CC BY
30
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
2022. Volume 59. Pp. 67-84
MSC2020: 49N79, 49N70, 91A24 © B.T. Samatov, A. Kh. Akbarov, B. I. Zhuraev
PURSUIT-EVASION DIFFERENTIAL GAMES WITH GR-CONSTRAINTS ON CONTROLS
In the paper, a pursuit-evasion differential game is considered when controls of the players are subject to differential constraints in the form of Gronwall's integral inequality. The strategy of parallel pursuit (briefly, n-strategy) was introduced and used by L. A. Petrosyan to solve simple pursuit problems under phase constraints on the states of the players in the case when control functions of both players are chosen from the class L^,. In the present work, the n-strategy is constructed for a simple pursuit problem in the cases when control functions of both players are chosen from different classes of the Gronwall type constraints, and sufficient conditions of capture and optimal capture time are found in these cases. To solve the evasion problem, we suggest a control function for the Evader and find sufficient conditions of evasion. In addition, an attainability domain of the players is constructed and its conditions of embedding in respect to time are given. Results of this work continue and extend the works of R. Isaacs, L. A. Petrosyan, B.N. Pshenichnyi, A. A. Chirii, A. A. Azamov and other researchers, including the authors.
Keywords: differential game, Gronwall's inequality, pursuit, evasion, optimal strategy, capture time. DOI: 10.35634/2226-3594-2022-59-06
Introduction
Advanced differential games constitute the theory of mathematical methods of control processes, including dynamism, control, awareness, fighting and other particular properties, and describe one of the most complex mathematical models of real processes having great practical significance. At present, there are hundreds of monographs on the theory. Nonetheless, completely solved problems of differential games are quite few. In the theory of differential games, problems of pursuit-evasion occupy a special place due to a number of specific qualities. The works of L. S. Pontryagin [1] and N.N. Krasovskii [2] have a great importance in the construction of the fundamental theory of differential games. The book of R. Isaacs [3] includes certain game problems which were investigated comprehensively and devoted for further research. Some game problems in [3] have been studied in details by L.A. Petrosyan [4]. In the book [4], the concept of strategy of parallel convergence (briefly, n-strategy) was first given and applied to solve the quality problems for the differential game. The n-strategy suggested in the book [4] served as the starting point for the progress of the pursuit method in games with multiple pursuers (see e. g. B. N. Pshenichnyi [5], B. N. Pshenichnyi, A. A. Chikrii, and I. S. Rappoport [6], A. A. Azamov [7], A. A. Azamov and B.T. Samatov [8], N.N Petrov [9,10], N.N Petrov and A. Ya. Narmanov [11], A.I. Blagodatskikh and N.N Petrov [12], A. A. Chikrii [13], N.L. Grigo-renko [14], L.A. Petrosyan [15,16], L.A. Petrosyan and Yu. G. Dutkevich [17], L.A. Petrosyan and B.B. Rikhsiev [18], L.A. Petrosyan and V. V. Mazalov [19], B. T. Samatov [20-24]).
In the theory of differential games, the problems are mainly considered for the cases when geometric, integral or mixed constraints are imposed on control functions (N.N Petrov [10], A.N. Dar'in and A.B. Kurzhanskii [25], D. V. Kornev and N. Yu. Lukoyanov [26], N. Yu. Sati-mov [27], G.I. Ibragimov [28,29]). Yet, differential type constraints on controls have been triggering a substantial interest in many applied problems such as technical, economical and ecological problems (J. P. Aubin and A. Cellina [30], J. S. Pang and D. E. Stewart [31]).
At the present time, there exists a large interest from the point of practical and theoretical view to investigate different type problems of optimal control theory for more complicated dynamical systems where the controls and system are subject to various kinds of stationary and non-stationary constraints (for instance, [28,32,33]). Differential games with phase constraints can be involved in such problems. In solving such problems, an attainability domain of players takes up a great importance. Constructing the attainability domain of players is considered as the significant result in the problems of avoidance of meeting [4] and in the conflict problems too [34-37]. A great deal of optimal control problems including linear and nonlinear dynamical systems were considered for the cases where stationary and non-stationary constraints are applied to controls. Furthermore, their applications were provided in details (for instance, [38,39]).
In the work of B. T. Samatov, G. I. Ibragimov and I. V. Hodjibayeva [40], the notion of Gr-con-straint (Gronwall type constraint) on controls of players, which expresses more general form of geometric constraints, is first introduced. In the present work, a pursuit-evasion differential game with Gr-constraints is considered for more generalized cases. In the paper, we will rely on Pontryagin's formalization [1] and use the method of resolving functions (see [5,6,13,14,27,40]), which allows using a much smaller amount of current information for the construction of the pursuer strategy. In order to solve the pursuit problem, the n-strategy of the Pursuer is defined and sufficient conditions of pursuit are shown and also, an attainability domain of the players, which contains the meeting points of the players, is constructed. Besides, in the evasion problem, a special strategy is suggested to the Evader and sufficient conditions of evasion are obtained.
§ 1. Formulation of the problems
Consider two objects, the Pursuer P and the Evader E, moving in the space Rn. Suppose that x and y are the locations of the objects respectively.
Let the movements of the Pursuer P and the Evader E be based on the following differential equations
P: X = u, x(0) = xo, (1.1)
E: y = v, y(0) = yo, (1.2)
where x, y, x0, y0, u, v G n > 2, x0 = y0. In the present paper, we propose a new set of controls of the Pursuer and the Evader described by the following Gronwall type constraints [40]
|u(t)|2 < p2 + 2fci/" |u(s)|2ds, t > 0, (1.3)
J 0
|v(t)|2 < a2 + 2^/ |v(s)|2ds, t > 0, (1.4)
0
respectively, where p, a are given positive numbers and k1, k2 are given non-negative numbers.
Here u is the velocity vector of the Pursuer and the temporal variation of u must be a measurable function u(-): [0, to) ^ Rn. We denote by UGr the set of all measurable functions u(-) that satisfy the Gronwall type constraint (briefly, Gr-constraint) (1.3).
Similarly, v is the velocity vector of the Evader and here the temporal variation of v must be a measurable function v(-): [0, to) ^ Rn. We denote by VGr the set of all measurable functions v(-) that satisfy Gr-constraint (1.4).
Since the problem (1.1)-(1.4) for the case k1 = k2 was studied enough completely (Samatov et al., [40]). In the present work, we will assume k1 = k2.
Definition 1.1. The measurable functions u(-) G UGr and v(-) G VGr are called controls of the Pursuer and Evader respectively.
Definition 1.2. For the pairs (x0, u(-)), u(-) G UGr and (y0, v(-)), v(-) G ¥,Gr the solutions of the equations (1.1), (1.2), that is, x(t) = x0 + u(s) ds and y(t) = y0 + / v(s) ds are
00 respectively called the trajectories of the Pursuer and the Evader on the interval t > 0.
The goal of the Pursuer is to capture the Evader, i. e., achievement of the equality x(t) = y (t) (Pursuit problem) and the Evader E strives to avoid an encounter (Evasion problem), i. e., to achieve the inequality x(t) = y(t) for all t > 0, and in the opposite case, to delay the instant of the encounter as long as possible. We use the following statement.
Lemma 1.1 (Gronwall'sinequality, see [41]). If |y(t)|2 < a2+2k /" |y(s)|2ds, then |y(t)| <
0
< aekt, where y (t), t > 0, is a measurable function and a, k are non-negative numbers. By Lemma 1.1, if u(-) G UGr and v(-) G VGr, then
|u(t)| < peklt, |v(t)| < aek21, t > 0. (1.5)
It can be easily presented that the converse is not satisfied, that is, the inequalities (1.5) do not mean the inequalities (1.3) and (1.4). To define the notions of optimal strategies of the players and optimal pursuit time, we will consider two games.
Definition 1.3. Let B (c; r) denote the ball of radius r and centered at the point c G . Then we say that the function u(t, v): R+ x B (0; aek2t) ^ B (0; pekl t) is a strategy of the Pursuer if u(t, v) is Lebesgue measurable in t for each fixed v, Borel measurable in v for each fixed t.
Let z = x — y, z0 = x0 — y0. Then from (1.1), (1.2) we have the equation in the form
z = u — v, z(0) = z0. (1.6)
For solving the pursuit-evasion problems, the following definitions are important.
Definition 1.4. The strategy u(t,v) is called winning for the Pursuer P on the interval [0,TGr] in the simple game (1.1)-(1.4) if for every v(-) G VGr there exists some moment t* G [0, TGr] that the solution z(t) of the following Cauchy problem, which follows from (1.6),
z = u(t,v(t)) — v(t), z(0) = z0, (1.7)
is equal to zero at this moment, i.e., z(t*) = 0, and this implies that x(t*) = y(t*). Here the number TGr is called a guaranteed capture time.
Definition 1.5. The strategy u(t, v) is called a parallel pursuit strategy, or n-strategy, if for every v(-) G VGr the solution z(t) of the Cauchy problem (1.7) can be represented as
z(t) = AGr (t, v(-))z0, AGr (0, v(-)) = 1,
where AGr(t, v(-)) is some scalar continuous function of t, t > 0, that can be called a convergence function in the pursuit problem.
Definition 1.6. A control function v*(-) G VGr for the player E is called winning in the simple game (1.1)-(1.4) if for every u(-) G UGr the solution z(t) of the Cauchy problem
z = u(t) — v*(t), z(0) = z0, (1.8)
is different from zero for all t > 0, i. e., z(t) = 0, t > 0.
This paper is devoted to solving the following problems for Gronwall type constraints on controls.
1. Solve pursuit problem in the simple game (1.1)-(1.4) with the Gronwall type constraints (briefly, Gr-Game of Pursuit).
2. Solve evasion problem in the simple game (1.1)-(1.4) with the Gronwall type constraints (briefly, Gr-Game of Evasion).
3. Construction of an attainability domain of the Pursuer.
§ 2. A solution of the pursuit problem in the case when p > a
In this section, we will construct optimal strategies of players and give a formula for optimal pursuit time.
Definition 2.1. If a) p > a and ki > k2, or b) p > a and k2 > ki, then the function
MGr(t,v) = v-\*Gr(t,v)to, A *Gr(t,v) = (v,to) + y/(v,£0 >2 + p2e^ - |tf, (2.1)
is called the nGr -strategy of the Pursuer in the simple game (1. 1)-(1.4), where = z0/|z0|, and (v, £0) is the scalar product of the vectors v and £0 in the space Rn.
Proposition 2.1. If a) p > a and ki > k2, or b) p > a and h2 > ki when t G [0, 9], 9 = fc lfc In then, for all v such that < aek2t, the function A Gr(t, v) is defined and nonnegative and besides, for uGr (t, v) in (2.1) the following holds
\uGr (t,v)| = pekl K (2.2)
L e m m a 2.1. Let one of the following conditions
(1) p > a, ki > k2;
(2) p > a, ki < k2, \z0 \ < A;
be valid. Then the following equation
with respect to t, has at least one positive root and we denote it byTGr, where
fca fci / 1 1 \ a p ^ ^
A = pk2-Haki-k2 ----¡-)+-j--J-. (2.4)
\ki k2 J k2 ki Proof. 1) Let p > a, ki > k2. Using the equation (2.3) we will form the function
and analyze it as follows:
a) Fi(0) = |z0| > 0;
b) the derivative of the function Fi(t) equals Fi(t) = aek2t — peklt < 0;
c) the function Fi (t) is decreasing when t > 9;
d) take the limit of the function F1 (t) as t ^ to
lim F.it) = lim eklt (f e^2"^ - f - MzJlIA^ = .qo. t^ ^k2 k1 eklt y
2) Let p > a, k1 < k2. Then the Pursuer should complete the pursuit on the interval [0,0] depending on the part b) of Proposition 2.1.
Then we will investigate the function F1 (t) as follows:
a) F1 (0) = |z01;
b) the function F1 (t) is decreasing on the interval [0,0]. Therefore, F1(t) at t = 0 must be negative for a positive root of (2.3) to exist. From this, we get |z0| < A (see (2.4)). □
Theorem 2.1. If one of the conditions of Lemma 2.1 holds, then the nGr -strategy (2.1) is winning in the Gr-game of pursuit on time interval [0, TGr].
Proof. Assume that the Pursuer applies the nGr-strategy for any control function of the Evader v(-) G VGr. Then in accordance with (1.6) we have the Cauchy problem
z = x — y = —AGr (t,v(t))&, z(0) = z0, z0 = x0 — y0,
and write its solution as follows:
1 /"t
z(t) = AGr(t,v(-))z0, AGr(t,v(-)) = l~ —J A*Gr(s,v(s))ds. (2.5)
Now we will study the behavior of the function AGr (t, v(-)) of t. Using the definition of the function AGr (t, v(t)), we have
AGr(t,v(-)) = 1 - — / (MsUo> + V(v(s),to)2 + P2e^-\v(s)\*) ds.
Since the function f(t,w) = \v(t)\w + ^\v{t)\2w2 + p2e2klt — \v{t)\2, w G [—1,1], is monotone increasing with respect to w for every t > 0. Therefore, from the second inequality in (1.5) we obtain
A Gr(tM')) < 1 " A [\peklS ~ k2S)ds = 1 - ^ (£ (efcli - 1) - f(efc2i - 1)) = A .(t).
|z0^0 |z0^ k1 k2 J
Clearly, the function A1(t) is monotone decreasing on [0,TGr] and A1(TGr) = 0 (see Lemma 2.1). Consequently, there exists some time t*, 0 < t* < TGr, such that AGr(t*, v(-)) = 0, and hence by (2.5), we obtain that z(t*) = 0. □
§ 3. A solution of the pursuit problem in the case when p < a
Let p < a, k2 < k1 be satisfied. Then the inequality peklt > aek2t doesn't hold on the interval |(). 0), where 9 = k2l_kl InHere we assume that the Pursuer begins the pursuit from
the moment t = 0 and the Evader moves with some control function v(-) G VGr from the initial
, e
state y0 to the state y (0) by t = 0, where y (0) = y0 + v(s) ds.
0
Lemma 3.1. For the Evader, the inclusion y(t) G SRo(y0) holds at every moment t, t G [0,0], where SR0(y0) is a ball whose center is on the point y0 and whose radius is R0 = a(ek2 e — 1)/k2.
Proof. Let v(-) e VGr. Then from (1.2) and (1.4) it proceeds that
,-e
r0
\y(t) - yo\ < / \v(s)\ ds < aek2Sds = a(ek20 - 1)/k2 = Ro J 0 Jo
for all t e [0,0].
Corollary3.1. Lemma 3.1 means that for the function z(t), the relation
max \z(t)\ < \zo\ + Ro = \z0\
O<i<0
is satisfied on time interval [0,0].
Definition3.1. We call the n^r -strategy of the Pursuer the function
□
uGr (t,v)
on time interval [0, TGr ], where
0,
if 0 < t < 0;
v - ag;(t,v)C0, if0 < t;
(3.1)
AGr(t,v) = (V,Q + \j(v,Co)2 + P2e2klt - M2
and TGr is the first positive root of the equation
CO
TTT, z* = x0-y(e),
\z0 \
Pklt _ °k2t + °k2e _ Pkld _ = Q ki k2 k2 ki
with respect to t.
Note that the equality (2.2) holds for the strategy (3.1) when t > 9.
Theorem 3.1. Let p < a and ki > k2 be valid. Then in the simple game (1. 1)—(1.4), the nGr-strategy (3.1) is winning on time interval (0, TGr], where TGr = 9 + TGr.
Proof. Suppose that the Pursuer applies the nGr-strategy (3.1) from the moment 9 when the Evader implements any control function v(-) G VGr. Then using the equations (1.1), (1.2) we obtain the equation
z = —AG; (t,v(t))CS, z(9) = zS.
Integrate both sides of this equation over [0, t]
z(t) = AGr (t,v(-))z,
(3.2)
where
1 ^
A*Gr(t,v(-)) = l~— / A*G*r(s,v(s))ds. \z0 \ Je
Now we will study the behavior of the function A*Gr(t, v(-)) of t. Using the definition of the function AG*r(t,v(t)), we have
Ahr(t,v(-)) = 1 - -i- T f(v(s),Q + V(v(s),&)2 + P2e2klS ~ K«)!2) ds.
\z0 \ Je v 7
*
Since the function f(t, ¡1) = \v(t)\fi + y/\v(t)\2fi2 + p2e2klt — \v{t)\2, ¡jl G [—1,1] is monotone increasing with respect to ^ for every t > 0. Therefore, from the second inequality in (1.5) we have
1 r*
^Gr(tM')) < I ~ TTJ / (pekiS-aeknds = F2(t)/\4\=A2(t), \zo\ Je
where F2(t) = |z0*l - £efcli + - + f^9.
Now we will show that there exists a positive solution for the equation
A2(t) = 0 (3.3)
on the interval [0,T£r]. For this purpose, we will analyze the function F2(t) = |z0\A2(t), that is, the function has the following properties.
a) F2(0) = |z0\ > 0.
b) It is obvious that < 0 for every t > 9, i. e., the function I-2d) is decreasing. Moreover, the limit of this function as t ^ to is equal to
hm F2(t) = - hm ek^ (^ + ^ ~ phek^-\z0\klk2 _ e(k2_kl)t\ =
Hence, we can conclude that the equation (3.3) has at least one positive root on the interval [0, +to).
Clearly, the function A2(t) is monotone decreasing on [0,T£r] and A2(T^r) = 0. Consequently, there exists some time tl, 0 < tl < TGr, such that A^r(tl, v(-)) = 0, and hence, by (3.2) it follows that z(tl) = 0.
Next, we will prove the admissibility of the strategies (2.1), (3.1) for all t > 0. Let v(-) G VGr be an arbitrary control of the Evader. According to (1.3) and (2.2), we obtain:
a)
\uGr(t, v(t))\2 = p2 + p2(e2klt - 1) = p2 + 2ki f p2e2klsds =
Jo
= p2 + 2ki / \uGr (s, v(s))\2ds, Jo
b)
\uGr(t, v(t))\2 = p2 + p2(e2klt - 1) = p2 + 2ki Tp2e2klsds =
e
= p2 + 2ki / \uGr(s,v(s))\2ds. e
□
§ 4. A solution of the evasion problem
D e f i n i t i o n 4.1. In the simple game (1. 1)—(1.4), we call the strategy of the Evader the function
vGr(t) = -aefc2tCo, t > 0. (4.1)
Theorem 4.1. Let one of the following conditions holds: 1) p < a, k2 > ki; 2) p > a, k2 > ki, \z0\ > A (see (2.4)). Then in the simple game (1.1)—(1.4), the strategy (4.1) is winning and the distance between the players changes according to the function
Proof. Let us assume that the Evader implements the strategy (4.1) for any control function u(-) G UGr of the Pursuer. Obviously, v *Gr (t) G VGr. Then for any u(t), it follows the Cauchy problem (1.8). For the solution z(t) of this problem, we can write the following estimations:
\z(t)\ =
rt .
3fc2S<
z0 — / v * Gr (s) ds
0
— \u(s)\ds > \z0\ + a / ek2s^0 ds — \u(s)\ds.
Using the first inequality in (1.5) we obtain \z(t)\ > Fi(t), where
1. Let p > a and k2 > ki.
a) The function Fi(t) is increasing on the interval \9. to), 9 = In
b) Fi(0) = \z0\ > 0.
If the Unction Fi(t) is positive at value t = 9, i. e., f (9) > 0 or \z0\ > A (see (2.4)), then this function is positive for all t, t G [9, to).
2. Let p = a and k2 > ki.
a) Fi(0) = \z0\ > 0.
b) The function Fi(t) is increasing on the interval [0, to).
Thus the function Fi(t) is different from zero and positive for all t, t G [9, to).
3. Let p < a and k2 > ki.
a) Fi(0) = \z0\ > 0.
b) The function f(t) is increasing on the interval \9. to), 9 = In ^ < 0. Thereby the function Fi(t) is positive for all t, t G [0, to).
4. Let p < a and ki = k2. This condition was studied in [40].
□
§ 5. Dynamics of the attainability domain
Suppose that p > a, ki > k2 are valid and the Pursuer applies the strategy (2.1) when the Evader implements any control function v(-) G VGr. In relation to the equations (1.1), (1.2), it follows that the trajectories of the players are defined, respectively, as
x(t) = x0 + uGr(s,v(s)) ds, y(t) = y0 + v(s) ds. 00
For the pair of (x(t),y(t)) we construct the following sets
W(t) = {w: aek2t\w — x(t)\ > peklt\w — y(t)\}, (5.1)
W(0) = {w: a\w — x0\ > p\w — yd}.
We can see that the inclusion y(t) G W(t) is valid for all t, t G [0, TGr].
t
0
0
0
Lemma 5.1. If p > a, k1 > k2, then the set (5.1) can be written as follows:
W(t) = x(t) + AGr(t, v(-))(R(t, zo)S + C(t, zo)),
where
R(t,Zo) =
pae
(fcl+fc2)t
p2 e2fcii — a2e2fc2i
|zo|, C (t,zo) = -
p2e2fcii — a2e2fc2i
Zo,
and S is the unit sphere whose center is at the zero point in Rn.
Proof. We introduce the denotation w — x(t) = w. Substitute w +x(t) for w and the set (5.1) will take the form
W(t) = {W + x(t): aefc2t\wD\ > peklt\wD + z(t)\} = = x(t) + {W: aek2t\wD\ > pefclt\wD + z(t)\}.
Hence,
W (t) = x(t) + W *(t),
where W*(t) = {w: aek2t\w\ > peklt\w + z(t)\}.
Here we square the both sides of the inequality in W*(t)
(5.2)
a2e2k2t|w|2 > p2e2klt(|w|2 + 2(w, z(t)) + |z(t)|2) , l2 2p2e2fci^ ,,, p2e2fcii|z(t)|2
M2 + z(*)> + — ■ vL . < 0
p2e2fcii — a2e2fc2i
|w|2 + 2 ( w
<
p2e2fcii — a2e2fc2i
p2e2kltz(t) p2e2k1t _ a2e2k2t
p2e2klt\z(t)\2 p2e2k1t _ cr2g2fc2t / p2e2klt — a2e2k2t'
p2e2kltz(t) p2e2k1t _ a2e2k2t
p2 e2fciiz(t)
+
<
Reduce the right-hand side of the last inequality to the canonical form
w +
p2 e2ki 4z(t)
p2e2fcii — a2e2fc2i
pae(fcl+fc2)i|z(i)| — p2g2fcii _ (T2g2fc2i '
<
We denote as
R(t,z(t))
pae
(fci+fc2 )t
p2e2fcit — a2e2fc2i
|z(t)|,
C (t,z(t)) = -
p2e2fcit
p2e2fcit — a2e2fc2t
z(t).
Consequently, using (2.5) we can write the set (5.2) in the form
W (t) = x(t) + C (t, z(t)) + R(t, z(t))S =
x(t) + AGr (t,v(-))
pae
(fci+fc2)t
p2e2fcit — a2e2fc2t
|zo|S -
p2e2fcit
p2 e2fcit — a2e2fc2t
zo
This finishes the proof.
□
Lemma 5.2. f the conditions of Lemma 5.1 are valid for all t G [0, TGr ], then the set W (t) is monotone decreasing with respect to t, i. e., Wi (t) D W2(t) for ti < t2, ti, t2 G [0, TGr].
2
Proof. To prove Lemma 5.2 we construct the support function (see Blagodatskikh [42])
F(W(t),^) of the set W(t), where ^ G Rn and = 1. Now take the derivative of F(W(t),^) in terms of t
dd
-F{Wit), V) = TF(x(t) + A(t, v(-))[R(t, z0)S + C(t, zo),ip]) = dt dt
= ({x(t),^) + A(t,vO)[R(t,*b) + (C (t^),^)] + + A(t,v())[R(t,z0) + (C(t,z0),^)] = $i M) + $2 M),
where
$i(t, = (x(t),^) + A(t, v(0)[R(t, z0) + (C(t, z0),^)], $2 M) = A(t,v())[R(t,z0) + (C7(t,z0),^)].
Now we prove that the inequality
d
= $1M) + < 0
dt
is true on t G (0,TGr]. To do this, we first show that $i(t,p) < 0. Multiply the expression (p2e2klt)/(p2e2klt — a2e2k21) to the both sides of the inequality \v(t) \2 < a2e2k21 and reduce to the simple form
\v(t)\2 <
a2e2k2t
p2e2kit — a2e2k21
(p2e2klt —\v(t)\2).
(5.3)
Considering the equality X(t,v(t))(X(t,v(t)) — 2(v(t),&)) = p2e2klt — \v(t)\2, the inequality (5.3) can be transformed into the form
\v(t)\2 <
a2e2k2t
p2e2klt — a2e2k21
X(t,v(t))(X(t,v(t)) — 2(v(t),^0))
a2e2k2t a2e2k2t |y(i)|2 + 2A(i,y(i))—-^——Mi),Co> < A2(i,y(i))-
p2e2klt — a2e2k21
p2e2klt — a2e2k2t
From this, we get
\v(t)\2 + 2A(t,v(t))
a2e2k2t
a4 e4k21
<
p2e2k1t _ a2e2k2t N^)' + A (t, v(t)) ^2e2fcli _ cr2g2fc2i)2 -2a2e2(kl +k2)t
< A2{t,v{t))-
pae
(p2e2klt — a2e2k2t)2
Hence, we have
v(t) +
a2e2k2t
p2 e2kl t — a2 e2k2t
^A(t,v(t))
< A(t,v(t))
pae
(kl +k2)t
p2e2klt — a2e2k2t'
According to the property of the support function, for any vector ^ G = 1, the
inequality
v(t) + A(t,v(t))
a2e2k2t
p2e2klt — a2e2k2t
M) <
v(t) + A(t,v(t))
a2e2k2t
p2e2klt — a2e2k21
£
is valid. Using this we can write the following
^ w / \ \ a2e2fc2t \ pae(fcl +fc2)t
v(t) + A(t, v(t))————£o, V ) < A(t, v{t))-
p2g2fcit _ a2g2fc2ts0' r / — y ' v p^g^kli — a2g2fc2t p2p2fci t \ P^p(fcl +fc2)t Mi), V) - AM(*)) ( 1 - -<M> < A(t,v(t))- P
p2g2fcit — a2g2fc2W Ns° r/ — v ' v p2e2klt — a2g2fc2t p2g2fcit pae(fci +fc2)t <«(i) - HtAt))M) + p2e.2t,, _ ^^</■■) - Mt.»«))^,,,^, < o
pae(fci +fc2)t p2g2fcit
^„2kt _2„2kot
p2g2fcit — a2g2fc2^ 01 p2g2fcit — a2g2fc2t (±(t),^) + À(t,v(-))[R(t,zo) + (C(i,zo),^)] < 0.
< 0
This inequality means that $1(t, < 0.
Now we will show $2(t,^) < 0. Firstly, the function $2(t,-0) is rewritten in the following form
$2 (t,^) = A(t,v(-))[R(t,zo) + (C7(t,zo),^)] = A(t,v(-))$a(t,^),
where
y_// pWo V
31 ' ^ \p2e2klt - a2e2k2t) \ \p2e2k^ - a2e2k2t) ' ^
Because of A(t, v(-)) > 0 on t G (0, TGr], it is enough to the prove $3(t, < 0.
First, calculate the second derivative in $3(t, and according to the conditions of Lemma 5.1, we have
p2g2fclt y ^ 2p2(J2g2(fc1+fc2)t(fc2 -k1)<
p2g2fcit — a2g2fc2t / (p2g2fcit _ a2g2fc2t)2 for all t G [0,TGr]. From this
p2g2fcit F 1 > 0.
p2g2fcit _ a2g2fc2t
According to the definition of the support function, the inequality (£o,^) < 1 holds, and multiply the expression (—(p2e2klt)/(p2e2klt — a2e2fc2t)) to the both sides of this inequality
2g2fcit \ ' / p2g2fcit P p
p2g2fcit _ a2g2fc2W XS0' r/ — \ p2g2fcit _ a2g2fc2t / '
Add the expression ((pag(fci+fc2)t)/(p2g2kit _ a2g2fc2t))' to the last inequality
p^g(fci+fc2)t \ ' / P2g2fcit p * 1 p y(M)<
p2g2fcit _ ff2g2fc2U \ p2g2fcit _ ^2g2fc21
/ nrr
pag(fci +k2)t \ / p2g2kit y pag(fci+k2)t(ki _ k2)
— i - i _ —
p2g2fcit _ a2g2fc2H lp2g2fcit _ ff2g2fc2H (pgfcit + agfc2t)2
(5.4)
This is negative for all t, t > 0, in accordance with the conditions of Lemma 5.1. Therefore the left-hand side of the inequality (5.4) is negative too, i. e.,
p^g(fci+fc2)t \ ' / p2g2fcit
P W P >(M)<0. (5.5)
p2g2fcit _ a2g2fc2^ \ p2g2fcit _ a2g2fc21
Multiply |z0| to the both sides of the inequality (5.5)
/ pae^+k^\z0\ V [f p2e2kltz0 V aytjW) — [p2e2k!t _ a2e2k2t J ~ \ I p2e2klt — a2e2k2t J —
So we can write the relation
= A(t,v(-))[R(t,zo) + (C(t,zo),-)] = A(t,v(-))$a(t,-) < 0
for the Unction $2(t,—). Hence, we have shown that the support function F(W(t),—) is decreasing for all — £ Rn, | = 1, i. e.,
d
— F(W(t),ip) < 0.
On the other hand, the set W(t) is monotone decreasing in t £ [0, TGr]. This completes the proof.
□
§ 6. Individual cases for the pursuit-evasion games with Gr-constraints
Pursuit game
№ Capture conditions Resolving Duration Guaranteed capture time
function of the strategy
1 h = k2 p > a A Gr(t,v) [0, oo) M 1 + TS)
2 h > k2 p > a [0, oo)
3 h < k2 p > a, M < A A hr(t,v) [0,0] the first positive root of F\ (t) = 0
4 h > k2 p = a [0, oo)
5 h > k2 p < a A &{t,v) [0, oo) the first positive root of F2(i) = 0
where
XGr(t,v) = {v,Co) + V{v,Co)2 + Se2kt, 5 = p2- a2, \*Gr(t,v) = (v,Co) + V(v, Co)2 + p2e2klt — |w|2,
A *G*r(t,v) = (v,Q + y/(v,&)2 + p?e™-\v\*.
Evasion game
№ Evasion conditions Strategy of the Evader Duration of the strategy The distance function between the players
1 h = k2 p = a v*(t) = -aek2Xo t £ [0, oo) \z(t)\>\z0\ + (a-p)(ekt-l)/k
2 h = k2 p < a
3 h < k2 p > a, \zo\ > A \z(t)\ > \z0\ - A
4 h < k2 p = a \z(t)\ > \zo\-tie^+f2ek2<-f + fi
5 h < k2 p < a
where
k2 fci /1 1 \ a p
A = pk2-H aki-k2 ---— + ---—.
\k i k2 J k2 ki
Acknowledgments. We wish to thank prof. A.A. Azamov for discussing this paper and for providing some useful comments.
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Received 24.01.2022 Accepted 18.04.2022
Bakhrom Tadzhiakhmatovich Samatov, Doctor of Physics and Mathematics, Professor, Department of Mathematical Analysis, Namangan State University, ul. Uychi, 316, Namangan, 116019, Uzbekistan. ORCID: https://orcid.org/0000-0002-0734-8507 E-mail: samatov57@inbox.ru
Adakhambek Khasanbaevich Akbarov, PhD student, Department of Mathematics, Andijan State University, ul. Universitetskaya, 129, Andijan, 170100, Uzbekistan. ORCID: https://orcid.org/0000-0001-9797-0430 E-mail: akbarov.adhambek@bk.ru
Bakhodir Inomzhon ugli Zhuraev, PhD student, Department of Mathematics, Andijan State University, ul. Universitetskaya, 129, Andijan, 170100, Uzbekistan. ORCID: https://orcid.org/0000-0002-6920-4314 E-mail: j.bahodirjon@bk.ru
Citation: B.T. Samatov, A.Kh. Akbarov, B.I. Zhuraev. Pursuit-evasion differential games with Gr-constraints on controls, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Uni-versiteta, 2022, vol. 59, pp. 67-84.
Б. Т. Саматов, А. X. Акбаров, Б. И. Жураев
Дифференциальные игры преследования-убегания при Сг-ограничениях на управления
Ключевые слова: дифференциальная игра, неравенство Гронуолла, преследование, убегание, оптимальная стратегия, время поимки.
УДК: 517.977
БОТ: 10.35634/2226-3594-2022-59-06
В этой статье исследуется дифференциальная игра преследования-убегания, когда на управления игроков налагаются дифференциальные ограничения вида интегрального неравенства Гронуолла. Отметим, что стратегия параллельного преследования (короче, П-стратегия) была введена и использована Л. А. Петросяном для решения задач простого преследования при фазовых ограничениях на состояний игроков для случая, когда функции управления обоих игроков выбираются из класса Ьж. В настоящей работе для решения задачи простого преследования построена П-стратегия, когда функции управления обоих игроков выбираются из различных классов с ограничениями типа Гронуолла и для этого случая найдены достаточные условия поимки и оптимальное время поимки. Для решения задачи убегания предлагается функция управления для убегающего и находятся достаточные условия убегания. Кроме того, построена область достижимости игроков и даны условия вложения ее по времени. Полученные результаты являются развитием и продолжением работ Р. Ай-зекса, Л. А. Петросяна, Б. Н. Пшеничного, А. А. Чикрия, А. А. Азамова и других исследователей, включая авторов этой работы.
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Поступила в редакцию 24.01.2022 Принята в печать 18.04.2022
Саматов Бахром Таджиахматович, д. ф.-м. н., профессор, кафедра математического анализа, Наман-ганский государственный университет, 116019, Узбекистан, г. Наманган, ул. Уйчи, 316. ORCID: https://orcid.org/0000-0002-0734-8507 E-mail: samatov57@inbox.ru
Акбаров Адахамбек Хасанбаевич, аспирант, кафедра математики, Андижанский государственный университет, 170100, Узбекистан, г. Андижан, ул. Университетская, 129. ORCID: https://orcid.org/0000-0001-9797-0430 E-mail: akbarov.adhambek@bk.ru
Жураев Баходир Иномжон угли, аспирант, кафедра математики, Андижанский государственный университет, 170100, Узбекистан, г. Андижан, ул. Университетская, 129. ORCID: https://orcid.org/0000-0002-6920-4314 E-mail: j.bahodirjon@bk.ru
Цитирование: Б. T. Саматов, A. X. Акбаров, Б. И. Жураев. Дифференциальные игры преследования-убегания при Gr-ограничениях на управления // Известия Института математики и информатики Удмуртского государственного университета. 2022. Т. 59. С. 67-84.
