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. 9
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(t)=q(A ) (t)+ 0

(t) = 0 e| t
,
( -
) :
;
q(A ) + |q 0 (A ) W (| ) = ;1: (11.10)
-
9
15, 66, 76]. ,
. -
. ,
.

254
-
-
. -
, (11.10)
. ,
10

. , -
-
.
(11.10) .
.
{ A : -
(11.10) -
.
(11.10) 15, 76, 94,
113].
, -
.
;
A(| ) + ( q(A ) + |q 0(A ) B(| ) = 0 (11.11)

q 0 (A ) B(s) (11.11)
;
D(s) = A(s)+ q(A )+ s
- -
D(s) , D(| ) = 0:
11

(11.10)
W (|!)
1
; q(a !) + |q0(a !) : , -
-
. {
, ,
10
A -
, (11.10).
( , -
, ),
.
D(s) (
11 - -
) D(|!)
;1
! +1 15, 76, 80].

255
!
1
; q(a) + |q0(a) { a. -
.
=!{ A=a{
.
-
.
-
-
( - { ,
, ). -
( ., , 15, 66, 76, 94,
113]).
11.3.4. .

79, 94]. -
k
W(s) = T 2s2 + 2 T s + 1 ,
0
0
-
u = c signx ( . 11.1).
_ -




. 11.1. .
. 11.2.2. .
-
( . . 74),
:

256
. 11.2. () -
( ).

x1(t) = x(t) x2 (t) = x(t):
_ -
G, . -
L ( . 11.2, ,
x . 10.2, . 234).
x0 2 L. -
x00 2 L -
L.
L x
_ -
x>0
_ u=c -
x < 0 u = ;c:
_ -
-
x(0) = g x(0) = 0
_
g0
x(0) = x(0) = 0 ( . 11.2, ).
_ 15, 76], -
x1 =
limt!1 x(t) = ku k{ ,u{ -
( u= c
; = ;ck x+ = ck).
x1 -
1
, ,; x(0)
12

maxt x(t) = x1 + x1 ;x(0) :
x(0) = 0
_
( x1 = ;ck x(0) = g x1 = ck x(0) = g 0),
12
max x(t) ; x1
= t x1 .

257
. 11.3. h = '(g).
g 0 = ;(1+ck); g h = ck + (ck ;g 0) = ck(1+ )2 + 2 g:
, '(g) -
'(g) = ck(1 + )2 + 2 g:
g0 = '(g0 )
g0 = ck 1 + : (11.12)
1;
0< <1 -
(11.12)
( -
, Ax = maxt x(t) g0 ). -
=1 -
( = 0). -
" " .
-
'(g) -
(g h). -
d'(g) : -
dg g=g0
0 < 2 < 1, , -
, (11.12).
x(0) = g 0 x(0) = g 00
. 11.3.
258
-
w(t) . p
w(t) = Tk exp ; T0 t sin T0 t ( t 1;
0), = 2

L
0
15, 76],
p2 T0 2 : -
1;
T0 W(s)
p
= 1T; :
2

0
-
. -
(t) x(t):
_ -
W (s) = T 2s2 + ksT s + 1 (t) u(t)
20
0
(t): , -
( . 11.1), -
(11.10) ; -
: q(A )+
0(A ) W (| ) = 1:
+|q
q(A) = 4c :
(t) = c sign (t): A
:
; 2 4ck|
A ;T 2 + 2 T | + 1 = 1:
0
0
1
= T0
A = 2ck0 :
T
,
Ax A
!= :
,
1 Ax = 2ck :
= T0 (11.13)
-
(11.12). -
-
. -
15, 76],
= exp ; p
, : -
1; 2
A = Ax ; Ax
,
259
. 11.4.
.

A = j AAj (Ax (11.12), Ax {
x
(11.13)). . 11.4,
(
ck) Ax Ax
(11.13) -
: , < 0:5
10%, -
, -
72, 87]. ,
.
, = 0:5
15%, = 0:6 25%.
, -
-
, -
(. 6 . 249). -
, - -
1
= T0 2
= 0:2 3.9, = 0:5 { 1.8,
= 1:0 { 1.25 . , -
, .



260
11.4.
( ,
..
) .
.
.. 90- XIX
.
{ -
-
.
( . 64]).
11.4.1.

;
x(t) = f x(t)
_ x(0) = x0 (11.14)
, f(0) = 0: x=0
{ . -
x(t) 0
(11.14).
x0 6= 0 .
. -
-
: -
t
. ,
.
, f(0) = 0:
x 6= 0
? , , -
f(x ) = 0:
x x(t) x:
, -
x(t) = x(t) ; x : x0 = x0 ; x x(t) = ; + x
x(t)
˜ x(t)
_x(t):
x(t) =
_ x(t) = f
_
; ;
˜
f x(t) = f x + x (t)
˜
f(0) = 0: -
x(t) 0:
, -
x (t)

261
;
( )
x(t) = f x(t) t
_
x0 = x0;
x(0) = x0
x x(t) = x(t) + x x(t) = x(t) + x (t)
_ _; _ -
x(t) = f˜ x(t) t
_
˜
f(0 t) = 0 t:
, -
(11.14) -
;
x(t) = f x(t) t
_ f(0 t) = 0:
12, 66, 76, 94].
1. ( -
t!1
) ">0
jjx0jj <
> 0,
jjx(t)jj < " 2
t > 0:




. 11.5. .
S jjxjj < -
, ,
S, S":
S S" {
1.
jj jj
( ).
S S" S 6= f0g ( . 11.5).
2.
, t 12, 79].
3.
, " ".
262
S
, . ,
. ,
" " .
( ) ,
.
4.
. -
, , .




. 11.6. .
2.
: 1) 2) -
,
jjx0jj <
>0 ,
limt!1 x(t) = 0 ( . 11.6).
S , -
, x0 = 0 { -
( S ): 2
3.
( ),
2, S = X {
.2
4. (
x0 2 S )
), >0 ,
-
S" ( . 11.7). 2
, -
263
. ,
, , -
, -
,
.




. 11.7. .

11.4.2.
-
.
.
,
,
.
(x G) G
x -
(x G) = inf xz 2G jjx ; xz jj:
,
5. G ,
">0 > 0,
(x0 G) <
x0 ,
(x(t) G) < " t > 0: 2
,
, -
.. -
, -
.

264
,
( . 11.8).




. 11.8. .
, (11.14) -
0 t < +1: -
, ,
. , -
, ,
, -
.
34, 94].
6. M -
fxg x(t0 ) 2 M
, -
x(t) 2 M ;1 < t < +1:
t0 ,
x(t0 ) , -
2
.
-
, -
28, 86].
7. M
( (11.14)), ">0
(x0 M) <
> 0, x0 ,
(x M) < " 2
t > 0:
-
.

265
-
.
. -

x = f(x)
_ (11.15)
= h(x) (11.16)
x 2 Rn , 2 Rnu , nu n, f(x) h(x) { -
. (11.15) x=x ( -
-
).
8. x=x (11.15) -
h(x), -
">0 (") > 0, ,
jx0 ; x j <
x0 , -
x(t) x(0) = x0
t0
jh(x(t)) ; h(x )j < " t 0: (11.17)
x=x h(x) ,
,
lim h(x(t)) = h(x ) (11.18)
t!1
x
h(x).
x=x
h(x), (11.15) t0
(11.18) -
x0 , x=x ( (11.15))
-
h(x).
, nu = n h(x) = x 8
. .. -
.
1957 . .. -
,
x = colfy zg h(x) = y. -
. 28, 29, 64, 86]. ,
266
h(x)
fx:h(x) = h(x )g,
.
. 2-
x1 = x1
_
(11.19)
x2 = ; 1+x22 :
_ 2x
1

x1 (0) = 1, x2 (0) = a
a(1 + e;2t ) :
(t) = et
x1 x2(t) = (11.20)
2

x2
h(x1 x2) = 1 +2x2
1

:
h(x1 x2 ) = ; 1 4x2 2 + (1 x2x12)2 = ;4h(x1 x2) 1 ; 4(1 x1 x2 ) :
2 22 2
_
+ x1 + x1 +1
_
, ;4h h ;3h 0 , , h(x1 (t)x2 (t)) ! 0
_ (x1 (t) x2(t)) ! 0 t ! 1.
h ,
x2 (0) = a 6= 0
(11.19)
S = f(x1 x2 ): h(x1 x2 ) = 0g = f(x1 x2 ): x2 = 0g.
(11.20). 2
,
, -
34, 64].
9. -
,
0t1
, .. -
0 1) ( . 11.9). 2
10. -
( ),
S ,S SS x0 2 S
, -
t <1( , x0),
x0 2 S . 2
tt
S -
, S{ .

267
. 11.9. .




. 11.10. .
S{ , -
( . 11.10).
-
,
, .
11.4.3.

.
.
,n=1

268
x(t) = f(x)
_ f(0) = 0: (11.21)
f(x) -
x 6= 0 . .
xf(x) < 0
, f(x) = 0
x = 0: .
(11.21).
V (x) = 1 x2 : ,
2
x 6= 0:
V (0) = 0 V (x) > 0 x = x(t)
(11.21). ,
;
V (x) = V x(t) : -
(11.21). -
_
V (x) = x(t)x(t) = xf(x),
_
_
.. V (x) -
f(x): -
_
V (x) (11.21).
_
x 6= 0
, V (x) < 0 -
t!1
, V (t) ,
jx(t)j
. ,
V (x)) x(t) ! 0
(
t ! 1) , (11.21)
. 13

, -
(11.21), ,
{ f(x).
.-
n:
-
. -
. -
, .. -
1892 . 60].
. -
13
, , 12, 34, 79, 97].

269
" ] , -
, -
. ... , -
, -
x1 x2 : : : xn t -
,
t , x1 x2 : : : xn
t, ."
..
. -
XX . ,
, -
, -
, .
-
.
,
.
11.4.4.
V (x) -
: 1) V (x) - -
X,
x
2) V (x) -
: V (0) = 0 3) V (x) , .. -
, : V (x) > 0
x 6= 0: 2
W(x) ,
;W(x) .
x=0 -
( ).
c
(10.12) ( n=1 (11.14)).



270
14


@V @V @V
_
V (x)=rx V (x)f(x)= @x f1 (x)+ @x f2(x)+ + @x fn(x):(11.22)
n
1 2

.
1. (.. ).
x2 -
V (x) ,
(10.12) , -
.
2. (.. -
).
x2 -
V (x) ,
(10.12) ,
.
3.
S( " ") 93, 94].
2
C>0 V (x) C -
S f0g 2 S -
f0g
S(. 2, . 263).
4.
( { ).
2
= X, V (x) ! 1 kxk ! 1
, .. -
.
, -
, -
64, 93]. V (x t) -
23]:
14
x 2 Rn
V (x) 1 n-
(.. -
@V T
V x): @xi = (rx V (x)) -
f(x) 2 Rm x 2 Rn m n- , -
@fj
@xi
A = faij g
V m n- m n- , -
@V -
@aTij
n
xT Hx 2x H xT H + HxT :
x2R

271
. 11.11.
.
t
W(x) ! 1 kxk ! 1
V (x t) > W(x) ,
.
5. (.. ).
_
V (x)
, V (x) > 0
.
, -
_
V (x) 0 V (x) ! 1 kxk ! 1
,
V (x) ! 1 kxk ! 1
{
_ x2S :
V (x) < 0 =
-
. (11.22),
_
V -
V -
_
. V (x)
V (x)
( . 11.11). -
_
V ( S)
V = const -
.
, ,
_ (x),
V (11.22), -

@V (x) f(x) = ;Q(x) (11.23)
@x
272
Q(x) { -
. (11.23) -
, ,
93]. , -
V ( ), f( ) , V = V (x t) f = f(x t)

@V (x t) + @V (x t) f(x t) = ;Q(x t): (11.24)
@t @x
-
( . 3, 23, 47, 73, 93], . 209).
( . 265, . 6)
-
94, 174].
-
-
_
, V (x) (
).
_
, V (x) 0 -
x
C, , V (x(t))
C
( -
). ,
_
V (x) , V (x) < 0 -
. , -
_
, V (x) = 0: -
54].
6. (- ).
V (x) { , -
C = fx : V (x) <
x C
Cg: ,
C C
_ M
V (x) 0: -
!: t
M:
C
, 3 -
= f0g,
6 -
( . 11.4.6.).
!-
; x(t) (11.21),
+



273
,
t ! 1: t!1
x(t) , -
;. , x(t) (
+

t 0), !- ;+ ,
. 15
: -
" ", -
{ , -
( -
) . -
, -
66, 76, 93, 94].
,
..
V (x) = xT Hx
H (
),
T
H = H > 0: -
( . 1-3 . 270) , -
V (x) ! 1 kxk ! 1
,
.
H = HT ,
( ) -
H:
, H ,
.
, f(x t) = A(t)x
-
V (x t) = xT H(t)x (11.24)

_
H(t) + A(t)T H(t) + H(t)A(t) = ;Q(t) (11.25)
M Rn ,
15 -
( ) .
.


274
Q(t) = Q(t)T 0 (> 0) { -
. , V = V (x) A(t) A Q(t) Q
16

(11.25), -

AT H + HA = ;Q: (11.26)
(11.25), (11.26) ( -
) .
53],
H -
(11.26), ( )-
A: 3, 30], A- -T
, (11.26) n n- H=H
{
Z1 T
eA t QeAtdt:
H=
0

Q = QT H = HT
0 0 - -
H A: Hx0 = 0
HAx0 = 0.
-
,, , -
. C. . 432
lyap MATLAB,
(11.26).
, ,
( ), -
(. . 11.4.6.).
8, 36,
78, 93] ( ), -
.
, 14
16
. 271.

275
( )
'( ) ( . 10.1, . 230),
" ",
.. ..
Z
T
V (x) = x Hx + # '( )d #2R
0

{ " "
{
k

XZ
k j
T
#j 2R
V (x) = x Hx + #j 'j ( j )d j = 1 : : : k:
j
j=1 0



56].
90, 93]. -
, ,
.
11.4.5.
-
36, 78, 110, 174]. ,
.
-

x k + 1] = f(x k]) k = 0 1 2 :::: (11.27)
, x=0
(11.27), . . f(0) = 0 x k] 0 -
(11.27). , -
x (.. , x = f(x )),
-
,
x k] = x k] ; x :
(11.27)

276
( . 270),
V (x):
110].
1. -
.
V (x)
x2
, (11.27)
x 6= 0)
V (x) 0 ( < 0 (11.28)
( -
) .
2. .
V (x) -
V (x) ! 1 k!1 -
.
, 1, 2, -
_
V 0 V (x) 0:
-
.

x k + 1] = Ax k] x 0] = x0 k = 0 1 2 ::: : (11.29)
V (x) = xT Hx H = H T > 0: -
V (x) = V (f(x)) ; V (x) (11.29).
f(x) = Ax ;
V (x) = (Ax)T HAx; xT Hx = xT AT HA ; H x:

{T
H=H >0 -

AT HA ; H = ;G G = GT > 0: (11.30)
, -
-
, (11.30)

277
47, 73]. C. ( . 427) -
dlyap
MATLAB-
(11.30).
, -
.
78].
f = f(x) x 2 Rn f 2 Rn -
,
kf(x) ; f(y)k Lkx ; yk, L < 1:
,
17

78] , f
{ , -
x (.. , x = f(x ) .
. 246), (11.27) x0

k
L
kx k] ; x k 1 ; Lk kf(x0) ; x0k:
, -
(11.27).
, (11.29) -
78].
-
72, 78] ( . .
6.10.2. . 161).
, -
, - -
, , , -
.
. 11.5.
-
-
30, 83, 94].
64].
f
, -
17
L < 1.
( . (10.14) . 237)



278
11.4.6.
1. . -
m -
l( . . 23). -
, " " . ,
.
'(t)
J '(t) + '(t)+ mgl sin '(t) = 0
_ J = ml2
{ 0{
( , '=0 "
"). J
'(t) + %'(t) + !0 sin '(t) = 0
_ (11.31)
2

qg
!0 = l :
H -

H(' ') = J _2
'2 + mgl(1 ; cos '):
_ (11.32)
x = colf' 'g
_ -
(11.31)
x1(t) = x2(t)
_ (11.33)
x2(t) = ;!0 sin x1 (t) ; %x2(t):
_ 2


V (x)
V (x) = _22x2 + ! 2 (1 ; cos x ) (x =
H(' ') _ , 1
0
colf' 'g).
_ , -
. 270
. 3, H(' ') = 0
_
' = 2k ' = 0 _
(k = 0 1 2 : : :), .
= fx : j'j V (x) <
2!0 g: -
2

(' '),_ H -
,
;
H = max' mgl(1;cos ') = 2mgl: -
p
x2 = !0 2(cos x1 + 1): , -
,
V (x) x = 0:
279
_
V (x(t)) (11.33).
(11.22), . 271,
_
V (x) = ;%x2: (11.34)
2

_ x2
V (x) 0
1 ( . 271)
.
%=0(
) %>0( ).
_ (x) 0 . .
% = 0 (11.34) , V
.
V (x) , -
, -
, .. -
. V (x) = C
0<C <H , -

x2 ; 2!0 cos x1 = 2(C ; !0 ): (11.35)
2 2
2

,
(11.33). , (11.33) t -
x2dx2 = ;! sin x1 dx1 -
2
0
(11.35). 18

-
.
, V (x) -
, t -
(11.35).
x = 0: (11.35)
C 6= 0 ,
,
( C). -
(
V (x)) %=0 . 11.5, . 262.
-
.
. . 122, . 5.3.1.
18



280
12, 79],
x 2R n
x = f(x)
_ Q(x) -
-
( ,
-
- f(x)), Q(x)
,
, -
t . -
@Q(x) f(x) = 0:
79] @x
(11.22) ,
, V (x) = Q(x)
_
V (x) 0 .
-
_
, % 6= 0: , (11.34) , V (x) = 0
x2 = 0: -
. , -
6 ! x2 = 0: -
, -
x2 (t) 0, .
M = f0g: t!1
,
M .. x = 0:
,
(11.33),
-
.
V (x)
. 11.11, . 272.
. 11.12.
2. . -
, , -
M(t): -
u(t) = M(t) : .
J
, (11.31)
'(t) + !0 sin '(t) = u(t): (11.36)
2


H
(11.32). -
281
; _;
. 11.12. V x(t) V x(t)
(11.33).

, -
H ( . 6, 64]).
;
u = ; H ; H )'(t)
_ (11.37)

;;
u = ; sign H ; H )'(t)
_ (11.38)
6, 64]. -
.(
A., . A.
. (A.15) . 410). (11.37), (11.38)
H , >0{ -
( (11.37) ,
(11.38) { " " ).
, -
, V (x) = x + !0 (1 ; cos x1 )
_ 2
2
2
2
x = colf' 'g,
_ .

__
V = '' + !0 ' sin ' = u':
_ _ (11.39)
2




(11.37) ,
;
_
V = ; H ; H '2
_ (11.40)

282
(11.38) {
;
_
V = ; sign H ; H j'j: _ (11.41)
_
H < H ' 6= 0
, _ V > 0:
'(t) 0
_ -
'(t) 0, (' = ' = 0)
_ -
.
, H=H: -
, ,
u(t) 0 ( ,
;
H '(0) '(0) = H )
_ ,
H(t) = H ( . (11.35), . 280).
_
H>H V <0 V{
= fx : V (x) 2!0 g ( . . 279),
2

, (-
'(t) 0)
,
H(t) H : , -
.
. 11.13 -
() ( ).




. 11.13. .
3. .
-
: 52]
;
x1(t) = !x2(t) + a2 x1(t) ; x1 (t);x1 (t)2 + x2 (t)2
_ (11.42)
x2(t) = !x1(t) + a2 x2(t) ; x2 (t) x1 (t)2 + x2 (t)2
_
283
!>0 a>0{ . x=0 -
. -
V (x) = x2 + x2:
1 2
(. 4, . 271) .
V (x) (11.42) -
_ 2 + x2 )(a2 ; x2 ; x2 ):
V (x) = (x1 2 ,
1 2
_ (x) > 0 kxk < a x 6= 0:
V ,
kxk > a
. -
V (x) < 0
kxkj a
, -
_ (x) = 0 kxk = a
. V ,
x1 (t) 2 + x (t)2 = a2 (11.42),
2


{ (11.42) .
. 11.14 . -
_
V (x) V (x) ( a=1! = ) -
V (x(t))
x0 = ;0:1 0]T ( ) x0 = ;3 0]T ( -
).
. 11.14 .




. 11.14. -
.
4. . -
174]. ,
x vr - -
vh , -
( -
, "-
284
"). xh yh xr yr , -
, . xr (t) = vr yr (t) = 0
_ _
yr (0) = 0: ,
, k>0
;
xh (t) = ;k; xh (t) ; xr (t)
_
yh (t) = ;k yh (t) ; yr (t) :
_
k
xh (t) + yh (t) = vh :
_ _
2 2
2

8
xh (t) ; xr (t)
> xh (t) = ; q
>_ vh
> ;
>
< xh (t) ; xr (t) + yh (t)2
2

yh (t)
> yh (t) = ; q
>_ vh :
> ;
>
: xh (t) ; xr (t) + yh (t)2
2



x = xh ; xr y = yh
8
> xh (t) = ; p x(t) v ;v
>_
< x(t)2 + y(t)2 h r (11.43)
> yh (t) = ; p yh (t)
>_ vh :
: x(t) + y(t)
2 2


, ? -
: -
(x0 y0 ) ?
(11.43). , (11.43) -
x=y=0( ).
, -
.
: V (x y) = x2 + y 2 :p
_
V (x y)=;2vh x2 +y 2 ;2vr x:
(11.43)
_
, vh > vr V (x y) < 0 ,
. , -
19

( { ),
( ).
x 6= 0
x=0
, ,
19
p
;vh x + y ; vr x < ;(vh ; vr )jxj < 0:
2 2



285
5. -
. 174]
8
> x1 k + 1] = x2 k] 2
< 1 + x2 k] (11.44)
> x2 k + 1] = x1 k] 2 :
:
1 + x k] 2

x = 0 x = colfx1 x2 g -
( , x = f(x ) x =0 ,
). V (x) = x2 +x2:
1 2
, (11.44),
V (f(x)):
V (f(x)) = ; x2 2 2 + ; x1 2 2 = ;x1 + x2 2 = ; V (x) 2 :
2 2 2 2

1 + x2 1 + x2 1 + x2 1 + x2
2 2

x 6= 0:
, V (f(x)) < V (x) -
2 . 277, -
, x=0
(11.44) .
11.5.
11.5.1.
,
: -
. -

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