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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.
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