<<

. 12
( 15)



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356
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357
, . -
x1 2 Rn
x2 2 Rn -
, ,
x1 = x2 : (13.1)
(13.1) -
-
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-
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358
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135, 153].
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-
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,
.
, -

359
-
, -
131, 154].
-
.
13.2.
13.2.1. . -


T T
p = ; @H + Bu q = @H
_ _ (13.2)
@q @p
p q 2 Rn { H=
H(p q) { ( )
u = u(t) 2 R m{ ( ), B { m n-
, m n: -
,
6]:
S = f(p q) : H(p q) = H g: (13.3)


H((p(t) q(t)) ! H t ! 1: (13.4)
(13.2) -
( . 106, 103], A.). ,
x = colfp qg
(13.4) (A.3) ( . 407),

Q(x) = 1 (H(p q) ; H )2 : (13.5)
2
_
Q{ (13.5) (13.2). -

T T
_ = (H ; H ) @H ; @H + Bu + @H @H =
Q(x) @p @q @q @p (13.6)
@H Bu:
= (H ; H ) @p

360
(. A. (A.16)) -
6]
T @H T
u = ; (H ; H )B @p (13.7)
T @H T
u = ; sign (H ; H )B @p (13.8)
>0{ .
6], -
-
. , -
, , -

p
X
Q(x) = iQi (xi ) (13.9)
i=1
0{ , Qi(x)
I
(A.10), .
(13.4), (13.9) -
109, 145, 147, 158] 64].
, (A.10) ,
H0 = H(p(0) q(0)) H -
. ,
, -
, > 0: ,
(13.7), (13.8) -
. 1

-
Q(x) = ky(x)k
Q(x) y(x) { -
2

( ) ,
.
-
, .. . -
| .
, ,
1
.

361
,
, ,
. " "
.
, -
.
.
64].
. .
6], -
: 2


J ' + mgl sin ' = u (13.10)
'{ (' = 0 ) -
,u{ ,Jml{
, -
(-
), g { .

H = 1 J '2 + mgl(1 ; cos '):
2_ (13.11)
, -
H .. (13.4). -
,
, -
H0 : (13.4) H =0
, ..
, H > 2mgl
.
-
(13.7), (13.8)
u = ; (H ; H )'
_ _ (13.12)
u = ; sign ((H ; H )') :
_ (13.13)
64] , -
H0 H -
, H
-
2
, . . 281, (11.36){(11.38).

362
, -
( (2k + 1 0)) k = 1 2 : : : (13.4)
.
13.2.2. : -

:
, -
' + 0(') = u (13.14)
' = '(t) { , u = u(t) { -
, (') 0 { .-
(13.14) x = colf' 'g.
_ -
:
( )
\ "{ -
1 '2 + (')
f(' ') : H(' ') = H g, H(' ') = 2 _
_ _ _
{ . H, -
:
,
, ( .,
., 12] { (') =
= !0 2 (1;cos ') ). :
(13.14)
?
-
:
12
(') = 2 !0 '2
.. , -
' + !0 ' = u: (13.15)
2




u(t) = sin !t (13.16)
! = !0 -
: -
,
'(t) = ; 2!t0 cos !t:
363
.
-
, . , -
, -
-
50, 129].
-
-
.
, -
.
: -
, -
? ,
u(t) '(t), . . -
.
, -
, (.
64, 108, 142, 145, 158] .
(13.14) (\ "\ ") -
, 108, 142, 145],
(13.12), (13.13), . . , -
.
(13.14) ( -
) , .. (13.14)
' + %' + 0 (') = u
_ (13.17)
%>0{ .
1 !0 '2 )
(13.17) ( (') = 2 2 -
, -
! 2 = !0 ; %2=4.
(13.16) 2

%>0 (13.16), (13.17)

A = %!0 (1 + O(%2))


1 '2 + !0 '2 = 1
2 2
H= 2_ 2 1 + O(%2 )]: (13.18)
2%
364
(13.17)
(13.12) (13.13),
. 64, 109] , (13.13), (13.17)
,
H=1 %
2
(13.19)
2
(13.13) , H H: -
(13.19) % (13.18),
, (13.12) (13.13) -
(13.17) ,
( ) ,
-
.
.
,
, -
1638 .
( -
{ ) .
11] "
, ",
-
.
,
( ) -
35]. (13.12), (13.13)
,
.
: -
(13.17) -
? , -
(13.19), -
.
64].
,
.
q
H( )
E( ) = (13.20)
365
H( ) { ,
.
E( ) 6= const
E( ) = const. -
,
( . 13.1).




. 13.1. (13.17)
% = 0:1 !0 = 10:
2


,
(13.16)
, -
( ) ,
. E( ) -
-
.
, (13.8) (
(13.8) ) - , -
. ,
(13.8) E( )
.

366
-
. , -
-
146].
13.3.
13.3.1.
( , )
{ 70-
. 61, 65, 68, 85]
, -
( -
, . .). -
-
, ,
, ,
, -
,-
, ,
.
, -
," " , .
, -
. ,
XX .,
,
,
0 t<1
y(t) + ! 2 y(t) = 0 (13.21)
(13.21)
y(t) = A0 sin!t + A1 cos!t (13.22)
p 2 2! T = 2 =!, -
A = A0 + A1 :
A1 = y(0) A0 = y(0)=!.
_ , (13.22)
, ..
367
y(0) y(0)
_ y(t)
t < 1.
0
(13.22) !=2
-
(13.21) -
!1 ::: !r . , !1 ::: !r -
( !0 ),
2 =!0 .
!i , -
, .
, -
.
, " " -
,-
(
)
-
.
XIX-XX . , -
.
{
{ . ,. ,
.. ,.. .. , 11, 18, 33,
52, 55]. -
{
, (
{ -
). -
, , -

y + (y 2 ; 1)y_ + ! 2 y = 0 (13.23)
>0
y + py ; qy + q0 y 3 = 0
_ (13.24)
p > 0 q > 0 q0 > 0
y + py + qy ; sign(y) = 0:
_ (13.25)
368
-
, (
) ,
(
) .
-
,
, .
-
-
. , -
. -
: , {
( . 11, 18, 33, 55, 94], . 11.2.3. . 246) -
, -
-
{ , .
XX . ,
:
{ -
.
. 175], 1963 .,
, -
, -
{ , -
3- :
(
x1 = (x2 ; x1 )
_
x2 = rx1 ; x2 ; x1 x3 (13.26)
_
x3 = ;bx3 + x1 x2:
_
(13.26) -
( , = 10 r = 97 b = 8=3 )
. -
( ) -
( ), -
( . 11.9 . 268). -
. .
187], " -
", . . 169], -
""
369
. -
60-70-
(.. , ..
, .. ),
( .. , .. ). -
, -
, , ,
. . 61, 70, 65, 85, 99, 133, 135, 159, 183]. -
-
, , {
. , -
, -
.
-
, -
. -
.
,
, .
, " "
-
, .
13.3.2.
-
, -
. .
-
x = F(x)
_ (13.27)
x = x(t) 2 Rn { , 0 t < 1.
1. Rn
(13.27), :
) ,
0
x(t) (13.32), , -
0
t !1(..
t 0
dist(x(t) ) ! 0 t ! 1, x(0) 2 0 , dist(x ) =
= inf y2 kx ; yk { x )

370
)
2
.
2. ,
, ,-
2
.
3. ,
.2
,
:
xk+1 = F(xk ) k = 0 1 2 ::: (13.28)
-
, " -
", " ",
: -
.
. , -
:
( ), -
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" " -
. .. 55], .2, -
,
-
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-
{
.
.
1. ( ) . 1982 .
. .
,
,

371
, .
(
x = p(y ; f(x))
_
y =x;y+z (13.29)
_
z = ;qy
_
xyz{ , -
f(x) = M0 x + 0 5(M1 ; M0 )(jx + 1j ; jx ; 1j). p=9 q=
14 286 M1 = ;1=7 M0 = 2=7 (13.29) -
.
2.
, -
, , -
(13.23){(13.26)
z(t) = Asin(!0 t): (13.30)

" "
.
-
,
n = 1.
3.
xk+1 = xk(1 ; xk ) xk 2 R1 (13.31)

F(x) = x(1 ; x), 65, 68]
0< < 4, 3 57. -
0
0 1]:
4.
xk+1 = fMxk g (13.32)
fAg -
A, M > 1.
(13.32)
{ , .
,
x0 , M, -
(13.32), , -

372
0 1] 92]. -
, -
,
0 1].
13.3.3.
.
,
, ..
. , -
, -
.
" -
" x(t) (13.27)
x(0) = x0 .
( , x(t))
d
dt x = W(t) x (13.33)
x = x ; x(t) W(t) = @F(x(t)) {
@x
(13.27) ( ), -
x(t). ,
F(x) , .. (13.27)
- . z = x(0),

(x0 z) = t!1 1 ln k kzk
x(t)k
lim t (13.34)
-
(13.33) z -
( )
z 34, 65, 68].
.. , -
(13.34) , -
z2 Rn
x0 x(t). ,
, -
: ::: n
1 2
2 Rn
zi i = 1 ::: n, (x0 zi ) = i = 1 ::: n.
i

373
.
1
>0 x(t), -
1
, , -
. -
1
, ,{ -
.
x = Ax
_ -
x(t) = 0, , 1 = maxi Re i(A),
. . j 1j (
) .
-
1




= 1 ln kx(t) ; x(t)k (13.35)
t
1

x(0) kx(0) ;
x(t) { (13.27)
x(0)k = , t{ , >0
.
(13.35) -
x0 , x(t). t
68].
. , x(t) -
T -
,
1 ln
T (13.36)
1
{ . ,
. -
-
, ,
,
( ).
-
, -
" ", " ". -
. N( ) {
.
( d) = lim N( ) d : (13.37)
!0

374
( , (13.37) -
{ -
). , df > 0 ,
( d) = +1 d < df ( d) = 0 d > df .
-
;df ,
. N( ) ,

df = ;lim log N( ) : (13.38)
!0 log
, , -
, df
0, 1 2, . , -
df { .
. . -
,
: -
df 2 07, df 2 81. -
, , -
df
,
, 2df + 1. -
,
df + 1.
, , -
df -
, -
2df + 1. -
-
65, 68].




375
13.3.4.
.
,
, -
( , ).
-
. , ,
" " ,
. :
( ),
, -
,
, " " -
. , -
, -
( ) . -
. -
.
1. -
.
, -
.
61, 65, 68, 85]:
{ -
{ , -
( , ),
{ , -
-
( , -
, . .)
{ -
( , {
{ ( ,
- , , 133]).
,
376
-
.
2. . -
92]. -
.
-
, , -
, -
,
.
( -
, . .). -
( , -
. .)
. , " "
-
-
. , -
, -
-
-
.
3. -
.
( ) -
. ,-
156, 159, 181], -
( ),
-
.
-
,
-
. -
,( . 135, 153]).
, { ( -
) -

377
. , -
-
: , ,
.. -
( ,.
13.1.). ,
(
61], {
- 133]). , , -
( , -
-
).
.
-
, , -
. ,
,
, (..
) -
. , , , -
( . . 13.2, . 13.4). ,
. -
,
.
13.4.

13.4.1.
( -
)
, 180] -
. , -
, , -
. -
, -

378
103],
.. 1966 . ( . B).
, -

dx = F(x u) y = h(x) (13.39)
dt
x = x(t) { n- u = u(t) { -
( ) y = y(t) {
, . -
( )
u(t) = Ufy( ) u( ) 0 tg u(t)2U, U{
, U= -
u,u] u>0.
jy(t) ; y (t)j < (13.40)
y (t)=h(x (t)) { , -
-
( ) x (t) (13.39) u(t) u. ,
x(t) , -
" >0 "- -
,
T". ( . -
1927 ., 45].
. , -
(13.39) , ..
x (t) " " u .
, y (t) y(t)
tk k=1 2 :::
-
. , -
Su ,
, ,
x(t) , ..
u 2U. Su
x0 = x(0) x(t).
, ( )
Su Su , (13.39), -
x 2 Su

379
P : S u U ! Su
x0=P(x u).
Su -
.
:
xk+1 = P(xk uk ) yk = h(xk ) k = 1 2 ::: (13.41)
uk 2U, xk = x(tk ) 2 Suk , xk ,
x. (13.41)
(13.39) tk x(t)
Suk , -
z 2 Rn;1 {
: u(t) = uk , tk t < tk+1.
S x0
z(x).
, z(x0 )=0, (13.41)
R n;1 :

yk = ˜ (zk )
˜
zk+1 = P (zk uk ) h (13.42)
P(0 0)=0, ˜ (0)=0.
˜ h (13.42)
{ :
yk+1 + : : : + an;2 yk;n+2 = b0uk + :: + bn;2uk;n+2 + 'k (13.43)
ai bi { -
(13.42) :
n;2
X
˜
B( ) =C( I;A);1 B C = @ h(0) i
B( )= bi
@z
A( ) i=0

n;1 + P ai i
n;2
A( )= 'k -
i=0
j'k j L'(1 + kAk)2n ( 2 + 2 ) (13.44)
z u


#k = b;1 col f1 a0k : : : an;2 k ;^1k : : : ;^n;2 k g 2 R2n;2
^ ^ b b
0k

( )
!k = col fy yk : : : yk;n+2 uk;1 : : : uk;n+2g 2 R2n;2

n
uk = #k;1k if j#T wk j u
Tw
(13.45)
uk k

380
-
, -
, . -
(13.43) -
x(t). , -
, ,
(
kxk;x(tk )k kzk ;z kk,
), " " -
. -
148]:
(
jyk+1 ; y j > y
1
jyk;i ; y(tk;i)j < i = 0::N ; 1
k+1 =
(0
#k ; sign(b0 )(yk+1 ; y )wk =jwk j2
#0k+1 = k+1 = 1
#k
0 = #0 T wk+1
uk+1 k+1 (13.46)
8 #0 ju0k+1j u2 k+1 = 1
> k+1
> #0 ; (u0 ; u)=jwk j
> k+1
>
< k+1
u0k+1 > u k+1 = 1
#k+1 = > #0 ; (u0 + u)=jw j2
> k+1 k
k+1
> u0k+1 < ;u k+1 = 1
>
:
#k k+1 = 0:
>0 { ,u{ -
y{ -
yk y " -
" x(t),
{ (13.43). -
-
: -
(13.42) ,
N-
8" > 0 9 > 0 : jyk+ij < i = 0::N ; 1 ) kzk k < ": (13.47)
-
.
148]. F (13.39) {
,h{ . -
,
381
1) x(t)
2) (13.42) N- N >0
3) b0 (13.43)
4)
:
!
n;2 n;2
X X
jb0j ; jbij > 0 jy j < u jb0j ; jbij : (13.48)
i=1 i=1
>0 , y<
0 0
> 0 , >0, 2(0,1) , (13.40) y
k >0 (13.39),
y
juk j< u.
(13.45), (13.46)
-
V (x) = k ; k . 2

-
, -
{
31, 149].
, 31].
13.4.2.

-
, . . -
192] 1952 . .
70].
_
X = A ; (B + 1)X + X 2 Y (13.49)
_
Y = BX ; X 2 Y
X{ Y{
AB{ ( -
).
X = A, Y = BA;1 , AB
, (13.49)
70],
. 13.2. .
tk Y (t)
382
. 13.2. .

k- ,
yk = Y (tk ). -
u(t) - ,
A tk -
yk : A = A0 + u(t), u(t) = uk tk t < tk+1.
A0 B .
Y (t) y -
u(t) tk . -
" { " (13.43)
yk+1 = ayk + buk + 'k (13.50)
a b{ , 'k {
.
-

uk = (y ; ^k yk )^;1
a bk (13.51)

uk , , -
^k ^k
ab

383
. 13.3. Y (t) .




. 13.4. -
.


384
. 13.5. Y (t)
(y =2.5).




. 13.6. u(t)
(y =2.5).

385
( >0{ -
)
^k+1 = ^k ; (yk ; y )yk
a a (13.52)
^k+1 = ^k ; (yk ; y )uk :
b b
. 13.3 Y (t) t
(13.40) y = 4:5 ( max Y (t) 3:55 -
). -
: A0 =2 B=5.2 X(0)=2
Y (0) = 2:5 =0.095, ^0 =1, ^0 =100.
a b
, -
68], A -
:
A = A0 + a cos(!t).
˜
68]: A0 = 0:4, B = 1:2 a = 0:05 ! = 0:81 :
˜ -
(.
13.4).
-
.
A = A0 + a cos(!t) + u(t), yk = Y (tk ):
˜ -
A0 B a !˜ . -
-
Y (t) y
u(t) tk .
. 13.5, 13.6 Y (t) u(t) -
t y =2.5 (max Y (t) 3:2 -
). -
: X(0)=0.5 Y (0)=1.0. -
, (13.40) -
y, y =3.5.
13.4.3.
-
-
.
, -
-



386
. 13.7. .




. 13.8. Y (t) (y = 6).


387
. 13.9. u(t) (y = 6).




. 13.10.
(y = 6).


388
. 13.11.
.




. 13.12. Y (t) (y = 2).


389
. 13.13.
(y = 2).

. 1976 . 68, 70]:
8
_
< X = ;Y ; Z
_ = X + AY (13.53)
: Y_ = BX ; CZ + XZ
Z
ABC{ . -
X=Y =
=Z =0 A,B,C -
(13.53) 70], -
. 13.7.
. tk {
, Y (t)
k- . -
yk =Y (tk ).
- , -
C tk -
yk : C = C0 + u(t), u(t) = uk tk t < tk+1.
A,B,C0 -
.
Y (t) y
u(t) tk . -
390
" { " (13.43) -

yk+1 + a1yk + a2 yk;1 + a3yk;2 = b1uk + b2uk;1 + b3uk;2 + 'k (13.54)
fai g3 fbig3 {
uk { 1 1
'k { .
-

uk = y + a1k yk + ^2k yk;1 + a3kyk;2 ; ^2k uk;1 ; ^3k uk;2]^;1 (13.55)
^ a ^ b b b1k

uk , , -
^i k ^i k
ab -
(13.54) ( >0{
):
^i k+1 = ai k ; #k yk;i+1
a ^ i=1 2 3 (13.56)
^i k+1 = ^i k ; #k uk;i
b b i=1 2 3
1
" - "
. -
-
-
,
.
" - " , -
. -
,
( , -
),
-
.
-
-
72], MATLAB.
. 13.8, . 13.9 Y (t)
u(t) t -
(13.40) y = 6 (max Y (t)=4.54
).
391
. 13.10. -
: A=0.38 B=0.3 C0 =4.5
X(0)=1 Y (0)=Z(0)=0 =0.005. "- "
^i j0 =0 (i = 1 2 3) ^1 j0 = 5 ^2 3 j0 = 0.
a b b
-
. , -
-
, y
. ,
, -
( -
). -
( -
u = 1:9):
-
- -
.
-
.
A = 0:38 B = 0:3 C0 = 1 Ymax 0:8
( . 13.11). C0 = 2:5
Ymax 1 2:3. , -
C .
(13.40) y =2, y
Y (t).
. 13.12, . 13.13
, -
.
13.5. -

( , -
) -
-
. -
. -
392
. -
-
.
151].
13.5.1.
. 13.1., -
x2 Rn z2 Rn
-
x(t) ; z(t) 0 t 0

kx(t) ; z(t)k ! 0 t!1 (13.57)
-
, ,
.
-
181, 136, 128, 156, 153, 135].
, -
, ,{ -
37, 112]. -
,
. ,
.
-
,
( ), -
.

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