Методы линейной оптимальной фильтрации

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0,:  7 7  9. !. +42,    (,-'' 7  D.;. / 2,

ISBN 5-211-03916-5


c "   5       , 2000


  1 2 3

4

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1  , &8  & 2. #  ' 1,2: W21Oa b] | &,  5 1  &   , &  && ' & - (('7 -( u(t) a  t  b &  2' &,  8  Zb
du
(t) dv(t)
(t)v(t) dt + u (u v)1 = dt dt a p 1 kuk1 = (u u)1 | ' W2 Oa b] W2;1 Oa b] | 5  &   , &  && ' '6   L2 Oa b] & 5  ' 8Zb 9 <
= kuk;1 = sup : u (t)v(t) dt j v(t) 2 W21Oa b] kvk1 = 1 u(t) 2 L2 Oa b] : v a

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Zb

hu vi = klim u
k (t)v(t) dt !1 a

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t=a

t=b

8

1       

2  &  (( ' 6' ; d dtu(t) 2 + u(t)  & I 2 2 2  4' 5 5 & 2 ' { ' / G(t s) 2 ,5 4 (( 5

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8 0 > < 1=2 '(a)  >b=a< a limh(t ; ) '" i = > 1=2 '(b) "!0 : '(t) a < =
 2 1  (' &        4-!2 5 (Q )

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-(, '837 l & 7 &,  7  Oa b] .
' C l Oa b] ' 

 dku(t)  t 2 Oa b] 0  k  l : kukC l = max tk dtk

#'  , 4 ' C l Oa b] '6

  :   8 ' & (' 

kukC l =

Xl k=1





dku(t)  t 2 Oa b]g: max f t dtk

*  C01O0 t]  Ct1O0 t] | '6     , & - (('7  5' O0 t]

-( x()  2 O0 t] 2 7 & 28 2   2 x(0) = dx()  =0 = 0 (  d 1 1   x(t) = dx() d  =t = 0 ), 5 1  &    W20O0 t]  W2t O0 t]| && 2 '6  C01O0 t]  Ct1O0 t] & ''

0Zt 0Zt 2 11=2 2 11=2 dx() dx() d A  kxk1t = @ d A  kxk10 = @ 0

d

0

d

k : : : k10  k : : : k1t | ' :7 &   7, W20;1O0 t]  W2;t 1 O0 t]| && 2 '6   L2 O0 t]

& 5 ' ''

Zt y
()v() d  v 2 W201 O0 t] kvk10 = 1g kyk;10 = sup f v 0

1.1         

Zt y
()v() d  v 2 W21tO0 t] kvk1t = 1g: kyk;1t = sup f v

9

0

Z, h: :i0  h: :it 1,' 1  ('     &   7 W201 O0 t] W20;1 O0 t]  W21t O0 t] W2;t 1 O0 t]:  2 :7 & &   3  8 ,'  & I0  It  1683 W20;1O0 t]  W201 O0 t]     W2;t 1O0 t]  W21t O0 t]: 9& I I0  It '6 &     I = jj   I0 = j0 j0  It = jt jt : %,'  & j j0  jt 168 L2O0 t]  W21O0 t] W201 O0 t] W21t O0 t]    ,  & j   j0  jt 168 W2;1O0 t] W20;1O0 t] W2;t 1O0 t]  L2 O0 t]:  2 & j j0  jt j   j0  jt 3  8 1 &: D = j ;1  D0 = j0;1  Dt = jt;1  D = (j  );1  D0 = (j0 );1 Dt = (jt );1 :    8  : & &,   . 1.1.

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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       

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 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]:
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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]  &    1132 ('  4 O25] 2 h\
() vi 3 66 h\1
2 () 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]

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 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 62  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() | &  ' ,  '' &32' 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

<   v() = Dt w(): # ' &    '' 5 &  v() : MOv()] = Dt MOjt Dt w()] = Dt MOw()] = 0 MOv()v
()] = Dt Dt MO(jt v())(jt v())
] = = Dt Dt MO(jt Dt w())(jt Dt w())
] = Dt Dt MOw()w
()] = Dt Dt (R min(t s)) = R( ; ) n <    Dt (min( )) = 1(t s) = 1 0  >    Dt (1( )) = ( ; ) 5  ( ; ) {(2 , ( ; ) 2 W2;1 O0 t]: <' 1,', v() | 1  4'.   1.5 2  ,    "   '     *   + , '           * "   . #  4' & "    ,  &' 2 "   *  !" '    . 0 '' ,  4 (1.1.6) , 5  v() 2 2 2  , 5 1 5 5  5 4', dx() = F()x() + v() x(0) = x  0    t (1:1:9) 0 d 5  x0 = x0(!) | 2 5  2   , ' '2 & ' ''', &,  2

(1.1.9) &' 2 113' ' .   1.6 3 *  (   +  x() x( !) "  '     '( )  (1.1.9),    '+     (1.1.9) (   (      1.1). * x()    283 (1.1.9), ' &   &  1.    , ,  (1.1.9) '6 ,&    ;Dt x() = F()x() ; Dt w() 5  v() = ;Dt w(): *   '  1   :5     &' jt  5  & ' ;x() = jt (F()x()) ; w(): <  jt Dt | 6    &,  & jt  : '7 , L2 O0 t] '  :

Z

Z

0

0

jt u() = ; u(s) ds  7 ',  x() = 

R

F(s)x(s) ds + w():

D5   '' 1.1, x() 2 L2 O0 t]     , F (s)x(s) ds & 6 CO0 t]  x()  '' 0 7 & 7 ( 6 &   O0 t] 2 8 1. #  '  &22  & 2. D ' , 8 818  &  ,'   837 & ' &,   & . #4 ,   2   ' , 8 7 ' 5 ' ( ,'3', & ', 7   ). # 2  '  6838     5  ' , 8 7 ' 5 ' ( 7   ).

1.1         

13

D   ',  2 & 2 ,     ',  7 ''  '  . #4&   &  2  ', 7 7  7 7 5  2 28 2 13'  2 & 2 '' 7 '    ' & 2 18 2. *  _ { 2  '. # 6  '' '   ' _ &  7  ,    u() , (  5 '6   U  &6  7 8   y() & 638 '6   Y: 9'',  7  ,    u()  '6 1 &,   (. #1 '6   U & &  ''  '    ', '6   2 , ' 162', &' { (, ',  3 &  2 2 '' ' &1 2'. ' 5, , 7 5 5  u()  &  ,    2 &  2 y(): J 7    ,  , 13' ,   7 5 ,   2,    &   ,'2  ', ..     22 x() 2 X 5  X | &     2  '. "'  818  ' '6 ,    ' & '7 (  7  U  4'   y() = H(u()) 5  u() { (2  7 , u() 2 U y() { , (  7   ' ''  H | &  ',   83 , 1   & '7 7 7 5  1   , 7 7 5   ' Y y() 2 Y: "'  '  8  ' , 8 1M  7 : ' : 1) &     2 X 2) &    7 7 5  U 3) &    7 7 5  Y 4) 4, 2, 837 7   7  5     22  ' (&  ' H ). 9&  ' H 1 &  28  : H = C (\) 5  (2 \    , U X  C , X Y ( \ : U ! X C : X ! Y ). # 1963 5  0. '  13 &   &22 '   ',   13&2.

  1.7 0  _ "  ' ( *   , 

1. 5   :   (     T |      ,      6 6  '  U       X     6 6  ( Y: 2. 0    6  !*+    \       : ) \       6   t0 ( 2 T t0 2 T )     T T X U      X x(t) = \(t u() x(t0) t0    t t 2 T)

   t0 ( 2 T t0 2 T) \    "      ('* { " '(   *   '  

 )7 ")  "6 t > t0  t 2 T t0 2 T   '(  u() (  2 T     '(   t0  t )  x(t0 )  '(       x(t) (  " ( "). 3) 5 6   "  C : T X ! Y    6   ( y() = C ( x()):    "& " ,,  &4  2  1 3,   1. K    , 8 6   ' (,   ,'   '. D ', ,  '     '6   T  2     .

  1.8 ,( *   _ '            (  -

 )     *  , *        T            6 ( ,  '     *      (*  )     *  , *  T |    + 6 ( . # '57 & 627  & 2 ,  '6  & '   '  ''  3    1 '6  ' &  2 2 '' ' 162'.

  1.9 ,( *   _ '        *  , *  )    X U  Y   *   7 ")  "  \ : T T X U ! X     6 t   7 )  "  C : X ! Y      "6 t:

14

1       


&'',  16 H ,  2 ',       2 && &&,

H(u1 () + u2 ()) = H(u1 ()) +  H(u2 ())   - &,    ,  u1 () u2() 2 U:   ' 1 '  '     &  '   '. " 1 ' 6 5 ,  7  ,    (& ) u() &   (,'2, &1,)  2 x(t0)  2 x()   5 ('  , &6 ) & x() &'38 &1, 2 H: * 4' &''   ' '6 6 &1,    y() =

Zb a

h( s)u(s) ds

(1:1:10)

2 5   & '7 ( U ,      ' h( s) ,'  m n: <, &',   8z 2 En h( s)z 2 L2 Oa b] & 1' 5'',  '6   & '7 &  U  6 & 6 &    L2 Oa b]  5 G15 (1.1.10) 2 ' . *   (1.1.10) 4  & , 2  & 2 '  ''. D   ',   5    2 5 '6   U 4 (2' , L2 Oa b]: <,  ' ,  7 &'    2  ' & 2 2 2 2 , 7   7  5    {( , 2 2 2 2 113 ( (  2 W2;1Oa b] ). # :' , 1 ,16 &     62 (1.1.10), &7 2   '8  '

'7 &7 7   & 7 ( (.. &  2 2 &1,  G&     ;).
 : &' ' 4 2 7 ' 7  '.

  1.10 ,( *   _ '  +      *  , * 

) T   

 (    "( +      6 ( )7 ") U '*       Z s            u() = u( + s)   6  s 2 T  ) \(t  x u) = \(t + s  + s x Z su)   6 s 2 T  )  "  C  '  t: #  4' 1   '  2,   ',    '. *:' &6',    &' 6 (1.1.10),   &  2 2 2 : '' 5 5 &    W2;1Oa b]: 91,' , hk ( s) k = 1 2 :::m  ' h( s)  &1', 1 h
k ( s) 2 W21 Oa b] & 1' 5''. "6   L2 Oa b] &  W2;1 Oa b]: "6   &  (('7 ( C 1 Oa b] &  L2 Oa b]       W2;1Oa b] <'1 1,', 2 815  u() 2 W2;1Oa b] 3   &      -( ui ()i=1 , C 1Oa b] 7 232 2 & ' 5 5 &     u() .. ku ; uik;1 ! 0 & i ! 1: 0 '' &      yi () =

Zb a

h( s)ui (s) ds:

D '

 7 1,  '6 ,&   

2 Rb 3 66 a h1( s)ui(s) ds 77 66 Rb 77 6 77 : h ( s)u (s) ds 2 i yi () = 66 a 77 66 ::: 4 Rb h ( s)u (s) ds 75 a

m

i

15

1.1         

* '  &   & i ! 1 ,  & ,2 &   1  (' ( '. . 7), 7 '

2 Rb h ( s)u (s) ds 3 lim i 66 !1 a 1 77 2
3 66 77 hh1
() ui b R h2( s)ui (s) ds 77 = 66 hh2 () ui 77 = y(): 66 lim lim y () = !1 a i !1 66 77 4
: : : 5 : : : 64 Rb 75 hhm() ui lim h ( s)u (s) ds i !1 m

(1:1:11)

a

*6',  : &  3    5 2 815 (  5  2 Oa b]: *  yk ()  yik ()    y()  yi ()    . <5 ,  & ,2     4-!2 5 2 k = 1 2 : : : m '' jyk () ; yik ()j = jhh
k () uiij  kh
k ()k1 ku ; ui k;1 ! 0 & i ! 1: %, :5 42, &   5 2 815 (  5 ,2  2 Oa b]   3     5  &   (1.1.11). <' 1,',   u 2 W2;1Oa b]  : ' ' h( s)  6 & 1' 5'' & 6 &    W21Oa b]  5 (1.1.10) 1 ' &'  &  (1.1.11). *6',   y() & 6 &    5'7  ' & G15 (  , Oa b]: * &  8 ' L2 Oa b]  (1.1.11), '':

0Zb m 11=2 0Zb m 11=2 X X kyk0 = @ jyk ()j2 d A = @ j < h
k () u > j2 d A : a k=1

a k=1

*''  & ' 68     Q . <5 

0Zb m 11=2 0Zb m 11=2 X X kyk0 = @ j < h
k () u > j2 d A  @ kh
k ()k21 kuk2;1 d A = a k=1

a k=1

0Zb m 11=2 X =@ kh
k ()k21 d A kuk;1 < 1 a k=1

  & 6 & s   u 2 W2;1 Oa b]: D    , y() 2 L2 Oa b]: *  : ' ' h( s) & &''   s & 6 '6   C 1Oa b]: <5 ,    7  ' & & 5 u(s) = z( ; s) 5  z |   , En  2 Oa b]  ( ; s) -  {(2 ,  5     '  {( ( '. . 8) 8 0 Zb  < a  > b < h( s)z( ; s) ds = : 12 h( )z  = a  = b h( )z a <  < b a  y() = h( )z & a <  < b: <' 1,', (8 h( )z   &   8  '   {(8 '' ' : " h( ) , 8 '&   &7  '  '. # (' 7 (1.1.10)  (1.1.11) 2  ' ''  ,    , (  7  

''  > s   '' '  < s: #  & 2   8 2  ',  2 7 ,    & & 47  & &837  7 5  ,    1 37. *:' 8,  h( s) = 0 &  < s: K    , 8   ' (,   3  '  ( ,' )  '.   1.11 ,  +  u()  "'  H '    !  (,

  , ! ),   ' !  (1.1.10),  -   + h( s) * , ( (1.1.10)    (     **  (1.1.11)). h
k ( s)

W21Oa b]

16

1       

  1.12 ,  +  x() !!  +   dx() = F()x() + G()v() d  v() | 6  ,  F()  G() | '  +, '  !  ! ,   +  x()   6   -        6  "   .   2 '' 5 & 2  &  '   ' ' 1 '  & ,   83 &  :

  1.13 :     ( *  

"  '  ,  *   :  *           " "  !!  +    dx() = F()x() + G()u() x(t ) = x  (1:1:12) 0 0 d  6   y() = C()x() (1:1:13)  F() G()  C() | '  +, u() | 6  , x0 |  *  (      ,  '     " "    . ;+ C()  ' + '    "  . <   + F ()     L2 Ot0 t]  + G()  C() {   W21 Ot0 t]: #6 ,  & 2 '8 &22 1 8 '   & 2'  ' 7  '. #  1  ' &1 ' 1 8 ' .  2 :5  ''   8  ' n -5 &2 : dx() = F()x() x(t ) =? (1:1:14) 0 d y() = C()x(): (1:1:15) " F() C()  m -' 7  5 y() | ,       Ot0  t]:

  1.14 = *      x(t)     

(1.1.14)-(1.1.15) '  "     Ot0 t]      '(   

 ' '   y()    Ot0 t]:

  1.15 >  *     x(t)      (1.1.14)-(1.1.15) "     Ot0  t]   '    "     Ot0 t]:   1.16 >  *  t    t0 * , (    x(t)   (1.1.14)(1.1.15)   "     Ot0 t]  *   '    " 

.

04  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). <5  y() = C()\
( t)x(t):

(1:1:18)

17

1.1         

'6' : 6  \
( t)C
()   , 52 &  7 ':

Zt t0

\
( t)C
()y() d

#  ' 1,

=

Zt t0

\
( t)C
()C()\( t) d x(t):

Zt

M(t t0) = \
( t)C
()C()\( t) d: <5 

t0

Zt t0

\
( t)C
()y() d = M(t t0)x(t):

(1:1:19)

(1:1:20)

 2 &  2 x(t)   (1.1.20)  6 1 ,4'. U    4 :5  2 3   5     5 , 5  ' M(t t0)  6 . " M(t t0) &  2 6' (1.1.19), ,  2 ' 1 8 ' . K '' 2   2 ' &2  n:

' 1.1 ,  "     (1.1.14)-(1.1.15)    Ot t0]  "6  

  ( , ( " + "   M(t t0)          (1.1.19), "     .     ,      .      ,   2      42 (1.1.20).
17 ' . '6'   (1.1.20)   x
(t)   & ,2 (1.1.18) & '

Zt t0

y
()y() d = x
(t)M(t t0)x(t):

(1:1:21)

%, :5      ,  ' M(t t0)   &  . *  M(t t0) 6  2. <5    2 72 1    x(t) 6= 0 ,  M(t t0)x(t) = 0:  2 5  13 2   & 2   (1.1.21) ,     {  2. <' 1,',  y() 0  '    Ot0  t]    &  .
  6 7  5 1   & x(t) 0: <   & 1 8 27 y()  2 Ot0 t] 3  8  , '7  22  ',  : &  &  8 1.15 ( &  1 8 ' ). <' ,.  2    2 &22 & 2'   '  ''   8  ' n | &2  1,  2  7 : dx() = F()x() + G()u() x(t ) = x : (1:1:22) 0 0 d

" F()  G() ,  2 7  , Ot0  t]: #7  5 u() ' ,'  p:
   2  ' x0 & & 5 2 ,  ',  6 '   2 x(t) { , . J  & 2   ', 1    & 2 Ot0 t] 1 & 283 ,    ' 1,', 1    2  ' &   .

  1.17 0   x(t0)      (1.1.22) '        Ot0 t]     Ot0 t]      !*+ u() *, ( x(t) = 0:

  1.18 >  *     x(t0)      (1.1.22)       Ot0  t]   '          Ot0 t]:

  1.19 0  (1.1.22) '              

     *  , *   *  t0    t * , (   (1.1.22) "          Ot0 t]:

18

1       

# :7 &  27   2 & 5 2  '  8. K      2  ' , ' & 2   &  & 2'   4 13   6 .  ,  , 13 4  2 (1.1.22) ,  2 ('  4

Zt

x(t) = \(t t0 )x(t0) + \(t )G()u() d: t0

U  x(t) = 0 

Zt t0

(1:1:23)

\(t )G()u() d = ;\(t t0)x(t0 )

 '62 1   :5      \
(t0  t) 7 ', 

Zt

t0

\(t0  )G()u() d = ;x(t0 ):

#  ' ' W(t t0 ) =

Zt t0

\(t0 )G()G
()\
(t0  ) d:

(1:1:24)

(1:1:25)

U  ' W(t0  t)  6 ,  -(2 u() = ;G
()\
(t0  )W ;1(t t0)x(t0 ) t0    t

(1:1:26)

2 2 2 4' ,  & 2.    , &  '     (1.1.26)  8   (1.1.24). <5 

Zt

Zt

t0

Zt

\(t0  )G()u() d = \(t0 )G()(;G
()\
(t0 )W ;1(t t0 )x(t0) d = t0

= ; \(t0  )G()G
()\
(t0  ) d W ;1 (t t0)x(t0 ) = ;W(t t0 )W ;1(t t0)x(t0 ) = ;x(t0): t0

Z  1   . D'' 2,   &  2 ' W(t t0) ,  2 ' & 2'   ' (1.1.22). D&     83  6 :

' 1.2 ,        (1.1.22)    Ot t0]  "6     ( , ( " +     W(t t0)          (1.1.25), "     .

,    : ' '6  O14]. D(' '  ' ( 5  ,,  &18 2 & ,    ' 3   2 42 ,   ( .

' 1.3 (' (-*+24]) 3*   !*+  f(x)      -

     "' L          E            6    , . .       !*+  F (x)      E  * , ( 1) F(x) = f(x)  x 2 L 2) kF kE = kf kL :

19

1.2       .  
 - 

' 1.4 (' ,. - .. 2]) 3*   !*+ 

f(x)      "      H    f(x) = (x u)H   (: :)H | *  '    H  -   u  '(     !*+   f:
 -  kf k = kukH : *  fH +  H0 H ; g |  3 5 1  &    1  (' h: :i: <5  ' ;. 0  '6 ('    83' 1,': ' 1.5 (//01 2 ,. - ..) 3*

 

!*+  f(x)        H ;    "      f(x) = hx vi  -   v 2 H +   '(     !*+   f:
 -  kf k = kvkH + : 3*   !*+  f(x)        H +     "      f(x) = hx ui ;  -   v 2 H   '(     !*+   f:
 -  kf k = kukH ; . ,    : '  7  '  1. 9'' 4 ,   3      2  ,   , &  5 O2]. 1.2           .      

91,' , x() x( !) t0    t ! 2 ] |  n | ' 5   &  ,2', & 63' 2 8 1 &    & 7 ( COt0 t]   ' MOx()] = mx () MO(x() ; mx ())(x() ; mx ())
] = Kx ( ) 5  t0    t C() | ' 1 8  (,') ,'  m n m  n 5  ' : '', v() v( !) | m {' 5    & '' ' 6 ' MOv()] = mv ()    ' Kv ( ) = MO(v() ; mv ())(v() ; mv ())
] &  x()  v()   . <  &  x()  v() 5  ,  &     7 &  8 &  8 O6,24]. K ' ' Kx ( ) & 1' 5'' & 6 L2 Ot0 t]   , v()  mv ()  : '   ' Kv ( ) '5 & 6 5 ' &    W2;1Ot0 t]: J   &'  (     83'. *  ,     & fy() t0    tg 2, & ' x() 4' y() = C()x() + v(): (1:2:1) <1 2 & 1 8 2' &  fy() t0    tg  8  x^() &  x()    2838 8 '''      41 '' '  = t : m(t) = inf fMO(z x(t) ; x^(t))2n ]j x^(t) = h

Zt

t0

h(t ) d g

(1:2:2)

5  z | &,   & 2  , En ,  622 5 1 2 & ' '' h(t ) ,'  n m 2 7 3   & ' (  x^(t) =

Zt

t0

h(t )y() d:

20

1       

,,   (  2 5  7 &  2 2 2  4' , 7 (  ,    2837 8 '''      41 O8,30]. ' 5, , & 2 & :' 8,  &  ',  ' & 8 ' '' & & 12  1  '  ' O30]. *6',  ,   (  :    48 5 5 5  2 {  2 #-F&(. *6  ' & &      2 #-F&(, &1,' ,   (   83' 1,'. #  '  &  xc() = x() ; mx ()  vc () = v() ; mv (): *  xc ()  vc () '8    '' & 5 &2 ,  7    & 8  2' &  x()  v(): #'  &  y() 1 '  & ,  & yc () = y() ; C()mx () ; mv () = C()xc() + vc ():   1 '  '  ,   (  2 &  xc () vc () yc ()  7' & :'  1,2 x() v() y():  1.2 (.1    3 .4 . 1) >  *    !*+ u()

Zt

t0

u()v() d = 0

 v() - !*+,      '* Ot0 t]  v() 0  -    '* .  2  35 &    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 & '. <5  818 ' h(t ) , '6   & '7 ' '6 &    83' 1,': h(t ) = h0 (t ) + hv (t ) 5   |  , hv (t ) | &,  2 ' ,    & '7 '. *1,' 6 m(t) :

Zt

MO(z x(t) ; (h0(t ) + hv (t ))y() d)2n] = MOz
x(t)x
(t)z ;

;2z
+2z


Zt Zt t0 t0

Zt t0

t0

x(t)y
()h
v (t ) dz ; 2z


Zt t0

h0 (t )y()y()h
v (t )dd + 2 z
+z


Zt Zt t0 t0

x(t)y
()h
0 (t ) dz+

Zt Zt

t0 t0

hv (t )y()y()h
v (t )dd+

h0 (t )y()y()h
0 (t ) dd]:

((2 : 6 &   &  2 &,  8   8   = 0 7 ',  t t

Z

Z

t0

t0

MO2z
f;x(t)y
() + h0 (t )y()y
() d gh
v (t )dz] =

21

1.2       .  
 - 

= 2MO(f;x(t)y
() +

Zt t0

h0 (t )y()y
() d g
z h
v (t )z)0] = 0

(1:2:3)

,   (: :)0 -  2 &,   L2Ot0  t]: <  ' hv (t )   z &,  ,  (1.2.3) 1  & 2 2 & 7 h0(t )    2837  8 MOx(t)y
()] =

Zt

t0

h0 (t )MOy()y
()] d:

(1:2:4)

K    '   #-F&(. J'' (1.2.4)  & y() 5 &  ' (1.2.1) 5  & '   Kx (t )C
() =

Zt

t0

h0 (t )C()Kx( )C
() d +

Zt

t0

h0(t )Kv ( ) d:

(1:2:5)

  (1.2.5) 13'  2 2 2  ' ; 5 ' & 5  , 4 5,  ,  ,    , ' 5, '6  & 6    & '7 '. J   &'  (  ,  &3 2,   4' 1 8 27 v() 1  1 ' 5  ' 4''  6  '    ,   ,   '   4' Kv ( ) = R()( ; )  R() | & 6  &  2 ' ( 2 8u 2 L2 Ot0 t] 9 > 0 2,  (u Ru)0  ckuk20 ), : ' R() { 5   (, R() |     1 5 4'. %' & 7   27 1  &  &   5' 42 ,   &'  (  +.
.  '5 ',
. #', 0.  ''  0. !8 . # , 8 :5 &5( 3    ,'  ' 5. D  & v() 4 (1.2.1) 1 ' ,  4'',  x() | & ,' 5 '.

  1.20 (25]) ? v()  " 6 y() '  +  ( " ),   

*    * +  + Kv ( )         {!*+ ,*. > -   + 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' 42 ,   &'  ( , & 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( !) '  * +  +  ,  * +  + '  *  '    . <' 1,',  4' '   & 1    '. # 8 , 5   4'     7 & 7, '8 2     .   2.4 0(   +  x( !)  y( !) ' +  ',  6 ' * +  + '  *  '     Kxy ( ) = Kxy ( ; ) . *  1  4' v( !) ( '. &   1.3) ' & 28 '     V () = V vij = const vij > 0 5  vij {: ' ' V . <5  5  2 ' &  2 2 ('  Kv ( ) = V ( ; ) .   2.5 2   v( !) '  +  "   ,   * +  + Kv ( ) = V ( ; ):
 ' & 8 &   5 1 5 4'.  2 :5, &'22 113 &1,  ; O6,26]    ' 5 1 5 4', & ': 1V Sv () = 2

Z+1

;1

1 V: (s)e;is ds = 2

(2:1:1)

23

2.2 



 1   5 (2.1.1) &' 2 113' '  O6, 8, 25]. <' 1,', & 2 &   5 1 5 4' & 2. #    :5,  ( 5    , 8 2 1 ' 4'' &  5 1 ' ',  &  '& 5 '8     6    . 2.2

#  

* ,   4, En { n {'      &      2' &,  ' (: :)n , L2(;1 1) {5 1  &    -(, &  7      &2' (;1 1)  5'7   ' & G15, ' k : k0   2' &,  ' (: :)0 . #  ' 1,2: W21 (;1 1) {&,  5 1  &   , &  && ' '6   &  (('7 -( u() 2 L2 (;1 1)  du() d 2 L2 (;1 1) &  2' &,  8 Z1
(u v)1 = (u
()v() + dud() dv() d ) d ;1

p kuk1 = (u u)1 {' W21 (;1 1) , W2;1(;1 1) {5  &   , &  && -

' '6   L2 (;1 1) & 5  '

Z1

kuk;1 = sup f u
()v() d j v 2 W21 (;1 1) kvk1 = 1 u 2 L2 (;1 1)g: v ;1

*        ,    . *  x()  v() { 5     &  ,'  n  m    ,   83'  ': MOx()] = 0 MOx()x
()] = Kx ( ; ) MOv()] = 0 MOv()v
()] = Kv ( ; ) MOx()v
()] = 0 : ' ' Kx ( ;) & 6 & &''    &    W21 (;1 1) ,  : ' ' Kv ( ; ) , &    W2;1 (;1 1) . %,'2 2 & fy()  2 (;1 t]g , 2, & ' x() 4' y() = Cx() + v() (2:2:1) 5   {' 1 8  ,'  m n , ,  232  ' &'',  x() { n { ',  v() { m {' . * y() { . <1 2   &  x()   = t ,    2838 8 '''      41: m(t) = inf fMO(z x(t) ; x^(t))2 ]j x^(t) = n

h

Z1

h(t ; )y() d g

(2:2:2)

;1

5  622 5 1 2 & ' & '' '' h(t ; ) ,  5 &' 2 '  1  (' ( '. &   1.2 . 11  . 15).  ' K  2 ,  (2.2.1)-(2.2.2) 2 2 2   #-F&(   Kxy (t ; ) =

Z1

h(t ; )Kyy ( ; ) d

(2:2:3)

;1

( '.    2 (1.2.4 )). #  ' 1,2: t ;  =   ;  = : <5  Kxy (t ; ) = Kxy () , Kyy ( ; ) = Kyy () , h(t ; ) = h( ; )    (2.2.3) ,&4 2    2  & 5   Kxy () =

Z1

;1

h( ; )Kyy () d:

(2:2:4)

24

2 


%5    (2.2.4)      2 4 4 7 27,  ', 5  4' v( !) 1     ', &,  283' &      5' 42 :5  2.  ' 4''  2 2  1  5   4'  4', & ' ' &  &    1 8 2 &  (,8 O30],    2 2 2 ', &  2'   Syy () = ()
(;) (2:2:5) 5  Syy () {' &  &    1 8 , () {', &     '    & 8    & &    '&   &'  . K    5 ,  ()   ;1() &  & &    {    (     8 (,   ,'' & ' (2' 7  ' & 2 (    '  &6

7 5 5 ). 0 '' 4 ,   &'  (  & &    2 (2.2.5) . U  v( !) {1  5    4',  ' Syy () &  (,8 (2.2.5) O8]. *  4' 1 8 27 v( !)     ' & ' x( !) , '        8 Kv ( ; ): *' ' Kv ( ; )  ,  & 2 &    1 8 2 Kyy () &  (,8 (2.2.5). %,     2 #-F&( ( '. . 20)  ,  17 ''   '  62 6 5 (  (2.2.2) 2 2 2 &      MO(f;x(t)y
() +

Zt

t0

h0(t )y()y
() d g
z h
s (t )z)0 ] = 0

5  hs(t ) {&,  2 & '2 ' ,'  m n ,  ' h0 (t )   2 688 5 (  (2.2.2) ( '. 1.2.3).  2 7 &  &  & 2 & '' 5 6 2 :    ,&4 2   (f;Kxy () +

Z1

h0 ( ; )Kyy () dg
z h
s ()z)0 = 0

(2:2:6)

;1

 = t ;   =  ;  , z {&,    , En : %5 &' 2 113' '  (&   1.2). *  : ' ' Kxy () Kyy () h0()  hs() & 8 &' 1135 &1, 2 ;. <5 , 5   ' *   2 O30], '':

1 Z1 0 Z1 z
@;Kxy () + h0 ( ; )Kyy () dA h
s () dz = ;1

;1

= z


1 2i

Z

+i1 ;i1

(Kx ()C
+ H0()Ky ())Hs
(;) dz

(2:2:7)

5  Kx ()  Ky () { &  &   &  x()  y() , H0() {;-1, ' h0() , Hs() {;-1, ' hs() z {&,   : ' , En  i {''2   ( i2 = ;1 .) <  &   8 & 2 &    1 8 2 &  (,8 (2.2.5),  4 (2.2.7) '6 &1,   83' 1,'

Z

+i1 ;i1

=

(Kx ()C
+ H0() ()
(;)) Hs
(;) d =

Z   Kx ()C
O
(;)];1 
(;) + H0() ()
(;) Hs
(;) d =

+i1 ;i1

25

2.3  ! 


=

Z   Kx ()C
O
(;)];1 + H0()() 
(;)Hs
(;) d = 0:

+i 1 ;i1

(2:2:8)

0     8 (2.2.8)   , (2.2.6). *  ' ' Kx ()C
O
(;)];1   7 5'7 Kx ()C
O
(;)];1 = A() + B(;)

(2:2:9)

5  A() | ' 1M 2   , '83 & 8    & &    ,'2 '&   &'  ,  B(;) |   , '83 & 8  &  & &   . <5      (2.2.8) 1  5 2  :

Z

+i1 ;i1

Z

+i1

fA() + H0(l)()g
(;)Hs
(;) d ;

;i1

B(;)
(;)Hs
(;) d = 0:

(2:2:10)

# 5 & , 63'   & &    &'  , 13 2  ,   : ' ' B(;)  
(;) : & &        (,  Hs
(;) | &,  2   '6 1 6      & &   . <' 1,', & ', 

Z

+i1

;i1

fA() + H0 () ()g
(;)Hs
(;) d = 0

(2:2:11)

2 &,   ' Hs
(;) . #      '' 5   2 ( '. . 20)  3   2 1 '  ;1() &  & &    (   & 8  ' 7 2 2   & &    '. . 24), , (2.2.11)  ,  H0() = A() ;1 (): (2:2:12) K  1  4 ,   &'  (  2 7 &  ' &  &    1 8 2, & 83 (,8 (2.2.5). " (  # (2.2.12)  4 45 &'2 & ,-,   7    & (, ' Kx () . *  76 2 '  () ,1 !..+  ' O36]  &  2 1    68   8 & . ' 5, & 2  ,2 ,15 &,6 5 (   '-!8  ,   ,  5     ' (  #. K  5'  '  4 ,   3.2.2. 2.3     

0 ''  ' (  #,  4  & , 2  & 2. # 1 O8, . 325] &   ',   17 '      2 (, &  &   5 & . 9'',   2 5'2 (2 K() (;1 <  < 1) &  (,8 ((,'),    '6 1 &    : K() = (i)(;i) = j (i)j2

R1

R1

5  (i) = b(s)e;is ds , jb(s)j2 ds < 1   &  &  &  8 0 0 2 2 2  (,   ,'  '  1   4',    7 ' ('835 (  (&   1.12). *6',  '6 &   &  (, 2 5 &   &  &  8, &  283 1 1- 8 (8 O27],  &   &7 8 (8 ('835 ( .

26

2 


*    & y() ' & 8 &   Ky () , 2 2838 2 1-  (,    ()  Ky () = PQ2m() 2n

5  P2m () {& ' & 2m , Q2n() { & ' & 2n , &' n > m . * ' Q2n()  P2m()  6    &  ,   Ky () { 2 (2 (    &  &  ). *,  & ,' 2 1 4 &, 2 &   2,   & 

   1   &   ( y() ,     ,  1  '3  &'7 ( & , 5  '6 ' 1 8 '3 )  ('2  & ,' 5  2 2. ' 5, Q2n()   6 13 2    & '    '  ,   :'   1   5 &    &  1   7  2. <' 1,', '5  Q2n() > 0 2 7    7  ,  P2m() {   ( P2m ()  0 , 2 7    7  ). 0 '' & ' Q2n() . K & ' '   '&    ( 2    7  ,  &, 4, '5  Q2n() > 0 ). <5  Q2n() = a2n( ; 1 )( ; 2 ) : : :( ; 2n) 5  fk jk = 1 : : : 2ng {  2 Q2n() = 0 k 2 C k = 1 2 : : : 2n Im k 6= 0 C | '&  2 &   , a2n > 0 , a2n | :(( & ' '  &  . %,  51 ,  ,    k   2 Q2n() = 0 ,  k | &26  k '&    , 6  :5  2. 9 8   ,   :(( & ' Q2n() |      ,   & ' # 2 Q2n a2nx2n + a2n;1x2n;1 + : : : + a1x + a0 = 0 '':

a0 =     : : :   2 R1 a2n 1 1 2 2 n n

; aa1 = 1 1 2 2 : : : n;1n;1n + 1 1 2 2 : : :n;1 n;1n + 2n

+1 1 22 : : :n;1n n + 11 2 2 : : :n;1n n + : : :+ +1 2 2 : : :n;1 n;1n n + 1 2 2 : : :n;1n;1n n 2 R1 ::: a2n;2 =   +   +   +   +   + : : : +   +   + : : : +   2 R1 1 1 1 2 1 2 2 2 n n 1 2 1 2 1 n a2n

; aa2n;1 = 1 + 1 + : : : + n + n 2 R1 2n

5  R1 {'6      7   ,  {&2 &262 '&  5  . %  '  & 6  '5  Q2n()  '&   &    C . *  k 6

& '   . <5  k+1 = k { 6  '   ,    ''          Re  .  k+2  k+3 = k+2  & 58 2 '' ''   Im  ,      ( Q2n() . <' 1,',   & ' Q2n()  & 58 2 ''      ''   '&   &    &'  . 5''  & 62  & ' Q2n()  '   &    C &      2.1.

27

2.3  ! 


6Im 

k+2 r

r

k

Re -

r

r

k+3

k+1

0 .2.1. 0 & 6  & '  '&   &   

C

+ 5  2 & ' P2m () . 9  4 ',  P2m() '6 '      . U  P2m () '     ,   '8 8  . *6',  :  6  . *  k {    ,     2  P2m () . <5  3         k+1 ,    & ' P2m () ' ' 2 & 2. * & 6',  k 6= k+1  57  . # :'  2 7  , 637 '6  k  k+1 '5  P2m () 1   ',  : &    8    P2m (): *  &  , ,  k = k+1   k '   2. *  ' '5  P2m ()   &,  2 7 '6   C : P2m () = b2m ( ; 1)( ; 2 ) : : :( ; 2m ) = b2m

2m Y

( ; k )

k=1

Q

5    k  k = 1 2 : : : 2m   , b2m > 0 b2m | :(( & 2m , | , &,  2. D5&&' '6  ( ; k )  83' 1,':  '  &  5&& '6   , 7,  7 2 k & 6 ' '''  2' ( Im k > 0 ).  : 5&&  ' 6 &   '6     ' 2' & ' P2m(): U  k {        2r ,  & 8 5&&  8' r '6       {  8. #

 5&&  6 2  '&   , 63 6 & &    ( Im  < 0 ). 91,' ,  23 2  &  5&&, , 1  2 : : : m ,    { m+1  m+2  : : : 2m . D   ',  m+k = (k ; ik ) = k + ik = k  k = 1 2 : : : m k  0: <5  p p P2m () = O b2m( ; 1)( ; 2) : : :( ; m )]O b2m ( ; m+1 )( ; m+2 ) : : :( ; 2m )] =

p

= O b2m

'6' '5 

m Y

p

( ; k )]O b2m

m Y

( ; k )]:

k=1 k=1 m pb Q ( ; k )  im . <5  2m k=1 m m p Y p Y m i b2m ( ; k ) = b2m (i ; 0k ) = (i) k=1 k=1

5  0k = ik = i(k + ik ) = ; + ik  k  0 . m p Q + 5, '6' b2m ( ; k )  (;i)m : <5  k=1

p

(;i)m b2m

m Y

p

k=1

( ; k ) = b2m

m Y

~ (;i ; 0k ) = (i)

k=1

(2:3:1)

28

2 


~ = (;i): '6  i  3  2 &  '&   &     90o . *6',  (i) <' 1,',   (i) '  0k = ;k + ik ,  0k = ;k ; ik { 6  (i)  (i ; 0k )(i ; 0k ) = (i)2 + (i)2k + 2k + k2 :

(2:3:2)

9'',  '6  (;i)  3  2 &  '&   &     ;90o . <5 ,   0 ~ k = ;k ; ik -  (i) ,   k = ;k + ik { 5  & :' 0

(i ; 0k )(;i ; 0k ) = (;i)2 + (;i)2k + 2k + k2 : ~ = (;i) ,    im (;i)m = 1 ,  %, (2.3.2)  (2.3.3)  ,  (i) P2m() = (i)(;i)

(2:3:3) (2:3:4)

 (i) & 6   & &   . * &2  5 2 & ' Q2n() , 7 ',  Q2n() = d(i)d(;i)

(2:3:5)

  d(i) 6   & &   . *  ' (2.3.4)  (2.3.5) 6 &  &   Ky () , 5  (i)(;i) =  (i) (;i) Ky () = d(i)d( (2:3:6) ;i) 5  (2 (i) = (i) d(i) '    & 8      '&   &   . <' 1,', (i) ,  &7 8 (8    (,   ,'   '. ;,2 &  &   & .     ' '  #     ,  ,  (, &  2 2  2 7  5 &' 5 ( ,  : &,  2 5 16  3  & '5 42. ' 5, ' ' #   2  4 ,   ( 

 1 7 4' ,'27    52, 2,  3'  5 8 &   && 2  '. 9  &  &  2 &'  &7  ' (  3 17 '    (,   ,'8  ,  :,    , 4, 65,    5  & 2 , .

3  ( !"#( $#% ) &%(* + (* !%& 3.1

# %   -&'

# 1960 5  0.  ' & 6  5' 42 ,   &'  (  2  7  7 5  7 7 &  . + 1961 5   '  0. !8   &1    5' 42 ,  2 & 5 ' O13]. 0.  '    ,' &   ,   &'  ( . 9 '  '' 5 6 2    2 & 2  '5 5 &  x( !) &'

('83 (  { ' 8  ', ,16 '8 1 ' 5  ' 4''. # ,  1 &    5' 42 ,   2   2 1 5 4' ,'27. + 5'  1 2  ,  K#". K &    5 4'  & 8,  '2  ,  13'  ,' & 2 5 &  &'38 '' 5  5     &  ('83' ( '    2 3  4 &1 ' 76 2 &' '   '. *1 ' & 2 ('835 (  &  8 4 4 2 7 7 &  , & 2 &   7 &  (,8 (2.2.5).
  6 2 1 4   & 7 ,   & 2 : ,    2  &4 4,   '57 27 ('83 (  ,   6    & 2  ' & 2. 0 '' &    4 ,   &'  ( , & 6 0.  ''  0. !8 .  2 1  2 &    42 ,  1 '  & ,  ' ,     7  37 5 1 7 &   O25,16] *        ,    . *    n {' & x( !) (' 2 &' && 2 p {'5 1 5 5  5 4' u( !) , 8  ' dx() = F ()x() + G()u() x(0) = x  (3:1:1) 0 d 5  &,  2 &' 2 113' ' ,  4  2 (3.1.1){ '  &  2 1.6 ( . 12) F() | ' ,'  n n : '' , L2 O0 t], G() | ' ,'  n p n  p 5  ' : '' ( Gk () 2 C 1 O0 t] Gk () | k -  1 ' G() ) 1  4' u( !) '        8 ' MOu()u
()] = Q()( ; )   2 O0 t] , Q() |      5 1 5 4' x0 x0 (!) | n {' 5   

, MOx0] = 0 , MOx0x
0 ] = P0 , P0 |   &  2 '' 2 ', & u( !)    x0(!) . %,'8  &  m {' & y( !) ,  2 O0 t] 2, & ' x( !) 4' y() = C()x() + v() (3:1:2) 5  C() | ' 1 8  ,'  m n m  n , 5  ' : '' v( !) | 1  m {' 5   4',     ' & ' u( !)  ' ' x0(!) ,   '  '    ' MOv()v
(s)] = R()( ; ) R() | & 6  &  2 '' 2 ' &2  m 5  ' : '', .. 2 8u() v() 2 L2O0 t] (u Rv)0 = (Ru v)0  (u Ru)0  ckuk20:

30

3  "   -#$ 

<1 2 & ,'2' fy() 0    tg  8  x^() 5 &  x() ,    2838   = t 8 m(t) = inf fMO(z x(t) ; x^(t))2 ]j x^(t) = n

h

Zt 0

h(t )y() d g

(3:1:3)

622 5 1 2 & ' '' h(t ) ,'  n m : '' , W21O0 t] . %5 (3.1.3) &' 2  6   4 (1.1.11). D   ',  1  & 6  &    ' Q()  '     1 5 4' (&   1.5)  2 2 2 17 ''  5 '6  1     ' ( 2 817 u 2 L2 O0 t] (u Qu)0  0 ),    4'  ' '6 1

6 ' 1 ' 4'' (&   1.20). 1. %,'    ' h(t ) ,    283  8 #-F&(,  2 2 2  ' K  2 (3.1.3)  '  Kx (t )C
() =

Zt 0

h(t )Ky ( ) d

(3:1:4)

5  Kx (t ) = MOx(t)x
()] Ky ( ) = MOy()y
()] = R()( ; ) + + C()Kx ( )C
()  5   &'  &   1.2. " Kx ( ) = MOx()x
()]  2 2  83' 1,':

5   113 ('  4 ( '. . 10)   22  ' x() &  2 2 6' Z x() = \( 0)x(0) + \( s)G(s)u(s) ds 0

 ' MOx()x
()] MOx()x
()] = \( 0)MOx(0)x
(0)]\
( 0) + = \( 0)P0\
( 0) +

Z Z 0 0

Z Z 0 0

\( s)G(s)MOu(s)u
()]G
()\
( ) dsd =

\( s)G(s)Q(s)G
()\
( )(s ; ) dsd:

9 8 ,  & ,2     {( , 7 ',   2 ' Kx ( ) 5 &  x() '6 1   & (' : Kx ( ) = \( 0)P0\
( 0) +

Z

min() 0

\( s)G(s)Q(s)G
(s)\
( s) ds

(3:1:5)

5  \( ) | ( ' 2 '  ' (3.1.1),    2832 (( '  8 d\( ) = F()\( ) \( ) = I  (3:1:6) n d In |  2 ' &2  n . 9'',  ,   ' Q(s) G(s) (5    7 : ' ), &   : ' ( ' 7 ' \( s) \
( s)  (3.1.5)  3    &,   (&  ' & ) ' Kx ( ) & &''   : *  22 6 Ky ( ) (3.1.4)   2     {( , & ' Kx (t )C
() =

Zt 0

h(t )C()Kx ( )C
() d + h(t )R():

(3:1:7)

  (3.1.7) 2 2 2 5 '  ' 5  , 4 5,  &  , 3  ,         & 48  ' ' ,'32' 7,  ', &   

31

3.1  "   -#$ 

' 3.1
  -   * +  + Kx( )  + C()  R() 0    t 0    t   L2 O0 t] .          h(t )   3 @ ! (3.1.7), -   *     !*+ ' L2 O0 t] .
 
  ,      . 91,' 
, u() = h (t )z , z { &,    , En  t R Au = z
h(t )C()Kx ( )C
() d  Ru = (z
h(t )R())
 f() = (z
Kx (t )C
())
 0

Bu = Au + Ru:

(3:1:8)

'62 (3.1.7)  z
  & ,2 4

  1,2, & ' Bu Au + Ru = f: (3:1:9) 9& B & 6  &    '' L2 O0 t] . K   ,   ' Kx ( )  R() . " Kx ( ) | '' 2   2,  R() | '' 2  & 6  &  . * 2 {   2 ' 5 &  x( !) ,  2 {  '     1 5 4'.    ,   Kx
( ) = Kx ( ) ,  2 817 u()  v() , L2 O0 t] '':

9 Zt 8
* 6 2 &    & B   ,    & A  & 6  &    R (Bu u)0 = (Au u)0 + (Ru u)0  ckuk20 (3:1:10) 5  c | & 6 2  . %,     (3.1.10)        42  2 #-F&(. *6' :. *  u1  u2 | 42 (3.1.9), &' u1 6= u2 . <5  Bu1 = f  Bu2 = f . #'    , & 5, & ' B(u1 ; u2) = 0 . %,     (3.1.10) '' 0 = (B(u1 ; u2 ) (u1 ; u2))0  cku1 ; u2 k20 ,    ku1 ; u2 k0 = 0  u1 = u2 . + : &  & & 68, &2' 4. U     ,. 6' & 3    42 (3.1.9) .  2 &,  5  v 2 L2O0 t] '' (f v)0  kf k0kvk0  ckBvk0 kf k0 . D 2 &,   (f v)0 = lf (v) '6  '  

 5 (  Bv . * ' F-!7 ( '. . 18)  4' lf (v) 

 L2O0 t]  & ' 0  ( '. . 19)  &   5 (  5 1 ' &    & ',  lf (v) = (u Bv)0 , 5  u | (   : ' L2 O0 t]: <' 1,', (f v)0 = (Bv u)0 = (v Bu)0 ,   B ''  &. %, &  5   &,    v  &  L2 O0 t]  ,  Bu = f: <' ,. ' 3.2 > -   + Kx( )C
() C()Kx( )C
() R() @@ OKx( )C
()] , @ d
@ OC()Kx( )C ()] d R()     *  ( : " )  O0 t] ,     h(t )   3 -@ ! (3.1.7)    '      ,     L2O0 t] .     ,      .  2 1,2 (3.1.8), ((' (3.1.9), 5  @ f
() = Z u
()C() @ OK ( )C
()] d + @u
() R() + u
() d R(): @ @ x @ d t

0

32

3  "   -#$ 

9 8 

@u
() = @f
() R;1() ; Z u
()C() @ OK ( )C
()] dR;1() ; u
() dR() R;1(): @ @ @ x d t

0

#,  '    :    . %52 &  6 &  7  0  t   & ,2 ,       (a ; b)2  2a2 + 2b2 , 7 '

+2

Zt @u
() @u()

Zt @f
()

0

0

@

Zt Zt Zt 0 0 0

@ d  2

@f() ;1 ;1 @ R ()R () @ d+

@ OK ( )C
()]R;1 ()R;1 () @ OC()K ( s)]

u
()C() @ x x @

C
(s)u(s) ddsd + 2

Zt 0

dR() ;1 ;1 u
() dR() d R ()R () d u() d:

 , &'22     4-!2 5, ''

( ) 2  dR() 2  @ 2 Zt @u() 2  @f() ; 1 2 2 ; 1 2 2
; 1 2       @ d  2  @ 0 kR k + kuk0  d  kR k + kuk0  @ C()Kx ( )C () kR k  0

5  kRk = supfj(u Rv)0j u v 2 L2 O0 t] kuk0 = kvk0 = 1g | &2 '. uv *   8 ' & 2   &  5     5,   

 2 Zt @u() 2 d =  @u()  < 1  @u() 2 L2O0 t]: @ @ 0 @ 0

) <  u() = h
(t )z , 5  z | &,    , En ,  : ' ' @h(t @ & 6 & &'  &    L2O0 t] . <' ,. %, ' 3.1  3.2  ,  4  2 #-F&( 3    & 6 & '' '6   &7 7 ' ( .  2     2 (   '-!8  1 2 ,    ' h(t )  & &' t . #2  :  1 &'38 &22    . 2. *6',  4 ,   &'  (  (3.1.1) - (3.1.3)    (:   ) 48  ,  &' 5 & 2   '   ' '   . %, ' 3.1  3.2  ,  : ' ' h(t ) | 42  2 (3.1.7) & (  ' t & 6 &    W21 O0 t]: 0 , 5 5 &  y( !) 2 28 2 (2' &    W2;1O0 t] ,   ' & 6  , 1 5 4' v( !) ( '. &   1.4). <' 1,', 5 6 

Zt

x^(t) = h(t )y() d

(3:1:11)

0

  &' 113' '  ( '. . 15, (1.1.11)). Z1 '     2'  2', '6' 1   (3.1.11)  &,    z 2 En . #  ' 1, H(t ) = ;z
h(t ) . D&    &    z
x^(t) =

Zt 0

z
h(t )y() d

Zt

Zt

0

0

= ; H(t )C()x() d ; H(t )v() d:

(3:1:12)

33

3.1  "   -#$ 

#  '  '(t ) , &  2' 2 6 5 (  5 t ,  4 (( 5  2 d'(t ) = ;F
()'(t ) ; C
()H
(t ) d   (3:1:13) d'
(t ) = '
(t )F() ; H(t )C() d   ' '(t )j =t = '(t t) = z 0    t: (3:1:14) *  Z"t   + u"() = A" u u()!" ( ; s)ds !" () = ";1 !0 "  MOx0] = 0 0 < " < 1 0

!0() = 0 &  < 0   > 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 04 (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 62     &''  & ' 48 &8 '' 5 6 2. <5  MO(z x"(t) ; x^"

(t))2 ] = n

Zt Zt 0 0

'
(t )G()MOu"()u
" ()]G
()'(t ) dd+

34

3  "   -#$ 

+

Zt Z t 0 0

H(t )MOv()v
()]H
(t ) dd + '
(t 0)P0'(t 0):

<  5    7 ' & & 68  &  u() , v()    x0 ,  ',  MO(z x"(t) ; x^"(t))2n ] = '
(t 0)P0'(t 0) + +

Zt Zt 0 0

'
(t )G()

Z"t 0

Zt 0

H(t )R()H
(t ) d+

!" ( ; s)Q(s)!" ( ; s) ds G
()'(t ) dd:

(3:1:16)

# ' 5'' &        (3.1.16) &'2' &   5 2 5  2 5 '' 6

1 0 Zt 1 Z"t 0Zt @ '
(t )G()!"( ; s) d A Q(s) @ !"( ; s)G
()'(t ) dA ds: 0

0

(3:1:17)

0

  &6',    ' h(t ) &  & &'   O0 t],  max kA;" h(t );h(t )k ! 0 

R"t

& " ! 0 , 5   2 O0 t]  A;" | &   2, A;" u() = u( + s)!" (s) ds  k:k | '0

sm n P P h2 :    , ,   ( ! ()  ,  R"t! (s) ds = 1  2 ', khk = " " ij i=1 j =1 0

&    & 4

Z"t Oh(t  + s) ; h(t )]!"(s) ds  max kA;" h(t ) ; h(t )k = max   0

 sup sup jh(t  + s) ; h(t )j ! 0 & " ! 0  2 O0 t] s 2 O0 "t]  s

  1    ,. R"t * '  &   & " ! 0 (3.1.17).  2,  !"(s) ds = 1 , & ' 0

1 0 Zt 1 Z"t 0Zt lim @ '
(t )G()!"( ; s) d A Q(s) @( !" ( ; s)G
()'(t ) dA ds = "!0 0

0

=

Zt 0

0

'
(t )G()Q()G
()'(t ) d

(2 '(t )  , O0 t]  2    8.  , &7 2  &       (3.1.16) & " ! 0 , '' MO(z x(t) ; x^(t))2 ] = '
(t 0)P n

Zt

0'(t 0) +

Zt 0

H(t )R()H
(t ) d+

+ '
(t )G()Q()G
()'(t ) d = m(t H): 0

(3:1:18)

35

3.1  "   -#$ 

J   &2  ' ,  &      x"(t)">0  x^" (t)">0 & " ! 0 7 2 2 & ' &    L2 O0 t]  x(t)  x^(t)    .
&'' &   ,  5 &' 5 & 2. *  ' 2  ' & 2 &   2 (( '  ' (3.1.13)   ' (3.1.14). <1 2   , & 2 H(t )  ' (3.1.13),(3.1.14),  '',  (3.1.18). <& '6 ('   ,8 4 '     ,   (   ,  5 &' 5 & 2   ' '   . ' 3.3 ('  .2.2 ) &   '(    !+ (3.1.1)-(3.1.2) -*  6   '*     H(t )     (3.1.13)-(3.1.14)      !*+  (3.1.18) {    '(       

      . 3. *6',  ,  5 &' 5 & 2 (3.1.13), (3.1.14), (3.1.18)   2  48  2 0 dP () = F()P () + P ()F
() + G()Q()G
() ; P ()C
()R;1()C()P() (3:1:19) d P()j =0 = P0  5  P0 = MOx0x
0 ] . D &'38   (3.1.13)  (3.1.19) &1,' 6 d O'
(t )P()'(t )] = d'
(t ) P()'(t ) + '
(t ) dP () '(t ) + '
(t )P() d'(t ) = d d d d = ;'
(t )F ()P ()'(t ) ; H(t )C()P ()'(t) + '
(t )F ()P ()'(t ) + '
(t )P()F
()'(t )+ +'
(t )G()Q()G
()'(t ) ; '
(t )P()C
()R;1()C()'(t ) ; '
(t )P()C
()H
(t ) = = ;OH(t )+ '
(t )P ()C
()R;1()]R()OH(t )+ '
(t )P ()C
()R;1 ()]
+ H(t )R()H
(t )+ +'
(t )G()Q()G
()'(t ) ,   &1    H(t )R()H
(t ) .  . &  22 &  4     '
(t t)P(t)'(t t) = '
(t 0)P0'(t 0) + '' '
(t 0)P0'(t 0) +

Zt

Zt 0

H(t )R()H
(t ) d +

Zt d
d O' (t )P()'(t )] d 0

Zt 0

'
(t )G()Q()G
()'(t ) d =

= '
(t t)P(t)'(t t)+ fOH(t )+'
(t )P()C
()R;1()]R()OH(t )+'
(t )P()C
()R;1 ()]
gd: 0

G 2   :5 62  &  (3.1.18)   5 '' 5 ,2 & H(t ) = ;'
(t )P()C
()R;1():

(3:1:20)

9 8   , , ,2 4  2 0 P () , '6  4 ,  &' 5 & 2. 4.
 ' ((   2 2  . #  ' 1, K() = P()C
()R;1()  ,&4'     (3.1.20)   H(t ) = ;'
(t )K() . <5  (3.1.13) &'  d'(t ) = ;OF
() ; C
()K
()]'(t ): (3:1:21) d

36

3  "   -#$ 

*  ( t) | 4 (( 5  2 d( t) = OF() ; K()C()]( t) d  '   ' (t t) = In , In |  2 '. # :'  4 (3.1.21)  '   ' '(t t) = z 2, ( t)    ' '(t ) = 
( t)z . D    , H(t ) = ;z
( t)K() . 9'',    ( t) 2 2 2 ( '  ',  H(t ) (('  & & ' 5'. % & ,2 1,2 (3.1.12), 7 ',  h(t ) = ( t)K(): <' 1,', : ' ' h(t ) & &' t & 6 '6   (('7 (,   x^(t) ,&4 2   x^(t) =

Zt 0

( t)K()y() d:

" O( t)];1 = (t ) 2 2 2 ( '   & & ' 5'    2 &26'  (3.1.21)  8. J,   x^(t) | 4 (( 5  2 d^x() = OF () ; K()C()]^x() + K()y() d   d^x() = F()^x() + K()OC()^x() ; y()] (3:1:22) d  '   ' x^(0) = MOx(0)] = MOx0] = 0 . D&    83 , . ' 3.4 ( , 52 6-*57. ) :  + * x^()  *     x()   (3.1.1)  '  6 (3.1.2),    !*+  (3.1.3),      !!  +    (3.1.22),  K() | * -!!+    !, K() = P ()C
()R;1 () P() | * +  + "* + *,       &** (3.1.19). %, ' 3.4 ,   5' 42 ,   &'  (  2  7 7 &    2 1 5 5  5 4' 1 8 27. #  5' (   '-!8  3  8   5  6   ' R() | '     1 5 4'. *& &  4 ,   (  2 1  45    4'  '  ,'27 & &'   , 6 &  &1  2 ,  0.  '. 0 ''  '  42 ,   &'  (  & '25 5  4' 1 8 27   '. 3.2   (   %   -&'

# &  3' ,    ' (   '-!8 , 5  &  u( !)  v( !)     '8      ,2. # :' &5( &    5' 42 , 

 &'  (  1  13 &  . *        ,    . # n {'  & x( !) '   2 '   ' dx() = F()x() + G()u() + L()f() x(0) = x  (3:2:1) 0 d 5  F() , G() , L() | ,   ', u( !) | 1  4', f() | ,  2 ' 2 ( 2) -(2, x0 (!) | n {'  . 0,'  ' F() , G()  L() 5    ,' 8  x() , u()  f()    . K ' ' F() & 6 L2 O0 t] ,  ' G()  L() 5  , f() 2 L2 O0 t] .

37

3.2  %!  "   -#$ 


1 8  2 & fy() 0    tg 2, x() 4' y() = C()x() + v() + g() (3:2:2) g() | ,  2 (2 , L2 O0 t], v() | 1  4', C() | ' ,'  m n 5  ' : ''. <1 2 & 1 8 2' fy() 0    tg  8  x^()  x()   = t ,    2838 8

8 9 Zt < = 2 ] x^(t) = h(t )y() d  m(t) = inf MO(z x(t) ; x ^ (t) ; (m (t) ; m ^ (t))) x x n  h : 0

(3:2:3)

5  z 2 En , mx (t) = MOx(t)] , m(t) ^ = MO^x(t)] . *  u()  v() { ,'     '8  83   MOu()] = mu () MOv()] = mv () MOx0] = mx0  MOu()x
0 ] = 0 MOv()x
0 ] = 0 MO(x0 ; mx0 )(x0 ; mx0 )
] = P0 P0  0 P0 = P0
 MO(u() ; mu ())(u() ; mu ())
] = Q()( ; ) Q()  0 Q() = Q
() MO(v() ; mv ())(v() ; mv ())] = R()( ; ) R()  0 R() = R
() MO(u() ; mu ())(v() ; mv ())
] = S()( ; ) 0     t: * '  2 1135 (   '-!8 .  2 :5  ' ,': u() = u1() + mu () , v() = v1 () + mv () , x0 = x01 + mx0 . *  u1() , v1 ()    x01 '8     .   (3.2.1) 1  '  dx1() = F()x () + G()u () + G()m () + L()f() x (0) = x  (3:2:4) 1 1 u 1 01 d  4 (3.2.2) &'  y() = C()x1() + v1 () + mv () + g(): (3:2:5) #  ' -(8 x2() ,    2838 (( '  8 dx2() = F()x () + G()m () + L()f() x (0) = 0: (3:2:6) 2 u 2 d #'   (3.2.6) , (3.2.4), & ' d(x1() ; x2()) = F()(x () ; x ()) + G()u () x (0) ; x (0) = x 1 2 1 1 2 01 d , 1,2 x1() ; x2() = xc() yc () = y() ; C()x2() ; mv () ; g() 7 ' dxc () = F()x () + G()u () x (0) = x  (3:2:7) c 1 c 01 d yc () = C()xc () + v1(): (3:2:8) %, 132 '   (3.2.1)-(3.2.3) &  &  7 ,'     '   (3.2.7)-(3.2.8), 5  &  u1()  v1 () '8       ,'   .
 '  2 (  2 '   (3.2.7)-(3.2.8). #  &  '    :& . %   c  1 1, 7 &  xc , yc , u1  v1 1 ' & . 1.
 ' ((    2 ' '&   &7  ( ( .   #-F&( '  MOx(t)y
()] =

Zt 0

h(t )MOy()y
()] d:

(3:2:9)

38

3  "   -#$ 

#  :5  2  ,   1.2. 0 ' 6 MOx(t)y
()]  & ,2 (3.2.8)  1138 ('  4 ( '. . 10  O25]), & ': MOx(t)y
()] = MOx(t)x
()]C
() + MOx(t)v
()] = Kx (t )C
()+ +MO\(t 0)x0v
()] + + 5

Zt 0

\(t )G()MOu()v
()]d = Kx (t )C
() + \(t )G()S():

(3:2:10)

MOy()y
()] = C()Kx ( )C
() + C()MOx()v
()] + MOv()x
()]C
() + R()( ; )

8 0    <
1=2G()S()  =  MOx()v ()] = \( s)G(s)S(s)(s ; )ds = : \( )G()S()  >  0 8 Z 0  <  < MOv()x
()] = S
(s)(s ; )G
(s)\
( s)ds = : 1=2S
()G
()  =  S
()G
()\
( )  > : 0 Z

#  ' '8 (8 L( )  8

8 C()\( )G()S()  >  < S
()G
()\
( )C
()  <  L( ) = : 1=2OC()G()S() + S
()G
()C
()]  = :

<5    #-F&( 1  '  Kx (t )C
() + \(t )G()S() =

Zt 0

h(t )fC()Kx( )C
() + L( )g d + h(t )R():

(3:2:11)

K   2 2 2  ' ; 5 ' 5  , 4 5 { 2 , . "6 &,  6,   3.1,  ' h(t ) ((' & t . * ((' (3.2.11) & t . <5  @Kx (t ) C
() + \(t ) G()S() = h(t t)C(t)K (t )C
()+ x @t @t +h(t t)L(t ) +

Zt @h(t ) 0

@t

C()Kx ( )C
() d +

<  L(t ) = C(t)\(t )G()S() & t >  , 

Zt @h(t )

@h(t ) @t L( ) d + @t R():

0

(3:2:12)

h(t t)C(t)Kx (t )C
() + h(t t)L(t ) = h(t t)C(t)OKx (t )C
() + \(t )G()S()]: #   7 17 {  2    2 #-F&(. <' 1,', h(t t)C(t)Kx(t )C
() + h(t t)L(t ) =

Zt 0

h(t t)C(t)h(t t)Ky ( ) d:

(3:2:13)

0 ''  8   (3.2.12). % & ,2 (3.2.7)     (,   ,'   ' MOu(t)y
()] = 0 7 ',  @Kx (t ) C
() + \(t ) G()S() = MO dx(t) y
()] = @t @t dt = F(t)MOx(t)y
()] + G(t)MOu(t)y
()] = F(t)MOx(t)y
()]:

39

3.2  %!  "   -#$ 

%, (3.2.9)  , 

@Kx (t ) C
() + \(t ) G()S() = Z F(t)h(t )K ( ) d: y @t @t t

0

*  ' 62(3.2.13)  (3.2.14) (3.2.12), ''

Zt

 

0

Zt

Zt @h(t )

0

0

F (t)h(t )Ky ( )d = h(t t)C(t)h(t )Ky ( )d +

Zt @h(t ) 0

O @t

(3:2:14)

@t Ky ( )d

; F(t)h(t ) + h(t t)C(t)h(t )]Ky ( ) d = 0:

<  Ky ( ) & 6  &  2 ',  2 h(t ) & '   @h(t ) = F(t)h(t ) ; h(t t)C(t)h(t ): (3:2:15) @t 2. #  ' ((    2   x^(t) . <  x^(t) 2 L2 O0 t] ,   { CO0 t]  '6  1138 &,  8 &  x^(t) : d^x(t) = h(t t)y(t) + Z @h(t ) y() d: dt @t t

0

% & ,2 (3.2.15)  ,  x^(t) = 7 '

Zt 0

h(t )y() d

d^x(t) = h(t t)y(t) + Z F(t)h(t )y() d ; Z h(t )C()h(t )y() d = dt t

t

0

  

(3:2:16)

0

= h(t t)y(t) + F(t)^x(t) ; h(t t)C(t)^x(t) = F(t)^x(t) + h(t t)(y(t) ; C(t)^x(t))

d^x(t) = F(t)^x(t) + h(t t)Oy(t) ; C(t)^x(t)] x^(0) = 0: (3:2:17) dt #  27 (3.2.15)  (3.2.17) ,     2 ' h(t t) . 3. *6',  h(t t) (  2, ' 41  2 P (t) . #  '  41  2 x~(t) = x(t) ; x^(t): (3:2:18) J&4'   #-F&( (3.2.11)   MOx(t)x
()]C
() + \(t )G()S() =

Zt 0

h(t )fMOy()x
()]C
()+

+C()MOx()v
()]gd + h(t )R(): *'2' ' ' & 5 2  '' 5 6 2. <5 ,  & ,2 1138 ('  4  (3.2.16), & ',  MOx(t)x
()]C
() + \(t )G()S() = MO^x(t)x
()]C
()+

40

3  "   -#$ 

2Zt 3 Z +M 4 h(t )C() \( s)G(s)MOu(s)v
()] dsd 5 + h(t )R(): 0

0

* ' &  5' &    :5 62   ,  & ,2 (3.2.18), 7 ' MO~x(t)x
()]C
() + \(t )G()S() =

Zt 0

h(t )C()\( )G()S() d + h(t )R():

(3:2:19)

* ' :' 4  &   &  ! t ''

Zt

MO~x(t)x
(t)]C
(t) + \(t t)G(t)S(t) = h(t )C()\( t)G(t)S(t) d + h(t t)R(t): 0

<  \( t) = 0 & t >  ( &7 2 (2 (,   ,'  ' )  \(t t) = In |   ' &2  n  MO~x(t)x
(t)]C
(t) + G(t)S(t) = h(t t)R(t): (3:2:20) *1,'  8 MO~x(t)x
(t)] : MO~x(t)x
(t)] = MO~x(t)fx
(t);x^
(t)g]+MO~x(t)^x
(t)] = MO~x(t)~x
(t)]+MO~x(t)^x
(t)] = MO~x(t)~x
(t)] = P (t)   MO~x(t)^x
(t)] = 0 { & '' 1 5 ' &  O26] ( 2  5 5  5 &  x( !) & 8 '''  &  41 & , ' ' &    1 8  Y = fy( !) 0    t ! 2 ]g ,  2 5  & x  Y Rt .. MO~x(t)^x
(t)] = 0 x~(t) = x(t) ; x^(t) x^(t) = h(t )y()d ). 0 <' 1,', , (3.2.20)  ,  h(t t) = OP(t) + G(t)S(t)]R;1 (t): (3:2:21) D 2, h(t t)   '  41  . 4. # ' 1138 &,  8 62 (3.2.18). % & ,2 (3.2.7), (3.2.17), '' d~x(t) = dx(t) ; d^x(t) = OF(t) ; h(t t)C(t)]~x(t) + G(t)u(t) ; h(t t)v(t) dt dt dt x~(0) = x(0) = x0 : (3:2:22) <' 1,',      41  2. 5. 9&  '   2 '   41  2 P (t) = MO~x(t)~x
(t)] . ((' P(t) & t ( 113' ' ):     dP(t) = M d~x(t) x~
(t) + M x~(t) d~x
(t) : (3:2:23) dt dt dt  d~x(t) 
0 '' 6 M dt x~ (t) . %, (3.2.22)  ,    M d~xdt(t) x~
(t) = OF(t) ; h(t t)C(t)]P(t) + MOG(t)u(t)~x
(t)] ; h(t t)MOv(t)~x
(t)]: (3:2:24) % & ,2 (3.2.18), '': MOu(t)~x
(t)] = MOu(t)x
(t)] ; MOu(t)^x
(t)] .  : MOu(t)x
(t)] = MOu(t)x
]\
(t 0) + 0

Zt 0

MOu(t)u
()]G
()\
(t ) d = 12 Q(t)G
(t)

41

3.2  %!  "   -#$ 

MOu(t)^x
(t)] =

Zt

Zt 0

MOu(t)y
()]h
(t ) d

=

Zt 0

MOu(t)v
()]h
(t ) d+

+ MOu(t)x
()]C
()h
(t ) d = 12 S(t)h
(t t): 0

" MOu(t)x
()] = 0 & t >  ,    2  '  '6 ,    & & 45 3  7 5 . <' 1,', (3:2:25) MOu(t)~x
(t)] = 21 Q(t)G
(t) ; 12 S(t)h
(t t): %,' 6 MOv(t)~x
(t)] . <  x~(t) = x(t) ; x^(t) ,  MOv(t)~x
(t)] = MOv(t)x
(t)] ; MOv(t)^x
(t)]: (3:2:26) J',  & ,2 ('  4  &   ,  6 ,  ,    (,   ,'   ' MOv(t)x
()] = 0 , 7 ' MOv(t)x
(t)] = MOv(t)x
]\
(t 0) + 0

MOv(t)^x
(t)] =

Zt

Zt 0

Zt 0

MOv(t)u
()]G
()\
(t ) d = 21 S
(t)G
(t)

MOv(t)y
()]h
(t ) d

=

Zt 0

MOv(t)v
()]h
(t ) d+

+ MOv(t)x
()]C
()h
(t ) d = 21 R(t)h
(t t): <' 1,',

0

MOv(t)~x
(t)] = 21 S
(t)G
(t) ; 12 R(t)h
(t t): (3:2:27) *  ' (3.2.27)  (3.2.25) (3.2.24), 5  MO d~xdt(t) x~
(t)] = OF(t) ; h(t t)C(t)]P (t) + G(t)O 12 Q(t)G
(t) ; 12 S(t)h
(t t)] ; h(t t)O 21 S
(t)G
(t); ; 21 R(t)h
(t t)] = OF (t) ; h(t t)C(t)]P(t) + 12 G(t)Q(t)G
(t) ; 12 G(t)S(t)h
(t t); ; 12 h(t t)S
(t)G
(t) + 12 h(t t)R(t)h
(t t): (3:2:28) + 5,
MO~x(t) d~xdt(t) ] = P(t)OF(t) ; h(t t)C(t)]
+ 12 G(t)Q(t)G
(t); 1 1 1



(3:2:29) 2 G(t)S(t)h (t t) ; 2 h(t t)S (t)G (t) + 2 h(t t)R(t)h (t t): D''2 (3.2.28) (3.2.29), 7 ',  dP (t) = OF(t) ; h(t t)C(t)]P(t) + P(t)OF(t) ; h(t t)C(t)]
+ dt +G(t)Q(t)G
(t) ; G(t)S(t)
h(t t) ; h(t t)S
(t)G
(t) + h(t t)R(t)h
(t t) P(0) = P0  (3:2:30) 5  h(t t) = OP(t) + G(t)S(t)]R;1(t) . D(' '  5' (  2 '   (3.2.7)-(3.2.8).

42

3  "   -#$ 

  2 76 2  '  d^x(t) = F(t)^x(t) + h(t t)Oy(t) ; C(t)^x(t)] x^(0) = 0 dt 5  h(t t) = OP (t) + G(t)S(t)]R;1 (t) ,  '   41 (  P(t)    2  8 0 (3.2.30). + 5'  2 113 (   '-!8  (4 ,  (3.2.1) - (3.2.3)).   2 76 2  '  d^x(t) = F(t)^x(t) + G(t)m (t) + L(t)f(t) + K(t)Oy(t) ; m (t) ; g(t) ; C(t)^x(t)] u v dt x^(0) = mx0  5  :((   2 (  K(t)  2 2 & ('  K(t) = OP (t)
(t) + G(t)S(t)]R;1 (t): "   41 (  P (t)    2  8 0 dP(t) = F(t)P (t) + P(t)F
(t) + G(t)Q(t)G
(t) ; K(t)R(t)K
(t) dt P(0) = P0 : 3.3 !    %   -&'

*  ' F , G  C , 7 23  ' 8 &   ,   &'  (  (3.1.1){(3.1.3),  ,'28 2  '',   4' u()  v() (;1 <   t) 2 28 2   ' ' 4' '  1 ' 4''. <5  '   '   ', ('83  & x() ,  '   1 8 2 fy() ;1 <   tg '6 ,&   : '   132 { dx() = Fx() + Gu() (3:3:1) d 5  F  G | & 2 ', MOu()] = 0 MOu()u
()] = Q( ; )   2 (;1 t]  Q |  2 '' 2   2 ' '   ,' { y() = Cx() + v() (3:3:2) 5  ' C | ' & 2' : '', MOv()] = 0 MOv()v
()] = R( ; )   2 (;1 t] R | & 6  &  2   2 '' 2 '. 0    (3.3.1)  (3.3.2) &'8 2 113' ' . * 7 & & 627 x()  y() 2 28 2 ' ' ' & ' O27,30]. J   (  , 8 2  83': Rt & 1 8 2' fy() ;1 <   tg 1 2  8  x^(t) = h(t )y() d &  ;1 x()   = t    2838 8 m(t) = inf fMO(z x(t) ; x^(t))2n ]j h
(t )z 2 W21 (;1 +1)g (3:3:3) h 5  z | &,    , En: %, ,   3.1 ,  ,   x^() 2 2 2 4'  2 d^x() = F x^() + KOy() ; C^x()] x^(;1) = 0 (3:3:4) d

43

3.4 '    ($%) 

5  K | :((   2 ( , K = P C
R;1 

(3:3:5)

 P |  2 41  2. <  2 7 &  P | & 22 ',  dP d = 0  ((    (3.1.19) 2 &  2   41 6  2  &'  FP + PF
+ GQG
; PC
R;1CP = 0: (3:3:6) <' 1,', &  , 5 (   '- !8  17 ' 4   '   (3.3.6). 0 ,2  5' 76 2 42  5  51 5  2  K#"  5   1. *:' 1  :((  '  42 (3.3.6)    (   4 2) 4 (( 5  2 0 dP () = F P() + P()F
+ GQG
; P(t)C
R;1CP(t) P (0) = P  (3:3:7) 0 d 5  P0 | &,  2 '' 2 & 6  &  2 '. 91   (3.3.7) 48  K#"  7 &, & P()   5    45 2 ,2. K    4 2 ,  &' 2 , P . # ,  7 2  3.3 &   , 7 '  42 ,   &'  ( 

' . # ,   2 2 & 2 (  &'   & 7 ,  3.3 {

'. 9     :7 7 & 7  2, , ' &  x() . # (  # & x() ,  2 , & 8 &   Kx () ,   '  (  {  ' (( 7   (3.3.1). K  & 7  :   ,   &  ' (3.3.1) '6  & 8 &   Kx () , 2   Kx () = (In  ; F );1GQG
(In  ; F
);1  5  In |  2 ' n -5 &2 . + & &  &   Kx () , ,  ' &   ,  #, '6 &  (,   ,'8 ' 8  ', &6 838  & x() . *7 8 (8   ' '6   & ,2 ' (,, 2  '7  ' ' , 6 2.3,  2 '5'7 { O31]. !  & 1    5' (  #   ' &   O30]. D   ',  (  #    (   ' &''     1 5 4' 1 8 27. K 2, ',  & 5 &   & , 2    (,   ,'  ( ,  ', (, ' &  &    1 8 2. K 5  '6   & '7 4. <' 1,',  5' (  # 2 2 2 "5 2,83'" 2 ,   &'  ( ,  7 & 2 &   ,' &  (,8 (2.2.5). + 5'  ' '   '  1    2 5 2 2 2 3  ' 1  5, 1 4' ,'27 1 1 '.
  5'  ' 1   1 2  ,  K#"  &:' 4 4 &' & 4 & 7 ,    '57 27   (  #. 3.4 *  "'( + 

K ' & 6 ! '  R7 ' 1965 5  O39]. # '   && &' (   '-!8  2 42 ,  5  2 ,   4' 1 8 27 1 . *        ,    . *    & x()    2  8 dx() = F ()x() + G()u() x(0) = x  (3:4:1) 0 d 5  u() | 1  5   4', Ou()] = 0 Ox0] = 0 Ou()u
()] = Q()( ; ) Ox0x
0 ] = P0

44

3  "   -#$ 

& u()    x0   . *,  2 (3.4.1) &' 2 113' ' . %,'8  &  fy() 0    tg , 2, x() 4' y() = C()x() + w() (3:4:2) 5  w() | m {' 4', w() '   2   ', ,16 ' 1 ' 5  ' 4'' v() dw() = N()w() + L()v() w(0) = w  (3:4:3) 0 d 5  v() |    u()  x0 , Ov()] = 0 Ov()v
()] = R()( ; ) w0 | m {' 5   , Ow0] = 0 , Ow0w0
] = S0 , ' R() & 6  &  ,  S0 |  . <1 2  8  x^()  x()   = t ,    2838 8 '''      41 m(t) = inf fMO(z x(t) ; x^(t))2 ] j x^(t) = n

h

Zt 0

h(t )y() d g:

(3:4:4)

!   R7  & 6  2 42 &   ,   838 & . # ' 1138 &,  8 &  y() ,  & ,2 62 (3.4.1)  (3.4.3), 7 ' dy( = dC() x() + C()F()x() + N()w() + C()G()u() + L()v(): d d #  ' 1,2:  dC()   F() = OF() N()] C () = + C()F() N()  y1 () = dy() 1 d d  x()   u()  V1() = OC()G() L()] G1() = OG() L()] x1() = w()  u1 () = v() : <5  ,  (3.4.1)-(3.4.4) '6 &&   83' 1,'. 0 4   22 x1 ()    2  ' dx1() = F1 ()x1() + G1()u1 () d x  (3:4:5) x1(0) = x10 = w00 :
1 8  2 & y1 () = C1()x1() + V1 ()u1 () (3:4:6) &'   MOx10] = 0 MOu1()] = 0 MOx10x
10] = P00 S0 = P10  0

 Q()

0    0 ( ; ) Q() 0 > 0: R() 0 R()

MOu1()u
1 ()] = 0 <1 2  8  x^1()  x1 ()   = t '',838     8 41. J  (3.4.5)-(3.4.6) 2 2 2 ,   &'  (    ' 1 ' 4'' ,'27   ',  5' 42   ,   3.2.
  ' &  5 '  2 2 2 ,  1 ,    ', '  832 4' w() . + & 1 & 2 ('835 (  (3.4.3) 2 &  w() ,1  ,   4' ,'27 { 5     &  &  &  8, & 83 (,8 (2.2.5) O27,26]. ' 5, &2 (( 2 ,' { : &'  & 7 7,  ,  ,  O33,25]  76  &  y1 () , .

45

3.5 '  ',!--$  3.5 *  '. -/' 

  5' ,1 1975 5  9'0  
8'' O42] 2 2, 5  4' 1 8 27 2 2 2 6 ' 1 ' 5  ' 4'' (&   1.20, '. . 21). 1. *        ,    . *  n {'   & x() = x( !)  2 O0 t] ! 2 ] 5 2 &' && 2 p {'5 5  5 4' u() , 8 ' 8  ' dx() = F ()x() + G()u() x(0) = x  (3:5:1) 0 d 1 8  2   & ,'  m ( m  n ) y() = C()x() + w() (3:5:2) 5  F () , G()  C() | '    83 ,' , &' : ' ' G()  C() | 5   (,  F() |  -& . 04 ,  4 (3.5.1) &' 2 '  &  2 1.6. 0 , 4' w() & 6 2 8   5 ' &    W2;1 O0 t],  4' w() '6 1 &      0  w() = v()  (3:5:3) 5  v() | 1  q {' 5   4' q < m: J   &'  (  , 8 2  83': 1 2 & 1 8 2' fy() 0    tg  8  x^()  x()   = t ,    2838 8 m(t) = inf fMO(z x() ; x^())2 ] j x^() = h

n

Zt 0

h(t )y() d g:

(3:5:4)


622 5 1 2 & ' & '' '' h(t ) ,    ', 2 7 5 (3.5.4) ' ' , z | &,    , En: %,   6   7 &  u() w()  5  x0: 2.*6',   4' w()   (3.5.3)   2 81 6  1  5   4'.    , &  1 8  2 m {'  & y() = C()x() + w1() (3:5:5) 5  w1() | m {' 6  1  4'   '  '    ' MOw1()w1
()] &' 5 '     4'   q q < m .  2 ' R1() ,     ''     , 3   5 2 m {'2 '  () O1] 2,  2  0 ::: 0 0 ::: 0 3 66 01 2 : : : 0 0 : : : 0 77 66 : : : : : : : : : : : : : : : : : : : : : 77 R1() = T () 66 0 0 : : : q 0 : : : 0 77 T
(): 66 0 0 : : : 0 0 : : : 0 77 4 ::: ::: ::: ::: ::: ::: ::: 5 0 0 ::: 0 0 ::: 0 9'',   ' T
()R1 ()T ()  5   5  2 4 q & 6 7  

1 2 : : :  q : #  ' ' H() ,'  m q 2  0 ::: 0 3 66 01 2 : : : 0 77 66 : : : : : : : : : : : : 77 H() = T() 66 0 0 : : : q 77  66 0 0 : : : 0 77 4 ::: ::: ::: ::: 5 0 0 ::: 0

46

3  "   -#$ 

5 ' H()   q (Rang H() = q): <5  R1() = H()H
()  y() = C()x()+H()v1 () , 5  v1() | q {' 1  5   4'   '  '    MOv1()v1
()] = = Iq ( ; ) , Iq |  2 ' &2  q . *  '     ' H() . 1. 91,' , N(H) = fx j Hx = 0 x 2 Eq g 2  ' H() . "6   N(H)  6

4    . K   , 5,    Hx = 0 '     4 Eq ,   Rang H()   ,'   x: 2." H
()H() |  6 2.    ,   H()x = 0 ,  H
()H()x = 0  N(H)  N(H
H) . D 5 ,   H
()H()x = 0 ,  x
H
()H()x = (Hx Hx)q = kHxk2q = 0: 9 8   ,  N(H
H)  N(H) . <' 1,', N(H
H) = 0 Rang H = Rang(H
H) = q: 3. " H
H | '' 2 &2  q .
 ' ''  : ' ~hij  h~ ji &,  2 ' H
 H :

X X h~ ij = hkihkj = hkj hki  m

m

k=1

k=1

5  hki | : ' ' H() . #  4' 1   & ,   83 &2 &  1 ' ( '  "-*, O1]).   3.1 ;+ H + '    " + * H ,  1: HH + = H + H 2: HH + H = H 3: H + HH + = H + : D&    ' 3.5 , * + H '  n m + H + = lim f(H
H + 2Im );1 H
g = lim fH
(HH
+ 2Im );1 g !0 !0

  . , "  n {    *  z -   x = H + z    *    

     6  *  , '6 !*+  kz ; Hxk2n . ,    ' '6  O1] . #  ' ' H1() = OH
()H()];1=2H
() . K ' ' q  ,  '7 . 9&  ' ' H1+ . D5   ' 3.5 H1+ () = lim H
()OH1()H1
() + 2Iq ];1 = H1
()OH1()H1
()];1 !0 1   H1()H1
() |  6 2 '. *  ' : 6 , ' H1() . <5  n H1+ () = fOH
()H()];1=2H
()g
OH
()H()];1=2H
()

H() (OH
()H()]
);1=2

o;1

= fOH
()H()];1=2H
()g
= H1
(): + 5, 2 ' H() 7 ',  H + () = OH
()H()];1H
(): 0 '' ' H()H + () .

47

3.5 '  ',!--$ 

  3.2 ;+  '      ,         2 =  . " H()H + () -  '&2,   OH()H + ()]2 = H()OH
()H()];1H
()H()OH
()H()];1H
() = = H()OH
()H()];1H
() = H()H + (): " Im ; H()H + () 6  '&2: (Im ; H()H + ())2 = Im ; H()H + () ; H()H + () + H()H + ()H()H + () = = Im ; H()OH()H
()];1H
() ; H()OH()H
()];1H
()+ +H()OH()H
()];1H
()H()OH()H
()];1 H
() = = Im ; H()OH()H
()];1H
() = Im ; H()H + (): D1   ,2  '& '   1   0.    . *  x 1   ,    83 1  ' ,8  , .. x = x ,    '&2 ', 2 =  . <5   &    A2 x = A(Ax) = A(x) = Ax = 2 x: <' 1,', x | 1    ' A2 ,    83 1  ' ,8,  ' 2:
 2 =      A2 x = Ax = x  2 =   ,'6 4 &  = 1    = 0: 9 8   ,  5  '& '     : Rang A = Trance A (  ' { Pn Pn ''  5 7 : ' , Trance A = aii = i , i | 1   , '  , i=1 i=1 aii | 5  : ' : '). % & ,2  '&  ' Im ; H()H + () , H()H + () , '' Rang OIm ; H()H + ()] = Trance OIm ; H()H + ()] = = m ; Trance H()H + () = m ; Rang H()H + () = m ; q: <' 1,', ' Im ; H()H + ()  6 m ; q  ,  '7 . D  ' , :7  ' H2() ,'  (m ; q) n: 9'' 6,   ' Im ; H()H + () 5   1' ' H()    OIm ; H()H + ()]H() = H() ; H()H + ()H() = H() ; H() = 0: " H2() 1,  ,  ' Im ; H()H + () ,     , H2()H() = 0  H2()H1
() = 0 ,   H2()H1
() = H2()H()fOH
()H()];1=2g
: %,     H2()H1
()  = 0 ,   ' H2()  H1()  ,  '. *H () 1 :' ' H () ,'  m m  6 2  '6

    6  2 &1,   y ()   H ()  1 1 y2 () = H2() y(): *'22 : &1,   1 8 2' (3.5.5), & '

y1 () = H1()C()x() + s() y2 () = H2()C()x()

(3:5:6)

5  s() = H1()H()v1 () = OH
()H()]1=2v1 () |  6  q {' 1  5   4'   '  '    ' MOs()s
()] = H
()H()( ; ):

48

3  "   -#$ 

%, ' &  ,  , 1 8 2 (3.5.5) :    1 8 2' (3.5.6) 4''   (3.5.1). *:' 5' 2   4''   (3.5.3). < 4' &   838 &8 :   1 8 2 &,  2 2  ,   , 7, &54  7 '42, &1 6 '6  '. K     & ' m ; q  ' 4' (3.5.3). 9   ,'2' &' ' & 6' ,   8 4'  6  '    . 9'',  & q = m ' & ' ,  (   '- !8 . 3. *1,' 1 8 2 (3.5.2) 4''   (3.5.3)   '  : y1 () = x1 () (3:5:7) y2 () = C1()x1() + C2 ()x2() + v2 (): *  ' C() ' 5 m . <5   '6 &    83' 1,': C() = OA() B()] 5  ' A() ' 58 18 ' A;1 () . #  '  6  &1,   ;1() ;;1()B()  S() =  (3:5:8) 0 In;m 5  In;m |  2 ' &2  (n ; m) . <5  &        y() = OA() B()]x() + w() = OA() B()]S()S ;1 ()x() + w() = OIn;m 0]S ;1()x() + w(): 91,' x() = S()() . #-(2 () 2 2 2 4'  2 d() = S ;1 ()OF()S() ; dS() ]() + S ;1 ()G()u() d d (3:5:9) ; 1 (0) = S (0)x0 = 0  &,  2  S() 3  ,   &   8 : ' ' 1 8 2 C() | 5   (. # 1 8 2 y() 2, & ' () 4' y() = OIm 0]() + w(): 0, 6'  ()     283   () = 1 ()  (3:5:10) 2 () & 1() ' ,'  m ; q ,  2() { (n ; m + q) . <' 1,', &'38 &1, 2 (3.5.9)  &  2 (3.5.10) 1 8 2 y() &  2 2   '   (3.5.7)     1 ()  2 () : y1 () = 1 () (3:5:11) y2 () = L2 () + v() 5  L = OIq 0] - ' ,'  q (n ; m + q) . J  (  '6 ('   &     () ,  (' 2  ' (3.5.9), 5  ,'2 '8  (3.5.11). 9 x^()  x() 2,  ^()

 () 4' x^() = S()^() . 4.
 '   #-F&(    '    ' h(t )    283 8 (3.5.4).  2 &32    ',  &  u()  v() '8     ,   , &'    & 6 , O0 t] t < 1 MOu()u
()] = Q()( ; ) , Q() | '' 2,  2 ' 5  ' : '', MOv()v
()] = = R()( ; ) ,

49

3.5 '  ',!--$ 

R() | '' 2, & 6  &  2 q {'2 ' 5  ' : ''. # x0 '     ,    & ' u()  v() , MOx0x
0 ] = P0 , P0 | '' 2,  2 '.
76  &'    '  '''      41,  6 '   ,   1.2, :    48  2 #-F&( (1.2.13)   (1.2.14). D '     ' ' ,   , ,  ' 5,  4' 1 8 27 '  (3.5.3), 5    (1.2.14) ,&4 2   Kx (t )C
() =

Zt 0





0  h(t )C()Kx( )C
() d + h(t ) 00 R()

(3:5:12)

5  ' Kx ( ) &  2 2 6' ( '. . 30) Kx ( ) = \( 0)P0\
( 0) +

Z

min() 0

\( s)G(s)Q(s)G
(s)\
( s) ds:

(3:5:13)

; ' 2 ' \( ) 0     t 4  ' (3.5.1)    2  8 d\( ) = F()\( ) \( ) = I : n d ! 5 2 5    : ' ' G Q   2 &   : ' ' F , (3.5.13)   &   : ' ' Kx ( )  3   , &  ',   5 &,   & 1' 5''. 91,': u() h
(t )z f() Kx
(t )z 5  z | &,   (    , En  C u C
()u C | &,  23     : ' 5 m {'55 &    : ' n {'5 &    5 6 &, &',   u 2 Lm2 O0 t] ,  C : Lm2 O0 t] ! Ln2 O0 t] 5  Lk2 O0 t] | k {' &    5'7   ' '  G15 -( &  7  O0 t] k 2 fm ng C  | &26  C &  2 &  5 4 &': C  : Ln2 O0 t] ! Lm2 O0 t]

Av

Zt 0

Kx
( )v() d

8v 2 Ln O0 t] C  AC u = 2

Zt 0

C()Kx
( )C
()u() d

0   0 0   Iq u 0 Iq u Rm u = Iq RIq u Iq R 0 Iq u = 0 R u 0 0  Zt Zt

Bu fMOy()y
()]g
u() d = C()Kx
( )C
()u() d + 0 R() u() 0

0

    0 J 0 0 : Bu = C  AC u + IqRIq u Rm 00 R() m 0 Iq

 3.1 0     

hu Bui  c1fkC
uk2;10 + kIq uk20g

(3:5:14)

 c1 |    * .
    (3.5.14) '6 ,&  6  :

hh
(t)z Bh
(t)z i  c1fkC
h
(t)z k2;10 + kJm h
(t)z k20 g 5  h
(t)z = h
(t )z = u(): < &       1  5  &'2 2  4'.

50

3  "   -#$ 

    ,      . % & ,2 (3.5.13) 

  4 1,2, & '

hu Bui ==

Zt 0

z
h( )

h
(t )z ds d d +

Zt 0

Zt 0

C()

z
h(t )

Zt

Z

minfg

Zt 0

\( s)G(s)Q(s)G
(s)\
( s)C
()

0

C()\( 0)P0\
( 0)C
()h
(t )z d d+

+ z
h(t )Rm h
(t )z d = k1 + k2 + k3: 0

9' &  5' (3.5.15) . *  \^ ( s) =

(3:5:15)

 0 s > 

\( s) s  :

<5  &     832 & 4

2 0 minfg 1 3 Zt 6Zt Z k1 = 4 z
h(t )C() B \( s)G(s)Q(s)G
(s)\
( s) dsC @ A C
()h
(t )zd 75 ds = 0

0

0

2 0 minfg 1 3 Zt 6Zt Z = 4 z
h(t )C() B \^ ( s)G(s)Q(s) G
(s)\^
( s) dsC @ A C
()h
(t )z d 75 d = 0

0

0

3 2Zt 3 Zt 2Zt = 4 z
h(t )C()\^ ( s)d5 G(s)Q(s)G
(s) 4 \^
( s)C
()h
(t )z d 5 ds  0

0

 c2 2 = maxfjG()Q()G
()j

0

Z t 0 Z t

1  2 @ \^
( s)C
()h
(t )z d  A ds n

0

0

(3:5:16)

5  0    tg , j : j | '2 ' En , z | &,    , En: <  \( ) | ( ' 2 '  ' (3.5.1),  ,   \( )  ,  \
( ) | ( ' 2 ' &26  '  (3:5:17) l ; dxd() ; F
()x () = C
()h
(t )z  &    x (t) = 0 4 (3.5.17) ,  2 ('  x (t) = 

Zt 0

\
(s )C
(s)h
(t s)z ds:

9& l (x ) 5''( 16 L2O0 t]  W20;1O0 t] ( '. & 1 O25] . 38). *:'

1 Zt 0Zt  2 c2 @ \^
( s)C
()h
(t )z d nA ds  c3 kC
h
(t)z k2;10 = c3kC uk2;10 0

0

5  c3 | 2 & 6 2  .

(3:5:18)

51

3.5 '  ',!--$ 

0 ''  5' (3.5.15). %'': k2 = =

Zt 0

Zt 0

z
h(t )

Zt 0

C()\( 0)P0\
( 0)C
()h
(t )z dd =

z
h(t )C()\( 0) dP0

Zt 0

\
( 0)C
()h
(t )z d 

 c4kP0k k\( 0)k2 kC
h
(t)z k2;10  c5 kC
h
(t)z k2;10 = c5 kC uk2;10: (3:5:19) J   &' 113     4-!2 5 (Q )O25]: 2 8u 2 W2;1 O0 t]  8v 2 W21O0 t] jhu vij  kuk;1kvk1 ,  k : k - &2 ', 4  5 | & 6   .  2 5 5'5 &    k3 = =

Zt 0



Zt 0

z
h(t )Rm h
(t )z d = (u Iq RIq u)0 = (Iq u RIq u)0 =



 z
h(t ) I0q R() 0 Iq h
(t )z d  c6kJm h
(t )z k20 = c6kIq uk20:

(3:5:20)

91,2 , 1 = maxf2c3 2c5 c6 g , & ' 1'    .  3.2 > + G()Q()G
() , R()  P0       ,    "   *  ' 2 L2 O0 t]       (' GQG
')0  cq k'k20 , (Jm ' RJm ')0  cr kJm 'k20   8z 2 En *  '   (z P0z)n  cp kz k2n   q  r  p -    * ,        c0 f kC
h
(t)z k2;10 + k Jm h
(t)z k20 g  hh
(t)z B(h
(t))z i c0 = const c0 > 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). <' 1,',    v() = = C
()h
(t )z 

Zt    k1  cq \
( s)C
()h
(t )z d 20  cq c0 21 kC
h
(t)z k2;10 = c0q kC uk2;10 s

5  c0q = cq c0 21: %, 62 (3.5.19)  & 6  &    ' P0  ,  k2 =

Zt 0

z
h(t )C()\( 0) dP0

Zt 0

\
( 0)C
()h
(t )z d

Zt     cp \
( 0)C
()h
(t )z d 2n : 0

+   v() = C
()h
(t )z  & ('  4 2 &265  2 x(0) =

Zt 0

\
(s 0)C
(s)h
(t s)z ds

,  & ,2  8       (1.1.5), 7 ', 

Zt    k2  cp \
( 0)C
()h
(t )z d 2n  cp c021 kC
h
(t)z k2;10 = c0p kC uk2;10 0

5  c0p = cp c021 : %, & 6  &    ' R()  4 & 7    2 k1  k2  ,  hu Bui  c0 q kC uk2;10 + c0p kC uk2;10 + cr kIq uk20  c0 fkC uk2;10 + kIq uk20g 5  & 22 c0 = minfc0q  c0p  cr g . G'' 3.1  3.2 &, 8,  & &     '' 3.2 &    &    : c0 fkC uk2;10 + kIq uk20g  hu Bui  c1 fkC uk2;10 + kIq uk20g c0 fkC
h
(t)z k2;10 + kJm h
(t)z k20 g  hC
h
(t)z BC
h
(t)z i  c1 fkC
h
(t)z k2;10 + kJm h
(t)z k20 g: <' 1,', &'38 1  (' hu Bui 5 ' &    W20;1 O0 t] '6

  ': kuk2H hu Bui kuk2H  kC uk2;10 + kIq uk20: Z, HO0 t] 1, &    ' k : kH . *    HO0 t] 17 ,   W20;1O0 t]  L2 O0 t] - 5 1  &   .

 3.3 0        h
(t) ; h
(t) 2 1 m z  = g(hn ) + 1 g(hm ) ; g( hn + hm )  n H 2 2 2 2



Zt ; g(h) = m(t) = inf fM z x(t) ; h(t )y() d "2n j h
(t )z 2 W21O0 t]g: h 0

53

3.5 '  ',!--$ 

*     6 2 '' 5 &  &    &  ,   &   ( g(h)  : n;2

g(h) = MO(z x())2   5 

k0 = z
MOx(t)x
(t)]z .

Zt 0

z
MOx(t)y
()]h
(t )z d +

Zt Z t 0 0

z
h(t )MOy()y
()]h
(t )z d d

g(u) = k0 ; 2hC f ui + hu Bui

' 3.6 > + G()Q()G
() , R()  P0       ,       3 -@ !        * !*+ ' HO0 t] .     ,      . * &  8 6 5, 2 815  5 n   2  hn(t ) ,  fg(h) j h
(t )z 2 HO0 t]g: 0  a  g(hn )  a + n1  a = inf h % & ,2     '' 3.3, 2 : ' hn(t )  hm (t ) , 1   &  2 g(h) '' kh
n (t)z ; h
m (t)z k2H  2(a + n1 ) + 2(a + m1 ) ; 4a = n2 + m2 ! 0 n m ! 1: <' 1,', &      fh
n (t )z g1 n=1 ( '  &    HO0 t] ,   &  HO0 t]  7  2  ' : ' h0 (t ): D3    42  2 #-F&( ,. *  h1 (t )  h2(t ) | 42  2 #-F&(, &' h1 (t ) 6= h2 (t ) . <5 ,  & ,2 4 '' 3.2, 7 ' kC
(h1 (t) ; h2 (t))
z k2;10 + kJm (h1(t) ; h2 (t))
z k20  c(2a + 2a ; 4a) = 0: <' 1,', &  ,  h1(t ) = h2 (t ) '  &    HO0 t] . %, 4 &,,  2 76 2 '&   &7  ' (  h0(t ) 17 ' 4   #-F&( (3.5.10). * :'   '& ' h0(t ) & &'     2 4 5 '  2' & 5     & 6 &    5'7   ' ( 6 &   ,   '& ' Kx ( ) C() R() 5  . 04 (3.5.10) 13'  {  &  2 , .
 & 7 527   7  ,  7 1  , 6,   2 &   5' 42 ,   &'  (  1, &'2 '  5 2, +.
. <7 .  ' ' '   2  5' 9'0  
8'. 5. 0 '' ' 9'0 -
8'. *        ,    . *  ,  ' 2  ' dx1() = F ()x () + F ()x () + u () (3:5:21) 11 1 12 2 1 d dx2() = F ()x () + F ()x () + u () (3:5:22) 21 1 22 2 2 d 5  0    t  1 , x1()  x2 () | n1 {  n2 {'  & , &  83  2  ', u1()  u2() | 1  5   4'   '  ', x1(0)  x2(0) | 5    , &' 2 8 2 O0 t]  < 1 '8 '  42: MOu1()u
1 ()] = Q11()( ; ) Q11() = Q
11() > 0 MOu2()u
2 ()] = Q22()( ; ) Q22() = Q
22() > 0 (3:5:23) MOu1()u
2 ()] = Q12()( ; ) = MOu2()u
1 ()]

54

3  "   -#$ 

MOx1(0)] = x10 MOx2(0)] = x20 MO(x1(0) ; x10)(x1(0) ; x10)
] = P11 P11 = P11
> 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]:   <1 2 & ,'2' fy1 () y2() 0    tg  8   x() = xx1() 2()

  = t    2838 8 m(t) = inf fMO(z x(t) ; x^(0) ; x^(t))2 ] j x^(t) = n

h

Zt 0

h(t )y() d g

(3:5:27)

5  622 5 1 2 & ' '' h(t )    283' 48: h
(t )z 2 HO0 t] ( '. & 4 ,   3.5) 2 8z 2 En  x^(0) {   5  22  x(0): *  ,    (3.5.24),  x1() 1 8  2  &'6 O0 t] ,  ,    2  76 8  x^2(t)  x2 (t) , 

 x (t)  Zt 1 x^2(t)

=

0





h0(t )y() d + x^0(0) x^0 (0) = xx^1(0)  2(0)

h0(t ) { &' 2 '&  2 &7 2 ' ( , x^0(0) {  42   5  22  x(0): #  2 ',  1 &' 8  x^0 (0) ,  2 ,2  , x1(0) .  2 :5 x^2 (0) &  '   x^2(0) = x20 + N(x1(0) ; x10): (3:5:28) " N (3.5.28) 1' , 1     2 41 & &    x^2(0) 1  '' . *  "(0) x2(0) ; x^2 (0) P(0) MO"(0)"
(0)]: <5  "(0) = x2 (0) ; x20 ; N(x1 (0) ; x10)   P (0) = P22 ; NP21 ; P21N
+ NP11N
 P12 = P21
: 9  ,  P (0)  5 '45 ,2 ( '  &: '5  2 ') '    ' ,   ' N = P21P11;1: <' 1,',  4     ' '     1 : x^2(0) = x20 + P21P11;1Ox1(0) ; x10]

(3:5:29)

P(0) = P22 ; P21P11;1P21
:

(3:5:30)

55

3.5 '  ',!--$ 

< , &   8,    x2 (0)  x1 (0) {  '  5  ,   4       x^2(0)    x2(0) 2 2 2    ''  6  MOx2(0)jx1(0)]   &  6' (3.5.29).
 ' &' 8 8  x^2(t) t > 0 . 91,' , xk(t) 5    & x2(t) ; x^2(0) . <5  MOkx(0)] = 0      (3.5.27) '' m(t) = inf fMOz xk(t) ; x^2 (t))2n ] j x^(t) =  h

Zt 0

kh(t )y() d g

(3:5:31)

5  622 5 1 2 & ' & '' ''-(2' hk (t ) , kh
(t )z 2 HO0 t] , 8z 2 En . %5 &' 2 '  42 (1.1.11). " kh0 (t )    2832 8 (3.5.31), 2 2 2,  ,  , 4'  2 #-F&( Zt
MOkx(t)y ()] = kh0(t )MOy()y
()] d  2 O0 t]: (3:5:32)

 y () 

0


1 8 2 y() = y1 () & 6 ' &,  8 &   L2 O0 t] W2;t 1O0 t]: 2 91,' , LO0 t] L2 O0 t] W2;t 1O0 t]: #  ' & # " d   f ()  f () ; F 0 1 11 d  f = f12 ()  F (f) ;C () I f () 2 1 q ;1 f1 () 2 L2 O0 t] , f2 () 2 W2t O0 t], ,'   f1()  f2 ()   m ; q  q    , &,  2 dd &' 2 113' ' . 9& F 2 2 2 ' & ' &', 1683' LO0 t]   &,   W2;t 1 O0 t] W2;t 1O0 t] ,   & L dd ; F11() - &  , L2 O0 t] W2;t 1 O0 t] ( '. '' 1.1). D3   &  & F ;1 , ,  ' 6'  L;1 0   \( 0) 0  ; 1 F C1()L;1 Iq = C1()\( 0) Iq  5  & L;1 | 1  L   : '7 y() , L2 O0 t] &  2 2 ('  L;1 y \( 0)y(0) , \( ) | ( ' 2 ' 4  2 Lx = 0: *  1 8  2 & s() 2, y() 4' " dy1() # ; F ()y () 11 1 s() = F (y) = : (3:5:33) d y2 () ; C1 ()y1 () <5 ,  2 (3.5.24), (3.5.25), (3.5.21)  (3.5.22), ,'2 s() '6 &       2 () + u1() : s() = FC11()x (3:5:34) 2()x2 () + v()  ()  1 91,' w() = uv() . <  Q11() > 0  R() > 0 ,  w() | 1  5   4'   '  '     8  () U ()  11 1 Rw Q (3:5:35) U1
() R() > 0  2 O0 t]: 0 ''   #-F&( MOkx(t)s
()] =

Zt 0

h1 (t )MOs()s
()] d:

(3:5:36)

56

3  "   -#$ 

<  s() = F y()  MOkx(t)fF y()g
] =

Zt

h1 (t )(MOF y()fF y()g
]) d:

0

 , '6' : 6  &,   : ' z 2 En  &' &  4  '2' ' ' &8 '' 5 6 2 M  & F (& M  F &  ), 5 

F MOy()kx
(t)]z = F

Zt 0

MOy()fF y()g
]h
1 (t )z d:

% & ,2   & F 7 ', 

FfMOy()kx
(t)]z ;

Zt

MOy()fF y()g
]h
1 (t )z d g = 0

0

 1 5 2 3   8 55 15 & F ;1  &,     z & ' MOy()kx
(t)] =

Zt 0

MOy()fF y()g
]h
1 (t ) d

 ,  &2 &   6, '': MOkx(t)y
()] =

Zt 0

h1 (t )MOF (y())y
()] d:

<' 1,',   (3.5.36) :     8 MOkx(t)y
()] =

Zt 0

h1 (t )F (MOy()y
()]) d:

D&    4

Zt 0

h1(t )F (MOy()y
()]) d

Zt

= Fm (h1 (t ))MOy()y
()] d 0

(3:5:37)

(3:5:38)

5  Fm x
= OF x]
2 8x 2 L2 O0 t] W2;t 1 O0 t]  F  | &26  F &,  
 F  = L0 ;CI1q ()  L | &26 L &, L ; dd ; F11
() . <' 1,',   (3.5.37) '6 ,& 

 

Zt 0

Fm (h1(t ))MOy()y
()] d = MOkx(t)y
()]:

(3:5:39)

#2   (3.5.39) , (3.5.32) 7 ', 

Zt 0

fkh0(t ) ; Fm (h1(t ))gMOy()y
()] d = 0:

(3:5:40)

57

3.5 '  ',!--$ 

*6', 

kh0(t ) = Fm (h1(t )):  2 :5   6',  4  2 (3.5.36) 3       . *   F ()  A() = C12() :

(3:5:41)

s() = A()kx2 () + w()  2  5 5  2 (3.5.36)

(3:5:42)

2

<5  (3.5.34) &' 

MOs()s
()] = A()Kx ( )A
() + Rw ()( ; ) 2 2 2 & 6  &  ',   ' Rw () & 6  &  ,  ' A()Kx ( )A
()  2. %, & 6  &    2   2 (3.5.36)   3          5 42. % & ,2     {( , & ' MOkx(t)s
()] =

Zt 0

h1 (t )A()Kx ( )A
() d + h1(t )Rw ()

 :   ; 5 ' 2-5  , 4 5 . '6'  2 (3.5.32)  (3.5.36)  &,    z , En 

 ' 1,2: u1() h
1 (t )z  Bs u

Zt 0

MOs()s
()]u() d 2 8u() 2 W21 O0 t]

f1 () MOs()kx
(t)]z  u0() h
0 (t )z  f() MOy()kx
(t)]z 

Zt

(3:5:43)

Bv MOy()y
()]v() d 2 8v() 2 W2(;m1 ;q) O0 t] W2(1 q) O0 t]

0 1 5  W2(m;q) O0 t] W2(q) O0 t] |   &,   m ; q {'5 5 5 &    W2(;m1 ;q) O0 t]  q {'5 &, 5 &    W2(1 q) O0 t]: ;1

 2 (3.5.32)  (3.5.36), '

 7 1,, &'8  :

Bu0 = f 

(3:5:44)

Bs u1 = f1 :

(3:5:45) G5 &,,  2 8u 2

    : Bs u = FBF u    { Bs = FBF   1 W2 O0 t]  f1 = F f -( f  f1 & 6 W21 O0 t] ( '. ,  3.1). 9'',  & F  5  2 8u 2 L2 O0 t] &       : kF  uk;10  c kuk0 5  c | & 6 2  . ' :5, & F  ' 5 1 &:   ;1  ;1
 (F  );1 = (L 0) (L ) I C1 () : q W21O0 t]



K  ,    '' 1.1. 0 ''  2 &,   (Bs u u)0 u | &,   : ' , W21: %,   & Bs  F  2 8u 2 W21   &     & 4: (Bs u u)0 = (FBF  u u)0 = hBF  u F  ui  ckuk20  (cc)2 kF  uk2;10:

58

3  "   -#$ 

<' 1,', & B & 6  &     ' &,   &   : (m ; q) '' &    L2(m;q) O0 t]  q {'' { W2(1 q) O0 t]     L2(m;q) O0 t] W2(1 q) O0 t]: *  '   (3.5.45) '  & Bs 5 &   FBF   '6' & 

6    F ;1: # ,  7 ',    (3.5.45) :     8: BF  u1 = f: (3:5:46) %, 6 (3.5.44), (3.5.46)  & 6  &    & B '',  u0 = F  u1  u0 '6   & ('  u0 = F  (Bs );1 F f: D4 (3.5.41) ,.  2  2  x^2(t) & ' (  &  '-!8 . *  22   (3.5.22) ,  8 (8 x1 () , & ' ((    2   22  ' x2 () dx2() = F ()y () + F ()x () + u () (3:5:47) 21 1 22 2 2 d 5  y1 () = x1() ,  2 -(2. D',  1 8  2 & s()  F ()   u ()  12 1 s() = C () x2() + v() : (3:5:48) 2 "'  6  x2(0) ,'2'   ' '' ' 6 ' (3.5.29),   & 8 & 5'   P(0) , (3.5.30). # & ,' 2 & &  1135 (   '-!8 . # ,  & ' ( 

  d^x2() = F ()^x () + F ()x () + K () dx1() + F ()x (); 22 2 21 1 1 11 1 d d ;F12 ()^x2() + K2 ()Oy2 () ; C1()x1() ; C2()^x2 ()] (3:5:49)

 x^2 (0)  2 2 & (3.5.29),  ' K() OK1() K2()] '  K() = fP()Q
() + OQ21() U2 ()]gR;w 1() (3:5:50) 5  ' Rw () '  (3.5.35). "   41 P()    2 (( '  8 0 dP() = F P() + P()F
+ Q (); 22 22 22 d

;OP()A
() + OQ21() U2 ()]]R;w 1()OP ()A
() + OQ21() U2 ()]]


(3:5:51) 5  P0 '  (3.5.30). # (( '   2  x^2() &    2 &2 (( 2 1 8 '5 & . Z1   8 ((  (3.5.49),  ' ,' & ('  x^2 () = () + K1 ()x1 () (3:5:52) 5  ' K1() &  2 2 , 62 (3.5.50). *  &   (3.5.52) (3.5.49)  3 & ' d() = fF () ; K ()F () ; K ()C ()g + fF ()K ()+ 22 1 12 2 2 22 1 d +F21() ; K1 ()F11 ; K1 ()F12()K1 () ; K2 ()C1() ; K2 ()C1(); ;K2 ()C2()K1 () ; dKd1 () gy1() + K2 ()y2 () (3:5:53) &' (0) = x20 + (P21P11;1 ; K1(0))x1 (0) ; P21P11;1x10: (3:5:54) <' 1,', 2 76 2  x^2() 17 '  &'5 8 -(8 ()  & ,2 42 (3.5.51), (3.5.30), (3.5.53)  (3.5.54),  ,' &' ('  (3.5.52). # , 8 &'',   5' 9'0 -
8' '6 1 &' 2 42 ,   (  4 & &    : ' Q11() & 6  &    &    ' P11     2.

4 0+)%(  ( $#%(

# &  3' ,    ' (   '-!8   5  1132. *     5' 42 , 7,   '7 4, 3  8   5    2,    '  4'  ', 1 8 27    8 '   7   . U  :   2 4,  ,   &'  (    2  &    2  42 17 '  & ,  ' 5 2,. 4.1 . +"       ( + "  0 +

+ 1"   ' 2

1. *        ,     . *  n {'  & x() = x(! ) ,  2 O0 t] , ! 2 ] 5 2 &' && 2 5 5 &  u() '&'   1 7 5  7 4' ( ,'6 6 7) , 8  8  '

L(x) dx() d ; F()x() = G()u() x(0) = x0 :

(4:1:1)

y() = C()x() + w()

(4:1:2)


1 8  2   & ,'  m  n

5  F () , G()  C() | '-(    837 ,' . K ' ' G()  C() | 5   (,  ' F() |  - & . 04 ,  4 &' 2

'  &  2 1.6,  , 4'



0 w() = v()



(4:1:3)

'8 '& , 5 5 &    W2;1O0 t] . 0    (4.1.2) &' 2 '  5 5 &    W2;1O0 t]   &, v() | 1  ( ,'6 6 ) q {' 4'. #  : '   ' &  w() & 6   &,  8 W2;1 O0 t] W2; 91,' m(t) = inf f (z x(t) ; x^(t))2 ] j x^(t) = n

h

Zt 0

h(t )y() d g

(4:1:4)

5  z | &,   & 2  , En 622 5 1 2 & ' '' h(t ) : '' , &, 5 &    W21 O0 t] (& &'  ),  5 &' 2   (1.1.11). J   &'  (  '6 ('    83' 1,'. *  ,   1 8 2 fy() 0    tg . <1 2  &      n {'7 

Zt

fx^ (t) = h (t )y() d g >0 0

(4:1:5)

60

4 ,)(   ! 

''-(2' h (t ) ,  : ' 7 & 817  > 0 & 6 &, ' &    W21 O0 t]  8,  lim MO(z x(t) ; x^ (t))2n ] = m(t) 8z 2 En: (4:1:6) !0  2 &32  47    ',  4' u()  w() '8          '6  1, &' MOx0] = 0 MOx0x
0 ] = P0 P0 = P0
 0 MOx0w
()] = 0 MOx0u
()] = 0 MOu()u
()] = Q()( ; ) Q() = Q
()  0 MOw()w
()] = Rw ()( ; )  2 O0 t] 0 0  5  Rw () = 0 R()   R() | '     1 5 4' v()    MOv()v
()] = = R()( ; ) R() | '' 2,   &  2 ' 5  ' : ''.
76  &'    '  '''      41,  6 '  , :    48  2 #-F&( Kx (t )C
() =

Zt 0

h(t )C()Kx( )C
() d + h(t )

0



0 0 R() :

(4:1:7)

D    42 :5  2 , ,   3.5.4. 0 '' , 5  48 2   2, &2 3.5.5. 2. 0  4     ,     (         . 0 ''   MOx(t)y
()] =

Zt 0

h (t )MOy()y
()] d + h (t )

(4:1:8)

5   | & 6   , &' 5 2,. % & ,2       7 &  u() v() 5  x0      {( ,   (4.1.8) '6 ,&   83' 1,': Kx (t )C
() =

 I



Zt 0

h (t )C()Kx( )C
() d + h (t )S ()

(4:1:9)

0 m;q 5  S () = 0 R() + Iq  Im;q  Iq |   ' &2  (m ; q)  q .   (4.1.9)   2  (4.1.7) 5'' h (t )  & (  '  &  2 1   ; 5 ' 5  , 4 5,  ,  , 2 2 2  &   , . J&4'   (4.1.9) & ('.  2 :5 '6' (4.1.9)  &,   (    z 2 En 

 ' 1,2: C()Kx (t )z = f(t ) h
(t )z = u (t ) B = A + R

Au =

Zt 0

C()Kx
( )C
()u() d

0



0 u(): Ru = 0 R()

9& A , R  B | ''     &  .   #-F&( (4.1.9)

& (' '  Bu + u = f (4:1:10) %  '    42 u ( )  2 (4.1.10).

61

4.1
 .! /(   $0

' 4.1 > -   + Kx(t )C
() C()Kx( )C
() R()     *-

   *  !*   t   > 0 ,          h (t )   (4.1.9),     *      , . . h
(t )z 2 L2 O0 t] . ,    '  5 ,    ' 3.1. 9'',  & B = B +I | & 6  &    ''  & 817  > 0 . )C
() , S ;1 ()  dR()    ' -   , ' 4.2 > + Kx(t )C
() , @Kx (t@ d   6  *   O0 t]     h (t )   (4.1.9)      
(t ) @h *        ' ,    8z 2 En @ z 2 L2 O0 t] . ,     5 ,    ' 3.2.  4.1 &     Bu + u = f (4:1:10)     !*+  ' (u) = kBu ; f k20 + hu Bui u 2 HO0 t] (9&   &    HO0 t]  &  ,    '' 3.2, '. . 51).     ,      .   K  2 (  '(u) ' 

B(Bu + u ; f) = 0: 9 8   ,  ( ' (u)  4  2 Bu + u = = f  5 '''. ;,' &    W20;1 O0 t] & '6     1  (' hu Bui , u 2 HO0 t] & ' &    H; O0 t]. * ' &'38 ,' 5 & j0 &    H0O0 t]  83' 1,':   u 2 H;O0 t] ,  j0 u 2 H0O0 t] . #  ' H0O0 t] ' kuk2B = hu Bui , (hu Bui > 0 , 8u 2 H; O0 t] u = 6 0) . *&  &    H0O0 t] & : ' 1,' HB O0 t] .
&'',  5  ' 4' , 8 4, '83 '' 8 ' O33].

' 4.3
        fh (t )g >0   3 -@ ! (4.1.9)   ! 0

6  (  *  !*   t )     HB O0 t] *        (4.1.7). >        3.3,   *  !*   t        

kC
h
(t)z ; C
h
0 (t)z k2;10 + kIm h
(t)z ; Im h
0 (t)z k20 ! 0   ! 0

0 0   h0(t )       (4.1.7),  Im = 0 Iq :

    ,      . ; ' (u) '

 7 1, '6 ,&    ' (u) = kBu ; f k20 + kuk2B: K ( , 5   '' 3.4,  5 '''  : ' u  2 283' 2 4'  2 Bu + u = f: <' 1,', &        ' (u )  ' (u0)   & 1:

kBu ; f k20 + ku k2B  ku0k2B 

(4:1:11)

5   > 0 , u0 | 5   4  2 (4.1.7),  & (' '  %, (4.1.11)  , 

Bu = f:

(4:1:12)

ku kB  ku0kB :

(4:1:13)

62

4 ,)(   ! 

<' 1,', '6   4  2 Bu +u = f (1,' 5 , D ) 5

HB O0 t]  , { 1 '& O24]. #  ' , D 818 1 7 238 2 & &      fu g >0 1 &   1,' uk0 : *6',  uk0 = u0  Buk0 = fk = f: * '

(4.1.11)  &   &  : lim fkBu ; f k20 + ku k2B g = kBuk0 ; f k20 = kfk ; f k20 = 0 !0    uk0 = u0  fk = f: %, 1 7 '  u  u0   ' D.!7 O24,25] & ': lim ku k  ku0kB   , :5      (4.1.13)  ,  ku ; u0 kB ! 0 &  ! 0 ,  !0 B   ' '   2 7 '  HB O0 t]: %,   7 '  &      fu g >0  u0 HB O0 t]   7 '  & 1  (' hu Bui HO0 t] ,  & 2   

'' 3.3, ' '  4 kC
(h (t) ; h0 (t))
z k2;10 + kIm (h (t) ; h0 (t))
z k20 ! 0 &  ! 0: <' &  8 ,. ' 4.4
     +  * fx^ ()g >0     *  (4.1.6), 6  '     t x^ (t) =

Z 0

D(h (t ))j  y() d

(4:1:14)

 h (t ) -      (4.1.9), D  j  - '  ( *  ,   ' W21 O0 t]  L2 O0 t]  ' W2;1O0 t]  L2 O0 t]     (.'  1.1).     ,      . %  ' ,  jM(z x(t) ; x^(t))2n ] ; (z x(t) ; x^ (t))2n ]j = jhh
0 (t)z Bh
0 (t)z i; ;hh
(t)z Bh
(t)z i + 2h(MOx(t)y
()])
z fh (t) ; h0 (t)g
z ij: 9'    &  7 5'7 &   : jhh
0 (t)z Bh
0 (t)z i ; hh
(t)z Bh
(t)z ij = j kh
0 (t)z k2B ; kh
(t)z kB j   (kh
0 (t)z kB + kh
(t)z kB )kh
0 (t)z ; h
(t)z kB  2kh
0 (t)z kB kh
0 (t)z ; h
(t)z kB ! 0 &  ! 0: (4:1:15) J    & ,      (4.1.13)  ' 4.3. 0 '' & 6 h(MOx(t)y
()])
z fh (t) ; h0 (t)g
z i = hKxy
(t)z fh (t) ; h0 ()g
z i = (^h) 5  ^h = fh (t ) ; h0 (t )g
z z 2 En . ; (^h) | & 6 ' (  ' t &    W2;1 O0 t] . #-(2
(t )z & 6 &, ' &    & &'  2 817 z 2 En :  2 K
(t )z Kxy
(t )z = Buxy. <5  '6  (8 u () 2 HB O0 t] ,'6     8, 8,  Kxy (^h) = hBu  ^hi = hu  B^hi . D&        p p jhu Bvij = jh Bu B vij2 = hu Buihv Bvi 8u v 2 HB O0 t]: % & ,2 : 42, 7 ' j(^h)j2  hu  Buih^h B(^h)i  ch^h B(^h)i ! 0 &  ! 0 (4:1:16) & ' 4.3, c | & 6 2  ,   1 2 (' hu  Bui u 2 HB O0 t] 5. %, 4 (4.1.15)  (4.1.16)   &     ' 4.4. 05 2,    #-F&( (4.1.9)      83 ,  ( .
  & x() '   2  ' (4.1.1).

63

4.1
 .! /(   $0


1 8  2 & fy () 0    tg 2, x() 4' y () = C()x() + v ()

(4:1:17)

5  v () |  & & 1 5 5  5 4'   '  '    MOv ()v
()] = S ()(

; ) S () =

 I

m;q

0



0 R() + Iq 

S () | ''2,        '. *  v () , u()    x0 &&   . <1 2 &   & 1 8 2' (4.1.17)  x^ (t) &  x(t)   2838 ''' (  m(t) = MO(z x(t) ; x^ (t))2n ] (4:1:18)

Rt

5  x^ (t) = h (t )y () d: %5 ,   &' 2   4 (1.1.11),      0 (4.1.1)  (4.1.18),   4, '  5 5 &   . 05 2, 2 ,  (  (4.1.1), (4.1.17), (4.1.18) 2 6 5 (  5  > 0 2 2 2 1 ,   '-!8 .  2 : ,  &    '     (' 3.3), 8 '6 ('    83' 1,'. ' 4.5 5( + *  *       (4.1.1) "  (4.1.17) -*  6   '*     H(t )        

d'(t ) = ;F
()'(t ) ; C
()H
(t ) (4:1:19) d  (    '(t )j =t = '(t t) = z z 2 En (4:1:20)

 -  H(t ) { '  !*+  l(H) = '
(t 0)P0'(t 0) +

Zt 0

H(t )S ()H
(t ) d +

Zt 0

'
(t )G()Q()G
(t)'(t ) d:

(4:1:21)

5*     H(t ) '     6  !*+ ! h (t )      H(t ) = ;z
h (t ):

(4:1:22)

,    : '  5 ,    ' 3.3. <6 &    ',    832 2, '6  4' '5  2 0  &' ' & ' H(t ) . + ', ' 4.6
 ( !*+ P ()        &** dP () = F()P() + P ()F
() + G()Q()G
() ; P()C
()S ;1 ()C()P() P (0) = P : (4:1:23) 0 d      H(t )  '( (4.1.19)-(4.1.21)          H(t ) = ;'
(t )P ()C
()S ;1 ():

(4:1:24)

,    '    & 2    ,   3.1.2.   &6',   x^ (t) '6 1 &      , 42 (4.1.14), 2  , 5 1 2 , '&   &7  ' h (t ) ,    283  8 #-F&( (4.1.19),   , 42 5 (( 5  2,  &3

   & .

64

4 ,)(   ! 

' 4.7 ,   + * x^(t) ,         (4.1.4),         +  * fx^ (t)g >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 <5  h (t ) = o(t )K ()  6 2  '6 ,&    x^ (t) =

Zt 0

h (t )y() d =

Zt 0

o(t )K ()y() d:

((2 ( 113' ' ) : 6 & t , &'22 (4.1.26)  ' 4.4, & ': d^x (t) = o(t t)K (t)y(t) + Z do(t ) K ()y() d = F(t)^x (t) + K (t)Oy(t) ; C(t)^x (t)] dt dt t

0

  7 ' (4.1.25). 4.2 . 1             ' 2 

   1"

+ 5' 42 ,   &'  (   ' 4'' ,'27 ,1 1977 5  #.*.    9.U. L' O12,3,25] . 1. *        ,     . D  n {' & x() 5 2 &' && 2 5 5 &  u() '&'   1 7 5  7 4' ( ,'6, 6 7 ) , 8  8  ' dx() = F ()x() + G()u() x(0) = x  0    t < 1: (4:2:1) 0 d
1 8  2   & fy() 0    tg ,'  m  n y() = C()x() + w() (4:2:2) 5  F () , G()  C() | '    837 ,' . K ' ' G()  C() | 5   (,  ' F () |  -& . 04 ,  4 &' 2 '  &  2 1.6 ( '. . 12),      (4.2.1) { '  5 5 &   . Q' w() & & 5 2    ',     : ' 5   ' Kw ( ) = MOw()w
()]  2 O0 t] & 6   &,  8 L2 O0 t] L2 O0 t],  '& 5  , 2 8 1 { &    5'7   '  O0 t] (. <1 2  &      n {'7 

Zt

fx^ (t) h (t )y() d g >0 0

(4:2:3)

65

4.2 1 ! /(   $0

'' h (t )  : ' 7 & 81'  > 0 & 6 &, ' &    W21O0 t],  8,  lim MO(z x(t) ; x^ (t))2n ] = m(t) 8z 2 En  (4:2:4) !0 5  m(t) = inf fMO(z x(t) ; x^(t))2 ] j x^(t) = n

h

Zt 0

h(t )y() d g

(4:2:5)

622 5 1 2 & ' '' h(t ) : '' , W21 O0 t] & &'  . Q' u() , w()    x0 '8          '6  1, &' MOu()u
()] = Q()( ; ) MOw()w
()] = Kw ( ) MOx0x
0 ] = P0 ' Q()  P0 | ''     &  , : ' ' Q() | 5   (. 2. + 5     '   4    2 ,    . *  f'i ()g1 i=1 | &,   5  1, L2 O0 t]  KN ( ) =

N X

ij =1

'i ()Cij 'j () = MN
()AN MN ()

5  Cij | ',  232 , :(( ; : ' ' Kw ( ) : MN
() = O'1()Im '2 ()Im    'N ()Im ] Im | m {'2  2 ', 2 C C  C 3 11 12 1N 6 7 C C    C 21 22 6 AN = 4           2N 75 : CN 1 CN 2    CNN





0
#  ' 1,2: W(t) = B(t) 0 AN  V () = OC()Z() MN ()] , Z() | ',  1  2 28 2  ,  '' 42' ' dx() = F()x() d B(t) = Z (0)P0fZ ;1

;1

(0)g
+

Zt 0

Z ;1 ()G()Q()G
()fZ ;1 ()g
d:

' 4.8 0        !*+ fx^ (t)g >0  (4.2.3),     =

= (N) "   *  N         '+, (       fx^ (t)g >0 *, (    (4.2.4). , *  ( N)  * -!*+ x^ (t) = x^ (N ) (t)          *   !!  +    d^x (t) = F (t)^x (t) + ;1OQ (t)C
(t) + Q (t)M (t)]Oy(t) ; V (t)v (t)] x^ (0) = 0: (4:2:6) 1 2 N dt 3 *  v (t)           *   !!  +    dv (t) = ;1OV (t)P1 (t) + V (t)W (t)]
Oy(t) ; V (t)v (t)] (4:2:7) dt v (0) = 0

(  dP1(t) = ;;1OV (t)P1 (t) + V (t)W(t)]
OV (t)P1(t) + V (t)W(t)] (4:2:8) dt P(0) = 0:

66

4 ,)(   ! 

;+ Q1 (t)  Q2(t)  " *  ( * +   Q2 (t)  P2 (t) = QQ
1(t) 2 (t) Q3 (t)     (  !!  +          dP2(t) = F (t) 0 P (t) + P (t) F
(t) 0 + G(t)Q(t)G
(t) 0 ; 2 0 0 2 0 0 0 0 dt ;;1P2 (t)OC(t) MN
(t)]
OC(t) MN
(t)]P2(t) (4:2:9) P 0  P2 (0) = 00 AN : <' 1,',  5' & 2  x^ (t) 5 &  x(t) & (  7   N &'  83  .   5 (  { : ((   2 (4.2.6)  (4.2.7).
,   :(( :7   7 ', 42 '  2 (4.2.8)  (4.2.9) & 0.
 :' , 8 2 &    2. *  :5 & &83 1 8 2 y(t)   & '  7   ' (4.2.7)  &  2'  v (t) . J' :   1 8 2 y(t) &  2' (4.2.6)  7 '  x^ (t) . * 1 1    5'  O12], O25, . 105-136].

5 &  %2). (3% !%"% %0+)%&

# 17 O16, 25, 3]  ,   4 ',  ,   &'  (  (,  & 2 &'  '  '''      41   5 & ) &    5 4' 1 8 27  &  .  2  42 ,1 5 2,83  5',  &    5 2, ' (   '-!8 . + 5'       5 2,  2 #-F&(,  ', '   2 Bu = f (5:0:1)  '  2 5 2,    #-F&( Bu + u = f  > 0 (5:0:2) 5   - &' 5 2,. 04 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  (,     ,  42  2 (5.0.2)  1 2, & :' 1 7 '  fu (t )g >0 ! u0(t ) &  ! 0   2      7 '  5 2, 7   48 ,  (  (' 4.4, '. . 62). 5.1      1          

# :' ,   &   2 '  7 '    &     42 , 

 &'  (   &1 65 , 2  7 7 7 &  & ,  5 2,835  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 MOx0x
0 ] = P0  P0 = P0
 P0  0 MOu()x
0 ] = 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()x
0 ] = 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] . <1 2 & 1 8 2' fy() 0    tg &  &       fx^" ()g >0 &  x()   = t ,    2838 48 lim lim MO(z x(t) ; x^" (t))2n ] = m(t) (5:1:4) !0 "!0 5 

Zt

m(t) = inf fMO(z x(t) ; h(t )y() d)2n ] h
(t )z 2 W21 O0 t]g h 0

z | &,   : ' , En , h(t ) | '&  2 &7 2 ', 5 &  '

6 &' 2 '  (1.1.11). ' 5.1
     +  * fx^" ()g >0  *  6    (5.1.4),    "    '   x^" () = F ()^x () + P ()C
()S ;1 ()Oy() ; C ()^x()] x^ (0) = 0 " " " " " " " d  P"()      (    &** dP"() = P ()F
() + F ()P () + G ()Q ()G
() ; P ()C ()S ;1 ()C
()P () P (0) = P  " " " " " " " " " 0" " " " " d  I  0 m ;q S" () = 0 Iq + R"()  Im;q , Iq | ( + '  (m ; q) (m ; q)  q q     ,  > 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

04  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). <5  B^u + ^u = f" ; f + (B ; B" )u"  ^u = u" ; u  4 :5  2   2 ''' (  L(u) = kBu ; ff" ; f + (B ; B" )u" gk20 + hu Bui u 2 W20;1 O0 t]: D&        L(^u )  L(0) , 5  ,  h^u  B^u i  kf" ; f + (B ; B" )u" k20  2(kf" ; f k20 + kB ; B" k2 ku" k2;10) (5:1:16) 5  ku" k;10 | ' W20;1 O0 t],  k  k | &2 ': kBk = supfhu Buij u 2 W20;1 O0 t] kuk;10 = 1g: u

9' ' kf" ; f k0  kB ; B" k . < 
(t ) ; C()OKx"(t ) ; Kx (t )]
gz f" (t ) ; f(t ) = fOC"() ; C()]Kx"  kC"() ; C()k < "  kf" ; f k0  "kKx" k + (" + kC"k)kKx" ; Kx kkz kn

0 Zt 11=2 5  k  k = @ j  j2 d A   j  j2 { ''    : ' '.

(5:1:17)

0

* ('  4 , (5.1.1) '':

Rt 0 t R x"(t) = \" (t 0)x0" + \"(t )G"()u" () d x(t) = \(t 0)x0 + \(t )G()u() d

(5:1:18)

0

&'

d\(t ) = F (t)\(t ) \( ) = I dt (5:1:19) d\" (t ) = F"(t)\" (t ) \"( ) = I dt I |  2 '. % & ,2 (5.1.18), (5.1.19), 7 ' 6 2 ,   7 ' 7 &  x()  x" () : Kx" ( ) ; Kx ( ) = \" ( 0)P0"\
" ( 0) ; \( 0)P0 \( 0)+ +

Z

min() 0

\" ( s)G" (s)Q" (s)G
" (s)\
" ( s) ds ;

Z

min() 0

\( s)G(s)Q(s)G
(s)\
( s) ds:

*'22     4-!2 5  4 &  ' 68, &   67 &1,  & ' kKx" ; Kx k  j\"(t 0) ; \(t 0)j jP0"j j\"(t 0)j + j\"(t 0)jf"j\"(t 0)j + jP0j j\"(t 0) ; \(t 0)jg+

5.1 2     /





71

+k\" ; \kkG"Q"G
" kk\"k + k\" k "kQ" G
" kk\"k + kGkO"kG"kk\"k + kGkk\" ; \k] : (5:1:20)   ' ' ,  k\"(t ) ; \(t )k: #  ' 1,2 o(t ) = \" ( ) ; \( ): D&      do( ) = F()o( ) + fF"() ; F ()g\"( ) (5:1:21) d o( ) = 0 o( ) = 0 &  < : '6' (5.1.21)  &,      z 2 En  1,' v( ) = o( )z , g( ) = \" ( )z 5  (5.1.21) &'  dv( ) = F()v( ) + fF () ; F()gg( ) " d v( ) = 0 v( ) = 0 &  < : D&      dv     kF vk;1t + k(F" ; F )gk;1t: d ;1t

#,  '    1   :  ,  & ,2 ,       j2abj  a2 + b2  & '

 dv 2    2(kFvk2;1t + k(F" ; F)gk2;1t) d ;1t

d: Dt &    L2O0 t]     W2;t 1 O0 t], &' kDt vk;1t = kvk0 2 817 5  & d v 2 L2 O0 t] O25]. <  & & 8 : ' ' \" ( ) \( )  v( ) & 6 & 1' 5'' L2 O0 t] ,  kvk20  2(kFvk2;1t + k(F" ; F)gk2;1t) (5:1:22)  ' kFvk2;1t '6   83' 1,'

kFvk2

;1t

 1

Zt Zt 0 0

jv( s)j2 dsd

Zt

1 = tlim jF(s)j2 ds: !1 0

(5:1:23)


    (5.1.23)   , &  2 ' 5 ' &   

Zt

kFvk;1t = sup fj y
(s)F(s)v(s ) dsj y 2 W21tO0 t] kyk1t = 1g y 0

 4

Zt

Zt

0

0

Zt

j y
(s)F(s)v(s ) dsj = j y
(s) dsd F ()v( ) ddsj = 0

0Zt Zs 11=2 @ j F()v( ) d j2 dsA 

 dy  = j; F()v( ) ddsj   ds  ds 0 0 0 0 0 0Zt Zs 0Zt 2Zs 3 11=2 11=2 Zs p  kyk1t @ 4 jF ()j2d jv( )j2 d 5 dsA  c1kyk1t @ jv( )j2 ddsA : Zt dy
(s) Zs

0

0

0

<5  (5.1.22), ' (5.1.23), ,&4 2  

kvk20  21

Z t Zs 0 0

0 0

jv( )j2 dds + 2k(F" ; F)gk2;1t:

(5:1:24)

72

5 2     /       )( 2

 2     (5.1.24) &    '' /

:   %(t)  (t)   (, 2 7 & 2 2    

Zt

%(t)  c %(s) ds + (t) c = const c > 0 0

 ' '      %(t)  ect(t) . <' 1,', &    42

kvk20  22ctk(F" ; F)gk2;1t  2"2kgk2;1t2ct 

 2 817 z 2 En  ,,  

Zt

kF" ; F k2  2"2 2ctk\" k2 = "2 ! 0 & " ! 0

5  c = 4(" + tlim jF"()j2 d)): %, :5       kQk  "+ +kQ" k kGk  " + kG"k !1 0  ,  kKx" ; Kx k  " jP0"j k\"k + (" + k\" k)f"k\"k + (" + jP0"j)" g+  h +(" + k\" k) "kQ" G
" kk\"k + (" + kG"k) "kG"kk\" k+

i

+(" + kQ" k)O"k\"k + (" + kG"k)" ] = "x ! 0 & " ! 0 ' (5.1.25)  (5.1.17) '6 ,&   

(5:1:25)

kf" ; f k  "kKx" k + (" + kC" k)"x = "f ! 0 & " ! 0: 9' ' kB" ; Bk . 91,': ^Su = OS" () ; S()]u() ,

(5:1:26)









0  S" () = 00 R 0()  S() = 00 R() "

Zt


( )C
() ; C()K
( )C
()]u() d ^Bu = OC"()Kx" " x 0

 &  '  ' k^S k  k^Bk: * &  8 & '  0j  u 2 L O0 t] kuk 6= 0 : (5:1:27) k^S k = sup j(uk^Su) 2 0 uk20 u *'22     4-!2 5    8 (5.1.27)   2 (5.1.3), & '

j(u ^Su)0j  k^S kkuk20  "kuk20  8u 2 L2 O0 t]  + 5

k^S k = kS" ; S k  ":

(5:1:28)

j(u Bu)0j  kC"Kx"
C"
; CKx
C
kkuk20 = k(C" ; C)Kx"
C"
+C(Kx" ; Kx )
C
; "+CKx"
(C" ; C)
kuk20   f"kKx"
C"
k + (" + kC"k)O"kC"k + (" + kKx" k)"]gkuk20 8u 2 L2 O0 t]  8 

k^B k  "kKx"
C"
k + (" + kC" k)O" + kKx"
k)"]:

(5:1:29)

73

5.1 2     /

*    (5.1.29)  & ,        (5.1.3), (5.1.25)  42 kC k  "+kC" k , kKx k  "+kKx" k . %,    (5.1.28)  (5.1.29)  ,  kB" ; Bk  "(1 + kKx"
C"
k) + (" + kC"k)O"xkC" k + ("x + kKx" k)"] = "B ! 0 & " ! 0: (5:1:30) *  ' (5.1.30)  (5.1.26) (5.1.16). <5  hu ; u"  B(u ; u" )i  2("2f + "2B ku" k2;10) 

MO(z x^ () ; x^" ())2n ] = hu ; u"  B(u ; u" )i  2 ("2f + "2B ku" k2;10) ! 0 & " ! 0: (5:1:31) %, (5.1.31)  (5.1.12)   (5.1.9).   ' &  5' &        (5.1.12). 1 '  ' 6 MO(z x(t) ; x^ (t))2n ]  x^(t) ,    2838 48

Zt  2
(t )z 2 W 1O0 t] 8z 2 En : m(t) = inf MO(z x(t) ; x ^ (t)) j x ^ (t) = h(t )y() d h n 2 h 0

<5 , &   67 &1, , '' MO(z x(t) ; x^ (t))2n ]  m(t) + MO(z x^(t) ; x^ (t))2n ] + 2fm(t)MO(z x^(t) ; x^ (t))2n ]g1=2:

Rt

(5:1:32)


 '  62 MO(z x^(t) ; x^ (t))2n ]: <    x^(t) = h(t )y() d 

Rt x^ (t) = h (t )y() d (5  &'8 2  6,   (1.1.11)),  0

MO(z x^(t) ; x^

(t))2 ] = n

Zt Zt 0 0

0

z
Oh(t ) ; h (t )]Ky ( )Oh(t ) ; h (t )]
z dd =

= hu0 ; u  B(u0 ; u )i (5:1:33)
5  u0 | 4 (5.0.1), u | 4 (5.1.15), Ky ( ) = MOy()y ()]:  2  1  (' (5.1.33) ' (5.0.1) , (5.0.2). # ,  & '   B(u0 ; u ) + (u0 ; u ) = u0  (5:1:34) 5  u0() & 6,  6 '   , W20;1 O0 t]      (5.1.34) &' 2 '  5 5 &   . *'' & B  68 (5.1.34)  1,' , v , & B  : ' u0 ; u     v = B(u0 ; u ): <5  '',  Bv + v = f: 04 :5  2   2 ''' (  J(v) = kBv ; f k20 + (v Bv)0  &        J(v )  J(0)    kBv ; f k20 + (v  Bv )0  2kf k20   

(5:1:35) (v  Bv )0  kf k20 : *  & 28 2   2 '' 3.2. <5  2 8v 2 L2 O0 t] &        (v Bv)0   c0 kvk20  4 (5.1.35) &'  : pc0kv k0  pkf k0 :  2 ,  v = B(u0 ; u )

74

5 2     /       )( 2

p & ' 6: pc0 kB(u0 ; u )k0  kf k0 : <  B | 5, & 6  &   & ( '. '' 3.1  3.2),  hu0 ; u  B(u0 ; u )i   c1 kf k20  c0kKx k2kC k2kz k2n c0 > 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) 62 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. G5 &,,  &        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,   1135 &&  2, O33,20,21], 4 ,   &'  (  1  ,    2  5 42 O33]. # 17 O33,22] 2 1 &' 5 2, & 58 2  & 1,   ,  2 2 '    41 , 2  7 7 7 {   ,&'     42. # 13'  :  ,   (&   1 4'   &' ),   5 , 2 & 2  51 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( &  42 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     42.  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 )dC
(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

< 

@Kx (t ) = @ MOx(t)x
()] = MO dx(t) x
()] @t @t dt &,  2 &  ' 6 &' 2 113' '  (.. '  5 5 &   ), , &  22 : 6 '  dx(t) dt 5 , , (5.1.1)   2     &  x()  u(t) & t >  7 ',  @Kx (t ) = F(t)K (t ): x @t <5  6 (5.2.9) '6 ,&   83' 1,'

Zt  @h (t ) 0

@t



; OF(t) ; h (t t)C(t)]h (t ) C()Kx ( )C
() d+

  + @h @t(t ) ; OF(t) ; h (t t)C(t)]h (t ) S ;1 () = 0: K     & 2 2   &   ,  @h (t ) = OF(t) ; h (t t)C(t)]h (t ) @t &' h (t t) = P (t)C
(t)S ;1 (t) 5  P (t) 4  2 0 (5.2.7) ( '. ' 4.7  ' 4.5). D42 (5.2.5)  (5.2.7) ,. ((' &    (5.2.8), & ' Zt @h (t ) 0


() d + @h (t ) S () + h (t ) = 0: C()K ( )C x @ @

(5:2:10)

9&  ' ((    2  2   (t ) .  2 :5 & ((' (5.2.10) & t: <5  v (t ) = @h @ v (t t)C(t)Kx (t )C
() +

Zt @v (t ) 0

@t

(t ) S () + @h (t ) = 0: (5:2:11) C()Kx ( )C
() d + @v @t @t

J'' & ' 5'' 6 Kx (t )C
() 5   (5.2.8).  2 42 (5.2.5)  (5.2.10), &  7 &1,  & '    5   

Zt h @v (t ) 0

@t

h

i

+ v (t t)C(t)h (t ) ; fF(t) ; K(t)C(t)gv (t ) C()Kx ( )C
() d+

i

(t ) + v (t t)C(t)h (t ) ; fF(t) ; K(t)C(t)gv (t ) S () = 0 + @v @t  ' 4    4,   ' S () > 0 ,  ' C()Kx ( )C
() |   2  5 5 &. <' 1,', @v (t ) = OF(t) ; K(t)C(t)]v (t ) ; v (t t)C(t)h (t ): @t

77

5.2
     )( 2

D4 (5.2.3) ,,  ' ,   v (t t) .  2 &  2 v (t t) ((' &  6 (5.2.6) @h (t t) = v (t t) = @P (t) C
(t)S ;1 (t) + P (t)C
(t) @S ;1 (t) : (5:2:12) @ @ @ " P (t)    2832  8 0 (5.2.7), '6 1 &   '   41  2 ,   &'  ( ,  :   22 x() (' 2  ' (5.1.1),  1 8  2 & y () , 2, x() 4' p (5:2:13) y() = C()x() + w() + () 5  () | m {' 1  5   4'   '  ',   MO()
()] = Im ( ;)     & ' u() w()  x0: 941  2 ,  (5.1.1), (5.2.13) &  2 2 6' x" (t) = x(t) ; x^ y (t) 5  x^ y (t) =  '   41  2

Zt 0

(5:2:14)

Dh (t )j  y () d

(5:2:15)

P (t) = MO(x(t) ; x^ y (t))(x(t) ; x^ y (t))
]: ((' P (t) &  :

  
  @P (t) = M ; @^x y (t) (x(t) ; x^ (t))
+ M (x(t) ; x^ (t)) ; @^x y (t)
: y y @ @ @ y (t) . %, (5.2.15) # ' &,  8 @^x@ @^x y (t) = Z Dv (t )j  y () d + (2p);1 Z Dh (t )j  () d: @ t

t

0

0

(5:2:16)

(5:2:17)

0 '' &  5' &    (5.2.16). *  22 5 (5.2.17), ''

 @^x (t)  Zt y
M ; (x(t) ; x^ y (t)) = Dv (t )j  MOy ()x
(t)] d+ @

0

Zt Zt p ; 1  +(2 ) Dh (t )j MO()^x y ()] d ; Dv (t )j  MOy ()^x
y (t)] d ; 0

Zt

0

;(2p);1 Dh (t )j  MO()^x
y(t)] d: 0

(5:2:18)

<   &  ()  x()   ,   5' (5.2.18) 13 2

0.  , 5   (5.2.13), MOy ()x
(t)] = C()Kx ( t) MOy ()^x
y (t)] =

Zt 0

MOy ()y
()]h
(t ) d = S ()h
(t )+

78

5 2     /       )( 2

Zt

+ C()Kx ( )C
()h
(t ) d = C()Kx (t ): 0

*         , (5.2.8). *  ' : 62 (5.2.18). <5  M

 @^x y(t) @

(x(t) ; x^ y (t))




p Z = (2 );1 Dh (t )j  MO()^x
y (t)] d: t

0

% & ,2 (5.2.15)  (5.2.13), 7 ', 

Zt

Zt

0

0

MO()^x
y(t)]= MO()x
()]C
()h
(t ) d + MO()w
()]h
(t ) d+

p Z p +  MO()
()]h
(t ) d = h
(t ) t

0

,     2     &  () , x() , w() ,  6 (      {( . <' 1,',  1 Zt  @^x (t) y
M @ (x(t) ; x^ y (t)) = 2 h (t )h
(t ) d: 0

*  22 : 6 (5.2.16), & '

@P (t) = 1 Z h (t )h
(t ) d @ 2 t

0

  '' v (t t) = <' ,.

Zt 0

;1 (t) : h (t )h
(t ) d C
(t)S ;1 (t) + P (t)C
(t) @S@

6 5   !"# $#%&  "*  %!%(" !%"%"

0,  & 23 & ' 42 ,   (   & 8  5' &  2  &1 65 42 2  ', &  '7  2' &1  5 & &    5 4' ,'27. #  ' 5   '7  &  ' &'' ,'   0.  ' O14], +.!   O37]  ..
 1 4   1   5' &   2 & (',  ,2 5  K#" 2 2 2    6 &1 '. ' 5,   1 8 27 &      4',  ,   (  2 2 2  &    2  42 17 ' &'2 5 2,8. 9  , & 7 1, & 237 &  76 2 &1 6    22  ' & 2 :

& ' &', 2 2 2 O25],  , 7 '   5' 42  1 7   7 4' 1 8 27. *1 ' 42 ,  (  2  ', &  '7  2'  7 &,  7

  27 &   , & 23 1 O3,16,17]. 6.1     (     %1     2  

# ' ,      2 ,  42 (( 7   &1  5 & 5'   2'
' , 5  & 2   & 6 &    1137 (. * 7  48 7 ,  ,1 17 O3,25,16]. *      ' &    En ,  ,'2 52 1   P 5 @P = ;

6    3      2 '  nk0  C 2 (P) | '6   6  (('7    ' '  ( u(x)  P  C12(P) | '6   ( u(x) 2 C 2(P ) 2 7 & 2 2    du(x) = 0 (6:1:1) d x2;

2 % !231=2 n n n P P P 5   | '  (&2'8 & 283'   ' i = a1 aij nj  5  a = 4 aij nk 5  j =1 i=1 j =1 nj |   '   &7 238 , 6 8  5 ; , 8 ' 8). D&    4 O31] n X

n X d cos(kn0  xj )aij (x) @x@ = a cos( xi) @x@ a d j i i=1 ij =1

d d | &,  2 & ' ,

2 0 1231=2 n n X X a = 64 @ aij (x) cos(kn0  xj )A 75 : i=1 j =1

80

6 4         

#  ' 1,2: L2 (P ) | 5 1  &    5'7   ' ( '  G15  P  (: :)0P  k : k0P |  2 &,    ' L2 (P )  W21(P) | &,  1   &    Z X n @u @v  + u(x)v(x) dP (u v)1P = @x @x P

i=1

p

i

i

   2 &,   W21(P)  kuk1P = (u u)1P | ' W21 (P ): D&        kuk0P  kkuk1P  8u 2 W21 (P ) k = const: 0 ''  P  ((  6 5 &2  n @ X @u(x) Nu = ; @x (aij (x) @x ) + c(x)u(x) ij =1

(6:1:2)

5  aij (x) = aji(x) 2 C 2(P) i j = 1 ::: n c(x) 2 C(P) C(P) | &    & 7 (  P c(x)  c0 2 8x 2 P c0 = inf fc(x) j x 2 P g > 0: x * & 5',  &    6     n X

ij =1

aij (x)i j  

n X i=1

i2 

(6:1:3)

5   | & 6 2  , i | &,   3     ( i = 1 ::: n). D42 (6.1.1)-(6.1.3) , 8 &    L2 (P) ,'  & B ,  Bu = Nu 2 8u 2 C12(P): 9& B ' & 8 L2 (P ) 1   &  2 D(B) 2 2 2 '' '  & 6 -&  ', .. &    42 (Bu v)0P = (u Bv)0P  2 8u v 2 D(B) (Bu u)0P  c k u k20P  2 8u 2 D(B) c = const > 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 ) . <5  6 (f u)0P  u 2 HB+ &  2 &  (  HB+ (&     , (6.1.4))  j(f u)0P j = jlf (u)j  kf k0P kuk0P  ckf k0P kuk+  f 2 L2 (P) u 2 HB+ (HB+  L2 (P)) k : k+ | ' HB+ . *& ' L2 (P) & '

  j(f u)0P j +  f 2 L kf k; = sup 2 (P ) u 2 HB  kuk+ 6= 0 : kuk+ u *  &&  ,  &    HB; . 9& B &     , HB+ HB; . *  O0 t] |  ,  &'2  2 O0 t]: 9&  ' 1   Q = P O0 t] L2 (Q) | &    ( u( x)  Q 16837 5' O0 t] &    En  7, 

Zt 0

k : k0Q | ' L2 (Q) .

ku()k20P d = k u k20Q < 1

0 ''  &    L2 (Q) ((  & @u + Bu u 2 D(L ) Lu @ 1

81

6.1  %     " /

5  D(L1 ) | '6   ( u( x) , 7 1   Q &  (('7 &  2 O0 t]  & &' x 2 P ( u( x) 2 HB+     2837   2' x) = 0: (6:1:5) u( x)j =0 = 0 du( d x2; #  ' 1,2: W201 (Q) | &&  '6   D(L1 ) & ' Z @u (6:1:6) kuk10Q = ( O( @ )2 + uBu]dQ)1=2 Q

k : k10Q (: :)10Q | '   2 &,   &, ' &    W201 (Q)  W20;1(Q) |

5  &   , &  && ' L2 (Q) & '  j(u v)0Qj  1 kvk;10Q = sup kuk10Q  v 2 L2 (Q) u 2 W20(Q) kuk10Q 6= 0 : u 91,' , D(L1 ) '6   ( u( x) '837 72 1   &,  8    ' '  &   & &' x ( u( x) 2 HB+     2837   2' x) = 0: (6:1:7) u( x)j =t = 0 du( d x2; W21t(Q) | &&  '6   D(L1 ) & ' (6.1.6). *  W2;t 1 (Q) 5  &    2 W21t(Q)  L1 | &26 &  L1   L1 u ; @u @ + Bu u 2 D(L1 ): 0 4' & L1  L1   &    W201 (Q)  W21t(Q)    . 0 4 & 1 ' 1, L  L :   6.1 D" "      '( Lu = f f 2 L2 (Q) (6:1:8)

 f | ' !*+, ' !*+ u( x) 2 W201 (Q)  *           !*+ fui ( x)g1 i=1 '    D(L1 )         kLui ; f k;1tQ ! 0 kui ; uk10Q ! 0  i ! 1:   6.2 D" "      '( Lv = g g 2 L2 (Q) (6:1:9)  g | ' !*+, ' !*+ v( x) 2 W21t (Q)  *            fvi ( x)g1 i=1 !*+ '    D(L1 )         kL v ; gk;10Q ! 0 kvi ; vk10Q ! 0 i ! 1:  6.1 , (6.1.8)  (6.1.9)       kLuk;1tQ  c1kuk10Q u 2 W201 (Q) c = const c > 0 (6:1:10) (6:1:11) kLvk;10Q  c2 kvk1tQ v 2 W21t(Q) c = const c > 0:     ,      . *  u( x) | 5  2 (2 , &, 5 &    W201 (Q)  v( x) 2 W21t(Q) . <5  &        (v Lu)0Q = (Lv u)0Q . <6 ' '  4

p

j(u Bv)0P j  (u Bu)0P (v Bv)0P :

82

6 4         

0 '' 6

 Z
@u( x) j(v Lu)0Qj = @ v( x) + OBu( x)]v( x) dQ = Q

 Z
@v( x) = ; u( x) @ + u( x)Bv( x) dQ  ckuk10Qkvk1tQ  Q

5  c = const c > 0   & ,      kuk0Q  c kuk10Q 2 817 ( u 2 W201 (Q): * &  8 ' 5 ' &     j(v Lu) j 0Q  kvk kLuk;1tQ = sup 1tQ 6= 0  c1kuk0Q  ckuk10Q kvk1tQ v u 2 W201 (Q) v 2 W21t(Q):  2 &,   ( u( x) 2 W201 (Q)     & ' &  ' &7 '. ,        (6.1.11)  5.  6.2 , '( (6.1.8)  (6.1.9)       c1 kuk0Q  kLuk;1tQ c1 = const > 0

(6:1:12)

(6:1:13)

c2kvk0Q  kL vk;10Q c2 = const > 0: (6:1:14)     ,      . #  ' & Jt L2 (Q)   (27 v( x) 2 W201 (Q) &  2 2 6'

Z

Jt v( x) b;1 (s)v(s x) ds = u( x): t



@u = 0 .. u 2 W21t(Q) . <' 1,', & Jt ;2 u( x) = Jt v &  = t   0  @ x2; &    W201 (Q)     W21t (Q)  b() = ;(1 + ) . 0 '' 6 Z @v Z @ Z @u @u ) dQ: (Jt v Lv)0Q = (u Lv)0Q = u( @ + Bv) dQ = u @ Ob() @ ] dQ + uB(b() @ (6:1:15) Q

Q

Q

J   ,  v( x) = b() @u @ . 9' 5  &   . *  5 '6 ,& 

  Z @ Z @ @u ] dQ ; Z b() @u 2 dQ: ] dQ = Oub() u @ Ob() @u @ @ @ @ Q

Q

Q

D5   ('  9 5 5 Z @ @u( x) ] dQ = Z u( x)b() @u( x) cos(kn  ) dS = 0 Ou( x)b() 0 @ @ @ Q

S

5  S = @Q | 5 1   Q . 0     8   , 5,  cos(kn0 )     2 4  & 7  P  P + T 5   |   ' f x1 = 0 x2 = 0 ::: xn = 0g .
,   v(0 x) 2 P  u( x) 2 P + T  v(0 x) = 0  u( x) = 0 . <' 1,', Z @ Ob() @u ] dQ = ; Z b() @u 2dQ: u( x) @ (6:1:16) @ @ Q

Q

83

6.1  %     " /

*1,'  5 &    (6.1.15). < 

Z Z Z @ @u db() Bu( x) dQ Ou( x)b() B u( x)] dQ = 2 u( x)b() B dQ + u( x) @ @ d Q

Q



Z Q

Q

Z @ Z @u 1 1 u( x)BOb() @ ] dQ = 2 @ Ou( x)b()Bu( x)] dQ ; 2 u( x) db() d Bu( x) dQ: Q

Q

% & ,2 ('  9 5 5, & ' Z @ Z Z Ou( x)b() B u( x)] dQ = u( x)b() B u( x) cos(k n  ) dS = ; b(0) u(0 x)Bu(0 x) dP 0 @ Q

S

P

(6:1:17)

(6:1:18)

( u(t x) = 0  cos(kn0  ) 6= 0 4  '6  7 P  P + T ). *   (6.1.16)-(6.1.18) (6.1.15), & '

Z   1 b(0) Z u(0 x)Bu(0 x) dP ; db() u( x)Bu( x) dQ: dQ ; (Jt v Lv)0Q = b() @u @ 2 d P

Q

% & ,2 2   & Jt  7 ' Z h @u 2 Z  @u 2 Z i Z  @u 2 1 1 1 (Jt v Lv)0Q = 2 @ + uBu dQ +  @ dQ + 2 @ dQ + 2 u(0 x)Bu(0 x) dP Q

Q

   6

Z Q

Q

P

 2 1 Z u(0 x)Bu(0 x) dP  0 ( + 12 ) @u dQ + @ 2 P

 &        (Jt v Lv)0Q  21 kuk21tQ . *' 1135     4-!2 5 ,  1 kuk2  (J v Lv) = (u Lu)  kuk kLvk  t 0Q 0Q 1tQ ;1tQ 2 1tQ kLvk;1tQ  12 kuk1tQ: #   & ,2 2   & Jt  & '

kuk21tQ =

Z

Z h @u 2

Q

@

i

+ uBu dQ =

Z

Z h @ Z

Q

i2

@ f;(1 + s)gv(s x) dQ+ t

+ u( x)Bu( x) dQ  O;(1 + )v( x)]2 dQ  kvk20Q : Q

Q

(6:1:19)

 8  kLvk;1tQ  c1kvk0Q .  2 &,  7 v 2 L2 (Q)     & ' &  ' &7 '.
    (6.1.14) '6 ,  5.

' 6.1 >    + * (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)    . <5  l() hu i0Q = hu Lvi = (u Lv)0Q = (Lu v)0Q = (f v)0Q : ; u( x)    283   2' (6.1.5), &  W201 (Q): <' 1,', 3   &      fui( x)g1 i=1 5  7 (,    2837 (6.1.5), 2,  kui ; uk10Q ! 0 i ! 1:  ,  & ,2 '' 6.1, 7 ',  kLui ; f k;1tQ ! 0 i ! 1: D3    42  5      ,. D3          42 ,  (6.1.9) , 8 2  5.   6.3 D" "      '( (6.1.8)  f 2 W2;t1(Q) ' !*+ 1 u( x) 2 L2 (Q) *, (           *6 !*+ fui( x)gi=1  u 2 W201 (Q)    6     kLui ; f k;1tQ ! 0 kui ; uk0Q ! 0 i ! 1:   6.4 D" "      '( (6.1.9)   ( ' W20;1(Q) ' !*+ v( x) 2 L2 (Q) *, (           !*+ fvi ( x)g1  v 2 W21t (Q) i i=1     6     kL v ; gk;10Q ! 0 kvi ; vk0Q ! 0 i ! 1:

' 6.2 >    + * (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 48 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    1135 42 ,  (6.1.8) &   8 , W2;t 1(Q) '  &  2 6.3. U       ,     (6.1.12). ,       '  5. 91' 2 &  &1 6' 48   , . 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 ()(Bs  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

(Bk  1)0P    (Bk  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   42 (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: <5 

P

Z

P +T

t(Buk (t x))uk (t x) d(P + T )

k () ) = 1 (Bu  u ) + 1 t(Bu  u (t x))  1 (Bu  u ) : (Buk  (2t ; ) @u@ 0Q 2 k k 0Q 2 k k 0Q 2 k k 0Q

86

6 4         

<' 1,',

k () )  1 (Bu  u ) + ( @uk ()  (2t ; ) @uk () )  (f (2t ; )) @u@ 0Q 2 k k 0Q @ @ 0Q

 21 (Buk  uk)0Q + c( @u@k ()  @u@k () )0Q  ckuk k210Q: J   ,  (2t ; )  1 0 < c  1 c |  .

*'22 113     4-!2 5     , 7 ' k () k  kf k k @uk () k  ckuk k210Q  kf k0Q (2t ; ) @u@ 0Q 0Q @ 0Q    &    4

0 12 2 k u k ku k4 k 10 Q kf k20Q  @c @u () A  @u () k 10Q = c2 kuk k210Q: k k 2 k k0Q k k0Q + (Buk  uk )0Q

@ @ *       ,   (Buk  uk)0Q  0: <' 1,', kf k0Q  ckuk k10Q   1    ,. ' 6.3
 fi (x)g1i=1 |        *6 !*+  L2 (P)  d (x) i = 0  yi () |    '( (6.1.23).  *, ( dx x2@P uk ( x) =

k X i=1

yi ()i (x) k = 1 2 :::

'        6.1    '( (6.1.8),         kuk ; uk10Q ! 0 kLuk ; f k;1tQ ! 0 k ! 1: 1     ,      . D5   '' 6.3 '6   ( fuk g1 ,k=1 5 W20(Q) 1 1 , 1 '& W20 (Q)  '6    1 7 238 2 & &      fukn gkn =1

W201 (Q): <5    &   W201 (Q) & &      fukn g1 kn=1 1 7  2  ' &   uk 2 W201 (Q) . *6',  uk | 4 ,  (6.1.8). '6' (6.1.21)  (8 '() 2 W201 (Q)  &  5 2  0  t & ' (Lukn  '()j (x))0Q = (f '()j (x))0Q  j = 1 ::: k: (6:1:24) <  kukn ; ukk10Q ! 0 k ! 1  5       (6.1.10) kLukn ; Lukm k;1tQ  kukn ; ukk10Q ! 0 n ! m: ;1 ;1 *      fLukn g1 kn =1 ( '  W2t (Q)  ' &  W2t (Q) . 91,' : ; 1 &  Luk: ;2 '()j (x) & 6 W2t (Q)  (6.1.24) '6 &' '  1  (', .. (6.1.24) &   ,   Lukn 2 W2;t 1 (Q) . * '  &   & k ! 1 (6.1.24)  ' &    25 &,  2 L2 (Q) & ' hLuk '()j (x)itQ = hf '()j (x)i0Q  j = 1 ::: k: K 6 '6 ,&    hLuk ; f '()j (x)itQ = 0: (6:1:25) #   &   '()j (x) 7 ' Luk ; f = 0 .. uk | 4  2 (6.1.8),   1    ,. D   ',  (6.1.25) &     ,   f 2 W2;t 1 (Q) . ' 6.4
 fig 1i=1 |        *6 !*+  L2(P) (  i(x) = 0  * -!*+ f( x)     W2;t 1(Q)  i (x) 2 W21(P)  d'dx x2@P ff"g">0 |         !*+ f !*+ yi" () |    '( k dyi" () + X yj"()(Bj  i )0P = (f"  i)0P  yi" (0) = 0 i = 1 ::: k: d j =1

87

6.1  %     " /

        kuk" ; uk0Q ! 0 kLuk" ; f k;1tQ ! 0 k ! 1 " ! 0  uk"( x) =

Pk y () (x) kf ; f k ! 0 " ! 0: i" i " ;1tQ

i=1

    ,      . J&4' 4 (6.1.21) 2 ( f  f"  @uk"  @  j 0P + (Buk" j )0P = (f"  j )0P   @uk  @  j 0P + (Buk  j )0P = (f j )0P : *  2 & 5 , 5 & '  @(uk ; uk")   j 0P + (B(uk ; uk") j )0P = (f ; f"  j )0P : @

(6:1:26)

R

'62 (1.1.26)  & Jt (yj" ; yj ) = O;(1 + s);1 ]Oyj"(s) ; yj (s)]ds  ''2 & j (j = 1 ::: k) t 7 '   @(uk ; uk")  J (u ; u ) +(B(uk ;uk") Jt (uk ;uk"))0Q = (f ;f"  Jt(uk ;uk"))0Q = (f ;f"  Jt(uk ;uk")0Q : t k k" @ 0Q 9& Jt &   : ' , L2 (Q) &    W21t(Q) &:' &        h @(uk@; uk")  Jt(uk ; uk")i0Q + hB(uk ; uk") Jt (uk ; uk")i0Q = hf ; f"  Jt(uk ; uk")i0Q    hL(uk ; uk") Jt(uk ; uk")itQ = hf ; f"  Jt(uk ; uk")itQ : D5   (6.1.26) hL(uk ; uk") Jt(uk ; uk")itQ  ckJt (uk ; uk")k21tQ: # 6     hf ; f"  Jt(uk ; uk")itQ  ckJt (uk ; uk")k21tQ  , &'22 113     4-!2 5, 7 ' kf ; f" k;1tQ  ckJt(uk ; uk")k1tQ  ckuk ; uk"k0Q (6:1:27)  & ' 6.3 kuk" ; u"k10Q ! 0 kLuk" ; f" k;1tQ ! 0 k ! 1: %' '  4 ( '. '' 6.2) c0 kuk" ; u" k0Q  kL(uk" ; u")k;1tQ  (6:1:28) <  kf ; f" k;1tQ ! 0 & " ! 0  &      fu" g">0 ( ' .    , kf"1 ; f" k;1tQ ! 0 & "1  " ! 0  , (6.1.28) ,  ku" ; u"1 k0Q ! 0 "1  " ! 0 . 91,' uk = "lim u" . *6',  uk = u .
    (6.1.13) &    2 7 u 2 L2 (Q) .. !0 kLu"k;1tQ  cku"k0Q    ,

kL(uk" ; u)k;1tQ = kf" ; f k;1tQ  cku" ; uk0Q: *7 2  &   & " ! 0 7 ' 0  ckuk ; uk0Q  kuk ; uk0Q  0  &:' uk = u: 9' ' kuk ; uk0Q . *'22     5 , 7 ' kuk ; uk0Q  ku ; u" k0Q + ku" ; uk"k0Q + kuk" ; uk k0Q:

(6:1:29)

88

6 4         

#12 "  k , 1 ku ; u"k0Q  3  ku" ; uk"k0Q  3  kuk" ; uk k0Q   3  (  > 0  | (   &,    ),   & ,2 4 (6.1.27)  ' 6.3, & '

kuk ; uk0Q <  ! 0 k ! 1 5  uk (t x) =

k X i=1

yi ()i (x) k = 1 2 :::

Dyi dydi () ; Fk yi () = Gk () Gk () 2 W2;1O0 t] yi ()j =0 = 0 i = 1 ::: k:

 2 ( y() 2 L2 O0 t] 3   &      5  7 ( fi()g1 i=1 ,  i(0) = 0 i = 1 2 ::: 

ky() ; i ()k0 ! 0 kDi () ; Gk ()k;10 ! 0 i ! 1: <' 6.4 ,. 6.2    +      2   2

0 '' ,    2 5   '7, &  '7 (( '  2' &1  5 & &    7 4' ,'27.  ,  , 4 , 

 (  :' , 13 5 2,  O33,25].
6 & 6 ' 76 2 &1 65 42 , ,       5 2, +.
. <7     ,   (   '7 &' :(('  ,   ( 

 '7, &  '7 1 ' (( '  2'. ,  2 7 '   5' 42 , . * ,   4, n {'' :    ' &    En ,  52 ,'2 1   P 5 @P . # 6   5 @P = ; 3      2 422 '  nk 0: L2 (P ) | &    5'7   ' '  G15 (  P (,     4 1  17 ,  1     '6  ,  ' &   (, & 63  ' '' &   ) (: :)0P |  2 &,   L2 (P ):
 '6   6  (('7 ( C 2(P ) ,  & B &   4' Nu = ;

n @ X @u(x) ) + c(x)u(x) (a (x) ij @x @x

ij =1

5  aij (x) = aji(x) 2 C 2 (P ) i j = 1 ::: n c(x) 2 C(P ) C(P) | &    & 7 ( Pn a (x)    Pn    | & 6 2  P c(x)  c0 2 8x 2 P c0 = inf fc(x) j x 2 P g > 0: ij i j i ij =1 i=1  , i | &,   3     ( i = 1 ::: n ). D3   52 &,  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 42' 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' ' . <1 2 & ,'2' fr( x1) 0    tg &   &       u^ ( x) &  u( x)   = t 2  & 2 2 4 lim MO(z A r ; u)2q ] = m(t) A r = u^ ( x) (6:2:4) !0 5  A |  &  &,   83 , &    1 8  W2;t 1(G) &     2 L2 (Q) z | &,    , En m(t) = inf MO(z Air ; u)2n] (6:2:5) A i

622 5 1 2 & ' ' & ' &' Ai    83' , W2;t 1(G) L2(Q): *    7 & 1,' , F(G Q): 04  ,  (  2  '  &  ' &'' :    48 &5  2 #-F&( A0Rr = Rur  (6:2:6) 5  A0 |  &,    283 8 (6.2.5) Rr |  2 & 5 &  r( x1 ) Rur | ,' 2 & 7 &  u( x)  r( x1): D&     83  6 . ' 6.5
        fA g >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 62 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 : 04 :5  2 13'   O33]. 05 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: <5  &        ;(z Ru z)n  ;(z A0Rru z)n  ;(z A Rru z)n     , (6.2.12)      (z Ruz)n  j(z A0Rru z)n j   inf f' (A) A 2 F(Q G)g = (z Ruz)n ; (z A0Rru z)n  0: A <' 1,' &      & fA g >0 ,  58 , &      f(z A Rru z)n g >0 &  ! 0: K &      ' 1 ,  ,, 3   & 

lim (z A Rru z)n : !0 <  (A)   (A) 2  > 0  7 A 2 F(Q G)  inf f(A) A 2 F(Q G)g  inf f (A) A 2 F(Q G)g A A      inf f(A) A 2 F(Q G)g  ; lim (z A Rru z)n : (6:2:13) !0 A D&    6     inf f(A) A 2 F(Q G)g  ; lim (z A Rru z)n  (6:2:14) !0 A

91

6.3 .  /    ! !  2

  , 42 (A) = lim  (A)  ; lim (z A Rru z)n  !0 !0    (A)   (A ) = inf f (A) A 2 F (Q G)g: A %,    (6.2.12)  (6.2.14)  ,  inf f(A) A 2 F(Q G)g = ; lim (z A Rru z)n : !0 A

# 8  , &    4  (A ) ; (z A A z)n = (A )  inf f (A ) A 2 F (Q G)g = ; lim (z A Rru z)n !0 A  (A )  inf f(A) A 2 F(Q G)g = ; lim (z A Rru z)n : !0 A <' 1,' 7 ',  inf f(A) A 2 F(Q G)g = ; lim (z A Rru z)n  (A )  inf f (A ) A 2 F (Q G)g = ;(z A Rru z)n : !0 A A *7 2  &   &        &  &  ! 0 & ',  inf f(A) A 2 F(Q G)g = ; lim (A ): !0 A

(6:2:15)

D    , &      4 fA g >0 &5  2 (6.2.7) 2 2 2 4' ,  ( . 6.3  0   1        

*  fi (x)g1 i=1 5 2  ' 5  7 ( &    L2 (P) &' & 2d i 8 2   2 d(x) x2; = 0 fi (x)g1 i=1 | 5 2  ' L2 (d): *6  ' & &  , 68 ' , ''      ' '7 (  & . U  u( x) 2 W2;t 1(Q)  -(8 u( x) '6 &     u( x) =

1 X

i=1

#i()i (x) #i() 2 W2;t 1 O0 t]:

   . *''  ( u( x) & jt  & ' jt u( x) = ( x) 2 L2 (Q)  ( x) = % & ,2 & Dt  7 ' u( x) =

1 X

i=1

1 X

i=1

Dt i ()i (x) = 

i ()i (x) i () 2 L2O0 t]:

1 X

i=1

#i()i (x)

(6:3:1)

5  #i() = Dt (): 91,' , #() = f#1() #2() #3() ::: #k :::g | 1  . <5 ,   (2 g( x) 2 W2;t 1 (Q)   '6 &     g( x) =

1 X

i=1

!i ()i (x) = !
()(x)

(x) = f1 (x) 2 (x) :::g | 1    ' , L2 (P) !() = f!1 () !2() :::g:

92

6 4         

 6.4 ,   ( "   A   ' W2;t1(Q)  L2(Q)        : 1 X Ag

i=1

#i()i (x) = u

(6:3:2)

"   ,  "6     ( , ( "  *  #()   

Z

#() = H( s)!(s) ds

(6:3:3)

0

 H( s) | " *  ( +  -   hij ( s)     s '    W21 O0 t]:     ,      .
17 ' . *  & A - & . <5  2 6 5 i = 1 2 ::: 6 #i() = (Ag i )0P 2 2 2 ' & ' ( '  W2;t 1(Q) (   &   & A   25 &,  2 L2 (P ) ). D5   113 ' 0     5 & 5 (   3' 5 1 ' &    O25],  ' ,  2  ( 5 1 ' &    W2;1 '  : \(u) = hu vi 5  v |  : ' , W21  , &  2' : '' v  & :' k\k = kvk1  7 ' #i () = (Ag i )0P = hg hi i (6:3:4) 1 5  hi( x) 2 W2t(Q)     . K ' hi ( x) '6 &     hi ( x) =

1 X

j =1

hij ()j (x) hij () 2 W21 O0 t]

fj (x)g1j =1 | 5 2  ' ( L2 (d): #6 (6.3.4) ' 4 & 5 42  (6.3.1), ,&4 2  83' 1,' 1 X

hi ( x) = h

j =1

!j ()j (x)

1 X

k=1

hik ()k (x)i =

1 X

j =1

h!j () hij ()i:

     .  ,  ,   & &1 ' 5 1 ' &    ' ' &  ,   5 O4,25],    & 2 2     1 X 1 X

j

i=1 k=1

hik xk j2  N 2

1 X

i=1

jx i j 2 

(6:3:5)

1 P

2 817 xi (i = 1 2 3 :::)    2837     jxij2 < 1  N | & 22 ,  232  i=1 & A: K &   & 1 2 & ,    ' 7 ' . * '  48 ,  ( . *1 6 ,  22  ' @u + Bu( x) = v( x) (6:2:2) Lu @ x) = 0 u(0 x) = u0 (x) du( (6:2:3) d x2; 1 '      '' uk ( x) =

k X i=1

#i()i (x)

(6:3:6)

6.3 .  /    ! !  2

di (x) = 0 # () { 7 2 2 , 4 fi (x)g1 5  1,

L (P) 2 i i=1 d x2;

8  @uk( x)  > < @  j 0P + (Bu j )0P = (v j )0P  Pk > : uk (0 x) = #i(0)i (x)

93

(6:3:7)

i=1

d d | &,  2 & ' . *  22 4 (6.3.6) (6.3.7), & '  2 2 &  2 #j () k d#j () + P #i()(Bi  j )0P = (v j )0P  d i=1 (6:3:8) #j (0) = (u0(x) j )0P  j = 1 2 ::: k: J  (6.3.8) 2 2 2 113 ,  4 2  ' 7 1 7 (( 7   &2  k:
1 8 2, 2 &1 6 , , & '  83' 1,' (r() i)0 = (S u i )0 + ( i )0 (6:3:9) (: :)0 |  2 &,   5 1 ' &    L2 (d): ! ' ,  1 8  2 & frm () 0    tg 2, #() 4'

Pk

rmi () = (r() i)0 = #j ()(S rhoj  i )0 + ( i )0 j =1 i = 1 2 3 :::m < 1:

(6:3:10)

#  ' 1,2. #() { k {'  : '' f#i()gki=1  F | ' ,' k k Fij = (S i  j )0P  Gv | ' ,' 1 k (Gv)i = (v i )0P  C | ' k m Clj = (S j  l )0  rm () |  1 8 , rml () = (r() l )0 w() | 4', wl = ( l )0  i = 1 2 ::: k j = 1 2 ::: k l = 1 2 ::: m: D '

 7 1, ,   &'  (  '6 ('  

 . #  & #() '   2  ' d#() = F#() + Gv() #(0) = #  (6:3:11) 0 d 5  Gv | 1  4'.
1 8  2 & frm () 0    tg 2, #() 4' rm () = C#() + w(): (6:3:12) <1 2   #^() &  #() ''  = t    2838 8 '''      41 m(t) = inf fMO(z #(t) ; Hrm (t))2k ]g (6:3:13) H 5  622 5 1 2 & ' ' &' H   83' , &    1 8 

&     2, H | &  2 2 (4.1.5), z 2 Ek : %'8 '   83  : MO#0] = 0 MO#0#
0 ] = P0  P0ij = (U0i  j )0P  i j = 1 2 ::: k MOGv()] = 0 MOGv()(Gv())
] = Q()( ; ) Qij () = (V i  j )0P  MOw()] = 0 MOw()w
()] = R()( ; ) Rql () = (W q  l )0 q l = 1 2 :::m R() | ' &2  m R()  0 U0 |   & &  u0 (x) V |   & &  v( x) W |   & &  ( x):

94

6 4         

" R() 6 2. *:' 2 42 ,  (  (6.3.11)-(6.3.13) 17 '  & ,   5', &    4.1. *6',  &1 6 4 ,   &'  (  (6.2.1)-(6.2.3) Pk u^ k( x) = #^ i()i (x) &  & ,  (6.3.11)-(6.3.13) & k ! 1 7  2    83 i=1 '  ' 48.  2 :5 6' 2 &'5 7  6 .  6.5 , (  6 t      lim MOkuk(t x) ; u(t x)k20P ] = 0 k!1

 uk (t x) {    (6.3.6),  u( x) |      (6.2.2), (6.2.3)   (*  = t:     ,      . D5   ' 6.4, ' '  4

Zt

sk (t !) = juk () ; u()j2d ! 0 k ! 1

(6:3:14)

0

2 & 7 ! 2 ]  

Zt

MOsk (t !)] = MO juk () ; u()j2d] ! 0 k ! 1: 0

G'' ,. J&4'   #-F&( 2 ,  (6.3.11)-(6.3.13) MO#(t)rm
()] =

hmk (t )MOrm()rm
()] d:

(6:3:15)

hmk (t )MOrm ()rm
()] d + h mk (t ):

(6:3:16)

05 2,    '  MO#(t)rm
()] =

Zt 0

Zt 0

  (6.3.16) '6 ,&  &'38 1 7 ',  &   & 1 2 & ,    7 ' . % & ,2  8 ' MO#(t)rm
()] & ' 1 8 ' Kkm (t )  83' 1,' 2 MO#(t)r
()] 0    3 m 0 0  5: Kkm (t ) = 4 ;1



 

K ' ,   W2t (G)  &,   83 , W2;t 1(G) L2 (Q):    , 1 Rt P 2 815 g 2 W2;t 1(G) &    4 g = yj ()j (x)  h^ (t) = Kkm (t )^y() d 5  j =1

0

1 P y^() |   ' yi () (2 h(t x) = h(g) = hj (t)j (x) hj |   

j =1

^h(t): <' 1,', ' Kkm (t ) ,  &  W2;t 1(G): 91,' : & , Rkm: + 5, 1 2 ' Kmm ( )+I( ;) & 2 & MOrm ()rm
()]+ I( ;) ,  & Rmm    83 , W2;t 1 (G) L2 (Q): *  2 h (t ) 0    3 mk Hmk (t ) = 4 0 0  5:



 

95

6.3 .  /    ! !  2

<5 ,  & ,2 1  ',   (6.3.16) ,&4 2  

Zt

(t )Kmm ( ) d + H (t ) Kkm (t ) = Hmk mk 0

 & ('

A km Rmm + A km = Rkm :

*6',  4  2 (6.3.17) ,  &1 6 4 (6.2.7). 6'   2 &  7 ( .

(6:3:17)

 6.6 0       lim kRur ; Rkm k = 0:

km!1

    ,      .  2 6 5 (  5 t & &  8 & '  kRur ; Rkm k = sup j(ORkur'k; Rkmk]'k )0Qj  ;1tG 0Q '

o

' 2 W2;t 1(G)  2 L2 (Q) k'k;1tG 6= 0 kk0Q 6= 0 : 9'    &    :5    

j(ORur ; Rkm ]' )0Q j = jMO(u )0Q(r ');1tG ; (uk  )0Q (rm  ');1tG]j , &'22 113     4-!2 5, 7 '



j(ORur ; Rkm ]' )0Qj  (MOku ; uk k20Q ]MOkrk2;1tG])1=2+



+(MOkuk k20Q]MOkr ; rm k2;1tG])1=2 k'k;1tGkk0Q : %, '' 6.5  ,  &  5' &    & k ! 1 ' 2  0: *6',  & m ! 1 MOkr ; rm k2;1tG] ! 0: 0 '' ' kr ; rm k;1tG: 9'',  ,   r  rm &'8 2 1  .  (r   )  i 0 i  m ' 5    rk '8  ri = (r i)0   rkm | rmi = 0 i > m: &        krk ; rkm k;1tG = kr ; rm k;1tG 5  r  rm &   113' 2 ' ; 1 1 X X jt r = (jt (r i )0i  j  rm = (jt rm  i )0i : <5 

i=1

i=1

krk ; rkm k2;1tG = k =2

1 X

i=m+1

X 1

i=m+1

(r i)0i k0G  2

1 X

i=1

(jt k(r i)0 i k20G =

k(r i)0i k2;1tG ! 0 m ! 1

     , 62 ( 2 ;. <' 1,', MOkr ; rm k2;1tG] ! 0 m ! 1: G'' ,.

 6.7 , *  !*   t         lim kR r ; R mm k = 0:

m!1

96

6 4         

    ,      . Z, R r 1,' & R r = Rr + I( ; )  R mm = Rmm + +I( ;): K & & 7  > 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. *  ' : 42 (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. <' 1,' , 7 '  42 &1 6 5 2,  ,   (   48  7  , . + 5' 42 &1 6 ,   '   4.1. *' 5 2, 1 2 5   O33,20,25]    5.1.

98

%+%

5 (

1. + 1 +. 05 2, &    2    . ".:
, 1977. {224 . 2. +7,
.%., / ,' %.". <2 7 & 5 1 ' &   . <' 1. F : %, . & F/ ,   5 1M 2 #3 4 , 1977. {315 . 3. !  v.+.,   #.*.,  . "'  1 & 65 : &'. 911 ,' &     67  '.  :
  ', 1982. <. 1. {301 . 4. !,  v.". 0, 6 & 1  ' (2' ' &267 & .  :
  ', 1965. {798 . 5. #   #.+. "  &'  (   '7 & 2  ' ' &&' // #&  & 2  ' ' &&'. ".: ", 1975. D. 58-94. 6. #  ' #.D.  2 ''  (,. ".:
, 1967. {436 . 7. / ( %."., Q  /.U. 9113 (. #&. 1-5. ".: ;,'5,, 1958-62. 8. /7' %.%., D7 +.#. <2 7 &  . ".:
, 1970- 1975. <. 1-3. 9. /  ;.!.,
     T.+. 04  7 ,  (   & 2 & &,   &'7 ' ' '   2 // + '   '7. 1966. ?10. D. 153-168. 10.   #.*., , 
.
. 9 5 2, 7 7 ,  7  1 // +
DDD0. 1974. 214. ?3. D. 528-531. 11.   #.*., , 
.
. 05 2,  ' 42 ,   13 // +
DDD0. 1975. ?5. D. 1045-1048. 12.   #.*., L 9.U. ; 2  5 2,2.  : /, 1977. {51 . 13.  ' 0.U., !8  0.D.
  ,   (    & ,2 // <7.'7. 1961. <. 83, D. , ?1. D. 123-141. 14.  ' 0.U., ; 1 *., +11 ". 9 & ''    '. ".: ", 1971. {400 . 15.  '5 +.
. %&    : &   7 7 &      // %, . +
DDD0. D.'., 1941. 5, ?1. D. 5-16. 16.   %.#. 9 4 ,   (  &    5 4' 1 8 27 // .'.6 . 1979. 31, ?4. D. 372-379. 17.   %.#. 9 &1 6' 4 113 , 
' 2 &1  7   // #   5   : "     &',2 67  '.  : %, . /, 1982. #&. 1. D. 27-39. 18.   %.#. 91   ,    2 5   '7 &' :((' // #  2  &  2 ''. 0 &1 .'6  .. 1.,  : %, . /, 1981. #&. 45. D. 21-24. 19.   %.#.,   ".#. 91 7   42 ,   &'  (   ' 4'' 1 8 27 // # . '    ' 11 7  K#". ".: "/, 1988. D. 19-33. 20. ", #.+. 05 2 '  42  &  7 , . ".:
, 1987. {240 . 21. ", #.+., /1 +.%. "  42  &  7 , . + 5'   &. ".: "/, 1992. {320 . 22. G +.D.  1  8 1 &' 5 2, & 8  ,&'    42 // T#"  ";. 1978. 18, ?6. D. 1363-1376. 23. G& 0.Q.   & &' 5 (   ' &  1 '   4' 1 8 27 // + '   '7. 1974. ?1. D. 35-41. 24. G8  G.;., D1  #.%. K ' ( 5  ,. ".:
, /0;"G. 1965. {520 . 25. G24 %.%.,   #.*., L 9.U. ; 2 4' .  :
  ', 1979. {232 . 26. *5 #.D. %5    &  2 7 (  7 & 6  &  8 &' 7 7  ' // + '   '7. <. 28, ?1. 1957.

5 (

99

27. *5 #.D., D %.
. D7   ((   '. ".:
, 1985. {560 . 28. 0  . 9   '' 5  ,. ".: ", 1966. {320 . 29. 021 -94  +.*. 91     (   ' // %, . +
DDD0. D.7.1. 1970. ?5. D. 203-212. 30. D 6 K., " 6. <2  2   &' 2,  & . ".: D 2,, 1976. {496 . 31. D' ".G., 4 2
.D., /  K.!. 9   (( 7   ''  (,. ".:
, 1962. {768 . 32. D  0.G. *'  &  "  2 &'  (  5  // 0 7  : . 1960. <. 5, ?11. D. 1751-1763. 33. <7 +.
., +  #.w. "  42 7 , . ".:
, 1979. {286 . 34. ;75  /.".  (( 5  5 5   2. <' 1 - 3. ".:
, /0;"G, 1966. 35. Q2 +.
. #2 . ".:
, 1980. {576 . 36. Anderson B.D. An Algebraic solution to Spectral factorization problem // IEEE Trans. Autom. Control. 1967. V. AC-12, ?4. P. 410-414. 37. Bensoussan A.V. Optimization of sensors location in distributed „ltering problem // Notes in Math., 1972. 294. P. 62-84. 38. Booton R.C. An optimization theory for time-varying linear systems with nonstationary statistical inputs // Proc. IRE. 1952. V. 40. P. 977-981. 39. Bryson A.E., Johansen D.E. Linear „ltering for time - varying systems using measurements containing coloured noise // IEEE Trans. Automat. Contr.. 1965. 10, ?1. P. 4-10. 40. Bucy R.C. Optimum „nite time „lters for a special non-stationary class of inputs // Internal. Report Nr. BBD-600, Jons Hopkins University, Applied Physics Lad., 1959. 41. Kalman R.E. A new approach to linear „ltering and prediction problems // Trans. ASME series D., J. Basic. Engg.. 1960. V. 82. P. 35-45. 42. O'Reilly J., Newmann M.M. Minimal-order observer-estimators for continuous time linear systems // Int. J. Control. 1975. 22, ?4. P. 573-590. 43. Swerling P. First - order error propagation in a stage - wise smoothing procedure of satellite observations // J. Astronautical sciences. 1959. V. 6. P. 46-52. 44. Wiener N. Extrapolation, interpolation and smoothing of stationary time series, with engineering applications. New-York: Wiley, 1949. 45. Yoshihawa T. On discrete - time Kalman „lter in singular case and akiud of pseudo-inverse of a matrix // Int. J. Control. 1972. 15, ?6. P. 1157-1163.

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