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Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta
2023. Volume 61. Pp. 94-113
MSC2020: 49N79, 49N70, 91A24 © B.T. Samatov, U. B. Soyibboev
DIFFERENTIAL GAME WITH "LIFELINE" FOR PONTRYAGIN'S CONTROL EXAMPLE
The main purpose of this work is to solve one of the main problems of Isaacs, i. e., a game with a "lifeline" for Pontryagin's control example when both players have the same movement dynamics. To solve this problem, the pursuer is offered a strategy of parallel pursuit (briefly, n-strategy), which ensures the fastest convergence of the players and the capture of the evader within a certain closed ball. In addition, for the differential game under consideration, an explicit analytical formula for the players' attainability domain is given and the main lemma is generalized (L.A. Petrosjan's lemma on monotonicity of the players' attainability domain with respect to embedding for a game of simple pursuit). Using this main lemma, we find conditions for the solvability of the game with a "lifeline" for Pontryagin's control example as well. For clarity, at the end of the work, examples are given for some special cases.
Keywords: differential game, pursuit, evasion, acceleration, strategy, guaranteed capture time, attainability domain, lifeline.
DOI: 10.35634/2226-3594-2023-61-06 Introduction
The fundamental theory of differential games was formulated by R. Isaacs [1], L. S. Pon-tryagin [2], N.N. Krasovskii [3], B.N. Pshenichnyi [4], L.A. Petrosjan [5], A. Friedman [6], W.H. Fleming [7], L.D. Bercovitz [8], A. A. Chikrii [9], A.I. Subbotin [10] and others.
In the theory of differential games, pursuit-evasion problems occupy a special place due to a number of specific qualities. One of them is expansiveness of applications of various methods and originality of the obtained results [1,2,4,5,11,12]. This quality was clearly apparent in the model problems. For example, R. Isaacs' example called "a game with a lifeline" [1, Problem 9.5.1] with simple dynamics of the players was solved by L. A. Petrosjan introducing a special strategy [5] called strategy of parallel approach (briefly, n-strategy). Later, n-strategy was efficiently applied to solve other kinds of pursuit games [9, 13-18]. Later on, A. A. Chikrii [9] worked out the resolving functions on the basis of combination of ideas of n-strategy and first direct method of L. S. Pontryagin [2].
A. A. Azamov based on B.N. Pshenichnyi's work [4] gave analytical formula for n-strategy suited for all cases of maximal velocities of the players and inducted a recurrent relation for attainability domain [13]. Such a relation turned out effective in solving of pursuit-evasion problems and "a game with a lifeline" with different variants of constraints for control functions of the players [14,15].
Another important differential game famous under the name "control example" was considered by L. S. Pontryagin and for this game, further important results were achieved in the works [11,12,16]. Differ from "a game with a lifeline", the Pontryagin control example deals with inertial objects and is therefore appreciably complicated. Equations of a pursuer and an evader in the example describe the motion of two inertial objects with allowance for friction forces. In [19], for this example, /-capture problem was solved by exchanging given differential equations to a linear normal system. In [2], Pontryagin's example with many players having the same dynamic and inertial resources was studied and proved that matching of the phase coordinates is a pursuit condition. In the work [12] of N.N. Petrov, a "soft" pursuit for the Pontryagin
example of n pursuers and one evader was considered and proved that matching of the phase coordinates and that of the velocities are pursuit conditions. In [20], D. A. Vagin and N. N. Petrov obtained sufficient conditions for the capture of at least one evader for the Pontryagin example with many participants and phase constraints imposed on the position of evaders with identical dynamic and inertial resources of players, and with all evaders using the same control.
The problems in the theory of differential games are basically studied in the cases where control functions of players are subject to geometric, integral or their mixed constraints (for instance, [21-25]). Nevertheless, various type constraints on controls have been provoking a considerable interest in a number of applied problems such as economical, biological and ecological problems.
At the present time, from the standpoint of applied and theoretical perspective there is a substantial enthusiasm to explore various type problems of optimal control for more complex systems when different types of stationary and non-stationary constraints are imposed on the controls (for example, [26-28]). Differential games under phase constraints might be involved in such problems where the attainability domain of players has a considerable significance. Construction of the attainability domain is regarded as the remarkable result in the problems of avoidance of encounter [5] and in the conflict problems [29-34] as well. In the work [35], the authors investigated the differential game with a "lifeline" for the inertial movements under geometric constraints on controls of players and the dynamics of the attainability domain of an evader was examined through finding solvability conditions of the pursuit-evasion problems in favor of a pursuer or an evader.
In this paper, pursuit-evasion problems and a "lifeline" game are considered for Pontryagin's control example when both players have the same movement dynamics. In the pursuit problem, an analogue of n-strategy is constructed and a sufficient condition of capture is determined. In the evasion problem, a constant control function is proposed as a strategy of an evader and a sufficient condition of evasion is found. Further, an explicit formula for the attainability domain of players is generated. Monotonicity conditions of the attainability domain with respect to embedding and a sufficient condition of defining a solution of the game with a "lifeline" are obtained.
§ 1. Formulation of the problems
We will consider a differential game of two players. Let in the Euclidean space Rn, a controllable player X (the pursuer) follow another controllable player Y (the evader). Assume that positions of the players X and Y are described by x and y, respectively in Rn.
Let movements of the pursuer and the evader be based on the following differential equations with initial conditions:
where x, y,u,v E n > 2, a > 0; xi0, yio are the initial positions of the players X and Y, and x2o, y20 are their initial velocity vectors. Here it is required that x10 = y10 and x20 = y20.
The parameter u in (1.1) is a control of the pursuer and it will be thought of as a measurable function u(-): R+ ^ Rn. On this vector-function, we put the geometric constraint (briefly, the G-constraint) of the form
where a is a given positive parametric number and it expresses the maximal value of acceleration of the pursuer.
X: x + axe = u, x(0) = x10, x(0) = x20, Y: y + ay = V y(0) = yi0, y(0) = y20,
(1.1) (1.2)
|u(t)| < a almost everywhere t > 0,
(1.3)
Likewise, the parameter v in (1.2) is a control of the evader and it will be regarded as a
measurable function v(-): R+ ^ Rn. On this vector-function, we put the G-constraint
|v(t)| < P almost everywhere t > 0,
(1.4)
where P is a given non-negative parametric number and it expresses the maximal value of acceleration of the evader.
Control functions u(-) and v(-) of the players depend on time t, t > 0. The set of all control functions u(-) (v(-)) satisfying the constraint (1.3) (the constraint (1.4)) is denoted by U (by V).
Definition 1.1. A measurable function u(-) G U (v(-) G V) is called an admissible control of the pursuer (of the evader).
By equations (1.1) and (1.2), for controls u(-) G U and v(-) G V, the triplets (x10,x20,u(-)) and (y10, y20, v(-)) pose the trajectories
x(t) = xw + — ( 1 - e~at) + - i u(s a V J a Jo
y(t) = yio + yf(l-e-at>j +-aj\[s)\ i-e
(s)^1 - e-a(t-s)^) ds, (s)^1 - e-a(t-s)^) ds
(1.5)
(1.6)
of the players X and Y, respectively.
Assume that a closed subset L called the "lifeline" is given in Rn. The main target for the pursuer is to catch the evader, as it were, to obtain the equality x(t*) = y(t*) at some t* > 0 while the evader stays in the zone Rn \ L. The main goal for the evader is to reach the zone L before being caught by the pursuer, or to sustain the relation x(t) = y(t) at each t G [0, and if this is not possible, then to extend an encounter time asap. We need to remark that the zone L does not restrict motion of the pursuer. In addition, it is supposed that the initial positions x10, y10 satisfy the conditions x10 = y10 and y10 G L at the beginning of the game.
It is known that if control functions of the players X and Y depend only on time t, t > 0, then they don't guarantee to solve the games of pursuit and evasion. Thus, the acceptable types of controls involve being strategies. Below we are going to present the key definitions and conceptions.
First off, let us write the notations
z(t) = x(t) - y(t), z(0) = Z10 = X10 - y10, ¿(0) = Z20 = X20 - y20-Then according to (1.1), (1.2), (1.7), we get the initial value problem
z + az = u - v, ¿(0) = z10, ¿(0) = z20 = 0.
(1.7)
(1.8)
Consequently, as an alternative to the game (1.1)-(1.4), we have generated the game (1.3), (1.4), (1.8), or for brevity, the game (U, V).
Later, we will use the symbol Be to denote the ball of a radius q centered at the origin.
Definition 1.2. It is said that a function u: V x Rn ^ U is a strategy of the pursuer if:
(1) for any control v(-) G V, the inclusion u(-) = u(v(-), ¿10) G U is satisfied. Here we term the function u(-) = u(v(-), ¿10) as an implementation of the strategy u(-);
(2) for every v1 (-),v2(-) G V and for each t, t > 0, the equality v1 (e) = v2(e) is fulfilled almost everywhere on [0,t], then u1(e) = u2(e) is true almost everywhere on the same interval, where uj(-) = u(v»(■), ¿10), i = 1, 2.
A special case of a strategy can be given by a mapping u: B^ x Rn ^ Ba, which is a Borel measurable function with respect to v, v G B^.
Definition 1.3. We say that a strategy u = u(v, ¿10) is a parallel pursuit strategy, or briefly, a n-strategy if, for any control v(-) £ V, the solution ¿(t) of Cauchy's problem
¿ + a,z = u(v(t),z10) — v(t), ¿(0) = ¿10, ,¿(0) = 0 can be transformed into the form
¿(t) = r(v(-),t)zio, r(v(-), 0) = 1,
where r(v(-), t) is a scalar function with respect to t, t > 0, and in general, this function is called an approach function in the game of pursuit.
Definition1.4. A n-strategy is said to be winning for the pursuer on the interval [0, T (u)] in the game (U, V) beginning from (¿10, ¿20) if, for any control v(-) £ V, there exists some time t* < T(u) such that ¿(t*) = 0. If this property is valid, then T(u) is called a guaranteed capture time.
Definition 1.5. A control function v*(t) : R+ ^ V is called a strategy of the evader if v*(t) is a Lebesgue measurable function with t, t > 0.
Now we are going to treat the game (U, V) from the perspective of the player Y.
Definition 1.6. A strategy v*(-) £ V is said to be winning for the evader in the game (U, V) beginning from (z10,z20) if, for any control u(-) £ U, the solution ¿(t) of Cauchy's problem
¿ + az = u(t) — v*(t), ¿(0) = ¿10, ¿(0) = 0, is not zero, i.e., ¿(t) = 0 at each t £ [0,
In this article, the following game problems will be individually investigated:
Problem 1.1. The problem of pursuit. To construct a n-strategy for the pursuer and to find a sufficient condition of capture.
Problem 1.2. The problem of evasion. To set an optimal strategy for the evader and to estimate how to alter the distance |z(t)| between the pursuer and the evader.
Problem 1.3. To construct an attainability domain of the pursuer (the set of capture points).
P r o b l e m 1.4. To solve the "lifeline" game.
§ 2. Solution of the problem of pursuit
In the present section, the n-strategy will be defined on the basis of the works [4,5,13-15,35] and a sufficient condition of capture will be taken.
If the players X and Y single out their admissible control functions u(-) £ U and v(-) £ V, respectively, then using (1.5)-(1.7), obtain the vector-function
z(t) = zw + i f ('u(s) - v(s)) (l - e-a(i"s)) ds. (2.1)
On account of (2.1), the pursuer aims for the attainment of the equality ¿(t*) = 0 at some t* > 0, while the evader strives to continue the relation ¿(t) = 0 for each t, t > 0. Consider the function
pit) = e~at + at - b, + (2.2)
a — p
The following assertion will be used to show the existence of capture time.
Proposition 2.1. Suppose a > P. Then the equation
p(t) = 0, t > 0, (2.3)
has one and only one positive root that will be denoted by T.
To construct the n-strategy, suppose that the pursuer is aware of the initial data ¿10, a, P and the value v(t) at the current time t.
Definition2.1. We say that the vector-function
^^,¿10) = v — y(v, ¿10)60, (2.4) is the n-strategy in the game (U, V), where
-y(v, z10) = {v, 60) + v>,£io>2 + a2-M2, (2-5) £10 = ¿10/1¿101, and (v, £10) is the scalar product of the vectors v and £10 in Rn.
It is worth noting that y(v, ¿10) in (2.5) is ordinarily termed a resolving function. Here are some elementary properties of (2.4) and (2.5).
Proposition 2.2.
a — P < y(v, ¿10) < a + P (2.6)
if and only if a > P.
Proposition 2.3. |un (v, ¿10)| = a holds for any v £ Bp. Definition2.2. For an arbitrary control v(-) £ V, the scalar function
T(v(-),t) = l--^— ft1(v(s),z10)(l-e-^-sAds (2.7)
a|¿lo| J0 v '
is called an approach function of the players X and Y if a > P.
L e m m a 2.1. Let a > P. Then for any control v(-) £ V: (a) r(v(-),t) monotonically decreases with t, t > 0;
(b) r(v(-),t) at each t £
where
0, TT is estimated as
^(i) < r(v(-),t) < r2(i), (2.8)
Proof. a) Determine the t-derivative of r(v(-), t) and from (2.6) it follows that
dm']>t] 1 (v(s),zw)e-^ds < ^(e"- - l) < 0.
ii, ,W),z ioje ^ 'ass—,-rie
dt ^10| J 0 ^¿101
b) Relying on the lemma about minimum-maximum in [36], we make the following estimates:
1 ^
r«),t) < 1 - -;-r min / -f{v{s),zw)(l-e-a{t-s))ds <
ajzioj
11
< 1 -
(t-~(l-e~at)) imn-y(v, z10) = T2(t).
V a v V v€B«
a|zio|
By the same lemma, we obtain
I\{t) = 1 - -¿—¡(t - -(1 - e-ai)) max7(v,z10) a|z10| V a v V veB^
1 ^
= 1-—-max /
a|zio| v(-)ev Jo
and the proof is complete. □
We can now formulate our main result for the problem of pursuit.
Theorem 2.1. If a > 0, then the H-strategy (2.4) will be winning for the pursuer on the
time interval
0,TT
, where T is the positive root of (2.3).
Proof. Let us first assume that the evader decides on any control v(-) e V and the pursuer utilizes the n-strategy (2.4). Then by virtue of (2.1) and (2.4), we have
z(t) = Zio - — [\l - e~a^)1(v(s),zlo)ds. a Jo
Since £10 = z10/|z101, rewrite (2.9) as
(2.9)
z(t) = r«),t)zio, (2.10)
where r(v(-),t) is the same as (2.7). Taking (2.8) into consideration yields
r«),t) < r2(t). (2.11)
Using (2.2), reduce r2(t) to the form
0 - a
r2(t) = ^r-Mt).
a2|zio|
From Proposition 2.1 it follows immediately that at a moment T, the equality p (t^ =0 is satisfied. We thus get r2 (T) = 0 and, in consequence, (2.11) indicates that there exists a
finite time T* e
0, TT
satisfying r(v(-),T*) = 0. For this reason, considering (2.10) we obtain z(T*) = 0, which completes the proof. □
§ 3. Solution of the problem of evasion
In this section, a constant function will be taken as a strategy of the evader and a sufficient condition of evasion will be given. Moreover, we will substantiate that a strategy of the evader is an optimal strategy and the time T stated in Theorem 2.1 is an optimal capture time.
Definition3.1. In the game (U, V), the control function
v*(t) = -Ö60 (3.1)
is called a strategy of the evader, where £10 = z10/|z10|.
Let us state our main result for the problem of evasion.
Theorem 3.1. (a) If a > 0, then the strategy (3.1) will be winning for the evader in the time interval 0, Tj;
(b) If a < 0, then the strategy (3.1) will be winning for the evader in the time interval [0, and for the distance between the players X and Y, the following estimate is satisfied:
i|z10|, if a = 0, |z( )l " (Jzial- ((a - 0)(e-at + at - l) /a2) , if a < 0.
Proof. (a) Let a > 0. Suppose that «(•) e U is an optional control of the pursuer and the evader applies the strategy (3.1). Then by means of (2.1), we have
z(t) = zw + - f u(s)(l-e~a{t-s))ds + ^ [\l - e~a{t-s)) ds. (3.2)
Taking (1.3) into account, the absolute value of (3.2) can be evaluated from below as follows:
r\ i , /.x. i #r>t
\z(t)\ >
Zl0 + ^l2 f\i - e-*-*)) ds - - I u(s)(l-e~a^)ds
a Jo a
>Ut-1-« 1
a|zio | ^ a / i a j o
- / u[ a .1 o
t
>
s I3»! 0+ ^ -1^ - - S / - <is
a|zio |
Based on the proof of Theorem 2.1, we can assert that the relation
|z(t)| > |zio|r2(t) > 0
is true for each t e 0, T^.
(b) Let a < 0. Then obtain the estimate as above once again and by (2.8), the following inequality is derived:
\z{t)I > |z10| - ——J— (e~at + at- 1) = r3(t). (3.3)
a2
Due to r3(0) = |zio| and r3(t) > 0 in the interval [0, it is evident that mint>o r3(t) = |zio|. This leads to z(t) = 0, which follows from (3.3), for all t, t e [0, The proof is complete.
§ 4. Attainability domain of players and its dynamics
In Theorem 2.1, we have proved that the pursuer seizes the evader through the n-strategy (2.4) at some point in Rn by the time T iff a > 0. Hence we are going to construct the set of capture points for the game (U, V).
The triplet (yio, y2o, v(-)), v(-) e V, produces the trajectory of the evader in the form (1.6) and the triplet (xio, x2o, «n(v(-), zio)), un(v(-), zio) e U, yields the trajectory
x(t) = + e~at^j uu(v(s),zw) (l - ds (4.1)
of the pursuer, where t e [0,T*], 0 < T* < T, and T* is the encounter time of the players X and Y, i.e., x(T*) = y(T*) holds. Accordingly, for each pair (x(t),y(t)) on [0,T*], we will consider the multi-valued mapping
ft(x(t),y(t)) = {^: - x(t)| > (a/0)|w - y(t)|>, (4.2)
o
o
and also, at t = 0 we have
^(xio,yio) = {ш: |ш - Xio| > (а/в)|ш - yio|}. (4.3)
Note that y(t) e Q(x(t),y(t)) for each t e [0,T*] is valid owing to |z(t)| > 0 on the interval [0,T*].
Remark 4.1. In fact, the trajectories x(t), y(t) and the set Q(x(t),y(t)) directly depend on how a control v( ) G V to be picked. For simplicity, we ignore this dependence.
L e m m a 4.1. For the set (4.2),
ft(x(t), y(t)) = x(t) + r(v(-), t) [Q(xio, yio) - xio] (4.4)
is satisfied, where r(v(-), t) is identical with (2.7) and
H(xio, yio) = xio - c(zio) + r(zio)B, (4.5)
( \ a2zi0 ( \ aP\Z™\ /-/I
фю) = r(zw) = (4.6)
and B is the unit ball centred at the origin in Rn.
Proof. Since z(t) = x(t) - y(t) (see (1.7)), it derives readily from (4.2) that the inclusion
ш G Q(x(t),y(t)) - x(t) is equivalent to
|ш|> (а/в)|ш + z(t)|. (4.7)
By reason of (4.2) and (4.7), the set Q(t) can be represented as
Q(x(t),y(t)) = x(t) + Q*(z(t)), ft*(z(t)) = {ш: |ш|> (а/в)|ш + z(t)|}. (4.8)
Let us confirm that the set Q*(z(t)) consists of a sphere. To do this, square both sides of (4.7) and make the following simplifications:
в2|ш|2 > а2(|ш|2 + 2(ш, z(t)) + |z(t)|2); (а2 - в2) |ш|2 + 2а2 (ш, z(t)) + a2|z(t)|2 < 0.
Divide both sides of the last inequality by (а2 - в2), i. e.,
. l2 / а2ф) \ а2|z(t)|2
Adding ( а2ф)/ (а2 - в2 Л to both sides of (4.9), we attain
a2z(tU ( а2ф) \ 2 ( tt2z(tn 2 а2|z(t) |2
а2 - в2/ \а2 - в2/ " \а2 - в2 J а2 - в2 а4 а2 \. . . |2 (ав|ф)|ч
(а2 - в2)2 а2 - в2У' W' V а2 - в2 or
а2г(£) ав |z(t)| ш + ^
а2 - в
< ^ (4.10)
So, the relation (4.10) is a sphere, which center is at the point c(z(t)) = —a2z(t)/ (a2 — 02) and which radius is equal to r(z(t)) = a0 |z(t)|/ (a2 — 02), thereby describing Q,(z(t)) in (4.8) as
Q,(z(t)) = {^: — c(z(t))| < r(z(t))}. (4.11)
Moreover, the set (4.11) can be written in the form Q,(z(t)) = c(z(t)) + r(z(t))B (see [37, p. 24]), and on that account it proceeds from (4.8) that
Q(x(t), y (t)) = x(t) + (z(t)) = x(t) + c(z(t)) + r(z(t))B. (4.12)
By (2.10) and (4.6), the functions c(z(t)) and r(z(t)) convert to the forms
c(z(t)) = —c(zio)r(v(-),t), r(z(t)) = r(zio )r«),t). (4.13)
Replacing (4.13) into (4.12) gives (4.4), and the proof is complete. □
Corollary 4.1. From Lemma 4.1 it may be concluded that at each t e [0,T,], the set Q(x(t),y (t)) is a ball of the radius r(zio)r(v(-), t) centered at the point x(t) — c(zio)r(v(-), t) and the set H(xio, yio) is a ball of the radius r(zio) centered at the point xio — c(zio).
L e m m a 4.2 (main lemma). Let
tf (x(t),y(t),x20) = il(x(t),y(t)) - —(1 - e~at).
a
Then ^(x(t2), y(t2), x2o) C ^(x(ti), y(ti), x2o) is true when ti < t2 for any ti, t2 e [0,T,]. P r o o f. It is not difficult to claim that (1.4) is equivalent to
0 2
(4-14)
From (2.5) we easily find
a2 — |v(t)|2 = Y (v(t),zio)(7 (v(t),zio) — 2(v(t), £io». (4.15)
Combining (4.15) with (4.14) and removing the brackets we get
20>(iUio) ( m w 02
K*)r + < V—(4.16)
Add (02y(v(t), zio)£io/ (a2 — 02))2 to both sides of (4.16) and simplify the right side of the result, i. e.,
2 20>(iUio) , , i027(v(t),zlo)Cio\2 ^ 02 2, .
K*)! +—I?-09 l\vv)-> zio) + --^- <-\v\t), Zio) +
a2 — 02 w' V a2 — 02 ) "a2 — 02
f 02 , 04 ^ 2 f \2
or
2
a0
< ^f^M^io). (4.17)
Clearly, the inequality
; 02 ^ .A ^ , 02
a2-/3
is valid for an arbitrary vector — e , | = l. From this and (4.17) we determine
a2 - 02 a2 - 0
Multiplying (4.18) by e-a(t-s), 0 < s < t, we take
«(*) + -TTJ,yM^), ¿10)60J ^ < Pa2_^2 <y(v(t),z10). (4.19)
Integrate (4.19) on [0, t] and obtain that
v(t) + T^L--27(v(t),z1o)Cio, Ip)< -^-¿j(v(t),z10). (4.18)
^ "v{s) +-J^iWs),zw)Zw) Ads< [\(v(s),z10)e-a^ds.
a2 - 02 a2 - 02 o
vis) h—J^—^y(vis),zm)tin 1 e u> ) as „ 1
a2 - 02 io io a2 - 02
(4.20)
From the left side of (4.20) we get the following equalities:
t 02 v[s' '
/0 \\
^ ^ ' a2
a2 - 02
v(s
o
a2 - 02
Y(v(s),zio)Ci^ e-a(t-s), -^ds = - l) Y(v(s),zio)£io) e-a(t-s), ^ds
= j*({v(s)-1(v(s),zlo)Cio)e-a{t-s\ -<p)ds + (^^10, j\(v(s),zl0)e-^-sUs. In view of the n-strategy (2.4) and the vector c(zio) in (4.6), ^om the last equality we obtain
t 02 v(s '
a2 - 02
Y(v(s),Zio)Ci^ e-a(t-s), ^ds =
(4.21)
lo / |zio | Jo
By virtue of r(zio) in (4.6), the right side of (4.20) can be expressed as
j^ 1(v(s),z1o)e~a^ ds = ^ jf 1(v(s),z1o)e~a^ ds. (4.22)
As a consequence of (4.20)-(4.22), we deduce that (J\n(v(8),z10)e-*-*d8, V) + (c(Zl0),^0~r(Zl0) l^f(v(s),zw)e-a^ ds < 0. (4.23)
Since, here, Q(x(t),y(t)) is mainly considered as the ball with a center and radius changing in time, it is easy to calculate its support function F(Q(x(t),y(t)), —) for any vector — e |-| = l (see [37]). Then taking account of (2.7), (4.1), (4.4)-(4.6) and on the strength of the properties of a support function, we take the t-derivative of F(Q(x(t), y(t)), —), i. e.,
d d
-F(i}(x(t),y(t)), 1>) = -F{x{t)+T{v{-),t)[r{z10)B-c{z10)], V) =
d d = -F{x{t)^)+F{r{z10)B-c{z10), ij)-T(v(-),t) =
dt \ a v a j 0
+ (r(zw)F(B, ip) - F(c(z10),VO)|(l - e~a^)dsy
+ ~e~at)+ll uu(v(s),z10)(l-e-a^)ds, ^ +
As F(B,^) = 1 and F(c(zio),^) = (c(zio),^) (see [37, p. 33]), from the latest equalities we arrive at the relation
jtF(Q(x(t),y(t)), = (e~atx2o,4>) + (^J\u(v(s),zl0)e-a^ds, ^ +
d
"ni''
0
|z10| ./0
(4.24)
Accommodating (4.24) we attain the following:
ft
e at X20.
+ ^ j0 «n(v(s),Z10)e-a(t-s) ds, ^ +
+ (c(zio), V) - r(zio) f 7{v{8)iZlo)e-<t-s)^_{e-atx3Dilp)i |z10| ./0
or
l-e-),
= / f* Uu(v(s), zw)e~a(-t~s'1 ds, A+(cM^)-r(zw) [\{v(s)) Zw)e-«(ts) ds.
\J0 / | z10| J0
As a result of (4.23), we see that
jtF(Q(x(t),y(t))~^f{ l~e~at), V) <0
holds for every ^ G = 1. This finishes the proof. □
By Lemma 4.2, we will construct an attainability domain of the evader.
L e m m a 4.3. The inclusion
yit) G n(x10,yw) + — (1 - e~at) (4.25)
a
holds for all t e [0, T,].
Proof. On account of Lemma 4.2, we derive the relation
Q(x(t),y(t)) - — (1 - e~at) C Q(xl0,yw),
a
or
Q(x(t),y(t)) C Q(x10,yio) + —(1 - e~at). (4.26)
a
Obviously, y(t) e H(x(t),y(t)) on the interval [0,T,] (see (4.2)) and therefore from (4.26) it proceeds the validity of (4.25), which completes the proof. □
Definition 4.1. We call the set
T
Six (xw,x2o,yw,f^ = U{n(x10,yio) + — (1 - e~ai) |
t=o a
the attainability domain of the pursuer in the game (U, V), where T is the guaranteed capture time (see Theorem 2.1).
Corollary 4.2. If x2o = y2o = 0, then the attainability domain of the evader consists of the set Q(xio, yio) (the Apollonius sphere) in (4.3).
§ 5. Solution of the "lifeline" game
In the present section, the game (1. 1)—(1.4) with the "lifeline" L will be solved to the advantage of both players X and Y.
Definition 5.1. In the "lifeline" game, the n-strategy (2.4) is said to be winning for the pursuer on the time interval [0,T] if, for an arbitrary control v(-) e V of the evader, there is some moment i e [0, T] such that:
(1) x(i) = y(i);
(2) y(t) e L at each t e [0,i].
Theorem 5.1. If a > 0 and (xio, x2o, yio, T) H L = 0, then the n-strategy (2.4) will be winning for the pursuer on the time interval [0, T] in the "lifeline" game.
Proof. The proof follows immediately from Theorem 2.1, Lemma 4.2 and 4.3. □
And now, we will consider the "lifeline" game from the point of view of the evader.
Definition 5.2. It is said that a control v*(-) e V is winning for the evader in the "lifeline" game if, for an optional control u(-) e U of the pursuer:
(a) there is some finite moment t such that y(i) e L and x(t) = y(t) for 0 < t < i; or
(b) x(t) = y(t) at each t e [0,
Let us take the sets
T>,y10) = {U: e~at* +aU = l + ^ ~ Vl°l, oj G Q(0)}, (5.1)
Qy(uj,x20) = {w*: uj* = c + - e~ai*), w G il(0)}. (5.2)
In the "lifeline" game, we call the set (5.2) the attainability domain of the evader.
Theorem 5.2. If a > 0 and (w, x2o) H L = 0, then there is such control v*(-) e V that is winning for the evader in the "lifeline" game.
Proof. According to the theorem, there exists at least one point w* e (w,x2o) H L satisfying the equality in (5.2). Then we fix the evader up with the constant control
w - yio ,,
|w - yio|
First off, let us demonstrate that by the control (5.3) the evader gets to the target point w* at some time t* e T* (w, yio). Indeed, due to x2o = y2o, by means of (1.6) we obtain
y (Q = yw + — (1 - e~ai*) + ^ (e~at* + aU ~ l) • (5.4)
a a2
Substituting (5.3) in (5.4) and allowing for the equality in (5.1) we achieve the desired result, i. e.,
y (Q = yw + — (1 - e~aU) +uo-ylo a v '
u*.
Now, we need to show the validity of the condition (a) of Definition 5.2, that is, the evader can escape by using (5.3) on the interval [0,t*]. Assume conversely, or, more precisely, for the pursuer there exists a control «*(•) e U by which the equality x (t*) = y (t*) occurs at some t* e [0, t*). Consequently, depending on (2.1), we get the vector
t*
z if) = z10 + - / (u*(s) - v*) (1 - e-^*--)) ds
(5.5)
It is easy to see that z (t*) = 0 holds, thanks to z(t) = x(t) — y(t) (see (1.7)). As a result, considering (1.3) we find the following estimations from the right side of (5.5):
z10
(1 — e-a(t*-s)) v* ds
<- I (l-e-a(t*-s))\u^s)\ds<^(e~at*+at*-l)
or
zw--1 (e~at* + at* — l)
a2 v y
<-2{e~at* + af - 1).
(5.6)
Let us introduce the notation A = (e at* + at* — 1) /a2 in (5.6). Then square both sides of (5.6) and because of |v*| = 0, we come to the quadratic inequality with A of the form
(a2 — 02) A2 + 2(zio,v*)A — |zio|2 > 0. As A > 0, from the latter inequality we derive 1
a2 — 02
z10,^)2 + (a2 - 02) |z10|2 - (z10,^)) < A = — (e~at + at* - l). (5.7)
Owing to t* < t* and taking the equality in (5.1) on board, it arises from (5.7) that
1
a2 — 02
zio,v*)2 + (a2 — 02) |zio|2 — (zio, v*n <
IW - yio\
0 '
(5.8)
In accordance with (5.3), we find by (5.8) that 0|w — xio| < a|w — yio|, i.e., u e ^(xio,yio) (see (4.3)), and this conflicts with our supposition. The proof is complete. □
Remark 5.1. According to the definitions of (xio, x2o, yio, T) and (w, x2o), we can conclude that (w,x2o) C (xio,x2o,yio,T) is reasonable in the "lifeline" game.
Theorem 5.3. Let a < 0. Then there is a control v*(-) e V, which is winning for the evader in the "lifeline " game.
Proof. The proof follows directly from Theorem 3.1.
§6. Examples
Below we will first address a problem for the game with the "lifeline".
□
o
*
*
t
t
1
a
o
o
Example 6.1. Let us analyze the game given by the following differential equations and initial conditions:
X: i + i = u, xio = (7,14), ¿2o = (0,1), |u(t)| < 2, t > 0, (6.1) Y: y + y = v, yio = (4, 8), y2o = (0,1), |v(t)| < 1, t > 0. (6.2)
In terms of (2.2) and Proposition 2.1, we get T « 7.71. On the basis of (4.5) and (4.6), we obtain
n(xio,yio) = {w =0^2): (wi - 4)2 + (^2 - 8)2 < 20}. (6.3)
It is obvious that the set (6.3) is equivalent to
Q(r) = {oor = : (uol - 4)2 + (a;2 - 8)2 = r2, r G [0, 2^5] }. (6.4)
Relying on (6.4) we determine the set (5.1) as follows:
T*(r) = {tr: e~tr + tr = 1 + r, r G (0, 2\/5
In view of (5.2) and (6.4), we generate the system
= ^
= + 1 - e-tr
for each r G (0,275]. From this system we find the set
QY(r,tr) = {w* = : ul = 8±^/r2-(o;i-4)2 + l - e~u, r G(^0,2^5 }
as the attainability domain of the evader in the game (6.1), (6.2) with the "lifeline".
Now, we will consider constructing the attainability domain for the case with multiple pursuers and one evader.
Example 6.2. Let the following game example be given:
Xj: ¿j + ai = Uj, x(0) = xjo, £¿(0) = 0, |uj(t)| < Si, t > 0, (6.5) Y: y + ay = v, y(0) = yo, y/(0) = |v(t)| < 1, t > 0, (6.6)
where a > 0, Si > 1, £¿0 ^ yo, i = I, m.
Based on Lemma 4.3, we can write the relation
m 0 y(t)ef)ni(xlo,yo) + -{l-e~at),
i=i
where
(xio, yo) = Xio - Ci(zio) + ri(zio)B, Ci(zio) = i2Zio/ (i2 - 1),
ri(Zio) = ¿i |Zio | / (i2 - 1) , Zio = Xio - yo-For the game (6.5), (6.6) we can define the attainability domain of the pursuers in the form
T
^Xi ,...,xm(xio ,yo,0,T) = U
t=o
where
' 0
• 1 a
i=i
T = min { t: e~at + at = 1 + -- (
i=l,m [ Si — 1 J
-at , „4_, , a"\zio\ \
Acknowledgments. The authors are greatly indebted to Professor A. A. Azamov for discussing this paper and for several helpful comments.
REFERENCES
1. Isaacs R. Differential games, New York: John Wiley and Sons, 1965.
2. Pontryagin L. S. Izbrannye trudy (Selected works), Moscow: MAKS Press, 2004.
3. Krasovskii N.N. Upravlenie dinamicheskoi sistemoi. Zadacha o minimume garantirovannogo re-zul'tata (Control by dynamical system. The minimum problem of a guaranteed result), Moscow: Nauka, 1985. https://zbmath.org/0589.93001
4. Pshenichnyi B.N. Simple pursuit by several objects, Cybernetics, 1976, vol. 12, issue 3, pp. 484-485. https://doi.org/10.1007/bf01070036
5. Petrosjan L. A. Differential games of pursuit, Singapore: World Scientific Publishing, 1993. https://doi.org/10.1142/1670
6. Friedman A. Differential games, New York: Wiley, 1971. https://zbmath.org/0229.90060
7. Fleming W. H. The convergence problem for differential games, II, Advances in game theory, 1964, pp. 195-210. https://doi.org/10.1515/9781400882014-013
8. Bercovitz L. D. Differential games of generalized pursuit and evasion, SIAM Journal on Control and Optimization, 1986, vol. 24, issue 3, pp. 361-373. https://doi.org/10.1137/0324021
9. Chikrii A. Conflict-controlled processes, Dordrecht: Springer, 1997. https://doi.org/10.1007/978-94-017-1135-7
10. Subbotin A. I. Generalization of the main equation of differential game theory, Journal of Optimization Theory and Applications, 1984, vol. 43, issue 1, pp. 103-133. https://doi.org/10.1007/BF00934749
11. Satimov N.Yu. Metody resheniya zadachi presledovaniya v teorii differentsial'nykh igr (Methods of solving pursuit problems in the theory of differential games), Tashkent: National University of Uzbekistan, 2019.
12. Petrov N. N. "Soft" capture in Pontryagin's example with many participants, Journal of Applied Mathematics and Mechanics, 2003, vol. 67, issue 5, pp. 671-680. https://doi.org/10.1016/S0021-8928(03)90040-2
13. Azamov A. On the quality problem for simple pursuit games with constraint, Serdica. Bulgariacae mathematicae publicationes, 1986, vol. 12, no. 1, pp. 38-43 (in Russian). http://www.math.bas.bg/serdica/1986/1986-038-043.pdf
14. Azamov A. A., Samatov B.T. The n-strategy: Analogies and applications, Contributions to Game Theory and Management, 2011, vol. 4, pp. 33-46. https://www.mathnet.ru/eng/cgtm177
15. Azamov A. A., Samatov B.T. n-strategy, Tashkent: National University of Uzbekistan, 2000.
16. Samatov B. T. Problems of group pursuit with integral constraints on controls of the players. I, Cybernetics and Systems Analysis, 2013, vol. 49, issue 5, pp. 756-767. https://doi.org/10.1007/s10559-013-9563-7
17. Petrov N.N. Simple group pursuit subject to phase constraints and data delay, Journal of Computer and Systems Sciences International, 2018, vol. 57, issue 1, pp. 37-42. https://doi.org/10.1134/S1064230718010094
18. Petrov N.N., Narmanov A. Ya. Multiple capture of a given number of evaders in the problem of a simple pursuit, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 2, pp. 193-198 (in Russian). https://doi.org/10.20537/vm180205
19. Grigorenko N. L. Matematicheskie metody upravleniya neskol'kimi dinamicheskimiprotsessami (Mathematical methods for controlling several dynamic processes), Moscow: Moscow State University, 1990.
20. Vagin D. A., Petrov N.N. Pursuit of a group of evaders in the Pontryagin example, Journal of Applied Mathematics and Mechanics, 2004, vol. 68, issue 4, pp. 555-560. https://doi.org/10.1016/jjappmathmech.2004.07.008
21. Kornev D. V., Lukoyanov N. Yu. On a minimax control problem for a positional functional under geometric and integral constraints on control actions, Proceedings of the Steklov Institute of Mathematics, 2016, vol. 293, suppl. 1, pp. 85-100. https://doi.org/10.1134/S0081543816050096
22. Ibragimov G., Ferrara M., Ruziboev M., Pansera B. A. Linear evasion differential game of one evader and several pursuers with integral constraints, International Journal of Game Theory, 2021, vol. 50, issue 3, pp. 729-750. https://doi.org/10.1007/s00182-021-00760-6
23. Ibragimov G.I. The optimal pursuit problem reduced to an infinite system of differential equations, Journal of Applied Mathematics and Mechanics, 2013, vol. 77, issue 5, pp. 470-476.
https://doi. org/10.1016/j .j appmathmech.2013.12.002
24. Samatov B. T. On a pursuit-evasion problem under a linear change of the pursuer resource, Siberian Advances in Mathematics, 2013, vol. 23, issue 4, pp. 294-302. https://doi.org/10.3103/S1055134413040056
25. Samatov B. T. The pursuit-evasion problem under integral-geometric constraints on pursuer controls, Automation and Remote Control, 2013, vol. 74, issue 7, pp. 1072-1081. https://doi.org/10.1134/S0005117913070023
26. Bannikov A. S. Some non-stationary problems of group pursuit, Izvestiya Instituta Matematiki i Infor-matiki Udmurtskogo Gosudarstvennogo Universiteta, 2013, issue 1 (41), pp. 3-46 (in Russian). https ://www. mathnet. ru/eng/iimi247
27. Bannikov A. S., Petrov N. N. On a nonstationary problem of group pursuit, Proceedings of the Steklov Institute of Mathematics, 2010, vol. 271, suppl. 1, pp. 41-52. https://doi.org/10.1134/S0081543810070047
28. Samatov B. T., Ibragimov G., Khodjibayeva I. V. Pursuit-evasion differential games with the Gronwall type constraints on controls, Ural Mathematical Journal, 2020, vol. 6, no. 2, pp. 95-107. https://doi.org/10.15826/umj.2020.2.010
29. Yuldashev T. K. Nonlinear optimal control of thermal processes in a nonlinear inverse problem, Lobachevskii Journal of Mathematics, 2020, vol. 41, issue 1, pp. 124-136. https://doi.org/10.1134/S1995080220010163
30. Vassilina G. K. Optimal control problem of stochastic systems, Lobachevskii Journal of Mathematics, 2021, vol. 42, issue 3, pp. 641-648. https://doi.org/10.1134/S1995080221030215
31. Ushakov V. N., Ershov A. A., Ushakov A. V., Kuvshinov O. A. Control system depending on a parameter, Ural Mathematical Journal, 2021, vol. 7, no. 1, pp. 120-159. https://doi.org/10.15826/umj.2021.L011
32. Filippova T. F. Estimates of reachable sets for systems with impulsive control, uncertainty and nonlinearity, Izvestiya Irkutskogo Gosudarstvennogo Universiteta. Ser. Matematika, 2017, vol. 19, pp. 205-216 (in Russian). https://doi.org/10.26516/1997-7670.2017.19.205
33. Munts N. V., Kumkov S. S. Numerical method for solving time-optimal differential games with lifeline, Matematicheskaya Teoriya Igr i Ee Prilozheniya, 2018, vol. 10, issue 3, pp. 48-75 (in Russian). https ://www. mathnet. ru/eng/mgta222
34. Munts N.V., Kumkov S.S. On the coincidence of the minimax solution and the value function in a time-optimal game with a lifeline, Proceedings of the Steklov Institute of Mathematics, 2019, vol. 305, suppl. 1, pp. S125-S139. https://doi.org/10.1134/S0081543819040138
35. Samatov B. T., Soyibboev U. B. Differential game with a lifeline for the inertial movements of players, Ural Mathematical Journal, 2021, vol. 7, no. 2, pp. 94-109. https://doi.org/10.15826/umj.202L2.007
36. Alekseev V. M., Tikhomirov V. M., Fomin S.V. Optimal'noe upravlenie (Optimal control), Moscow: Nauka, 1979.
37. Blagodatskikh V.I. Vvedenie v optimal'noe upravlenie (lineinaya teoriya) (Introduction to optimal control (linear theory)), Moscow: Vysshaya Shkola, 2001.
Received 02.11.2022 Accepted 20.01.2023
Bahrom Tadjiahmatovich 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
Ulmasjon Boykuzi ugli Soyibboev, PhD student, Department of Mathematical Analysis, Namangan State University, ul. Uychi, 316, Namangan, 116019, Uzbekistan. ORCID: https://orcid.org/0000-0003-0546-427X E-mail: ulmasjonsoyibboev@gmail.com
Citation: B. T. Samatov, U. B. Soyibboev. Differential game with "lifeline" for Pontryagin's control example, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2023, vol. 61, pp. 94-113.
Б. T. Саматов, У. Б. Сойиббоев
Дифференциальная игра с «линией жизни» для контрольного примера Понтрягина
Ключевые слова: дифференциальная игра, преследование, уклонение, ускорение, стратегия, гарантированное время поимки, область достижимости, линия жизни.
УДК: 517.977
DOI: 10.35634/2226-3594-2023-61-06
Основной целью данной работы является решение одной из основных задач Айзекса, а именно игры с «линией жизни» на контрольном примере Понтрягина, когда оба игрока имеют одинаковую динамику движения. Для решения этой задачи преследователю предлагается стратегия параллельного преследования (кратко П-стратегия), обеспечивающая скорейшее сближение игроков и поимку убегающего в пределах некоторого замкнутого шара. Кроме того, для рассматриваемой дифференциальной игры приводится явная аналитическая формула для области достижимости игроков и обобщается основная лемма (лемма Л. А. Петросяна о монотонности относительно вложения области достижимости для игры простого преследования). Используя эту основную лемму, мы находим условия разрешимости игры с «линией жизни» и для контрольного примера Понтрягина. Для наглядности в конце работы приведены примеры для некоторых частных случаев.
СПИСОК ЛИТЕРАТУРЫ
1. Айзекс Р. Дифференциальные игры. М.: Мир, 1967.
2. Понтрягин Л. С. Избранные труды. М.: МАКС Пресс, 2004.
3. Красовский Н. Н. Управление динамической системой. Задача о минимуме гарантированного результата. М.: Наука, 1985.
4. Пшеничный Б. Н. Простое преследование несколькими объектами // Кибернетика. 1976. Т. 12. № 3. С. 145-146.
5. Petrosjan L. A. Differential games of pursuit. Singapore: World Scientific Publishing, 1993. https://doi.org/10.1142/1670
6. Friedman A. Differential games. New York: Wiley, 1971. https://zbmath.org/0229.90060
7. Fleming W. H. The convergence problem for differential games, II // Advances in game theory. 1964. P. 195-210. https://doi.org/10.1515/9781400882014-013
8. Bercovitz L. D. Differential games of generalized pursuit and evasion // SIAM Journal on Control and Optimization. 1986. Vol. 24. Issue 3. P. 361-373. https://doi.org/10.1137/0324021
9. Чикрий А. А. Конфликтно управляемые процессы. Киев: Наукова думка, 1992.
10. Subbotin A. I. Generalization of the main equation of differential game theory // Journal of Optimization Theory and Applications. 1984. Vol. 43. Issue 1. P. 103-133. https://doi.org/10.1007/BF00934749
11. Сатимов Н. Ю. Методы решения задачи преследования в теории дифференциальных игр. Ташкент: Изд-во НУУз, 2019.
12. Петров Н. Н. «Мягкая» поимка в примере Л. С. Понтрягина со многими участниками // Прикладная математика и механика. 2003. Т. 67. Вып. 5. P. 759-770. https://elibrary.ru/item.asp?id= 17296313
13. Азамов А. О задаче качества для игр простого преследования с ограничением // Сердика. Бъл-гарско математическо списание. 1986. Т. 12. № 1. С. 38-43. http://www.math.bas.bg/serdica/1986/1986-038-043.pdf
14. Azamov A. A., Samatov B. T. The П-strategy: Analogies and applications // Contributions to Game Theory and Management. 2011. Vol. 4. P. 33-46. https://www.mathnet.ru/rus/cgtm177
15. Azamov A. A., Samatov B. T. П-strategy. Tashkent: National University of Uzbekistan, 2000.
16. Саматов Б. Т. О задачах группового преследования при интегральных ограничениях на управление игроков. I // Кибернетика и системный анализ. 2013. Т. 49. № 5. С. 132-145.
17. Petrov N.N. Simple group pursuit subject to phase constraints and data delay // Journal of Computer and Systems Sciences International. 2018. Vol. 57. Issue 1. P. 37-42. https://doi.org/10.1134/S1064230718010094
18. Петров Н. Н., Нарманов А. Я. Многократная поимка заданного числа убегающих в задаче простого преследования // Вестник Удмуртского университета. Математика. Механика. Компьютерные науки. 2018. T. 28. Вып. 2. C. 193-198. https://doi.org/10.20537/vm180205
19. Григоренко Н. Л. Математические методы управления несколькими динамическими процессами. M.: Изд-во Моск. ун-та. 1990.
20. Вагин Д. А., Петров Н. Н. Преследование группы убегающих в примере Понтрягина // Прикладная математика и механика. 2004. Т. 68. Вып. 4. С. 623-628.
https://elibrary.ru/item. asp?id= 17670019
21. Корнев Д. В., Лукоянов Н. Ю. К задаче управления на минимакс позиционного функционала при геометрических и интегральных ограничениях на управляющие воздействия // Труды Института математики и механики УрО РАН. 2015. Т. 21. № 2. С. 87-101. https://www.mathnet.ru/rus/timm1173
22. Ibragimov G., Ferrara M., Ruziboev M., Pansera B. A. Linear evasion differential game of one evader and several pursuers with integral constraints // International Journal of Game Theory. 2021. Vol. 50. Issue 3. P. 729-750. https://doi.org/10.1007/s00182-021-00760-6
23. Ибрагимов Г. И. Задача об оптимальном преследовании, приводимая к бесконечной системе дифференциальных уравнений // Прикладная математика и механика. 2013. Т. 77. Вып. 5. С. 657-666. https://elibrary.ru/item.asp?id=20406054
24. Саматов Б. Т. О задаче преследования-убегания при линейном изменении ресурса преследователя // Математические труды. 2012. Т. 15. № 2. С. 159-171. https://www.mathnet.ru/rus/mt234
25. Саматов Б. Т. Задача преследования-убегания при интегрально-геометрических ограничениях на управления преследователя // Автоматика и телемеханика. 2013. Вып. 7. С. 17-28. https://www.mathnet.ru/rus/at5465
26. Банников А. С. Некоторые нестационарные задачи группового преследования // Известия Института математики и информатики Удмуртского государственного университета. 2013. Вып. 1 (41). С. 3-46. https://www.mathnet.ru/rus/iimi247
27. Банников А. С., Петров Н. Н. К нестационарной задаче группового преследования // Труды Института математики и механики УрО РАН. 2010. Т. 16. № 1. С. 40-51. https://www.mathnet.ru/rus/timm526
28. Samatov B. T., Ibragimov G., Khodjibayeva I. V. Pursuit-evasion differential games with the Gronwall type constraints on controls // Ural Mathematical Journal. 2020. Vol. 6. No. 2. P. 95-107. https://doi.org/10.15826/umj.2020.2.010
29. Yuldashev T. K. Nonlinear optimal control of thermal processes in a nonlinear inverse problem // Lobachevskii Journal of Mathematics. 2020. Vol. 41. Issue 1. P. 124-136. https://doi.org/10.1134/S1995080220010163
30. Vassilina G. K. Optimal control problem of stochastic systems // Lobachevskii Journal of Mathematics. 2021. Vol. 42. Issue 3. P. 641-648. https://doi.org/10.1134/S1995080221030215
31. Ushakov V. N., Ershov A. A., Ushakov A. V., Kuvshinov O.A. Control system depending on a parameter // Ural Mathematical Journal. 2021. Vol. 7. No. 1. P. 120-159.
https://doi.org/10.15826/umj .2021.1.011
32. Филиппова Т. Ф. Оценки множеств достижимости систем с импульсным управлением, неопределенностью и нелинейностью // Известия Иркутского государственного университета. Сер. Математика. 2017. Т. 19. С. 205-216. https://doi.org/10.26516/1997-7670.2017.19.205
33. Мунц Н. В., Кумков С. С. Численный метод решения дифференциальных игр быстродействия с линией жизни // Математическая теория игр и её приложения. Т. 10. Вып. 3. С. 48-75. https://www.mathnet.ru/rus/mgta222
34. Мунц Н. В., Кумков С. С. О совпадении минимаксного решения и функции цены игры быстродействия с линией жизни // Труды Института математики и механики УрО РАН. 2018. Т. 24. № 2. С. 200-214. https://doi.org/10.21538/0134-4889-2018-24-2-200-214
35. Samatov B. T., Soyibboev U. B. Differential game with a lifeline for the inertial movements of
players // Ural Mathematical Journal. 2021. Vol. 7. No. 2. P. 94-109. https://doi.org/10.15826/umj.202L2.007
36. Алексеев В.М., Тихомиров В. М., Фомин С. В. Оптимальное управление. М.: Наука, 1979.
37. Благодатских В. И. Введение в оптимальное управление (линейная теория). M.: Высшая школа, 2001.
Поступила в редакцию 02.11.2022
Принята к публикации 20.01.2023
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
Сойиббоев Улмасжон Бойкузи угли, аспирант, кафедра математического анализа, Наманганский государственный университет, 116019, Узбекистан, г. Наманган, ул. Уйчи, 316. ORCID: https://orcid.org/0000-0003-0546-427X E-mail: ulmasjonsoyibboev@gmail.com
Цитирование: Б. T. Саматов, У. Б. Сойиббоев. Дифференциальная игра с «линией жизни» для контрольного примера Понтрягина // Известия Института математики и информатики Удмуртского государственного университета. 2023. Т. 61. С. 94-113.
