CC BY
35
Литература
1) Болтянский В.Г., Яглом И. М. Геометрические задачи на максимум и минимум// Энциклопедия элементарной математики: Книга V. - М.: Наука, 1966. - С. 270-348
2) Васильев Б.Н., Гутенмахер В.Л. Прямые и кривые. - М.: Наука, 1978. - 160 с.
3) Далингер В.А., Симонженков С.Д. Сборник прикладных задач на экстремум: учебное пособие для учащихся школ и классов математического профиля.- Омск: ООО ИПЦ «Сфера», 2007.- 60 с.
4) Тихомиров В.М. Рассказы о максимумах и минимумах.- М.: Наука, 1986.- 192с.
DISCRETE PLAYING OF PERSECUTION WITH LEVEL OF BRIGHTNESS OF DIGITAL IMAGE
DESCRIBED BY SECOND ORDER EQUATIONS
Mamatov M. Sh.
National University of Uzbekistan named after M. Ulugbek,
Tashkent 100174, Uzbekistan, DOI: 10.31618/ESU.2413-9335.2019.1.61.8
ABSTRACT.
The work is devoted to the study of a class of discrete pursuit games with a digital image level, which is described by systems of second-order equations. Sufficient conditions are obtained for the possibility of completing the pursuit in discrete games with boundary conditions. When solving the problem of pursuit with the level of a digital image, Chebyshev polynomials of the second kind are used.
Keywords: pursuit, pursuit, evader, terminal set, pursuit control, runaway control
1. INTRODUCTION
The approach of applying a two-dimensional second derivative in the tasks of improving the brightness of a digital image is reduced to the choice of a discrete formulation of the second derivative and to the subsequent construction of a filter mask based on this formulation. Isotropic filters are considered, the response of which does not depend on the direction of inhomogeneities in the image being processed. In other words, isotropic filters are invariant to rotate, in the sense that rotating the image and then applying the filter produces the same result as the initial application of the filter and then rotating the result.
The simplest isotropic operator based on derivatives is the Laplacian (the Laplace operator), which in the case of a function of two variables, is defined as
ô fa
ôx2 + ôy2
V2z = (1)
Since derivatives of any order are linear operators, then the Laplacian is also a linear operator. To apply this equation in digital image processing, it must be expressed in a discrete form. There are several ways to set the Laplacian in discrete form based on the values of the neighboring pixels. The following definition of a discrete second derivative is one of the most commonly used formulas. Taking into account that there are two
variables and notation z(x, y)|^x y ^ = z ( X, y j ) = Z j, for the partial second derivative with respect to X
z ( x-+i, y ) " 2z ( x-, yj ) + z (xM, y ) = zl+w - 2z-j + ZMJ
К * , y, )
, we get
ô 2 Z
dx2 , x
(x -, yj)
and, similarly for the partial second derivative y , we get
a2 z
ôy
= z (xi, yJ+i ) - 2 z (xi, y J )+ z (x I, yj-1 ) = zji - 4, + zi j-1.
( xi, yj)
The discrete formulation of the two-dimensional Laplacian given by (1) is obtained by combining these two components
V Z| (x,y. )=-4 Z ( x i, yj )+[ Z ( x-+1' yj ) + Z ( xi-!' yj )+ Z ( x i, yj+1 ) + Z ( x i, yj-)-
4Z ■ + Z. , ■ + z. 1 ■ + Z. • 1 + Z ■ 1
^i,J *i-1,J ^I+1,J ^I,j-1 1 *i,j +1-
So, as the Laplace operator is essentially the second derivative, its use underlines the discontinuity of the brightness levels in the image and suppresses areas with weak changes in brightness. This results in an image containing grayish lines at the site of contours and other gaps superimposed on a dark background without features. But the background can be "restored", while maintaining the effect of sharpening achieved by the Laplacian. To do this, simply add the original image and the Laplacian. It should be remembered which of the definitions of the Laplacian was used. If a definition using negative central coefficients was used, then obtaining the effect of increasing the sharpness, the Laplacian image should be subtracted, not added. Thus, the generalized algorithm of using the Laplacian for image enhancement is as follows.
z(x.,y.)-V2z(x.,y.), если w(0,0)<0,
g (x i, y J ) =
z(x.,y.) + V2z(x.,y.), если w(0,0)>0.
Where g(x, y) - processed image, z(x, y) - input image, w(0,0) - value of the central coefficient of
the Laplacian mask. Hence it is clear that in order to improve the image as already noted, we must change the ie drive Laplacian
y2 z (X, y J ) =- + zi-u + zi+u + j + z ¿+i.
Given the presence of noise, we obtain a model example of discrete games, the process of pursuing the equations described (show. [1] - [3])
—4 J + Zr-1, J + Zr +1, J + \ J "I + \ J +1 =~\ J + V, J,
\ J
<P,
V, J
<g,g < p,
z0,J - 0, zm+1, J
J - 0, z,0 - 0, zue= 0, (*)
i = 1,2,...,m,j = 1,2,...,0-1, where the left side of the equation is a discrete analog of Laplacian
d2z d2z
—7 +--7 image brightness functions
ar dyL
z — z(x, y), а zi j — image brightness at a point, (x2-, y j ) т.е. zt j — the value of the brightness levels of the image of the corresponding pixels (i, j), u j , v j — control parameters. Without loss of generality, it is convenient to assume that if either i — 0, either i — m +1, either j — 0, either j — в, that z^ j — 0, those the image is bordered by pixels with zero brightness levels. Harassment is considered complete if z2- j satisfy the condition: 5 < zt j <5 + s, i0 < i < i, j < J < J1 где 1 < i0, i'1 < m,1 < j0, j < в — 1 for some preset 5 > 0, s > 0. This means that zt j the value of the brightness levels of the image in the predetermined pixels was in a certain segment. Using boundary conditions, when i —1,2,..., m from (*) get the system
—4z1, j + z0, j + z2, j + z1, j -1 + z1, j+1 — —u1, j + V, j,
"4z2, j + z1, j + z3, j + z2, j—1 + z2, j+1 — —u2, j + V2, j,
—4zi, J + zi—1, J+zi+1, J + ~ziJ—1 + ziJ +1 — — UiJ +V,J,
—4zm—1,J + zm—2,j + zm,j + zm— zm—1, J+1 — —um—1, J +vm—1, J,
—4zm,J + zm—1, j + zm+1, J + zm, j—1 + zm, J+1 — —um, J + Vm, J.
Denoting,
T T
zJ — (z1,j,z2,j,...,zm,j) ,Uj — (u1,j,u2,j,.. ,um,j) ,Vj — (V,j,V2,j,.,vm,j)
%J, z2,J,
-mj
J
UJ
J '
J
m, J
we have
—zj_! + Czj — zj+1 — Uj —Vj,1 < J <в — 1,
z0 — 0, ze— 0
(**)
where Zj G Rm h u. — pursuer control parameter; Uj - escape player control parameter:
u ■ e Rm, Vj e Rm components that satisfies the condition, square - three-diagonal matrix [2]
u
i, J
<P,
V
J
< G, G < p, and C — Jacobi
■
T
C =
1 4
0 ^
14
M ^ Rm the terminal set that the game ends.
The tasks of pursuit and escape for various classes of differential and discrete games are the subject of numerous studies [4-16]. General questions of the theory of discrete games are considered in monographs [4], [5]. In [6], discrete differential games with information lag were studied, in [7] - [10], the relationship between differential games with distributed parameters and discrete ones was studied. The pursuit and escape problems for linear discrete games were studied in [11] - [14], the escape problem for nonlinear discrete games was studied in [15], [16]. In these papers, the motion of points is described by discrete first-order equations.
The purpose of this work is to study a new class of game problems with discrete second-order equations. For this class of discrete games, sufficient conditions are obtained for the possibility of completing the pursuit when the position of the object is given in the boundary moments. To solve this problem, Chebyshev polynomials of the second kind are used [2], [3].
2. PROBLEM STATEMENT
Instead of the game (**), we will consider a more general discrete game, which is point motion z m dimensional Euclidean space Rm described by equations
-zj_l + Czj _Zj+1 = Uj -Oj,1 < j <0_1, (1)
z0 = db, ze=4e (2)
where j - step number, C — mxm - constant square matrix, u, O - control parameters, u - chase parameter, O - escape parameter, Uj e P ^ Rm, Oj e Q ^ Rm, P and Q - non-empty sets, parameter u selected as a sequence u = u(.) = (ui,u2,...,u^—i),uj e P, j = 1,2,...,0 — 1, parameter O - as a sequence
o = o(.) = (O,O2,...,O0_i),Oj e Q, j = 1,2,...,0 — 1. In addition, in Rm allocated terminal set M.
Purpose of the pursuing player zj on the set M , fleeing player tends to put it.
Definition. We will say that in the game (1), (2) from the "boundary" position (^, <Pq) can complete the persecution for N <0 steps, if in any order l\,l>2,. . .,v'N_i escape control can build such a sequence T¡1,w2,...,¿7y_| prosecution management, what's the solution z = z(.) = (z0,~Z\, z2,..., Zy_|, zN) the equations
_Zj_y + CZj _Zj+1 = uj _Vj, 1 < j < N_1,
Zo = O0, ze=Ve, with some d < N hits on M : Zd G M.
Let a discrete game be described by equations (1), (2). Through - Un (x) denote the Chebyshev polynomial of the second kind of degree n :
sin(n +1) arccos x
Un (x) H
sin arccos x
2Í
x2-1
x |< 1
\n+1
|x WX2-1J -|x WX2 - 1J
-(n+1)
(3)
I x |> 1.
From here you can easily get the following: U_2 (x) = —1, U— (x) = 0, U3 (x) = 1, U (x) = 2x, etc. In the monographs [2], [3] there are the following recurrence relations
Un+2( x) = 2xUn+j( x) - Un ( x) , n > 0, U0( x ) = 1, U (x) = 2x.
1
Further we denote by Un (x) Chebyshev matrix polynomial from a matrix X, determined by the recurrence formulas (4). From here from (3), (4) we get: U-2(X) = E, U-1(X) = 0 , U0(X) = E, U—(X) = 2X, where E - single, and 0 - zero matrix. Let be u = u (J) = Uj , V = v(J) = Vj , 1 < J < N - preset controls.
If matrix C such that Uq_—(— C) non-degenerate matrix, then solutions of equation (1) with boundary condi-
'2
tions Zq = ^q, Zq determined by the formula (see [2], chap. I, § 4, equation (46))
z0 + ^1C \(uk ~Vk )
z„ = uq- ^ l c juq-,-1 ^ l c +u- ( 2c ( 2c ^
Q-l
1
zQ+T UQ-k-11 1C \(uk-"k) k=n V2 J
(5)
3. FORMULATION OF KEY RESULTS
Assumption 1. M = Mq + Mi, where Mq - linear subspace Rm ; Mi - subset of subspace L - orthogonal complement Mq in Rm. Through n denote the operation of the orthogonal projection of Rm to L, and through A + B and A * B - algebraic sum and geometric difference of sets, respectively [17]. Let be
M11 + M1 2 = M1 and
Wain) = £ nUQ-1 ( 1C\uQ-n-1 f 1C Juk-1 ( 1C J(p * Q )-M1,1 ; W1,2(n) = X nUQ-1 f1CJUn-1 f1CJUQ-k-1 f1CJ(P*Q)-M1,2,
(6)
In here U„
1
— C | - Chebyshev matrix polynomial. V 2
Assumption 2. Let there be such n = n0 <0 — 1, that
—n U——1 f 1C]Uo—n0 —1 f 1C]zo
and
-n UQ-1 f 2 C K -11 2 C J Zq
eWuino)
6W1,2(no).
(7)
Theorem 1. If assumptions 1, 2 are fulfilled, then in the game (1), (2) from the "boundary" position (zo, zo) may complete the pursuit of N(z0, z0 ) < n steps. Let be 1 < n < 0 — 1, W21 (0) = —M11, W22 = —M1 2,
Wu(n)= fi
u(k)eO 1 < k < n-1,
W22{n)= n
u(k)eO
w2a (n-i)+uu(2c) ^ [2c) Ue~k~i [2c)■v{ki)
Assumption 3. Let there be such n = no < Q -1, that
-n
uqQ-1 i 1C JUQ-no-1 f 1C IZq
n < k <Q-1.
^ W> i(no)
and
-П
uQ111С к -, 11С 1 Zq
eWWno).
(9)
Theorem 2. If assumptions 3 are fulfilled, then in the game (1), (2) from the "boundary" position (z0, Zq) may complete the pursuit of N(z0, Zq) < n steps.
и—1
Пусть an{-) = {aba2,...,an_x :ak>0, ^a(k) = 1},
k=l e-\
Pn(-) = {Pn,Pn+i,...,Pe-i-Pk>О, XP(k) = \}
к=n
and
n-1
W,(an ( • )) = £
к=1
У
akMxx + ПЦ- |1С lUo-n-i | 1С jUk-1f 1С IP
л
V
1
*nUQ-1 f 1С jUQ-n-1 [ 1С k-1 f 1С jQ
Q-1
w2(Pn ( • )) =£
к=n
+nuQ-1 f 1с jun-1 f 1с juQ-k-1 f 1с j p j
1
1
Set
-nU,"-1 f 1С jUn-11 1С \Ue-k-11 1С IQ
W3 1(0) = M11, W3 ,1 (n) = U wa(•)), 1 < к < n-1,
ak( •)
W3,2(0) = Mu, W32(n) = U W1P(•)), 1 <к <Q-1.
pk( •)
Assumption 4. Let there be what n = no < Q -1, what
-П
-П
U*--1 [ 1С jUQ-no-1 f 1С j -О
U--1 (1С К -1 V1С j *
^ W3,1(n0):
GW3i(no).
(10)
(11)
Theorem 3. If assumptions 4 are fulfilled, then in the game (1), (2) from the "boundary" position (z0, Zq) may complete the pursuit of N(z0, Zq) < n steps.
4. CONCLUSION
Summarizing the obtained results, we conclude that the discrete pursuit game (1), starting from the "boundary" position (2) Zr jrer, can be finished for N(z0, Zg) < n steps. Thus, to solve the game problem of
pursuit of the form (1), (2), we used the Chebyshev polynomials of the second kind of the form (3) and the recurrent relation (4), the formula (5). The set (6) is a discrete analogue of the so-called first integral of L. S. Pontryagin [17], the inclusion of (7) gives the first sufficient conditions for the possibility of completing the pursuit of the task. The set (8) is a discrete analog of the second integral of L. S. Pontryagin, the inclusion of (9) gives the second sufficient conditions for the possibility of completing the pursuit of the task. Set (10) is a discrete analogue of the third method of N. Yu. Satimov [14], and the inclusion of (11) gives the third sufficient conditions for the possibility of completing the game. In Theorems 1-3, sufficient conditions are obtained for solving the corresponding problems in this form. It should be noted that many problems of mathematical physics can be approximated using problem (1), (2).
ATTACHMENT Proof of Theorem 1. From (6) and (7) it follows that such
a(k) i
n
u(k)eQ
f
1 c iu
í
n n Ue\
u(k)eQ
v 2 O
0-no -1
v 2C >-i
2C JUno-1
1C W -1
J C j( P-o(k)), 1 < k < no-1,
1Cj(P-v(k)), no <k <0-1,
b1 G Mu , b2 GMU,
that
-n
Ue-1 [ 1C \Ue-n0-1 | 1C Izo
-n
Um
í
2 c K -1
í
C I zc
n0 -1
= Z a(k) - b , k=1 -1
= Z a(k) - b2-
k=nn
(12)
Let be V = V (k) , 1 < k < 0 — 1 - arbitrary admissible control of the evader; pursuing player management u = u (k ) let's build as a solution to the following equation
a(k ) =
n U---1
Í
1C IU
Í
n U¿1
v 2 r1
0-no -1
v 2C >-1
J C \(u (k ) -v(k ) ),
1 < k < n0 -1;
(13)
V
-C \Un -1 2 J no 1
1
C |U
0-k -1
1
C ](u(k) - v(k) ).
It is clear that these equations have optional solutions a(k), because v(k) G Q and u (k) G P. Substituting v = Vk = V(k) and u = Uk = u (k) in (1) and applying the formula (5), will get
z(n0) = U0-1
1
C |U
e-n0 -1
1
C
V2
n0 -1
1
z0 + Z Uk-1 1C I(uk -vk)
k=1 v2 J
+U
0-1
-C lu
n0 -1
[ 1C ^
V 2 J
0-1 [ 1 j - -zo + Z U0-k-1 -C I(uk-vk)
k=n0 v2 J
Hence, applying to both sides of the equality the design projection operator and from equality (12), (13), we
have
nz(no) = n
U0-1
■C lu
0-no-1
j ^ , [ 1
■C\ zo +Z U
0-1
k=1
C lu
0-no-1
■C I-
-u
k-1
1C \(uk-vk)
2
+n
U011 1CjUno-1 [ 1C Iz0 +
0-1 í 1 + 0-U0-1 1
k=n
C Iu
no -1
■C Iu
=n
U--1 [1c Iu
0-k-1
0-no -1 I ^ C I z<
CI
-vk )
o
+n
U¿1
/
v
2
C Iu
no-1
C I z¿
v 2
0-1
+ Z a(k) = n
k=«a
u--1
no -1
+ Z a(k) +
k=1 /
v
1C IU0-n-1
CI z
v 2
-n
Uh 11C ]U0-no-111C Iz
o
+ b1 +n
U0-11 1C ]Uno-1 [ 1CI z0
<
1
2
1
1
-n
^x 11 c k-x (1C1 z„
+ b2 = b2 e Mx x + Mx 2 = Mx.
From this inclusion we get that nz(n0) e Mx and, means, z(no) e M. Q.E.D. Proof of Theorem 2. By the condition of the theorem and from (8), (9) there are such vectors
rl c\(P-u(*)), x < k < ng-x,
a(k) e
nuglx
-C \U,
0 - n0-X
nu
0-x
i
-c \u
n0-X
1
2 c Ju0--x
iCj(P-u(k)), no <k <0-X,
and bx e Mx x, b2 e Mx 2, that
-n
U0-x I \C Wno-x I xC jzo
U0-x (xC H-x ( xCJ z0
-n
From here we get that
gW21(ri) + a(n0 -\) + a(n0-2) + ... + a(n), x < n < n0-x,
eW22{n) + a{0-\) + a{0-2) + ... + a{n),no <n<6-1.
-n
x
U- I xC )U0-no-x I x C j zo
-n
Ug-x
i
-C \U
f
no-x
x
2 C Jzg.
no-x
= Z a(k)-bx ,
k=x 0-x
= Z a(k) "b2-
k=no
(14)
Now let v = Vk = u(k) , x <k <0- x - arbitrary admissible control of the evader; control of the pursuing player u = ûk = U (k) , U (k) e P are constructed as solutions of the following equation
a(k) =
nU
0-x
nU0lx
2 C jU0-no-x
xC j(û(k)-y(k)),x < k < no-x, v2 J
-C U
v2
no-x
x
x
(15)
x C j(û(k)-v(k)), no < k <0-x.
v2 J
v 2 c ,
Substituting u = uk = U(k) , v = vk = ) in (1), (2), considering (14), (15) and simultaneously applying formula (5) and the design operator, we have
nz(no) = n
U-x
x
x
C \U
0-no-x
x
no x x
Cj zo + Z U-x
k=x
C \U
0 - no-x
x
C \.
'Uk-xI 2C j(Uk-V]i)
+n
U0-x I xC K-x I xC \ z0 +
0 -x
+ Z U0
0-x
f
k=n
-C \U,
no-x
■C U
0-k-x
C\
-"k )
=n
u- | xc \u
+n
2 C K - x
x
0-no -x I x C \ z
0-x
o
no -x
+ Z a(k) +
k=x
C \ zn
+ Z a(k) = bx + b2 e Mx x + Mx 2 = Mx.
k=no
From here we get that z(no ) e M. Theorem 2 is proved.
Proof of Theorem 3. Instead of inclusion (11), bearing in mind (10), we consider the equivalent inclusion
x
2
1
x
—n U0—1 f 1C JUo—„0—1 ( 1C I zo —n U0—1 ( 1C V— 1 f 1C j zo
GW1(ann (•)),
e^P (•)),
existence a^•) = {al,a2,•••,a«0—1}, P(•) = {Pn0,Pn0 +i,---,p0-i} follows from (11).
From here we get
—n
U0-11 1c j U0—n0—1 ( 1C Iz
0
n0 —2
^Z
k=1
akmu +nu0iI1 cn—1 [ 1c jx
'0—1
U—111C jP*nUo—11 1C jUo—n0—111C jUk—1 f 1C 10
+
(
+
an0—1M1 +nU0—111C jU
1
1
V
0—n0 —11~C \U«—i \ ~C \P
n0—21 2
* n U0—1
C Iu
0—n0—1
C •
n 0 —2
C 10,
(16)
—n
U—1
2 c k—1
'iC 1 z
0
0—1 i Z
k=n0 +1
PkM 1,2 +nu—
-C IU
n0— 1
1C
2 ,
1
f 1 j * If 1 I f 1 I ,1 XU0—k—111C IP*nU0—11 1C IUn0 —11 1C I• U0—k—111C 10
+
(
+
pnMh2 +nU0—1f1C jUn0 —1 ( 1C j • U0—n0 —1 ( 1C j P^
* n^i
1c iu
V 2
'n 0 —1
1C IU
V 2
'0—n0—1
(2C10.
Now let U[,i>2 ,...,Vq_i . U/. <eO . \ <k <6 — \ arbitrary sequence. By virtue of (16) there is such cx j
and B„ that
^n0
—n 1
U01,1C IU
0—n0—11 ~ C 'z'
0
+ nU—111 C]U0—n0—1 (1C jx
n0 —2
U —2 CjUn0 —1 G Z 2 ' k=1
1
1
akMxx + nU0—111c ju0—n0—11 1c juk—11 ^c jP-
*nU0—1 (1CjU0—n0 —1 (1C j• Uk—1 f 1C 10
+
(
+«n0—M1,1 +nU0—1,1 c IU
V
1 j f 1 j j p-n -i I — C I U„ _71 — C IP 0 n0 1 ( o I n0 2 (2 j
n0
(17)
—n
U—1I2 C ]Un0—112 z0
+ nU0—11 1C jUn0—11 1C Wn0—11 2
C I x
xu
0—1
n0 G Z
k=«0 +1
PkM-12 +nU(
0—1
1C IU
n0—1
2 C jU
0—k—1
2 c j p *
* nU
0—1 f1C U—1 f1C jU0—k—1 f1C j 0
+^n0Mu +nuQ f1 c jun0 _ | 1 c \-Uq_« _ |1 c j p^
Control u«q _ e P and u e P let's build as a solution to the following equation
nu
e_1
- c
v 2 ,
U
+nUQ-i
1 c |U
v2
Q_n0 _1
Q_« _1
1 c |U
v2
- c
v 2 ,
f
U
n0 _2
- c
v 2 ,
Uno _1 +
nu
e_1
1 c
v 2 ,
U
n0
1
- c
v 2 ,
V- c J °n0_1 = an0_1a1 , ^l 6 M1'1' / 1 \
U
e_no _1
1 c
v 2 y
un +
n0
+nu
e_1
1 c |U
v ^
no _1
1 c lu
v ^
e_n0 _1
1 c =Pnb1, b eMu .
Further, by virtue of (11) and (17) we have
_nue_11 ~ u
e_no _1
c 1 z
no _2
+ z
k=1
akm1,1 + nu_11 c J.
ue_n_11 1 c k_111 c ]P*nue_1 f1 c W«_111 c ]uk_1 [1 c \Q
2
_nu,
e_1
.2
2c k _1
f VI - c z,
12
2
e_1
2
2
+ an0 _1a1,
e_1
+ e_
k=no +1
f
PM12 +nu
e_1
1
v 2 c y
Un0_111 c jUe_k_1 f 1 c jP*nU_1 f 1 c V _111 c jUe_k_1 f 1 c \Q
+ Pn0 b
Similarly, if control vn, on+1 becomes known, then the above described method can be constructed to
control u
no -2, uno +1, providing inclusion
_nu
e_1
c |U
e_n0 _1
c 1 z0
n0 _3 k=1
akM11 +nU
e_1
c I-
•U
e_n_H 1 c jUk_111 c\P*nUe_1 f1 cJ^^ f 1 cju^ f 1 c \Q
+an0_1a1 +an0_2a2 , a2 eM1,1
PM2+nue_1 1
+
_n
Um
1 c V* _1
c 1 z
e_2
e_1
^ z
k=«0 +1
2 c
•u«o_1 I 2 cU
f 1 I * if 1
e_k_111 c \ P *1 1 c U
«0_1|2 c U
e_k _11 1c IQ
+
+ +1^2 , b2 eM1
1,2,
etc. so get
(a«0 _1a1 + a«0 _2a2 + • • • + a1a«0) + (P«0 b + P«0 +1b2 + ■ ■ ■ + A«0 +e_1be_1)
G
nz« = «0
^ (an0 _1 + an0 _2 + ■ • • + a1)M1,1 + (Pn0 + Pn0 +1 + ■ ■ ■ + Pn0+e_1)M1,2 = M1,1 + M1,2 = M1 >
from here we have
z„. e M.
1
The theorem is proven completely. Comment. If in (*) the game is considered finished when the average
value
1 г1=го Л=Jo +M
zt,jЛо ^1 ^hJo ^ J^ Ji:z = v~ Z Z zuj,1 ^ib^m1 ^ Jo>Ji ^-1
Jo
satisfies the condition 5 < z < 5 + £. Then a corresponding change can easily generalize Theorems 1-3.
Bibliography
[1] Godunov S.K., Ryabenky V.S. Difference schemes. - M .: Science, 1977, 440 p.
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[3] Pashkovsky S. Computational applications of polynomials and Chebyshev series. - M .: Science, 1983, 487 p.
[4] Isaacs R. Differential games .- M .: Mir, 1967, 500 p.
[5] Azamov A.A. Foundations of the theory of discrete games. - Tashkent .: Niso Poligraf, 2011, 160 p.
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[13] Fazylov A.Z. The existence of the nucleus of a convex set in linear discrete control systems // Mathematical Notes, 1995, vol.58, №1, p.119-126.
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[15] Satimov N. Yu. The problem of runaway for one class of nonlinear discrete games // Techn. Cybern. Izv. Academy of Sciences of the USSR, 1973, v. 9, No. 6, p.45-48.
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DISCRETE PLAYING OF PERSECUTION WITH LEVEL OF BRIGHTNESS OF DIGITAL IMAGE DESCRIBED BY SECOND ORDER EQUATIONS_
Mamatov M.SH.
The work is devoted to the study of a class of discrete pursuit games with a digital image level, which is described by systems of second-order equations. Sufficient conditions are obtained for the possibility of completing the pursuit in discrete games with boundary conditions. When solving the problem of pursuit with the level of a digital image, Chebyshev polynomials of the second kind are used.
Keywords: pursuit, pursuer, evader, terminal set, pursuit control, evasion control
УДК 519.854.3_
МЕТОД ПОСТРОЕНИЯ ПРИБЛИЖЁННОГО РЕШЕНИЯ ИНТЕРВАЛЬНОЙ ЗАДАЧИ
ЧАСТИЧНО-ЦЕЛОЧИСЛЕННОГО ПРОГРАММИРОВАНИЯ_
Мамедов Князь Шираслан1, Мамедли Нигяр Октай2
1 Доктор физико-математических наук профессор Бакинского Государственного Университета и 1 Завлаб. Института Системных Управлений,
2Докторант Института Системных Управлений Б01: 10.31618/ЕБи.2413-9335.2019.1.61.2
АННОТАЦИЯ.
