3Uchinchi renessansyosh olimlari: zamonaviy vazifalar,
innovatsiya va istiqbol Young Scientists of the Third Renaissance: Current Challenges, Innovations and Prospects
A DIFFERENTIAL GAME OF i -CATCH UNDER EXPONENTIAL DECREASING CONSTRAINT IN TERMS OF ACCELERATION
A. A. A3aM0B
DSc, professor, V.I.Romanovskii Institute of Mathematics Academy of Sciences of the Republic of Uzbekistan.
M. A. Turgunboyeva PhD student, Namangan State University Uzbekistan.
Theory of Differential Games looks into conflict problems in systems which are expressed by differential equations. As a result of the growth of Pontryagin's maximum principle, it became apparent that there was a link between optimal control theory and differential games. Actually, problems of differential game describe a generalization of optimal control problems in cases where there are more than one player.
The study of differential games was initiated by American mathematician R.Isaacs. His research was published in the form of a monograph [5, p. 340] in 1965, in which a great number of examples were considered, and theoretical questions were only affected. Differential games have been one of the basic research fields since the 1960th and their fundamental results were gained by L.S.Pontryagin [9, p. 551], N.N.Krasovsky [3, p. 520]
The problem for the case of ^ -approach [8, p. 272] was first studied by Indian mathematician Ramchundra. Analogous effects in the case of geometrical constraint were considered in the works of Pshenichnyi [10, p. 484], Chikrii [3, p. 56], Petrosjan [11, p. 140], Satimov [12, p. 26], Grigorenko [4, p. 20], Azamov [1, p. 38], Azamov and Samatov[2, p. 33], Samatov [13], Khaidarov [7, p. 574].
In this paper, we investigate P -catch problem for a differential game of two players, a pursuer and an evader, in an inertial motion with exponential with decreasing constraints in terms of acceleration. We suggest an approach strategy, which is based on the method of resolving function, for a pursuer and by means of the strategy, new sufficient conditions of £ -catch are defined.
i—i n
Consider two players, a pursuer P and an evader E, moving in the space u . Suppose that x and y are the locations of the pursuer and the evader respectively.
Let movements of the pursuer and the evader be expressed by the differential equations and initial conditions
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Uchinchi renessans yosh olimlari: zamonaviy vazifalar,
innovatsiya va istiqbol Young Scientists of the Third Renaissance: Current
X = U, x(0) = x0, X(0) = X! ^
y = v, y(0) = y0, y(Q) = y^ (2)
n" >
correspondingly, where x>y>u>v e u » n ~ z? are the initial locations of the
players in which it is supposed that lx° y°l> ^^ > 0; Xl'yi are their initial velocity vectors accordingly; the acceleration vectors u v serve as control parameters of the players respectively, and they varies in respect to the time t - °.
The controls u and v will be picked as Lebesgue measurable functions
u() • [0, —» □ an(j v(). [0, +00) □ accordingly. We represent by ^1 a set off
all the measurable functions u () such that fulfill the geometric constraints (briefly, G-constraint)
-kt
(3)
|u(t )|< ae for almost every t - ° where a>k >°.
In the same way, we indicate by ^ a set of all the measurable functions fulfilling the G-constraint
|v(t)\ < Pe for almost every t - ° (4)
where P> k > °.
If "(•) e Ul and vQg^ then by yirtue of the equations ( i )^ (2), the triplets (x°, X1, u ()), (y°, yu v()) define the motion traj ectories
x(t ) = x0 + x1t + J (t - s)u ( s)ds,
0
t
y(t ) = Уо + y it + J (t - s)v(s)ds,
(5)
(6)
of the players correspondingly.
Definition 3.4.1. The measurable function (vOe ^)iscanecl an
admissible control of player P (of player E).
Note that the pair 0f the introduced classes defines a differential game.
That the main target of the pursuer is to approach the evader at the range at ^ > 0 - catch problem), that is, to reach the inequality
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23
Uchinchi renessansyosh olimlari: zamonaviy vazifalar,
innovatsiya va istiqbol Young Scientists of the Third Renaissance: Current _g ChaLges^nsand^s
- y(H)\ <* (7)
for some ^ > 0. But the basic target of the evader is to prevent occurring the above mentioned inequality, i.e. to keep the relation (evasion problem) ^ ^ >
forall ^ * ~ ^, and if it is impossible, the evader strives to delay the instant of catch.
Let's introduce the following denotations:
z(t) = x(t) - y(t), z0=x0-y0, z(0) = xl-yl.
Then the equation (1), (2) come to the unique Cauchy's problem in the form
z = u-v, z(0) = z0, z(0) = z1 ^
Note that we will consider the differential game (1)-(4) for the case
Zj =mz0,m gD
Definition 3.4.2. For a-fl, we say the function
u (t, v ) = v -Â( z0, v)
-kt
ae z0 + vt ae U +Z(t,v)£
(9)
n
an approach strategy or ' -strategy of the player P in the differential game (1)-
(4) of ^ _ catch, where,
h = |z0| —£
Ä(t, v) =
h
(v,zo) + ae ktl+ v,zo) +
-kt
ae
,2/ 2 -2kt 2 I |2
+ h [a e I — v
and is the scalar product of the vectors v and z° in □ . Moreover, the
function z°'v ) is generally titled the resolving function. Theorem 3.4.1. Let one of the conditions
a> ß, m <
k a—ß)
k (a\z0\ -pt) + (a-P)P
a = P, m < 0
be satisfied. Then the strategy (9) guarantees to occur the relation (7) on the
interval in the ~ catch problem (l)-(4), where ^ is a guaranteed approach
time and it is the first positive root of the equation
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Tashkent Institute of Economics and Pedagogy
Uchinchi renessans yosh olimlari: zamonaviy vazifalar,
innovatsiya va istiqbol Young Scientists of the Third Renaissance: Current _Challenges, Innovations and Prospects
e
■kt
akt 4 b, b = 14
a-ß a-ß
REFERENCES
, , , k2(|z0|-£) |z0|mk b,b = 1H---—--,a = 1-J—1-
1. Azamov A. (1986) On the quality problem for simple pursuit games with constraint. SerdicaBulgariacae math. Publ.Sofia, Vol.12, No.1, p. 38-43.
2. Azamov A.A., Samatov B.T. (2010) The P-Strategy: Analogies and Applications. The
Fourth International Conference on Game Theory and Management, St. Petersburg, Vol.4, P. 33-47.
3. Chikrii A.A. (1997) Conflict-Controlled Processes. Kluwer, Dordrecht, DOI 10.1007/978-94-017-1135-7
4. Grigorenko N.L. (1990) Mathematical Methods of Control for Several Dynamic Processes. Izdat. Gos. Univ., Moscow.
5. Isaacs R. (1965) Differential games. John Wiley and Sons, New York. 340 p.
6. Krasovsky N.N. (1985) Control of a Dynamical System. Nauka, Moscow, 520 p. (In Russian)
7. Khaidarov B.K. (1984) Positional l-catch in the game of one evader and several pursuers, Prikl. Matem. Mekh.,48, No. 4, p. 574-579.
8. Nahin P.J. (2012) Chases and Escapes: The Mathematics of Pursuit and Evasion. Princeton University Press, Princeton. 272 p.
9. Pontryagin L.S. (2014) Selected Works. MAKS Press, Moscow. 551 p. (In Russian)
10.Pshenichnyi B.N. (1976) Simple pursuit by several objects. Cybernetics and System Analysis. Vol. 12, No. 5, 484-485. DOI 10.1007/BF01070036
11.Petrosjan L.A. (1993) Differential games of pursuit. Series on optimization. Vol.2. World Scientific Poblishing, Singapore.
12.SatimovN.Yu. (2003) Methods for Solving the Pursuit Problem in the Theory of
Differential Games. Izd-voNUUz, Tashkent. (In Russian) 13.Samatov B.T. (2013) Problems of group pursuit with integral constraints on controls of the players II. Cybernetics and Systems Analysis, Vol. 49, No. 6, P. 907-921. DOI 10.1007/s10559-013-9581-5
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