THE DIFFERENTIAL GAME “COSSACKS–ROBBERS” ON TIME SCALES Текст научной статьи по специальности «Математика»

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta

2023. Volume 62. Pp. 56-70

MSC2020: 49N79, 49N70, 91A24 © E. S. Mozhegova, N. N. Petrov

THE DIFFERENTIAL GAME "COSSACKS-ROBBERS" ON TIME SCALES

In finite-dimensional Euclidean space, we address the problem of simple pursuit of a group of evaders by a group of pursuers in a given time scale with equal opportunities for all participants. The set of controls of each participant is a sphere of unit radius with its center at the origin. The goal of the group of pursuers is to catch all evaders. The goal sets are the origin. The goal of the evaders is the opposite one, namely, for at least one evader to avoid capture. Conditions for solvability of the local and global problems of evasion and the upper and lower estimates of the minimal number of evaders avoiding a given number of pursuers from any initial positions are obtained.

Keywords: differential game, group pursuit, pursuer, evader, evasion problem, time scale. DOI: 10.35634/2226-3594-2023-62-05

Introduction

The work of R. Isaacs [1] laid a foundation for the theory of two-player pursuit-evasion differential games, which has grown to be a profound and insightful theory in which various approaches to analysis of conflict situations [2-8] are proposed. A natural generalization is the situation of conflict interaction of a group of pursuers and a group of evaders, in which the goal of the group of pursuers is to catch a given number of evaders and the goal of the group of evaders is the opposite one.

Reference [9] addressed the problem of simple pursuit of a group of evaders by a group of pursuers with equal opportunities for all participants. The goal of the pursuers was to catch all evaders, and the goal of the evaders was the opposite one. Sufficient conditions for the solvability of local and global evasion problems were obtained and the upper and lower estimates of the minimal number of evaders evading a given number of pursuers from any initial positions were made. An improved variant of the lower estimate was presented in [10]. A generalization of the results of [9] to linear differential games with a constant matrix and a nonstationary matrix was provided in [11] and [12], respectively. In [13] a proof was given of the existence of a price of the game in nonlinear differential games with many participants in a finite time interval and with payoff functions of special form.

Sufficient conditions for the capture of all evaders by a group of pursuers in the nonstationary differential game of simple pursuit in a convex compact set with integral restrictions on the controls of the players were obtained in [14]. In [15], the problem of a group of evaders avoiding a group of pursuers was considered under the assumption that the motion of all participants is simple and that integral restrictions are imposed on the control. It is shown that, if the total energy of the pursuers is smaller than or equal to the total energy of the evaders, an evasion from an encounter occurs.

Reference [16] is concerned with the differential reach-avoid game between two opposing teams in a convex domain consisting of a target domain and a playing zone. The evasion team, which is initially located in the playing zone, strives to send as many team members as possible to the target region, while the pursuit team whose members are initially distributed both over the playing region and over the target region, strives to prevent that by capturing the evaders. The problem under investigation is that of assigning specific pursuers to chase the evaders in such a

way that the number of evaders that can be caught before they reach safely the target region is maximized.

Reference [17] treats the problem of pursuit of a group of evaders by a group of pursuers in a probabilistic setting. For each evader, a probability matrix is introduced which estimates the probability of a specific evader being in a particular position. These probabilities are used to construct a forecast of the location of the evaders. The pursuers coordinate their actions and try to decrease the probability matrix.

Reference [18] considers the conflict interaction of a group of pursuers and a group of evaders. It is shown that, if in this conflict the evasion of at least one evader occurs in an infinite time interval, then, with "weak" pursuers being added, the evasion from an encounter will occur in any finite time interval.

Reference [19] is concerned with the dynamical game for two teams: a lady and body-guards against gangsters. The goal of the gangsters is to capture the lady, and the goal of the lady and the body-guards is to prevent that from happening. The body-guards try to intercept the gangsters before they come into immediate proximity to the lady. An approach to solution for a linear system and a quadratic criterion is demonstrated.

Reference [20] addresses the problem of pursuit of several evaders by several pursuers in a convex compact set on a plane. Using a Voronoi partitioning, conditions for the capture of all evaders are obtained.

Reference [21] develops a strategy for cooperation of several unmanned surface ships for pursuing evaders in the presence of a dynamical obstacle ship. To solve the pursuit problem, the pursuers are divided into a pursuit group and an ambush group. The pursuit group drives the evaders into the ambush region and, together with the ambush group, completes encircling the evaders.

Reference [22] is concerned with the differential game of pursuit of a group of evaders by a group of pursuers in three-dimensional space under dynamical perturbations of the environment. It proposes a method for distributing the pursuers into groups each of which catches their own evader. Conditions for solvability of the pursuit problem are given in terms of sets of attainability of the participants.

References [23-25] examine the problem of the capture of a given number of evaders in recurrent differential games, games with fractional derivatives and games on a given time scale under the condition that the evaders use program strategies and that each pursuer catches no more than one evader. Sufficient, and in some cases necessary, conditions for solvability of the pursuit problem are obtained.

Sufficient conditions for the capture of a given number of evaders by a group of pursuers are obtained in [26], implying that each evader is to be caught by a given number of pursuers.

In this paper, we address the problem of conflict interaction of a group of pursuers and a group of evaders in a differential game on a given time scale with simple motion and equal opportunities for all participants. We obtain sufficient conditions for solvability of the local and global evasion problems, and the upper and lower estimates for the minimal number of evaders evading a given number of pursuers from any initial positions.

§ 1. Auxiliary definitions and facts

In this section, we recall some basic facts from the theory of time scales. All results presented below can be found, for example, in [27,28].

Definition 1.1. A nonempty closed subset T c R1 such that supt = is called a

teT

time scale.

Definition 1.2. Let T be a time scale. A function a: T ^ R1 of the form

a(t) = inf{s e T | s > t}

is called a translation function.

Definition 1.3. A function f: T ^ R1 is said to be A-differentiable at a point t e T if there exists a number y e R1 such that for any e > 0 there exists a neighborhood W of the point t such that the inequality

|f (a(t)) - f (s) - y(a(t) - s)| < e|a(t) - s|

holds for all s e T H W. In this case, the number y is said to be the A-derivative of the function f at the point t. The A-derivative of the function f at the point t will be denoted as f A(t) = yD e f i n i t i o n 1.4. A function f: T ^ Rn, f (t) = (f (t),..., fn(t)) is said to be A-diffe-rentiable at a point t e T if all functions f 1,..., fn are A-differentiable at the point t.

Let T be a time scale, E c T. Denote R(E) = {t e E | a(t) > t}. Then the set R(E) is no more than countable.

D e f i n i t i o n 1.5. A set E c T is said to be A-measurable if the set

E = E U U (t,a(t))

teR(E)

is measurable in the sense of Lebesgue.

Definition 1.6. A function f: T ^ R1 is said to be A-measurable on the A-measurable set E if the function f of the form

№) = if(t)' ' e E

\/(ti), t e (ti,a(ii)), t e R(E),

is measurable on the set E.

Definition 1.7. A Unction f: E ^ R1, E c T is called to be A-integrable on the 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 / f (s)As, assuming

E

i f(s)As = i fd^, Je JE

where ^ is the Lebesgue measure.

§ 2. Formulation of the problem

Suppose we are given a time scale T, t0 e T.

In the space Rk (k ^ 2) we consider a differential game involving n + m players: n pursuers P1;..., Pn and m evaders E1,..., Em. The motion of the players is governed by the laws

xA = Ui, Xi(t0) = x0, Ui e V, (2.1)

yA = Vj, yj(to)= Vj e V, (2.2)

where Xj, yj, x0, y0, Ui, Vj G Rk, i G I = {1,..., n}, j G J = {1,..., m}, V = {v G Rk: ||v|| ^ 1}. Assume that x0 = y0 for all i G I, j G J.

The goal of the group of pursuers is to catch all evaders. The goal of the group of evaders is to prevent this, i. e., to allow at least one of the evaders to evade an encounter.

Let z0 = (x0,..., x^, y0,..., ym). Denote this game by r(n, m, z0). Let a be a partitioning of the interval [t0, that has no accumulation endpoints. Let a = {t0, ti,...}, with t G T for all l. Denote Ts = [ts, ts+1) n T, Ts = [t0, ts) n T, where {ts} are the elements of partitioning a.

Definition 2.1. A piecewise-program strategy Vj of an evader Ej, which corresponds to the partitioning a, is a family of maps bj, l = 0,1,..., j G J, that associate to the quantities

(ti,xj(ti), i G I,ys(t),s G J, minmin ||xj(t) - ys(t)|| ) (2.3)

V teT1 i /

a A-measurable Unction vj (t) defined for t G Tl and such that vj (t) G V for all t G Tl.

Definition 2.2. A piecewise-program counterstrategy U of a pursuer Pi, which corresponds to the partitioning a, is a family of maps cj, l = 0,1,..., j G J, that associate to the quantities (2.3) and to controls vj(t), j G J, t G Tl, a A-measurable function uj(t) defined for t G Ti and such that uj (t) G V for all t G Ti.

Definition 2.3. In the game r(n, m, z0) an evasion from an encounter occurs if there exist a partitioning a and piecewise-program strategies V1,..., Vm of evaders E1,..., Em such that for any trajectories x1 (t),..., xn(t) of pursuers P1,..., Pn there exists a number p G J such that

yp(t) = xi(t) Vi G I, t G T, where yp(t) is the trajectory of the evader Ep that takes place in this situation.

Definition 2.4. A capture occurs in the game r(n, m, z0) if there exists T > t0, T G T such that for any partitioning a and any strategies V1,..., Vm of evaders E1,..., Em there exist piecewise-program counterstrategies U1,..., Un of pursuers P1,..., Pn corresponding to the partitioning a such that there exist time instants t1 ,..., Tm G [t0, T) n T and numbers s1,..., sm G I for which the following equations hold:

yj (Tj) = xSj (Tj), j G J,

where xsj (t), Sj G I, yj(t), j G J, are the trajectories of players Psj, Sj G I, Ej, j G J, that take place in this situation.

§ 3. The local evasion problem

In this section, we present some sufficient conditions for an evasion from an encounter in the game r(n, m, z0).

For a hyperplane H, we denote by H + and H- closed half-spaces defined by this hyperplane.

L e m m a 3.1. Suppose there exists a hyperplane H such that

a) x0 G H- for all i G I;

b) y0 G H+.

Then an evasion from an encounter occurs in the game r(n, m, z0).

Proof. Let q be the unit vector of the normal of the hyperplane H which is directed to H +, and let ui(t), i G I, be arbitrary controls of the pursuers. Define the control of the evader E1,

assuming v1 (t) = q for all t G T. Define the controls of the other evaders arbitrarily. From the systems (2.1) and (2.2), we obtain

yi(t) = y? + K(t)q, xi(t) = x? + i Ui(s)As, where K(t) = i As.

./to ./to

Define the functions

1 /,t = ^t) I u'(s)As-

Then ||ui(t)| ^ 1 for all t G T. Next, we have (z? = x? - y?),

= + ^ lk° " K(t)q\\ - K(t) =

= y/\№\\*-2K(t)W,q) + K*(t) - Kit) > 0,

since (z?, q) ^ 0 for all i G I. This proves the lemma. □

Corollary3.1. Let H (t) = H +(t-t0)q. Then for any controls ui(t), i G I, of pursuers Pi, i G I, and for all t G T the inclusion xi (t) G H -(t) holds, where H -(t) is a closed half-space defined by the hyperplane H (t) such that H- C H- (t).

Proof. Indeed, for all t G T and all i G I the following inequality holds:

(Xi(t) - y(t), q) = (z?,q) + K(t) [(Ui(t), q) - 1] ^ (z?, q) ^ 0. We note that y(t) G H(t) for all t G T. This proves the corollary. □

L e m m a 3.2. Suppose there exist hyperplanes H1, H2 and a number l G I such that

a) Hi||H2, H2+ C H+;

b) x? G H2+, x? G H- for all i G I, i = l;

c) y? G Hi, y? G H2;

d) the projections of all points x?, i G I, y?, y? onto the hyperplane H1 are pairwise different. Then an evasion from an encounter occurs in the game r(n, m, z?).

Proof. Let q be the unit vector of the normal of the hyperplane H1 directed to H+. Suppose that w1 (t) = y? + (t — t?)q, w2(t) = y? — (t — t?)q, d1 (t) is the distance from the point w1 (t) to the hyperplane H1, d2 (t) is the distance from the point w2(t) to the hyperplane H1. We note that d1(t?) = 0, d2(t?) > 0, d1 is an increasing function, and d2 is a decreasing function. Two cases are possible.

1. There exists t G T for which d1(r) = d2(r). Then we define the controls of evaders E1 and E2 as follows. Assume that v1 (t) = q for all t G T.

v (t) i-q, t g [t?,t] n t,

V2 ( ) \ q, t G (t, n T.

Let us define the controls of the other evaders arbitrarily. We prove that an evasion from an encounter occurs in the game r(n,m, z?). It follows from Lemma 3.1 that xi(t) = y1(t) for all i = l, t G T. In addition, by virtue of Lemma 3.1 and Corollary 3.1, x^(t) = y1(t), x^(t) = y2(t) for all t G TT .It follows from Corollary 3.1 that xi (t) = y2(t) for all t G T, i = l. We show that on the set (t, n T the pursuer Pi catches no more than one of the evaders E1 and E2. For all t ^ t we have y1(t), y2(t) G H1(t) = H + (t - t?)q. Therefore, if a capture of E1(E2) by the pursuer Pi occurs at the time instant T1, then, at the time instant T1, the conditions of Lemma 3.1

will be satisfied for the evader E2(£1 ). Thus, we have proved that an evasion from an encounter occurs in the game r(n, m, z0).

2. For all t G T the following inequality holds: di(t) = d2(t). Then there exist n, t2 G T, t1 < t2 for which (ti,t2) n T = 0, d1(r^1) < d2(r^1), d1(r2) > d2(r2). Let us define the controls of the evaders E1 and E2 as follows. Assume that v1(t) = q for all t G T.

If d1(r2) > d2(T1), then we assume

= q(di{r2) ~ d2{r0)

T2 - T1

q,

t G [to ,T1) n T,

t = T1,

t G [t2, n T.

If d2(T1) > d1(r2), then we assume

V2 (t) =

q,

-^d2(n) - d1 (t2))

T2 - T1

t G [to,T1) n T,

t = T1,

t G [t2 , n T.

Let us define the controls of the other evaders arbitrarily. We note that, if d1(r2) > d2(T1), then

Q < di(r2) - d2(ri) < di(r2) - dijn) _ ^

T2 - T1

T2 - T1

Similarly, if d2 (t1) > d1(r2), then

Q < d2(Ti) - di(r2) < d2(ri) - d2{r2)

T2 - T1

T2 - T1

1.

Therefore, the function v2(-) satisfies the condition ||v2(t) || ^ 1 for all t G T. We prove that in the game r(n, m, z0) an evasion occurs in this case as well. By virtue of Corollary 3.1, pursuers Pi, i G I, i = l, catch none of the evaders E1 and E2. On the set TT1, the pursuer Pi catches none of the evaders E1 and E2. If the pursuer Pi performs a capture of one of the evaders E1 and E2 at the time instant t ^ t1, then at the time instant t the pursuer Pi will be on the hyperplane H(t) and hence, by virtue of Lemma 3.1, the pursuer Pi will not be able to perform a capture of the second evader. Thus, an evasion from an encounter occurs in the game r(n, m, z0). This proves the lemma.

Corollary 3.2. Suppose that in the game r(n, m, z0) there exist hyperplanes H1; H2 and sets I0 C I, J0 C J such that

a) H1IIH2, H+ C H+;

b) | J0| ^ |I0| + 1, where | J | denotes the number of elements of the set J;

c) x0 G H+, i G I0, x0 G Hf, i / I0;

d) y0 G H+ n Hf, j G J0;

e) the projections of all points x0, i G I, j j G J0, onto the hyperplane H1 are pairwise different.

Then an evasion from an encounter occurs in the game r(n, m, z0).

Proof. Assume that J0 = {1,...,l}. Let q be the unit vector of the normal of the hyperplane H1 which is directed to H+. Let be one of the points j j G J0, that is nearest to H1. Suppose that w1(t) = y0 + (t -10 )q, wj (t) = y0 - (t - t0)q, j G J0, j = 1, and let dj (t) be

q

the distance from the point wj(t) to the hyperplane Hi. Define the control v1 of the evader Hi, assuming v1(t) = q for all t G T. Define the controls of evaders Ej, j G J0, j = 1, as follows. 1. If there exists a time instant Tj G T for which d1(r?) = d1(r?), then we assume

V (t)

9, 9,

t G [io,Tj] n T, t G (Tj, +œ) n T.

2. Suppose that the inequality dj (t) = d1(t) holds for all t G T. Then there exist Tj, T2j G T,

Tij < T2j, for which (Ty , T2j) n T = 0, d1(Ty) < dj(Tj), d1(T2j) > dj(T2j). Then, if d1(T2j) > dj (Ty-), we assume

9,

vj(t) =

q(di(T2j) - dj (Tij))

T2j - Tij

If dj (ry ) > d1(T2j), then we assume

t G [to ,Tij) n T, t = Tij >

t G [T2j, +œ) n T.

Vj (t)

-g(dj(Ty) -di(T2j)) T2j - Tij 9,

t G [to, Tij) n T, t = Tij >

t G [T2j, +œ) n T.

Define the controls of the other evaders, Ej and j / J0, in an arbitrary way. By virtue of Lemma 3.2, the function Vj(■) satisfies the condition ||vj(t)|| ^ 1 for all t G T. By virtue of Lemma 3.2, pursuers Pj, j / 10, catch none of the evaders Ej, j G J0. Each of the pursuers Pj, j G 10, can catch no more than one of the evaders Ej, j G J0. Therefore, by virtue of condition b) of the corollary, at least one of the evaders Ej, j G J0, will avoid a capture. Consequently, an evasion from an encounter occurs in the game r(n, m, z0). This proves the corollary.

Remark 3.1. The strategies of the evaders constructed in proving Corollary 3.2, which guarantee an evasion from an encounter, possess the following property. Let H be a hyperplane that is parallel to the hyperplane H1 and passes through point y(°, where is one of the points j G J0, that is nearest to H1, and let H(t) = H + (t — t0)q. Then the evader E1 is on the hyperplane H(t) at each time instant t G T. The motion of each of the other evaders Ej, j G J0, j = 1, consists of two stages. In the first stage, each of evaders Ej, j = 1, moves along the normal —q to the hyperplane H so as to be on the hyperplane H (t) at some time instant t. In the second stage, evader Ej, j G J0, j = 1, moves along the normal q to the hyperplane H and is thus located on the hyperplane H(t) for all t > t.

§ 4. The global evasion problem

Theorem 4.1. For any natural number p and any natural number m ^ p ■ 2p + 2, an evasion from an encounter occurs in the game r(2p + 1, m, z°).

Pro o f. Let n = 2p + 1, I sitions of the pursuers and let that all points ,

oo

yi >

yO,.

,y0

' k>m

{1,..., n}, J = {1,..., m}, x0,..., xn be the initial po..., ym be the initial positions of the evaders. Assume are pairwise different. Let q be a unit vector such that

(q, x^ — x"g) = 0 for all a = ft, (q, x0 — y0) = 0 for all i, j, and (q, y0 — y0) = 0 for all r = s.

Suppose that Hi

Hn are hyperplanes with the normal q for which xO G H for all i G I,

and Hi , H+ are closed half-spaces defined by the hyperplane H, i G I, with H+ c Hi+1 for all i = 1,..., n — 1. Assume that q is directed to H+.

q

If at least one of the points y0 belongs to H+ U Hf, then, by virtue of Lemma 3.1, an evasion from an encounter occurs in the game r(n,m, z0). Further, let yj0 G Hf n H+ for all j. The theorem will be proved by the method of mathematical induction with respect to p.

1. p =1. Define the sets J1 = {j | y0 G H2- n H3+}, J2 = {j | y0 G Hf n H2+}. Construct auxiliary controls Vj(t) for evaders Ej, j G J. For the group of evaders Ej, j G J\, Vj(t) are controls constructed in accordance with Corollary 3.2 with respect to the hyperplane H3 and the normal vector q directed to III. For the group of evaders Ej, j G J2, Vj(t) are controls constructed in accordance with Corollary 3.2 with respect to the hyperplane H1 and the normal vector -q directed to Hf.

Suppose that y^ is one of the points y0, j G J1, that is nearest to the hyperplane H3, and Ha is a hyperplane that is parallel to H3 and passes through y^. Next, suppose that y° is one of the points y0, j G J2, that is nearest to the hyperplane H; let H be a hyperplane that is parallel to H and passes through y°; let Ha(t) = Ha + (t - t0)q, H (t) = H - (t - t0)q; let t1 be the first time instant when all evaders Ej, j G J1, reach the hyperplane Ha(^) when using controls Vj(t), j G J\, let r2 be the first time instant when all evaders Ej, j G J2, reach the hyperplane Hpfa) when using controls Vj(t), j G J2; let r = max{ri, r2}; da(t) be the distance from the point wa(t) = ya(T) + (t - t)q to the hyperplane H„(t); let d^ (t) be the distance from the point w^(t) = y^(t) - (t - T)q to the hyperplane H„(t). Note that t G T, da(T) = 0, d^(t) > 0. Two cases are possible.

1.1. There exists a time instant t0 G T for which da(T0) = d^(t0). We define the controls of

evaders Ej, j G J, as follows: va(t) = q for all i G T, and Vj(t) = Vj{t) for all t G T, j G J\,

j = a

For j G J2, we assume

Vj (t)

^■(t), te[t0,r0]r\T, q, t G (T0, n T.

1.2. For all t G T, t > t, the inequality da(t) = d^ (t) holds. Then there exist time instants t 1,

t2 G T, t < t 1 < t2, such that da(T^ < d^(tda(T2) > d^(t2). Define the controls of evaders Ej, j G J, as follows.

If j G J\, then we assume Vj(t) = Vj(t) for all t G T. If da(r2) ^ dpij1) and j G J2, then we assume

(Vj(t), te[t0,rl)r\T,

g(da(r') - d.jr1)) t = t2 - t 1

q, t G [t2, n T.

If d^(t 1) > da(T2) and j G J2, then we assume

Vj(t), i G [to, t1) Pi T,

Vj(t)=<j t = ri i t2 - t 1

q, t G [t2, n T.

2. Now assume that the strategies V1,..., Vm of evaders E1,..., Em are constructed for all

p < r.

3. Construct the strategies V1,..., Vm of evaders E1,..., Em for p = r. Let n1 = 2r-1 + 1. J1 = {j I y0 G H" n H+}, J2 = {j | y0 G Hf n H+1}. By virtue of the induction assumption, in the game r(ni, | J\\, where z° = = ... ,x°n), y° = (G Ji), the strategies Vj, j G Ji, of evaders Ej, j G Ji, are defined. Similarly, in the game r(n, | J21, z0),

where z° = (x°,y°), x° = (x±,..., x°ni), y° = (y^j G J2), the strategies Vj, j G J2 of evaders Ej, j G J2, are defined. Next, suppose that y^ is one of the points y0, j G Ji, that is nearest to the hyperplane and is a hyperplane that is parallel to and passes through y^,

y0 is one of the points yj, j G J2, that is nearest to the hyperplane H1. Also, suppose that Hg is a hyperplane that is parallel to H1 and passes through y°; Ha(t) = Ha + (t — t0)q; Hg (t) = Hg — — (t—t0)q; / | G T is the first time instant when all evaders Ej, j G J\, reach the hyperplane Ha(t\) when using strategies Vj(t), j G J\, t2 G T is the first time instant when all evaders Ej, j G J2, reach the hyperplane Pg(i2) when using strategies Vj(t), j G J2; t = max{i!,i2}; da{t) is the distance from the point wa{t) = ya(t) + (t — t)q to the hyperplane Ha(t); dp{t) is the distance from the point wp(t) = yg(T) — [t — t)q to the hyperplane Ha(t). Note that t G T, da(t) = 0, dp(t) > 0. Two cases are possible.

3.1. There exists a time instant t G T, t > t, for which da(t) = dp{t). Then we define the strategies V1,..., of evaders E1;..., Em in the game r(n, m, z0) as follows. For all j G J1, we assume Vj(t) = Vj(t), t G T. If j G J2, then we assume

fe(i), i£Mlni,

|q, t G (i, n T.

3.2. For all t G T, t > i, the inequality da(t) = dg(t) holds. Then there exist time instants r1,

r2 G T, i < r1 < r2, (r1,r2) n T = 0, for which d«(r1) < dg(i2), d«(r2) > dg(r2). Define the controls of evaders Ej, j G J, as follows. If j G J\, then we assume r:J(l) = Vj(t) for all i G T. If d«(r2) > dg(r1) and j G J2, then we assume

Vj(t), te[t0,ri)nT,

VjVt) - \ -----, t = Ti,

i2 — i1

q, t G [i2, n T.

If dg(r1) > d«(r2) and j G J2, then we assume

Vj(t), i G [to, Ti) Pi T,

i2 — i1

q, t G [i2, n T.

Let to G T be the time instant when all evaders Ej, j G J, reach the hyperplane Ha(r0). Using the induction and Lemma 3.2, we find that on [t0, r0] n T pursuers P1;..., Pn catch no more than (p — 1)2p + 1 evaders. By virtue of Lemma 3.2, on [r0, n T pursuers P1,..., Pn catch no more than 2p evaders. Therefore, the total number of evaders that pursuers Py ..., Pn catch is no more than p ■ 2p +1. This proves the theorem. □

Remark 4.1. In accordance with the constructed strategies, the motion of the evaders occurs as follows. Originally the phase space is divided into layers by parallel hyperplanes in such a way that in each hyperplane there is one pursuer and inside the layer formed by two neighboring hyperplanes there are no pursuers. In the first step, the evaders that lie in the layer formed by two neighboring hyperplanes move so that all of them are on a hyperplane parallel to these hyperplanes. In the second step, the evaders situated on two neighboring hyperplanes come together, and so on. Before the last step, all evaders lie in two parallel hyperplanes. In the last step, the evaders come together so as to be in one hyperplane, and they keep moving along the normal to this hyperplane with the maximum velocity.

Define the function f: N ^ N

f (n) = min{m | in the game r(n, m, z0) an evasion from an encounter occurs from any initial positions z0}.

Theorem 4.2. There exists a constant C1 > 0 such that for any natural n, n =1, the following inequality holds:

f (n) ^ C1n ln n.

P r o o f. It follows from Theorem 4.1 that for any natural p the following inequality holds:

f (2p + 1) ^ p ■ 2p + 2. Let n G N, n = 1. Take a natural number p such that

2p-1 < n < 2p + 1.

Then

f (n) ^ f (2p + 1) ^ p ■ 2p + 2 ^ C1n ln n,

where Ci = This proves the theorem. □

Let Int A and co A denote the interior and the convex hull of the set A, respectively.

T h e o r e m 4.3 (see [29, p. 7]). Suppose that in the game r(n, 1, z0)

y0 G Int co {x1,..., XI}.

Then a capture occurs in the game r(n, 1, z0).

Theorem 4.4. There exists a constant C2 > 0 such that for any n G N the following inequality holds:

f (n) ^ C2n ln n.

This theorem is proved along the same lines as the theorem of [9] using Theorem 4.3.

Corollary4.1. For any natural number l there exist natural numbers n, m and a vector of initial positions, z0, such that m - n > l and a capture occurs in the game r(n, m, z0).

Corollary 4.2. For any natural number l there exist natural numbers n,m such that in the game r(n,m,z0) an evasion from an encounter occurs for any z0, and in the game r(n + 1, m + l, z1) a capture occurs for some z1.

The proofs of the last two corollaries are identical to those of the corresponding corollaries in [9].

Remark 4.2. We note that the results of [9] are a consequence of the results of this paper for

T = R1.

Funding. This research was funded by the Ministry of Science and Higher Education of the Russian Federation in the framework of state assignment No. 075-01483-23-00, project FEWS-2020-0010.

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16. Yan Rui, Shi Zongying, Zhong Yisheng. Task assignment for multiplayer reach-avoid games in convex domains via analytical barriers, IEEE Transactions on Robotics, 2020, vol. 36, issue 1, pp. 107-124. https://doi.org/10.1109/TR0.2019.2935345

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21. Sun Zhiyuan, Sun Hanbing, Li Ping, Zou Jin. Self-organizing cooperative pursuit strategy for multi-USV with dynamic obstacle ships, Journal of Marine Science and Engineering, 2022, vol. 10, issue 5, article 562. https://doi.org/10.3390/jmse10050562

22. Sun Wei, Tsiotras P., Yezzi A. J. Multiplayer pursuit-evasion games in three-dimensional flow fields, Dynamic Games and Applications, 2019, vol. 9, issue 4, pp. 1188-1207. https://doi.org/10.1007/s13235-019-00304-4

23. Petrov N.N., Solov'eva N.A. Multiple capture of given number of evaders in linear recurrent differential games, Journal of Optimization Theory and Applications, 2019, vol. 182, issue 1, pp. 417-429. https://doi.org/10.1007/s10957-019-01526-7

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29. Petrov N.N. The problem of simple group pursuit with phase constraints in time scales, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 2, pp. 249-258 (in Russian). https://doi.org/10.35634/vm200208

Received 21.07.2023 Accepted 23.09.2023

Elena Sergeevna Mozhegova, Post-Graduate Student, Department of Differential Equations, Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia. ORCID: https://orcid.org/0009-0007-6833-6968 E-mail: mozhegovalena@yandex.ru

Nikolai Nikandrovich Petrov, Doctor of Physics and Mathematics, Professor, Department of Differential Equations, Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia. ORCID: https://orcid.org/0000-0002-0303-3559 E-mail: kma3@list.ru

Citation: E.S. Mozhegova, N.N. Petrov. The differential game "Cossacks-robbers" on time scales, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2023, vol. 62, pp. 56-70.

Е. С. Можегова, H. Н. Петров

Дифференциальная игра «казаки-разбойники» во временных шкалах

Ключевые слова: дифференциальная игра, групповое преследование, преследователь, убегающий, задача уклонения, временная шкала.

УДК: 517.977

DOI: 10.35634/2226-3594-2023-62-05

В конечномерном евклидовом пространстве рассматривается задача простого преследования группой преследователей группы убегающих в заданной временной шкале с равными возможностями всех участников. Множество управлений каждого участника — шар радиуса единица с центром в начале координат. Целью группы преследователей является поимка всех убегающих. Целевые множества — начало координат. Цель группы убегающих противоположна, то есть предоставить возможность хотя бы одному из убегающих избежать поимки. Получены условия разрешимости локальной и глобальной задач уклонения, а также оценки сверху и снизу для наименьшего числа убегающих, уклоняющихся от заданного числа преследователей из любых начальных позиций.

Финансирование. Исследования выполнены при финансовой поддержке Министерства науки и высшего образования РФ в рамках государственного задания № 075-01483-23-00, проект FEWS-2020-0010.

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20. Pierson A., Wang Zijian, Schwager M. Intercepting Rogue Robots: An algorithm for capturing multiple evaders with multiple pursuers // IEEE Robotics and Automation Letters. 2017. Vol. 2. No. 2. P. 530-537. https://doi.org/10.1109/LRA.2016.2645516

21. Sun Zhiyuan, Sun Hanbing, Li Ping, Zou Jin. Self-organizing cooperative pursuit strategy for multi-USV with dynamic obstacle ships // Journal of Marine Science and Engineering. 2022. Vol. 10. Issue 5. Article 562. https://doi.org/10.3390/jmse10050562

22. Sun Wei, Tsiotras P., Yezzi A. J. Multiplayer pursuit-evasion games in three-dimensional flow fields // Dynamic Games and Applications. 2019. Vol. 9. Issue 4. P. 1188-1207. https://doi.org/10.1007/s13235-019-00304-4

23. Petrov N.N., Solov'eva N.A. Multiple capture of given number of evaders in linear recurrent differential games // Journal of Optimization Theory and Applications. 2019. Vol. 182. Issue 1. P. 417-429. https://doi.org/10.1007/s10957-019-01526-7

24. Петров Н. Н., Нарманов А. Я. Многократная поимка заданного числа убегающих в задаче с дробными производными и простой матрицей // Труды Института математики и механики УрО РАН. 2019. Т. 25. № 3. С. 188-199. https://doi.org/10.21538/0134-4889-2019-25-3-188-199

25. Petrov N. N. Multiple capture of a given number of evaders in the problem of simple pursuit with phase restrictions on timescales // Dynamic Games and Applications. 2022. Vol. 12. Issue 2. P. 632-642. https://doi.org/10.1007/s13235-021-00387-y

26. Благодатских А. И. Синхронная реализация одновременных многократных поимок убегающих // Известия Института математики и информатики Удмуртского государственного университета. 2023. Т. 61. С. 3-26. https://doi.org/10.35634/2226-3594-2023-61-01

27. Guseinov G. Sh. Integration on time scales // Journal of Mathematical Analysis and Applications. 2003. Vol. 285. Issue 1. P. 107-127. https://doi.org/10.1016/S0022-247X(03)00361-5

28. Cabada A., Vivero D. R. Expression of the Lebesgue Д-integral on time scales as a usual Lebesgue integral; application to the calculus of Д-antiderivatives // Mathematical and Computer Modelling. 2006. Vol. 43. Issues 1-2. P. 194-207. https://doi.org/10.1016/j.mcm.2005.09.028

29. Петров Н. Н. Задача простого группового преследования с фазовыми ограничениями во временных шкалах // Вестник Удмуртского университета. Математика. Механика. Компьютерные науки. 2020. Т. 30. Вып. 2. С. 249-258. https://doi.org/10.35634/vm200208

Поступила в редакцию 21.07.2023

Принята к публикации 23.09.2023

Можегова Елена Сергеевна, аспирант, кафедра дифференциальных уравнений, Удмуртский государственный университет, 426034, Россия, г. Ижевск, ул. Университетская, 1. ORCID: https://orcid.org/0009-0007-6833-6968 E-mail: mozhegovalena@yandex.ru

Петров Николай Никандрович, д. ф.-м. н., профессор, главный научный сотрудник, кафедра дифференциальных уравнений, лаборатория математической теории управления, Удмуртский государственный университет, 426034, Россия, г. Ижевск, ул. Университетская, 1. ORCID: https://orcid.org/0000-0002-0303-3559 E-mail: kma3@list.ru

Цитирование: Е. С. Можегова, Н. Н. Петров. Дифференциальная игра «казаки-разбойники» во временных шкалах // Известия Института математики и информатики Удмуртского государственного университета. 2023. Т. 62. С. 56-70.