X -
62-50:531.8

. . ,
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-
-
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, , - .

. , , [1-5]. , , , [6, 7].
, [8, 9]. , . (, q, ), . . . -
, . , [1]. . . .

,
%
R = f ( + )&, (1)
0
; .
1 ^ q(t0) = ..., q0J, () = [1>---> 0~\
1 ^ q(t1) = ^ . qln], 5^) = [5,51]
= [ ..., ,,] = [1, ..., ] [3]
1
5 ' R = J (8T + 5 'A)dt
= 0,
(2)
5! :
5 = ^3 5$,, (3)
3=1
Q = [01 (, , ),..., Q3 (, , )] ; = [1, ..., ] :
. (4)
(2)
_ __ = 0
dt 4 qa 3
s = 1,n; t = to, qs(to) = qQs; t = ^ qs (tl) = qis
(5)
- F(q, q, , ^. : = (, <), Gu (, ) R q(t)eRn , <1
*1 =f , , , ^)d^ ^ 1 (6)
<0
(5) (4).

(6) .
%
J = + J1 =f [( + ) + ^ , (7)
X , XeR1.
(#) Gu . -
(7), (5J > 0, 5^ > 0) .


(, , , X) =^ [03 (, , ) + Vs (, )}3, (8)
8=1
5
3 = , -
5<1
(6).
(, <, , )
(, , , X) (9)
,
(, , ) = ^ (, , (, )), (10)
(, )^
q(t), 4^) R2n, (, q) Gu.

Q(q, ) = X {3(, q)qs~Vs}, s = s
(11)
(q, q) , 8 < 0, (11) (8):
max_ (, q, u, X) = ^.3<2 < 0. (12)
u,(q, q)eGu
: u(q, q) Gu (q, q) (7) (4), (q, q) R2n, u(q, q) Gu.
0(q, q, u, X) = max_ {(XQ + V)q}, (13)
u,(q, q)eGu
:
X = Xq = const, (14)
t = t0, t = t1
-T) + F = 0. (15)
|j,(q, q) . 1, 2.
(13) , . , .
t
t
0

, . [8] , - .
:
1 91
= (_,-, 0 < #< = 2, 5 = f udg;
9
(0) = ; (0) = ; q(tl) = 0; (%) = 0;
0 < < , (16)
11 ; . :
J1 = f (1 + = f (1 + ku)F(q)dt, k > 0. (17)
- , <__1.
k
< --1 .
()
'-
(, ) = w(a) = w
(18)
w(a) = , ^-1 < < 1; w(a) = --1, < -k-1; w(a) = 1, > 1.
(, ) (16) (15):
dT 2 1 dF(q)
= = ^±-------------------.
dt X + X + dt
(19)
. 1. () (): q0 = -10; qo = 10; k = 0,5; X = 2; L = 2; [-2, 7]; q0 = -20; qo = 10; k = 0,25; X = 0,125; L = 2; [-4, 7]
)
. 2. : ; (21); ;
dF
=
- dT
= , [10], -
d^

|4 ) = _ |2 + \^~,
-
u(q, q) = w
( lqlq12+ q 1
1 rn q\ ( + kq)
(20)
(19) ,
X + kq > 0. (21)
. 1, , .
- [1, 1],
(. 2, ), t = * (. . 2, ) -

.

. . , , [8, 9]. .
1
. [1-3]. u(t) Gu (7) [1, 2],
18 f OMAOO-A
4, 2010

AJ = [( + ) + ^ = [( + ) + ]
10
^1 + [(5 + 5 ) + 5 ^. (22)

q) . (3),
5 = ^53,
3=1

(23)
= [( + ) + F] ^0 +
-Vsqs . (24)
, (28)
J = [( + ) + ^ + X
0 3=1 <?8

t1

0
3=1
dt


+ X
^
+

-V3 8qdt. (25)
(25) .

0
.
=1 ~ 0
3=1 0
*
- 1 ^0> =-21< .
3=1 .
(26)
(26) , ( ) q0s, q1s , [10].
(26) (25)
AJ = [( - ) + F]^t\t^ +
∞ *= %
53 + ^. (27)

-X -

- X
d -_- _
^3
^ 4 ^

( ) + = 0 t = t = (28)

X


-X

<!

- V 53. (29)
^) Gu, () . , (28):
d + + + +^
dt . ^.
5^.(30)

53 (#) = ^ (t) t = , (31)
, :
Qs = Qs (, );
(,, ) (32)
:
^ = 0, X = 0 = const dt
(33)
q(t) qs(t), (^), ) .

82J = AJ - AJ = /> ( - )
=1 1 ^
( - ) - ( - (^ )1 + (^ - )[^&. (34)
<1
:
- = (> )-( ) +
+ {'-"> (,,-,) + (1 (^-) (35)


( - ) ( - ) (, qs, 43)
dt ^
( - qs ) +
^ ^
( q, <),
<1
-(2 - <1 ) = 0- (36)
0
8 = 1



0
OMOO-BAE
.
<1
23 = -'/) + (^ V >0. (37)
8=1 <
[t0, ^] . te[t0, ) : ^) = u(t). . te [, + 5^| us(t) u(t), us(t) ^) . te[ + 5t, t1] us(t) = u(t).
+8^
52 J = ' ^ [(^^ + ) (Qs + ^ )]^ ^ +
s=1
f (Vs - Vs )Sqsdt.
s=1 +St
(38)
, () [, + 5] , , [ + 5, 1] (36)
t = + 5^ 5( + 5^ = (q - q)8t,
5q( + 5t) = (q - q)5t, (39)
5t ^ 0, q ^ q,
4 ^ 4, 4 ^ 4.
lim 2 J = - [(Q + Vs) - (IQs + Vs )]qs8t2 +
s=1
f (V* - Vs )5qsdt > 0.
(40)
s=1
, 52J : 0 < < t < 52J = 0, t >
-
^ = ( V (41)
=1
t = :
S2J = [(# + ^) - ^ )]35#2 > 0. (42)
3=1
(41) 5qs(t) (36). t , 5t 0 83 () = [5 () - 3 ()] > 0.
ttt
t >
lim S2J(t) = 82 J(t) = const > 0. (43)
t^x

lim 2 J | St2 = 6t>0
= -(#)} + ) -
8=1
- ((8 (#)) + 8 8 > 0. (44)

(, X) = [<3( (#)) + Vs(45)
&
< t < t1 ^) ^(),()) ), q(t1)), () = ^(), q()) [7].
,
((, 4), 4, 4, X) =

= _ \-KQiUs 4 4)) + ] 4 (46)
(4,4)^=1

> 4) = X-1 { 3 (, 4)48-^ } (47)
.
2
. (47) (8),
d
-----------------= Q(us).
dt dq3 dqs
(48)
, t = V t = . (15).
Fs =dL X(T-)
dqs
(49)
, 3 = ,
3
8
----= 3 .
4
(48)
2G
OMOO-BE
4, 2010


dps
Qs , * 1,
dt
(50)
.
dps
#
(51)
[10] ,

1. . ., . ., . . // . 2009. . 54. 4. . 450-457.
2. . . // . . . 2003. 2..86-92.
3. . . . .: . - .-. ., 1961. 824 .
4. / . . . . .: , 1987. 712 .
5. . . . . .: , 1983. 392 .
6. . . .: ; , 1960. 400 .

, = 1, ,
X.

\
' ' , = 1, ,
(53)
^ |
Ls , .
7. . . // . . . 2003. 3. .46-51.
8. . ., . ., . . // . 2007. 5. . 52-61.
9. . ., . ., . . // - . 2007. 12. . 34-40.
10. . . . .: , 1990. 416 .
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(http://www.elibrary.ru) , ( ), , , ().