v(s))As ^ 5 J As - J As, K(t) = J As + J As.
ÎQ Ti(t) T2(t) Ti(t) T2(t)
The last two relations imply the validity of the inequality
i+¿ v 7
t t
max min / A(a7, v(s))As ^ max / min A(a7, v(s))As. (4.5)
Aen°(J) jeA^ v Aen°(J)7 jeA J)
to to
T2(t)
Next, we have
t
For any nonnegative numbers ya(A € QU(J)) we have
1 v^ , (m - 2)(m - 3)
max 7a ^ — > 7a, where N0 = H----.
Aen°(J) No aÄJ) 2
Therefore,
t t max I mm A(a,j, v(s.....
f min A(a,7, v(s))As ^ —t- f V^ minA(a,7-,v(s))As
J jeA V " N0 J jzA
to to Aen°(J)
t
^ —- / max min A(cu,v(s))As ^ —- / max min A(a,7, v(s))As. N0 7 Aen°(J) jeA v j' No J Aen°(J) jeA v j'
to T2(t)
Hence, from (4.4) and (4.5) we obtain
t
max min / A(a,7, v(s))As ^ —- / i-en°(J) jeA J y 37 y " No J
5 f As> S 6K(t) - ß0
Since
Aen°(J) jeAj w w/ No J No 1 + ó
to T2(t)
lim K (t) =
it follows from the last inequality that there exists T0 € T for which the following inequality holds:
T°
A( j
max min / A(a,-,v(s))As ^ 1, Aen°(J) jeA J j t°
which implies the validity of the statement of the lemma. This proves the lemma. □
Lemma 4 [17]. Let a1,...,am,p1 € Rk be such that for each q € J = {1,..., m} the vectors {a^, i € J \ {q},p1} form a positive basis Rk. Then
ö = min maxi max min A(a^, v), (p1 , v)) > 0. vev Aen(J) ieA J
Lemma 5 [17]. Let a1,... am,p1 € Rk be such that for each q € J = {1,..., m} the vectors {a^, i € J \ {q},p1} form a positive basis Rk. Then there exists T° > t°, T° € T such that for any admissible control v(-) of evaders there is A = (a, ß) € ^(J),
T° T°
J A(aa, v(s))As ^ 1, J A(aß,v(s))As ^ 1.
t° t°
Theorem 2. Let r = 1 and suppose that there exists j € {1,2} such that for any q € /
0 € Intco{ zjj ,i € / \ {q},p^. Then a two-fold capture occurs in the game r(n, 2).
Proof. By virtue of Lemma 5
t
T° = min !t € T | t > t°, inf max min / A(z°-,v(s))As ^ 1
I 1 v(-) AeH(J) ieA y ij
t°
is finite. Let v(-) be an admissible control of evaders. Define the functions
hi(t) = 1 - f A(z°j, v(s))As.
t°
t°
Let pursuer Pj construct a control as follows. If the inequality hj(t) ^ 0 is satisfied at time t € T, then we assume
Uj(t)= v(t) - A(zjj,v(t))zjj.
If t € T is the first time instant for which hj(r) = 0, we assume that A(z0j,v(t)) = 0 for all t ^ t.
Let t € T be the first time instant for which hj(T) < 0, and let the inequality hj (t) > 0 be satisfied for all t € T, t < t. Define the number
t* = sup{t € T | hi(t) > 0}.
Then (t*,t) n T = 0. Indeed, if there existed a time instant t € (t*,t) n T, then the inequality h*(t) > 0 would be satisfied, which is impossible by virtue of the definition of the number t* . In this case, we assume
zH(t) = v(t) - A*(z° ,V(r))z° , where A*(z° ,V(r)) = ^to) . =
^(Ti ) Ti t Ti
We note that in this case A*(z0j,v(t)) ^ A(zij,v(t)) and therefore u(t) € V. Then
T*
1 - r X(z^,v(s))As- r \*(z°pv(s))As = ht(T*)- i MillAs = 0_ J to Jt* Jt* T Ti
Then from the definition of the controls of the pursuers and the system (3.3) it follows that for all t € [t0, T0] n T the equalities Zj(t) = zjhi(t), i € I, hold.
From Lemma 5 and the definition of the controls of the pursuers it follows that there exist numbers l, m € I such that h^(T0) = 0, hm(T0) = 0. This implies that pursuers p and Pm perform a capture of evader Ej. Consequently, a two-fold capture occurs in the game r(n, 2). This proves the theorem. □
Theorem 3. Let r = 1 and suppose that there exists a set I0 C I, |I0| = n — 2 such that for all l € I0
0 € Intco {z0i, Z02, i € I0 \ {l},pi}. (4.6) Then a two-fold capture occurs in the game r(n, 2).
Proof. By virtue of Theorem 1, it follows from condition (4.6) that for all l € I0 the set 1,
{z° , z02, i € i0 \ |1},Pi} forms a positive basis Rk. Denote c = y0 — y0. Since
z°2 = x0 - y0 = x0 - y0 + c = z° + c,
for all l € I0 the positive basis Rk forms a set {z01, i € I0 \ |l},c,pi}.
We assume that I0 = {1,..., n — 2}. It follows from Lemma 2 that there exists a number p > 0 such that for all l € I the vectors {w0, i € I \ {l},p1} form a positive basis Rk, where
z01, if i € I0,
wi = <( z°-i2 + pc, if i = n — 1, z^2 + pc, if i = n.
Hence, by virtue of Theorem 1, we find that for all l € I
0 € Intco {w0, i € I \{l},p^.
It follows from Lemmas 2 and 3 that the number
T0 = min111 > t0, t € T, inf max min / A(w0, v(s))As ^ 1 I 1 v(.)Aen°(i) jeA Jto j
is finite. Let v(-) be an admissible control of the evaders. Define the functions
t
hi(t) = 1 — / A(w0, v(s))As.
Jto
Let pursuer Pj construct a control as follows. If the inequality hj(t) ^ 0 is satisfied at time t € T, then we assume
Uj(t) = v(t) - A(w0,v(t))w0.
If t € T is the first time instant for which hj(t) = 0, then we assume that A(w0,v(t)) = 0 for all
t is the uisi time instant ioi which hj(t ) — 0, then we assume mat A(w0 t ^ T.
Let t € T be the first time instant for which hj(t) < 0, and let the inequality hj(t) > 0 be satisfied for all t € T, t < t. Define the number
t* = sup {t € T | hi(t) > 0}.
Then (t*,t) n T = 0. Indeed, if there existed a time instant t € (t*,t) n T, then the inequality hj(t) > 0 would be satisfied, which is impossible by virtue of the definition of the number Tj*. In this case, we assume
i(T) = v(r) - A*(w°,v(t))w°, where A*K°,t'(r)) = , =
CT(Ti ) — Ti t — Ti
We note that in this case A *(w°,v(t)) ^ A(w0,v(t)) and therefore uj(T) € V. Then
1- r \(ulv(s))As- fT \*(w®,v(s))As = hi(r*) — T = 0.
Jto A* 7r; t — Ti
Then from the definition of the controls of the pursuers and the system (3.3) it follows that for all t € [t°, T] n T the following equalities hold:
Zi1(t) = Zi°1hi(t), i € 1°,
zn-12(t) = z^-12hn-1(t) — pc(1 — hn-1(t)), Zn2 (t) = z^2hra(t) — pc(1 — hn(t)).
From Lemma 3 and the definition of the controls of the pursuers it follows that there exist numbers l, m € 1 such that
hi(T°) = 0, hm (T°) = 0. (4.7)
Also, the following cases are possible.
1. l, m € 1°. In this case, pursuers p and Pm perform a capture of evader E1, which implies that a two-fold capture occurs in the game r(n, 2).
2. Condition (4.7) is satisfied for A = {n — 1,n}. Then
Zn—12 (T°) = —pc, Zn2 (T° ) = —pc. (4.8)
We prove that in this case the following inclusion holds for any l € 1°:
0 € Intco{Zi1(T°),Zi2(T°),i € 1° \ {l},P1>. (4.9)
Let l € 1°. We have
Zi1(T°) = 4hi(T°), Zi2(T°) = Zi1(T°) + c = Zi1(T°)hi(T°) + z°2 — 4.
Therefore,
4 = TT£t, 4 = -¿2 (T0) +
0 _ ^l(To) _ _ ^l(To)(l-/7,(To))
Since the set {z01, z02, i € /0 \ {1},p1} forms a positive basis Rk, the positive basis Rk is formed by the vectors
W ^(To) +-Wo)-' 0 V { },Pl
From the condition hj(T0) € (0,1], for all i € /0 we find that the positive basis Rk forms a set
{zj1 (T0), Zj2(T0),i € /0 \{1},P1}.
By virtue of Theorem 1, the last relation implies the validity of (4.9). From equations (4.8) we obtain
Zn-12(T0) = -p(y1(T0) - y2(T0)), Zn2(T0) = -p(y1(T) - ^(T)).
By virtue of Lemma 1, we find that
0 € Intco{zj2(T0), i € /0 \ {1},P1}.
Taking T0 to be the initial time and using Theorem 2, we find that there are pursuers Pr and Pq, r = q, that perform a capture of evader E2. This proves the theorem. □
Example 1. Let k = 2, x0 = (3;1), X2 = (1; -2), x] = (5; -2), x4 = (1;3), x0 = (2; -3),
y0 = (0;0), y0 = (6;0), P1 = (0; 1), P1 = 100.
Then the condition for capture from Theorem 2 is not satisfied, and the condition for capture from Theorem 3 is satisfied for /0 = {1, 2, 3}.
Theorem 4. Let D = Rk and suppose that there exists j € {1,2} such that for any q € /
0 € Intco{z0j,i € / \ {q}}.
Then a two-fold capture occurs in the game r(n, 2).
This theorem is proved along the same lines as Theorem 2 using the results of [16].
Theorem 5. Let D = Rk and suppose that there exists a set /0 c /, |/0| = n - 2 such that for all l € /0
0 € Intco {zj01, zt02, i € /0 \ {l}}.
Then a two-fold capture occurs in the game r(n, 2).
This theorem is proved along the same lines as Theorem 3 using the results of [16].
Theorem 6. Let r > 1 and suppose that there exist p € Rk, p € R1, /0 c /, |/0| = n - 2 such that D c {x € Rk | (p, x) ^ p} and
0 € Intco {z01, z02, i € /0 \ {1},p}.
Then a two-fold capture occurs in the game r(n, 2).
The validity of this theorem immediately follows from Theorem 3.
5. Conclusion
In the problem of a simple pursuit by a group of pursuers of two coordinated evaders on a given time scale, we obtained sufficient conditions for a two-fold capture, provided that the evaders didn't move out of a convex polyhedral set. To solve the problem, we used the method of resolving functions. The results obtained can be used in the study of new problems of conflict interaction between groups of pursuers and evaders on time scales.
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