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519.6 517.5 62-50 !! 22.19 61 ".#., %.#. " &' ( / * . #.+. ", . { ".: %, - "/, 2000. { 102 . 01 & 23 ' ' 4 2 , &' ( 2 2 5 ( 4' , & . 2 , 6 ' , 7 5' ' ( '5( & 5 2,83 ' 4 2, &' ( '-!8 2 & , & 2 &' , & , 5 1 8 2 ' 4'', 6 ,1 &1 6 ' 4 2 , 2 5 '7, & '7 (( ' 2' ' &, '. "5(2 7 , & 47 , 837 2 &1 '' 2 5 ( 5 7 4' . 9 '6 1 & , & & 7 4 2 , ' ' 11 , : & ' , & 1 , & 2, : . %, 3 & ( & 6 ;b=a < a limh(t ; ) '" i = > 1=2 '(b) "!0 : '(t) a i 5 i | , , 83 & 6 ', 25 &, 2 1 (' 83 ' & . * 1 ' &, 7 5 7 & '6 ,' 2 '5(27 O25, 3, 7]. * F (t) | 2 ' &2 n : ' ' , L2 O0 t]: 0 '' (( 6 l(x) dx( ) (1:1:2) d ; F ( )x( ) 5 x( ) 2 C01O0 t]: 0 c2 > 0 c1 c2 { . (1:1:3) 2 &26 5 (( 5 6 2 l (x) ; dxd ( ) ; x ( )F ( ) x( ) 2 Ct1O0 t] (1:1:4) ' ' 4 c0 1kl (x)k;10 kxk0 c02 kl (x)k;10 (1:1:5) c01 c02 | & 6 O25].
10
1
# & '6 C01 O0 t] L2 O0 t] 2 815 x 2 L2 O0 t] 3 fxi ( )g1 i=1 | & -( , C01O0 t] ( ' 2 & ' k:k0 7 232 2 x0: 5 , (1.1.3) , kln (x) ; lm (x)k;1t ckxn ; xm k0 ! 0 & n m ! 1 ;1 . . & ln (x)1 n=1 7 2 1 , W2t O0 t]{ & & , ' & L(x): 9& L(x) & : ' 7 L2O0 t] , kLk = klk: 9& L , 2 4 ' (& 6 ') & l & & . 1.1 1 L l (1.1.2) C0 O0 t] (1.1.3). L ! " L2 O0 t] W2;t 1O0 t]: L l , (1.1.4) Ct1O0 t] (1.1.5). L ! " L2O0 t] W20;1 O0 t] O25]: 1.1 & '( ) dx( ) = F ( )x( ) + v( ) x(0) = 0 v( ) 2 W ;1O0 t] (1:1:6) 2t d " ' * -!*+ x( ) 2 L2 O0 t] * fx( )g1 n=1 ' C01O0 t] *, ( kxn ; xk0 ! 0 kLxn ; vk;1t ! 0 n ! 1: D ' , : ' ' F ( ) v( ) & 6 CO0 t] (' , 4 ' ' ' & 2 1.1 & 8. U (' , 4 2 , x(0) = x0 , &'38 ,' x^( ) = x( ) + x0 , 2 (1.1.6). %, , : ' ' F( ) v( ) & 6 & COt0 t] 4 , 4 & 2 2 (' 4
Z
x( ) = \( t0 )x(t0 ) + \( s)v(s)ds t0
(1:1:7)
5 \( s) | ( ' 2 ' ', 2 2 d\( s) = F( )\( s) \(s s) = I n d 1 83' ': \( s)\(s ) = \( ) \;1( ) = \( ): U : ' ' F( ) & 6 & L2 Ot0 t] v( ) , W2;1 Ot0 t] & 113 2 (' 4 O25] 2 h\ ( ) vi 3 66 h\12 ( ) vi 77 6 : : : 77 2 Ot t] x( ) = \( t0 )x(t0 ) + 66 h\ ( ) (1:1:8) 0 64 j : : : vi 775 h\n ( ) vi 5 \j ( ) = \j ( s) ' \( s) h\j ( ) vi | 1 2 (' 3 ' & fW21Ot0 ] L2Ot0 ] W2;1Ot0 ]g 2 2 81 & fvk (s)g1 k=1 , L2 Ot0 ] , kv ; vk kW2;1 t0 ] ! 0 & 2 2 4 ':
Z
h\j ( ) vi = klim \j ( s)vk (s) ds: !1 t0
11
1.1
1.2 , + K( ) ( 2 Ot0 t]) * n n - Kij ( ) 2 W21Ot0 t] i = 1 2 ::: n j = 1 2 ::: n " + h() ' n m - Rt h () 2 W ;1 Ot t] j = 1 2 ::: n k = 1 2 ::: m K( )h() d " jk
2
"' :
0
t0
2 hK ( ) h i 66 hK12 ( ) h11i Zt 6 :: K( )h() d = 66 hK :( ) 64 i : : : h1i t
::: ::: ::: ::: ::: hKn ( ) h1i : : :
3
hK1 ( ) hk i : : : hK1 ( ) hm i hK2 ( ) hk i : : : hK2 ( ) hm i 77 :::
:::
:::
77
hKi ( ) hk i : : : hKi ( ) hm i 77 0 5 ::: ::: ::: hKn ( ) hk i : : : hKn ( ) hm i Ki ( ) | * + K( ) hk () | "+ + h() hKi ( ) hk i | " ! fW21 Ot0 t] L2Ot0 t] W2;1Ot0 t]g: 0 '' 2 &22, & , ' 4 ' (& 1 ,5 '' , & ' '6 ,' 2 O8,27,3] ). 91,' , (] F P ) 2 & , ' & 83 2
4 ' (, | & ' ' 5 6 2. * x( ) x( !) { & ( 2 O0 t] ! 2 ]) 1 ( 5 2 8 1 (& ) & 6 & W2;t 1O0 t]: " ' 6 Ox( )] 2 2 & (' Ox( )] = Dt Ojt x( )] 2 ' 2 2 O0 t] Ox( )x()] = Dt Dt O(jt x( ))(jt x()) ] 5 & Dt ,'( 16 L2 O0 t] W2;t 1O0 t] jt 1 ' & .
1.3 (27]) 0( + v( !) * + !*+ , {!*+ ,*,
Ov( )] = 0 Ov( )v ()] = V ( )( ; ) V ( ) = V ( ) V ( )z 2 COa b]
8z 2 En ' " . "6 V ( ) & {( , 2 ' 1 5 4' v( !): % 1 5 4' & 2 1 & 8 '' 8 '. 2 & , , 2 8z 2 En 8 t 2 Oa b] 2 &, (z ( ))n 0: ' 5, ( ; ) = 0 & 6= ' V ( ) '6 1 V () pVV( )V (): & 2 Ov( )v ( )] 1 5 4' 1 , 5 , 2 7 5 1 ,7 7 . ! 4' 3 (, , 2 5 , 17 ' 1 2 '3 . *2 1 5 4' 2 2 2 ' ' 1 , 1 2 & 2 . * '6 5 4 1 4 ' 4 & &1 6 2 5 & 1 ' 4'. 0 & '6 & 2 ,' 1 ' 4'' 5 , 5 ' 6 , 2' 5' , & ' , 2 5 & & , ' . *2 " ' " ' ' 5 & , & ,' 5 5 & 1 ' 4'' 6 6 ' 1 2 1 , 1 & .
1.4 (27]) 0( + v( !) ' " , ( !! + ( + ' .
(( &' 2 113 ' ' 1 4' 2 2 2 113 ' ' & '. 0 , 1 5 4' 2 8 1 & 6 5 ' & W2;1O0 t]: 91,' , w( !) 2 O0 t] ! 2 ] & 83' ': 1) w( ) | & ' , '' &3 2' 2) w(0) = 0 &
12
1
3) 2 817 & 637 O0 t] ^w = w( ) ; w() ' 5 & , 8 ' Kw ( ) = Rj ; j R | -& 2 ' 4) & 2 7 ! 2 ] 1 ( w( ) & , (( ' & 1' ' ' 8 81' 5 ' ' 1 8 8 5) MOw( )] = 0 MOw( )w()] = R min( ): * w( !) 283 :' 2', , 2 ' ' & '. (( 2 113 ' ' 1 w( !) & ' & v( !) v( )
1 ( 5 & & 6 5 ' & W2;1 O0 t]: : 2 8'( ) 2 Ct1O0 t] ' '
Z d' hv 'i = ;hw d i = ; w ( ) '( ) d d : t
0
* t t0 * , ( x(t) (1.1.14)(1.1.15) " Ot0 t] * ' "
.
0 4 2 (1.1.14), , , (( 7 , 2 815 2 Ot0 t] '6 & x(t) = \(t )x( ) (1:1:16) 5 \(t ) | ( ' 2 ' 4 ' (1.1.14). " \(t ) 2 ' d\(t ) = F(t)\(t ) \( ) = I (1:1:17) n dt 5 In | 2 ' &2 n: " \(t ) 6 2 \;1 (t ) = \ ( t): ' , (1.1.16) x( ) & ' (1.1.15). * !*+ u( )
Zt
t0
u( )v( ) d = 0
v( ) - !*+, '* Ot0 t] v( ) 0 - '* . 2 3 5 & W21 Ot0 t] L2Ot0 t] W2;1Ot0 t] : '' '6 (' 83' 1,': 1.3 > * * -!*+ u( ) 2 W2;1Ot0 t] " ! hu vi = 0 v( ) 1 * !* * -!*+ ' W2 Ot0 t] v( ) 0 '* t0 t: > * * -!*+ v( ) 2 W21 Ot0 t] " ! hv ui = 0 u( ) * !* * -!*+ ' W2;1Ot0 t] u( ) 0 ( 6 ' Ot0 t]:
Rt
* 622 5 ( m(t) 5 2 & , h(t ) = h0(t ) ( x^(t) = h0 (t )y( t0 & '. - + Kv ( ) L2 Ot0 t] v( ) " ' ( + . > Kv ( ) = R( )( ;) + R( ) + , 8u 2 L2 Ot0 t] (u Ru)0 0 ( - ( " + * R( ) ), v( ) " ' " .
2 ( !" #( $ #% ) &%(* + (* !%&. #% %
# :' , & 17 ' & 83 ' & 2 ( , 7 ( O27,8,35], 6 5' 4 2 , &' ( , & 6 .# ' O44,30]. , 5 &' 2 ( # . 2.1 ! "
D' &2 , & , 7 7 5 ' ( , 2 ' ). # 7 ( , 8 & 7 7 & ((). 2.1 0( !*+ x( !) ' + * , ( * , * + + ' * ' , mx ( ) = mx = const Kx ( ) = Kx ( ; ): 2.2 0( !*+ x( !) ' + '* , * ( ' * ' . %, & 2.1 2.2 . 7 & ' '' ' & 5
5 &2 , 7 ,' ' , 2 2 2 & '6 ' '6 7 & , 7 4' ' . 2.3 0( !*+ x( !) ' * + + , * + + ' * ' . 0 5 vij {: ' ' V . 1 , !0 ( ) - 1 (( '2, 2 8 27 Rt 0 t , 2 O0 t] (2 !" ( )d = 1 . ;8 u" ( ) , 8 ' u( ): 0 (! & 18 ('8 1 & 7 2 '6 O25].) #' 2 (3.1.1) 1 ' ' ' & ' u"( ) : dx( ) = F ( )x( ) + G( )u ( ) x(0) = x : (3:1:15) " 0 d 0 4 (3.1.15) , " , &:' 1 ' 1, 5 , x"( ): # -(2 u" ( ) { 1 (( '2 x"( ) 2 2 2 4 ' (3.1.15) 1' ' . 2 (3.1.14), ' ' Zt d z x" (t) = ' (t t)x"(t) = ' (t 0)x0 + O d ' (t )x"( )]d : 0
% & ,2 2 (3.1.13) (3.1.15), ,&4 ' d O' (t )x ( )] = d' (t ) x ( ) + ' (t ) dx"( ) = ;H(t )C( )x ( ) + ' (t )G( )u ( ): " " " " d d d % 52 : 6 & 7 0 t , & ' z x" (t) = ' (t 0)x0 + # ' 4 z x^" (t) = ;
Zt 0
Zt 0
O;H(t )C( )x"( ) + ' (t )G( )u"( )] d :
Zt
H(t )C( )x"( ) d ; H(t )v( ) d : 0
D ' , z fx" (t) ; x^"(t)g = '(t 0)x0 +
Zt 0
' (t )G( )u"( ) d +
Zt 0
H(t )v( ) d :
#, ' 1 :5 6 2 &' ' & ' 4 8 & 8 ' ' 5 6 2. 0
, ' " '( , c0 f kC uk2;10 + kIq uk20 g hu Bui: , . 9 ' , 5 '7 (3.5.15). % & ,2 & 6 8 & ' G( )Q( )G ( ) , (3.5.16), 7 '
3 2Zt 3 Zt 2Zt k1 = 4 z h(t )C()\^ ( s) d5 G(s)Q(s)G (s) 4 \^ ( s)C ( )h (t )z d 5 ds 0
0
0
Zt Zt 2 ^ cq \ ( s)C ( )h (t )z d 0 = cq \ ( s)C ( )h (t )z d 20 : s
0
9' ', (' 4 2 2 l (x) ; dx( ) d ; F ( )x( ) = v( ) x(t) = xt v( ) 2 L2Ot0 t] & xt = 0 2 2 2 &26 ' (3.5.1), ' : x( ) = \ (t )xt +
Zt
\ (s )v(s) ds
52
3 " -# $
' 5, xt = 0 & ' x( ) =
Zt
\ (s )v(s) ds
& (1.1.5). 0 MO(x2(0) ; x20)(x2 (0) ; x20) ] = P22 P22 = P22 > 0 MO(x1(0) ; x10)(x2 (0) ; x20) ] = P12 P12 = P21
6 Q( ) > 0 , , 2 8u( ) 2 L2O0 t] (u Qu)0 ckuk20 c > 0 P > 0 { 2 8z 2 En (z P z)n c0 kz k2n c0 > 0: " fFij ( )gij =12 2 , : ' , & 637 L2O0 t]: 1 8 2 & 2 2 & , y1 ( ) = x1 ( ) (3:5:24) y2 ( ) = C1( )x1 ( ) + C2( )x2 ( ) + v( ) (3:5:25) 5 0 t < 1 v( ) | q {' 1 5 4' ' ' MOv( )v ()] = R( )( ; ) ' R( ) | '' 2 & 6 & 2. * u1( ) u2( ) v( ) ' ' x1(0) x2(0):
u ( )v () U ( ) M 1 = 1 ( ; ): u2( )v ()
(3:5:26)
U2 ( )
K ' ' R( ) fQij ( )gij =12 | 5 , . . ' 8 ''' & 8 & 8 &, 8, C1( ) C2( ) | & O0 t]: 0 2 O0 t]: 0 '' # -F&( MOkx(t)s ()] =
Zt 0
h1 (t )MOs( )s ()] d :
(3:5:36)
56
3 " -# $
0 d^x ( ) = F( )^x ( ) + P( )C ( )S ;1 ( )Oy( ) ; C( )^x ( )] d x^ (0) = 0 0 t < 1 (4:1:25) *, ( lim fMO(z x(t) ; x^ (t))2n ] ; MO(z x(t) ; x^(t))2n ]g = 0 8z 2 En: !0
, . * 22 (4.1.24) (4.1.22) 7 ', z h (t ) = ' (t )P( )C ( )S ;1 ( ) . 91,' P ( )C ( )S ;1 ( ) = K ( ) & ' ' ' (t ) 5 , ' (t ) = z o(t ) , 5 o(t ) | ( ' 2 ' ' (4.1.25), do(t ) = OF(t) ; K (t)C(t)]o(t ) o( ) = I: (4:1:26) dt 0 (5:0:2) 5 - &' 5 2,. 0 4 u 2 (5.0.2) 2 ''' ( '(u) = kBu ; f k20 + hu Bui > 0 (5:0:3) & 2 '& 8 & 7 8 '-(8 5 &' 5 ( . * fu (t )g >0 4 (5.0.2) & ! 0 7 2 '& & 7 ( u0 (t ) , 283 8 (5.0.1) ( '. ' 4.3) , 83 4 , &' ( ( ' 4.4). # , 6 ' , 4.1 5' , '& & 7 (, , 4 2 2 (5.0.2) 1 2, & :' 1 7 ' fu (t )g >0 ! u0(t ) & ! 0 2 7 ' 5 2, 7 4 8 , ( ( ' 4.4, '. . 62). 5.1 1
# :' , & 2 ' 7 ' & 4 2 ,
&' ( &1 6 5 , 2 7 7 7 & & , 5 2,83 5 5', & 5 , 4.1. * , . * n {' & x( ) ( 0 t < 1 ) 5 2 ' dx( ) = F ( )x( ) + G( )u( ) x(0) = x (5:1:1) 0 d 5 u( ) - 1 5 4' 83' ': MOu( )] = 0 MOu( )u()] = Q( )( ; ) Q( ) = Q ( ) Q( ) 0 , 2 O0 t] x0 | 5 , MOx0] = 0 MOx0x0 ] = P0 P0 = P0 P0 0 MOu( )x0 ] = 0: 0 (5.1.1) &' 2 ' 5 5 & . 1 8 2 m {' & fy( ) 0 tg 2, x( ) 4 ' y( ) = C( )x( ) + w( ) (5:1:2) 5 C( ) | ' 1 8 , w( ) | 5 6 1 4', 0 ( ; ) MOw( )] = 0 MOw( )u()] = 0 MOw( )x0 ] = 0 MOw( )w()] = 00 R( )
68
5 2 / )( 2
5 R( ) | ' q {' 5 1 5 5 5 4', R( ) > 0 R( ) = R ( ) ( 0 q m n ). #' '6 7 7 7 ^ = fF ( ) G( ) Q( ) P0 C( ) R( )g , & 7 &1 6 ^" = fF"( ) G"( ) Q"( ) P0" C"( ) R"( )g: " Q" ( ) P0" R" ( ) | '' & . "6 ^" , & 4 : max fkIi ; Ii"k Ii 2 ^ Ii" 2 ^" g " (5:1:3) i
5 " > 0 | 2 , k k | '2 ' L2 O0 t] . 0 | '+. , ' 5 , ' 4.4, & :' x^" (t) =
Zt 0
Dh" (t )j y( ) d
(5:1:5)
, h" (t ) | 4 2 # - F&( Kx" (t )C" () = MOx"( )x" ( )] ,
Zt 0
h" (t )C"( )Kx" ( )C" () d + h" (t )S" ()
(5:1:6)
5 Kx" ( ) = x" ( ) | 4 (( 5 2 (5.1.1) 7 ' ' ^" . (5.1.6) '6 ,& (' B" u" + u" = f" (5:1:7)
69
5.1 2 /
5
(t )z z 2 En u" (t ) = h" (t )z f" (t ) = C"()Kx" Zt B" u = C"()Kx" ( )C" ( )u( ) d + 00 R"0( ) u( ):
(5:1:8)
0
0 4 2 (5.1.7) '', (
'" (u) = kB" u ; f" k20 + hu B"ui: %' ' ' 5.2 D+ * ( '( !+ ' 2 "2 + "2 ku k2 MO(z x(t) ; x^" (t))2n ] MO(z x(t) ; x^ (t))2n ] + 2 f B " ;10 ! m(t) ! 0 " ! 0 " ! 0 (5:1:9) C k) + (" + kC"k)O"xkC" k + ("x + kKx" k)"] "f = "kKx" k + (" + kC"k)"x "B = "(1 + kKx" " "x = " jP0"jk\"k + (" k\" k)f"k\"k + (" + jP0"j)" g + " kG"Q"G" kk\" k+ h i + "kQ"G" kk\" k + (" + kG" k "kG" kk\"k + (" + kQ"k)("k\" k + (" + kG"k)" ) (" + k\"k) "2 = 2"2ect k\" k c > 0 c = const \"( ) | ! + (5.1.1) 6 ^" . > 3.2, c0kuk2;10 hu Bui c1 kuk2;10 c0 c1 | * + * : p p MO(z x(t) ; x^" (t))2n ] f m(t) + c0kKx kkC kkz kn+
r q 2
2
+ ("2f + "2B kPx^" kkz k2n g2 ! 0 ! 0 " ! 0 " ! 0
p
(5:1:10)
p
MO(z x(t) ; x^" (t))2n ] f m(t) + c0("x + kKx" k)(" + kC"k)kz kn+ r q 2 + 2 ("2f + "2B kPx^" kkz k2n g2 ! 0 ! 0 " ! 0 " ! 0 (5:1:11) Px^" (t) = MO^x" (t)^x" (t)] . , . # 6 (t) = MO(z x(t) ; x^" (t))2n ] &1 ' ' x^ (t) & ,' 25 &, 2, 5 , & ,2 4-!2 5, & ' (t) MO(z x(t) ; x^ (t))2n ] + MO(z x^ (t) ; x^" (t))2n ]+ +2fMO(z x(t) ; x^ (t))2n ]MO(z x^ (t) ; x^" (t))2n ]g1=2 (5:1:12) , x^ (t) | 4 , &' ( & 7 7 7 7, x^ (t) = h (t ) { 2 8 Kx (t )C () =
Zt 0
Zt 0
Dh (t )j y( ) d
h (t )C( )Kx( )C () d + h (t )S ()
(5:1:13)
(5:1:14)
70
5 2 / )( 2
Kx ( ) = MOx( )x()] , x( ) | 4 (5.1.1). 2 7 7 (5.1.14) ' : Bu + u = f: (5:1:15) 9 ' 6 MO(z x^ (t) ; x^" (t))2n ] . 2 & 2 (5.1.5) (5.1.13) 2 x^" (t) , x^ (t) 1, 2 (5.1.8), 7 ', MO(z x^ (t) ; x^" (t))2n ] = hu ; u" B(u ; u" )i: # ' , (5.1.7) (5.1.15). 0 c0 = const: 0 J' u0 u B 7 , 2', & ': MO(z x^(t) ; x^ (t))2n ] c0 kKxk2 kC k2kz k2n: (5:1:36) * ' (5.1.36) (5.1.32), ' ' MO(z x(t) ; x^ (t))2n ] m(t) + c0kKx k2 kC k2kz k2n + p p
p
p p
(5:1:37) +2f c0 m(t)kKxkkC kkz kng = f m(t) + c0 kKx kkC kkz kng2: < & '6 4 (5.1.12). J' 22 (5.1.12) 6 2 MO(z x(t) ; x^ (t))2n ] MO(z x^ (t) ; x^" (t))2n ] ' (5.1.31) (5.1.37), 7 ' p pp MO(z x(t) ; x^" (t))2n ] O m(t) + c0kKx kkC kkz kn]2 + 2 ("2f + "2B ku" k2;10)+ r q p p0 p +2O m(t) + c kKxkkC kkz kn] 2 ("2f + "2B ku" k2;10 = r q p p0 p = f m(t) + c kKx kkC kkz kn + 2 ("2f + "2B ku" k2;10 g2: (5:1:38) # & (5.1.38) & 8 : kKx k | ' ' 5 x(t) , kC k | ' ' 1 8 2, '5 1 & , ' &1 6 ' ' ^" &:' 7 17 ' 8. G 5 &,, & kKx k ("B + 1)kKx" k (5:1:39) k k (" + 1)k" k: (5:1:40) ' 5, & B | & 6 & , ku" k210 ckPx^" kkz k2n (5:1:41) 5 Px^" = MO^x" (t)^x" (t)] z Px^" z = hu" B"u" i cku" k2;10: * (5.1.39),(5.1.40) (5.1.41) & 8 (5.1.38), & ' (5.1.11). (5.1.10) , (5.1.38) (5.1.31). < ' & 8 ,. 5.2 + +"
* 4 , &' ( 1 , - 1 & , 2 7 7 7 7 , 8 2 51. # :7 27, &' 5 2, 1 5 8 2, 113 5 && 2, O33,20,21], 4 , &' ( 1 , 2 5 4 2 O33]. # 17 O33,22] 2 1 &' 5 2, & 58 2 & 1, , 2 2 ' 41 , 2 7 7 7 { ,&' 4 2. # 13 ' : , (& 1 4' &' ), 5 , 2 & 2 5 1 7 ' , & O22]. ' :7 ' 2 2 2 , 18 1 , 1 , &' ' &' 5 2, { O33,22] &:' 7 &' 2 5' & 1' 1 &' 5 2,, ', 2, O20], ''' '67 O33]. # :' &5( & 4 2 2 1 ,&' 5 , 2 &' 5 2, & 4 , &' ( .
75
5.2 )( 2
, & ' O33,22]. 3 *( *' '( '+ k " ' '( > 0 , ' 6 * !*+ @u (t ) J() = @ : ;10 (t ) 7 2, 1, , 4 : # @u @ Bu + u = f
@u (t ) @u (t ) + = f ; Bu : B
@ @ (5.2.2) & 83' 1,': (( 2 & (5.2.1) ' '
(5:2:1) (5:2:2)
(t ) @u (t ) B @u @ + @ + u = 0
, '6' & 4 ,' ' ' u 6 ' , (5.2.1). 2 (5.2.1) (5.2.2), & B | '' { ' 8 4 2. 2 , &' ( 5 2 (5.2.1) (5.2.2) '6 (( ', &, 2 & & . ' 5.3 3 *( '( '+ k " ' '( > 0 , ' 6 * !*+ @h (t ) J() = @ z ;10 ' z | ' - ' En , ( @h @(t ) = v (t ) 6 '
@v (t ) = OF(t) ; K(t)C(t)]v (t ) ; v (t t)C(t)h (t ) (5:2:3) @t
v (t t) =
Zt 0
;1 (t) h (t )h (t )d C (t)S ;1 (t) + P (t)C (t) @S@
(5:2:4)
6 + h (t ) ( !! + @h (t ) = OF(t) ; K(t)C(t)]h (t ) (5:2:5) @t h (t t) = P (t)C (t)S ;1 (t) = K(t) (5:2:6) + P (t) | &** dP (t) = P (t)F (t) + F(t)P (t) + G(t)Q(t)G (t) ; P (t)C (t)S ;1 (t)C(t)P(t) (5:2:7) dt P (0) = P0: , . * (( & t # -F&( Kx (t )C () =
Zt 0
h (t )C( )Kx ( )C () d + h (t )S ()
(5:2:8)
76
5 2 / )( 2
& '
@Kx (t ) C () = h (t t)C(t)K (t )C ()+ Z @h (t ) C( )K ( )C () d + @h (t ) S (): (5:2:9) x x @t @t @t t
0
7 ', @Kx (t ) = F(t)K (t ): x @t 0:
(6:1:4)
# ' 1, (u v)B = (Bu v)0P 2 8u v 2 D(B) . *& ' D(B) & :' 2' &, 8. * & 1,' , HB . *' ' : & , &, HB = HB+ & ' & HB+ L2 (P) 5 & HB; . * f(x) 2 L2 (P ) . + * (6.1.13) (6.1.14), " 6 !*+ f 2 L2(Q) g 2 L2(Q) " " '( (6.1.8) (6.1.9) 6 W201 (Q) W21t (Q) .
84
6 4
, . 0 '' (
lf (v) = (v f)0Q : % & ,2 , '' 6.2, 7 '
jlf (v)j = j(v f)0Q j kvk0Qkf k0Q ckLvk;10Q . . lf (v) | & ( Lv v 2 D(L1 ) . ;1
0 4' & ' F-!7 : ( W20 (Q): * 113 ' 0 O25] 2 5 & 5 ( , & 5 & W20;1 (Q) 3 (2 u 2 W201 (Q) 2, l( ) = hu i0Q 2 8 2 W20;1(Q) 5 hu i0Q | 1 2 (' & 7 W201 (Q) W20;1 (Q) . * = L v 5 v u | 5 (, 283 2' (6.1.7) (6.1.5) . + * (6.1.13) (6.1.14), " 6 !*+ f 2 W2;t1(Q) ;1
" " 6.3 '( (6.1.8)7 " 6 g 2 W2o (Q) " " 6.4 '( (6.1.9). , . * L2 (Q) & W2;t 1 (Q) &:' 2 81 f 2 W2;t 1(Q)
L2(Q) 3 & ffi g1 i=1 2832 4 8 kfi ; f k;1tQ ! 0 i ! 1 . * ' 6.1 2 81 f 2 L2(Q) 3 4 u( x) , (6.1.8) ' & 2 6.1. % & ,2 (6.1.13), 7 ' kLui ; f k;1tQ = kfi ; f k;1tQ ckui ; uk0Q: * 7 2 :' & & i ! 1 ' 3 113 5 4 2 , (6.1.8) & 8 , W2;t 1(Q) ' & 2 6.3. U , (6.1.12). , ' 5. 91' 2 & &1 6 ' 4 8 , . 0 '' , (6.1.8), 5 & 2 f 2 L2 (Q): U &1 6 4 1 ' uk ( x) =
k X i=1
yi ( ) i (x) k = 1 2 :::
(6:1:20)
5 (x) | & 2 ' 2 ' 5 7 ( L2 (Q) 2832 8 (6.1.1), 6 2 yi ( ) 7 2 , 4 k ( @u @ j )0P + (Bu j )0P = (f j )0P (6:1:21) (uk (0 x) j )0P = yj (0) j = 1 ::: k
85
6.1 % " /
, & ,2 (6.1.20), & ', dyi ( ) + Pk ys ( )(B s j )0P = (f j )0P s=1 d yi (0) = 0 i j = 1 ::: k:
(6:1:22)
K '6 ,& ' (' dy( ) = F y( ) + G ( ) y(0) = 0 k k d 5 y( ) Gk ( ) | 1, y( ) = fy1( ) y2 ( ) ::: yk( )g Gk( ) = f(f 1 )0P (f 2 )0P ::: (f k)0P g Fk | '
(6:1:23)
2 (B ) ( B ) 3 1 1 0P 1 k 0P 5: Fk = 4
(B k 1)0P (B k k )0P 9' ', 4 (6.1.22) &' 2 ' & 2 6.1. 6.3 , " !*+ f 2 L2(Q)
kf k0Q ckuk k10Q: j ( ) . D''2 & , . '6' 1 4 2 (6.1.21) (2t ; ) dyd j (j = 1 ::: k) 52 & 0 t 7 '
Zt Zt @uk ( ) X k k X dy ( ) j (2t ; ) j )0P d + (Buk (2t ; ) dyj ( ) j )0P d = ( 0
@
d
j =1
0
Zt
k X
0
j =1
= (f( x)
j =1
d
j ( ) ) d (2t ; ) dyd j 0P
8 , & ,2 4 (6.1.20), & ' k ( ) (2t ; ) @uk ( ) ) + (Bu (2t ; ) @uk ( ) ) ( @u@ k @ 0Q @ 0Q (Luk (2t ; ) @u@ k ( ) )0Q = (f (2t ; ) @u@ k ( ) )0Q: k ( ) )0Q : D& 0 '' 6 (Buk (2t ; ) @u@
k ( ) ) = 1 ( @ fO2t ; ]Bu u g) + 1 (Bu u ) : (Buk (2t ; ) @u@ 0Q k k 0Q 2 k k 0Q 2 @
*' 22 (' 9 5 5, ' ' @ fO2t ; ]Bu u g) = Z (2t ; )(Bu )u cos(kn ) dS = ( @ k k 0Q k k 0
Z
S
= ; 2t(Buk (0 x))uk(0 x) dP + u(0 x) = 0: 0: ij i j i ij =1 i=1 , i | &, 3 ( i = 1 ::: n ). D3 5 2 &, 2 @aij (x) 8x 2 P & 28 2 5 2 @xi du = 0 d x2;
5 d=d | &, 2 & ' : 9& B '' & 6 & . * Q = P O0 t] 5 O0 t]| &' 6 ' . 0 '' L2 (Q) (( @u + Bu( x) & '6 D(L1 ) ( u( x) 0 t & L1u @ x 2 P & (( '7 & 6 & (( '7 & x 28 2' u(0 x) = du d x2; = 0:
6.2
)
0 0
Z , W201 (Q) 1,' && D(L1) & ' Z 2 1=2 kuk10Q = ( O( @u @ ) + uBu] dQ) : Q
89 (6:2:1)
*' ' W201 (Q) , &, & . * W201 (Q) L2 (Q) & ' & W20;1 (Q): W21t(Q) | && & ' (6.2.1) 5 7 ( v( x) 2837 4 2' v(t x) = dv(x) d x2; = 0: * W2;t 1 (Q) | 5 & , & & W2;t 1 (Q) L( Q): 0 4 & L1 @v + Bv( x) W201 (Q) 1,' , L , L | & &26 L: L v ; @ 5 7 (27 v( x): * , . * & , q {' 5 u( x) (' 2 ' @u + Bu( x) = v( x) Lu @ (6:2:2) x) u(0 x) = u0 (x) u( (6:2:3) d x2; = 0
5 v( x) | 1 5 4' , ' ', , v( x) 2 W2;t 1(Q): %,' 2 &, 2 2 O0 t] 2, u( x) 4 ' r( x1) = S u + ( ) r( x1) 2 W2;t 1(G) 5 S | & & , 83 , & 2 L2(Q) & ,' L2 (G) G = d O0 t] d | '6 , & S & 1 P ( d En ) ( ) | 4' ,' (&' , 6 1 5 4'). 0 1 8 (6.2.2) &'8 2 ' 5 5 & , 4 (6.2.2), (6.2.3) { 113 ' ' . 0 A 2 F (G Q) * (6.2.4), 6 ' A Rr + A = Rur (6:2:7) | '+, > 0:
90
6 4
, . 0 '' 6 '(A) = MO(z Ar ; u)2n ] 5 A 2 F(Q G) z 2 En: K ( '6 ,& , ' '(A) = (z AA Rr z)n ; 2(z ARru z)n + (z Ruz)n (6:2:8) 5 A | & , 2, & ' A 4 ' A r = (Ar) Ru | 2 & 5 & u( x): %, & 5 6 2 2 '(A) , '', & 5 ' . # ' 1, 2 (A) = (z AA Rr z)n ; (z ARru z)n : 9' ', inf f (A) A 2 F(Q G)g = ;(z A0 Rru z)n (6:2:9) A 5 A0 2 ARr = Rru : 0 4 :5 2 13 ' O33]. 0 5 2,83 ( ' ' (A) = (z Ru z)n + (A) + (z AA z)n 6 , 17 ' '', , (A) = (A) + (z AA z)n : (6:2:10) K 2 ( ' (A) ' (6.2.7). K ' 4 . 91,' ^ = A ; A A | 4 (6.2.7), ,&4 ' 6 (6.2.10), & ,2 : 1, . * 67 & 1, 7 ' (A) ; (A ) = (z AA z)n + (z ^^z)n (z ^^z)n 0 & (A) (A ) + (z AA z)n (A ): (6:2:11) 622 5 (A) inf f (A) A 2 F(Q G)g = ;(z A Rru z)n : (6:2:12) A * & 1 > 2 > 0: 0 & ' W21 (G) ( C(G) | & & 7 ( G ). * & 8 & ' jhOR ; R ]g gij r mm 1 (G) kgk g 2 W = 6 0 : kR r ; R mm k = sup 1tG 2t kgk1tG g 9 ' . * 67 & 1, , 7 ' jhOR r ; R mm ]g gij = jMOhr gi2 ; hrm gi2]j = = jMOhr ; rm gihr + rm gi]j (6:3:18) 5 r ( ) | 1 8 2, 2, u( x) 4 ' r ( ) = S u+ +w ( ) w ( ) | 1 6 4' ' ' ' & ' MOw ( ) w ()] = (W + I)( ; ): 1 8 2 rm ( ) 2 2 5. K & 6 4 2 , , 5' 7 1 6. *' 22 113 4-!2 5 & (6.3.18), & ' jhOR r ; R mm ]g gij (MOkr ; rm k2;1tG](MOkr ; rm k2;1tG])1=2kgk21tG ! 0 & m ! 1 5 , '' 6.6, & r rm 6 , r rm : ' 6.6 , * !* > 0 t lim MO(z u^ km ; u^ )2n ] = 0 z 2 En : (6:3:19) km!1 , . %, 2 (6.3.17) 7 ', A km = Rkm (Rmm + I);1 : & , &' 8 8 u^ km (t x) = A km r = Rkm (Rmm + I);1 r = Rkm (R mm );1r: 2 2 (6.2.7) 5 u^ (t x) = A = Rur (Rr + I);1 r = Rur (R r );1r: D5 :7 (' ' ' 6 (6.3.19) '6 ,& MO(z u^ km ; u^ (t x))2n ] =
2
= MO z fRkm(R mm );1 (rm ; r)g + fRkm (R mm );1 ; Rur (R r );1 r n]
2 MO(z Rkm(R mm );1(rm ; r)2n ]+ +MO(z fRkm(R mm );1 ; Rur (R r );1 gr)2n] :
(6:3:20)
*' 22 4-!2 5 & ' 5 '' & , & ' MO(z Rkm(R mm );1 (rm ; r)2n ] kz k2nkRkm (R mm );1 k2MOkrm ; rk2;1tG] ! 0 & k m ! 1 & '' 6.6. 9 ' 5 ' . *1 ' ' Rkm (R r );1 & & 1, , &' ' 4-!2 5, ' ' MO(z fRkm(R mm );1 ; Rur (R r );1 gr)2n] 2 kz k2n kRkmk2 k(R mm );1 ; (R r );1 k2+
+k(R r );1 k2 kRkm ; Rur k2 krk2;1tG ! 0 k m ! 1
5 ''' 6.6 6.7. * ' : 4 2 (6.3.20) & ' 6 '.
97
6.3 . / ! ! 2
' 6.7 , * !* t lim lim MO(z u^ km(t x) ; u(t x))2n] = m(t)
!0 km!1
u^ km (t) | " '( !+ (6.3.2),(6.3.3). , . *1 ' ' & ,' 25 &, 2 u^ (t x) & ' 2 & 1, : lim lim MO(z u^ km(t x) ; u(t x))2n] = !0 = lim lim !0
km!1
km!1 MOf((z u^ km(t x) ; u^ (t x))n + (z u^ (t x) ; u(t x))ng2 ]
lim lim MO(z u^ km(t x) ; u^ (t x))2n]+ !0
+2f lim
km!1
km!1 MO(z u^km(t x) ; u^ (t x))2n]MO(z u^ (t x) ; u(t x))2n]g1=2+
+MO(z u^ (t x) ; u(t x))2n] = m(t)
5 ' 6.6. ': ?8> ,:?'- @::
. .
0
