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ОСНОВНОЙ РАЗДЕЛ
Doliyev O.B.
Namangan Institute of Engineering Technology Republic of Uzbekistan, Namangan city Mirzamahmudov U.A.
Namangan Institute of Engineering Technology. Republic of Uzbekistan, Namangan city
THE STRATEGY OF PARALLEL PURSUIT FOR DIFFERENTIAL GAME OF THE SECOND ORDER
Abstract. In this work is considered a differtial game of the second order, when control functions of the players satisfies geometric constraints. The proposed method substantiates the parallel approach strategy in this differential game of the second order. The new sufficient solvability conditions are obtained for problem of the pursuit.
Keywords. Differential game, geometric constraint, evader, pursuer, strategy of the parallel pursuit, acceleration.
Introduction
Let P and E objects with opposite aim be given in the space Rn and their movements based on the following differential equations and initial conditions
P: x = u,x -kx0 = 0, u <«,(1)
E: y = v, y -ky0 = 0, |v| <0, (2)
where x, y, u, v e Rn; x - a position of P object in the space Rn, x0 = x(0), x = i:(0) - its initial position and velocity respectively at t = 0; u -being a controlled acceleration of the pursuer, mapping u: [0, w) ^ Rn and it is chosen as a measurable function with respect to time; we denote a set of all measurable functions u (•) such that satisfies the condition |u| < a by U. y - a
position of E object in the space R", y0 = 3^(0), y1 = - its initial position and velocity respectively at t = 0; v - being a controlled acceleration of the evader, mapping v : [0, w) ^ Rn and it is chosen as a measurable function with respect to time; we denote a set of all measurable functions v (•) such that satisfies the condition v| <0 by V.
Research Methods and the Received Results. Definition 1. For a trio of (x0,x,u()),u() eU, the solution of the equation (1), that is,
t s
x(t) = x0 + xYt + (r)drds is called a trajectory of the pursuer on interval
0 0
t > 0.
Definition 2. For a trio of (y0, y ,v(-)), v(-) eV, the solution of the
t s
equation (2), that is, y(t) = y0 + yxt + ||v(r)drds is called a trajectory of the
0 0
evader on interval t > 0.
Definition 3. The pursuit problem for the differential game (1) - (2) is
called to be solved if there exists such control function u (•) e U of the pursuer
for any control function v(-) e V of the evader and the following equality is
carried out at some finite time t *
x(t *) = y (t *). (3)
Definition 4. For the problem (1)-(2), time T is called a guaranteed pursuit time if it is equal to an upper boundary of all the finite values of pursuit time t * satisfying the equality (3).
Definition 5. For the differential game (1) - (2), the following function is called n-strategy of the pursuer ([3]-[4]):
u ( v ) = v-A( v )4, (4)
zn
ч2 2 I |2
42 + a - v
where 4 = —, ¿(v,4) = (v,4)+J(v,^,)
(v, ^ ) is a scalar multiplication of vectors v and in the space Rn. Property 1. If a > (3, then a function X(v,) is continuous, nonnegative and defined for all v such that satisfies the inequality \v\ < (.
Property 2. If a > (, then the following inequality is true for the function
*( v,4 ):
a - |v| < A( v,£0)< a + |v|.
Theorem. If one of the following conditions holds for the second order differential game (1) - (2), that is, 1. a = ( and k < 0; or2. a > ( and k e R, then by virtue of strategy (4) a guaranteed pursuit time becomes as follows
(z0 k + yj z0 2 k2 + 2 z01 ( a - () ) / ( a - (), agar k ^ 0 va a > (,
-1/ k, agar k < 0vaa = ($,
T =
V2| *o|/(a-P)
agar к = 0 va a > p.
Conclusion
Proof. Suppose, let the pursuer choose the strategy in the form (4) when the evader chooses any control function v(-) e V. Then, according to the equations
(1) - (2), we have the following Caratheodory's equation:
z = -A(v(t))^,z(0)-kz(0) = 0 Thus the following solution will be found by the given initial conditions
t s
z (t) = z0 (kt +1) -4 Jj"^(v(r),4) drds
0 0
or
|z (t)| = |zo I (kt +1) - j j ((v(r), 4o ) +
0 0^
4o)2 + «
drds
According to the properties 1- 2, we will form the following inequalities
t s
|z (t)| < |z01 (kt +1) - ||(a - \v(r)\)drds ^
0 0
\z (t )|< \z0\(kt +1) +12(0 - a) / 2.
If we say f (t,a,k,a,0) = a(kt +1)- — («-0), a
zr
(5), then we
will check its properties 1. Let be a = 0.
1.1. If k > 0, then the function f (t, a, k,a, 0) = a (kt +1) is always continuous and isn't equal to zero (Fig-1).
(Figure-1)
1.2. If k = 0, then the function f (t, a, k, a, 0) = |z01 is a linear function (Fig-2).
(Figure-2)
2
1.3. If k < 0, then the function (6) is decreasing and it equals to zero at 1
t * =- - (Fig-3). k
(Figure-3) 2. Let be a > (.
2.1. If k > 0, then the function (6) is equal to zero at
T = z
(|Zo|к + 1/|zo|2 к2 + 2|Zo|( a -p) j/( a-P)(Fig-4).
(Figure-4)
In this case, a maximal time of unapproach is equal to t0 = |z01 k / (a — () and therefore, a maximal distance between them equals to
f (to) = (21Zo|( a -p) + \Zo|2 к2) / 2( a - p)
2.2. If k < 0, then the function (6) decreases monotonically, and this function turns to zero at time T as in the case 2.1 (Fig-5).
tit)
\M
(Figure-5)
2.3. If k = 0, then the function (6) becomes in the form
¿2
t
f (t,a,k, a,() = a -~( a -() and the pursuit time equals to the following
,a (Fig-6):
T =
2
zn
a -p
f(t) |zo'
0 t t
(Figure-6)
In conclusion, the relation (3) is true at some time t* according to the inequality |z(t)|< |z0|(kt +1) +12(fí- a)/2 and properties of (5), and it is
determined that a relation t* < T is correct, i.e., the pursuit problem is solved. Proved.
Summary
In the theory of differential games, issues of chase and escape occupy a special place in Aloxi. One of them is the breadth of implementation of various methods, as well as the specificity of the results obtained. This feature is especially obvious in model questions. In accordance with the condition given in the lemma, the theorem is conditioned and provides a proof. In the theory of differential games, questions in which geometric, integral and their joint constraints are imposed on controls have been sufficiently studied. This article includes new control classes in a control function called delimitation, using Gromwell's lemma. The chase-escape problem in a second-order differential game was studied, and new adequacy conditions were proposed for the pursuer and the evader.
References:
1. Gronwall T.H. Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math., 1919, 20(2): 293-296. https ://zbmath.org/authors/gronwall.thomas
2. Azamov A.A. About the quality problem for the games of simple pursuit with the restriction, Serdika. Bulgarian math. spisanie, 12, 1986, - P.38-43. https://www.researchgate.net/publication/268829581_On_the_quality_problem_ for_simple_pursuit_games_with_constraint
3. Azamov A.A., Samatov B.T. n-Strategy. An Elementary introduction to the Theory of Differential Games. - T.: National Univ. of Uzb., 2000. - 32 p. https://cajmtcs.centralasianstudies.org/index.php/CAJMTCS/article/view/89
4. Azamov A.A., Samatov B.T. The n-Strategy: Analogies and Appli-cations, The Fourth International Conference Game Theory and Management, June 28-30, 2010, St. Petersburg, Russia, Collected papers. - P.33-47. https://zenodo.org/records/7495576
5. Azamov A., Kuchkarov A.Sh. Generalized 'Lion Man' Game of R. Rado, Contributions to game theori and management. Second International Conference "Game Theory and Management" - St.Petersburg, Graduate School of Management SPbU. - St.Petersburg, 2009. - Vol.11. - P. 8-20.
https://dspace.spbu.ru/bitstream/11701/1233/1/Vol2.pdf
6. Azamov A.A., Kuchkarov A.Sh., Samatov B.T. The Relation between Problems of Pursuit, Controllability and Stability in the Large in Linear Systems with Different Types of Constraints, J.Appl.Maths and Mechs. - Elsevier. -Netherlands, 2007. - Vol. 71. - N 2. - P. 229-233.
https://www.researchgate.net/publication/245144708_The_relation_between_pr oblems_of_pursuit_controllability_and_stability_in_the_large_in_linear_system s_with_different_types_of_constraints
7. Barton J.C, Elieser C.J. On pursuit curves, J. Austral. Mat. Soc. B. - London, 2000. - Vol. 41. - N 3. - P. 358-371.
8. Borovko P., Rzymowsk W., Stachura A. Evasion from many pursuers in the simple case, J. Math. Anal. And Appl. - 1988. - Vol.135. - N 1. - P. 75-80.
9. Chikrii A.A. Conflict-controlled processes, Boston-London-Dordrecht: Kluwer Academ. Publ., 1997, 424 p.
10. Fleming W. H. The convergence problem for differential games, J. Math. Anal. Appl. - 1961. - N 3. - P. 102-116.
11. A. Friedman. Differential Games, New York: Wiley, 1971, - 350 p.
12. Hajek O. Pursuit Games: An Introduction to the Theory and Appli-cations of Differential Games of Pursuit and Evasion. - NY.:Dove. Pub. 2008. - 288 p.
13. Isaacs R. Differential Games, J. Wiley, New York-London-Sydney, 1965, 384p.
14. Ibragimov G.I. Collective pursuit with integral constrains on the controls of players, Siberian Advances in Mathematics, 2004, v.14, No.2, - P.13-26.
15. Ibragimov G.I., Azamov A.A., Khakestari M. Solution of a linear pursuit-evasion game with integral constraints, ANZIAM Journal. Electronic Supplement. - 2010. - Vol.52. - P. E59-E75.
16. Krasovskii A.N., Choi Y.S. Stochastic Control with the Leaders-Stabilizers. -Ekaterinburg: IMM Ural Branch of RAS, 2001. - 51 p.
17. Krasovskii A.N., Krasovskii N.N. Control under Lack of Information. - Berlin etc.: Birkhauser, 1995. - 322, p.
