Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://new.math.msu.su/department/opu/sites/default/files/attached_files/Pont_PM.pdf Äàòà èçìåíåíèÿ: Thu Feb 21 16:11:18 2013 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 04:27:01 2016 Êîäèðîâêà: IBM-866 Ïîèñêîâûå ñëîâà: m 63

..

-


.. , .. , ..

Mo c 2004


.. , .. , ..

, . - . , . í. , , . í , .. . , - . , , , . . 24 .

: .. , .. ,

, vraimax@mail.ru, dmitruk@member.ams.org, nikolai@osmolovskii.msk.ru.

c .., .., .., 2004 .


1. ç1.1. ç1.2. ç1.3. ç1.4. ç1.5. ç1.6. ç1.7. ç1.8. A B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 13 17 21 33 ?? 35 38 4

2. . í ç2.1. ç2.2. ç2.3. ç2.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 51 56 62

3. : ç3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ç3.2. L () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ç3.3. vraimax (t, x, u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ç3.4. max (t, x) . . . . . . . . . . . . . . . . . ç3.5. í - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ç3.6. . . . . . . . . . . . . . . . . . . . . . . . . ç3.7. : - . . . . . . . . . . . . . . . . . . . . . . 4. ç4.1. ç4.2. ç4.3. ç4.4. ç4.5. ç4.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ??

5. . ç5.1. ç5.2. ç5.3. ç5.4. . . . . . . . . . . ............... . . ............... ............ ............ ............ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?? ?? ?? ??

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166



(). .. í . [1] , , . .. ... , [1], , í , , . í . , , . ; " " (, , 2001), . , , , . ; , , . , , -III. , -, , -, . , , . , , , . , , , , 4


5 . .
1

í

1 , : , . , , , , , . A. A ( v -, ) , , . , ( ) "" "" , . 1 .. . , ; , . "" . A , , , , A , . , , (" ") . 2 í . . , : , . , í ( ). 2 . ( ,
1 [7], . [15], . .


6 .) í. 1 2 " ( )", . , 3, . . , ( , ) - ( ). C L , : ( í L ), , , - - L . , í . , , ( í) . í . 2, 3, . 4 ( , ) . " ", . ( ), , , . , , 5 (); , . , ... ... 1, 2 3 , 2000 2001 . 1, 2 , 3


7 . 4, 5 ; 3, 4 2 2004 . , , , , . , ; . .. , .. , . , , , . , 04-01-00482, , -304.2003.1. .. , .. 2004 .


1


1.1 A B

1. . , . , A B ? . . (x1 , x2 ) . (u1 , u2 )(t) . . u1 = u1 (t) u2 = u2 (t) , x1 = u1 (t), è x2 = u2 (t). è

u2 + u2 const . t , t [t0 , t1 ] . 1 2 [t0 , t1 ] . t1 - t0 . , , , . x = y , y = u(t), x = (x1 , x2 ) , y = (y1 , y2 ) , u = (u1 , u2 ) . y . , t1 - t0 min; x(t0 ) = a0 ,
0

x = y, y (t0 ) = b0 ,

y = u(t),

u2 + u 1

2 2

const,

x(t1 ) = a1 ,

y (t1 ) = b1 .

b0 b1 ( ) [a , a1 ], , , . 1
1 [a0 , a1 ] , x , y u . x , y u ( (x(t1 ), y (t1 )) ) . [1], .

8


1.1. A B

9

. , : (, ), , . . - , ( !) . 2. . x = f (t, x, u), u U. (1.1) U Rm , u = u(t) t . , U . u = (u1 , . . . , um ) . x = (x1 , . . . , xn ) . , . x = f (t, x, u) u = u(t) . t0 x(t0 ) = a u(t) [t0 , t1 ] , x(t) ( f ) x = f (t, x, u(t)), x(t0 ) = a.

f , fx ft . x(t), u(t) [t0 , t1 ] , (1.1) ( ), x(t) : [t0 , t1 ] Rn , u(t) : [t0 , t1 ] Rm , (..) [t0 , t1 ] x(t) = f (t, x(t), u(t)), u(t) U.

, u(t) . ( u(t) , ). w = (x, u). (x(t), u(t) | t [t0 , t1 ]) p = (t0 , x0 , t1 , x1 ) , x0 = x(t0 ), x1 = x(t1 ) . Q R1+n+m , f fx ft , Q, (t, x, u) Q . (1.2)

, (x(t), u(t)) .. [t0 , t1 ] . , , Q ( ) , (t, x(t), u(t)) .. [t0 , t1 ].


10

1.

, Q Q. Q , (" ") ; . Q . , Q = R1+m+n .

3. : A . : , J = F0 (t0 , x0 , t1 , x1 ) min, x0 = x(t0 ), t0 , x0 , t1 , x
1

x1 = x(t1 ),

0, K (t0 , x0 , t1 , x1 ) = 0,

F (t0 , x0 , t1 , x1 )

(x(t), u(t) | t [t0 , t1 ]) x = f (t, x, u), u U.
2+2n

[t0 , t1 ] a priori . F0 : R ‡ F :R K:R
2+2n

R,

Rk , Rq , Rn ,

F = (F1 , . . . , Fk ), K = (K1 , . . . , Kq ), f = (f1 , . . . , fn ).

2+2n

f : R1+n

+m

p = (t0 , x0 , t1 , x1 ), : J = F0 (p) min, F (p) x = f (t, x, u), 0, K (p) = 0, u U. (1.3) (1.4)

, F0 , F, K C 1 , , , f fx ft . U Rm . 2 F0 , , . (1.3) . , (1.4), . , f Q , . , F0 , F, K
, ( [1]), U Rn , . , .
2


1.1. A B

11

P R2+2n . , . F0 (p) = t1 - t0 , , ; t0 , t0 = 0 ( Ki (p) = t0 i ). , x(t0 ) = a x(t1 ) = b , . : Ki = x0 - a = 0 Kj = x1 - b = 0 i, j . K = 0 t0 , t1 , t0 t1 , . . , x(t0 ) = a (x(t1 )) = 0 , . K (p) = 0 . ,
t
1

J = F0 (p) +

(t, x, u)dt,
t0

t x . y : y = (t, x, u) J = F0 (p) + y1 - y0 , y1 = y (t1 ), y0 = y (t0 ) . , , : x = f (t, x, u), y = (t, x, u), u U. , . , A , . (x(t), u(t) | t [t0 , t1 ]) A , x(t) [t0 , t1 ] , u(t) [t0 , t1 ] . , . , , () . , . . , , . A ( ) . A , ( , ) .. (. [1]). , í u U. , U , , . , , "". . .


12

1.

, . , .. . . 4. B . , A , B . : J = F0 (x0 , x1 ) min, F (x0 , x1 ) x = f (x, u), , , x0 = x(t0 ), x1 = x(t1 ). 0, K (x0 , x1 ) = 0,

u U,

F0 , F, K R2n , f fu Rn+m . [t0 , t1 ] - . , B A , f t , .. x = f , , F0 , F, K t0 , t1 . B . , (x(t), u(t) | t [t0 , t1 ]) B , (x(t - ), u(t - ) | t [t0 + , t1 + ]) B . B B . dt f (t, x, u) d = 1, , x(§) u(§) x0 , x1 . A, A . x = t = t( ) - . : x = x( ), u = u( ) . (1.5)

t0 = t(0 ), x0 = x(0 ), t1 = t(1 ), x1 = x(1 ). : A J = F0 (t0 , x0 , t1 , x1 ) min, F (t0 , x0 , t1 , x1 ) 0, K (t0 , x0 , t1 , x1 ) = 0,

dt dx = f (t, x, u), = 1, u U, d d t( ), x( ) , u( ) , [0 , 1 ] , t0 , x0 , t1 , x1 (1.5). , A - , B . A A ? (x(t), u(t A. í (t0 , x0 , t1 , x1 ) , 0 = t0 , 1 = t1 , t( ) = . (t( ), x( ), u( ) A (t0 , x0 , t1 , x1 ) , (1.5).

-

) | t [t0 , t1 ]) - x0 = x(t0 ), x1 = x(t1 ). | [0 , 1 ]) -


1.2.

13

, (t( ), x( ), u( ) | [0 , 1 ]) A (t0 , x0 , t1 , x1 ) . t( ) = + const . , , , , .. A , t( ) = . t0 = 0 , t1 = 1 . (x(t), u(t) | t [t0 , t1 ]) A (t0 , x0 , t1 , x1 ) , x0 = x(t0 ), x1 = x(t1 ). A A . , B , A . (, !) , . t : . t = t( ) B , . . dt d = v , v ( ) - ( ) , , t = t( ) í ( ) . , t , .. ..; v -. ( , v ( ) u(t) .) v - .

1.2

B

1. v -. B : x = f (x, u), u U, (1.6) ;

(x(t), u(t) | t [t0 , t1 ]), t0 < t1 x(t) , u(t) . B : dx = v f (x, u), d dt = v, d u U, v 0.

(1.7)

. (1.7) (t( ), x( ), u( ), v ( ) | [0 , 1 ]) , t( ), x( ) , u( ) , v ( ) , (1.7) [0 , 1 ] , 0 < 1 - . (1.6) - t0 , t1 x0 = x(t0 ), x1 = x(t1 ) . (1.7) 0 , 1 x0 = x(0 ), x1 = x(1 ), t0 = t(0 ), t1 = t(1 ) . , v ( ) , .. [0 , 1 ]


14

1.

, , v ( ) . B . 1. (1.7) v (1.6) x0 , x1 . . v ( ) : [0 , 1 ] R ~ , t ~ , t( ) ~ dt( ) = v ( ), ~ d ~ t( ) = t0 + ,
0

~ t(0 ) = t0 ,


v (s)ds. ~
0

~ t( ) : [0 , 1 ] [t0 , t1 ]

- . 1 , 2 , . . . , s ~( ), . t
s

M0 =
1

k ,

M+ = [0 , 1 ]\M0 .

, M+ () . ~ t(k ) = {tk }, , ~ N0 = t(M0 ), ~ N+ = t(M+ ). ~~ , (t) : [t0 , t1 ] [0 , 1 ], , t( (t)) = t t [t0 , t1 ], . . ~ ~ ~ ~ t( ). t( ) "", . , , t1 , . . . , ts , N0 . tk kk k := [0 , 1 ], k = 1, . . . , s , k ~-1 (tk ). tk . (tk ) = 0 , k = 1, . . . , s. t ~ (t) . ~ , 1, . ~ t( ) , (t) v ( ) , . ~ ~ 1. u( ) : [0 , 1 ] Rm ~ , u( ) U .. [0 , 1 ]. u(t) = u( (t)), t [t0 , t1 ]. ~ ~~ u(t) , u(t) U .. [t0 , t1 ] . . N+ . (t) ~ , , u(t) U .. . k = 1, . . . , s ~ t . N+ = [t0 , t1 ]\N0 .

N0 = {t1 , . . . , ts },


1.2. 2. x( ) : [0 , 1 ] R ~
n

15

, (1.8)

dx( ) ~ = v ( )( ) .. [0 , 1 ], ~~ d

( ) : [0 , 1 ] Rn ~ , v ( ) - ~ - . x(t) = x( (t)), t [t0 , t1 ]. ~~ x(t) : [t0 , t1 ] Rn - , x(t0 ) = x(0 ), ~ , dx(t) = (t) .. [t0 , t1 ], dt (t) = ( (t)), t [t0 , t1 ]. ~~ . a) , x(t) . N+ (t) , , x(t) = x( (t)) ~ ~~ ( x(§) ~ ). , , x(t) tk N0 ( , (t) ~ ). x(t) . , (1.8) x( ) k (k = ~ 1, . . . , s), v ( ) = 0 k . ~ x(0 ) = x(1 ), ~k ~k k = 1, . . . , s. (1.11) (1.10) x(t1 ) = x(1 ). ~ (1.9)

tk N0 . x(t) tk . def def x(tk - 0) = x( (tk - 0)) = x( (tk )) = x(tk ), ~~ ~~ (1.12)
k . (tk ) = 0 . , ~ ~

x(tk - 0) = x(0 ). ~k ,
k x(tk + 0) = x( (tk + 0)) = x(1 ), ~~ k (tk + 0) = 1 . (1.11)-(1.14) , ~ def

(1.13)

(1.14)

x(tk - 0) = x(tk ) = x(0 ) = x(1 ) = x(tk + 0). ~k ~k , x(t) tk , , , [t0 , t1 ] . ) (1.9) (t0 ) = 0 . , ~ . (t) , ~
def

x(t0 ) = x( (t0 )) = x(0 ). ~~ ~


16

1.

1 M+ , 1 ~ t( ) ( s ), (t1 ) = 1 , , ~ x(t1 ) = x( (t1 )) = x(1 ). ~~ ~
ss s 1 s = [0 , 1 ] , (t1 ) = 0 ( ~ ). , x(t1 ) = x( (t1 )) = x(0 ). (1.11) x(0 ) = ~ ~~ ~s ~s s ) = x( ). , x(t ) = x( ). x(1 ~ ~1 ~ 1 ~( )) = x( ) [0 , 1 ] ( ), , x(t ~ ~ ~ (1.9) , t(0 ) = t0 t(1 ) = t1 . def

) , x(t) = x( (t)) (1.10). ~~ N+ ( 1:1 M+ ). , (t) ( ~ ~ t( ) ). ~ dt = v, d v = v () > 0, d ~ 1 =, dt v

dx(t) dx( ) d ~ ~ 1 = = v ( ) = ( (t)) = (t). ~~ ~~ dt d dt v , (1.10) , N+ , N+ , (1.10) [t0 , t1 ] , . ~ 1. (t( ), x( ), u( ), v ( ) | [0 , 1 ]) ~ ~ ~ ~ ~ (1.7). t0 = t(0 ), t1 = t(1 ) , (t) : [t0 , t1 ] ~ ~ [0 , 1 ] t( ), . x(t) = x( (t)), u(t) = u( (t)), t [t0 , t1 ] . ~~ ~~ 1, u(t) , u(t) U .. [t0 , t1 ]. 2, x(t) [t0 , t1 ], x(t0 ) = x(0 ), x(t1 ) = x(1 ), ~ ~ dx(t) = f (x(t), u(t)) .. [t0 , t1 ]. dt , 2, dx( ) ~ = v ( )f (x( ), u( )), ~ ~ ~ d , (x(t), u(t) | t [t0 , t1 ]) f (x( (t)), u( (t))) = f (x(t), u(t)). ~~ ~~ (1.6).


1.3.

17

~ 2. B B , . B : J (x0 , x1 ) min, F (x0 , x1 ) 0, K (x0 , x1 ) = 0, x = f (x, u), u U,

x0 = x(t0 ), x1 = x(t1 ) , x(t) , u(t) - [t0 , t1 ] , t0 , t1 - . ~ B B : J (x0 , x1 ) min, F (x0 , x1 ) dx = v f (x, u), d dt = v, d 0, K (x0 , x1 ) = 0, v 0, u U, 1

x0 = x(0 ), x1 = x(1 ) , x( ), t( ) , u( ) , v ( ) (!), [0 , 1 ] , 0 , 1 . x t , u v . v ~ 0 = t0 B B , ~ B B . ~ 1 ( B B ). ~ B . = ~ , B ~ ~ í B

= (x(t), u(t) | t [t0 , t1 ]) ~( ), x( ), u( ), v ( ) | [0 , 1 ]) (t ~ ~ ~ x(0 ) = x(t0 ) , x(1 ) = x(t1 ) . ~ ~ .

~ ~ . B . ~ B ~ ~ = (t ( ), x ( ), u ( ), v ( ) | [0 , 1 ]) ~ ~ ~ : J (x (0 ), x (1 )) < J (x(0 ), x(1 )). ~ ~ ~ ~ ( ). 1 B : = (x (t), u (t) | t [t0 , t1 ]) , x (t0 ) = x (0 ), ~ x (t1 ) = x (1 ). ~ B ( ~ ~ B ) J (x (t0 ), x (t1 )) < J (x(t0 ), x(t1 )). , B . , ~ B B . , .

1.3



.


18

1.

1. . X, Y - . g : X Y x0 , A : X - Y , g (x0 + h) = g (x0 ) + Ah + y (h) h , ~ y (h) 0 h 0. ~

A x0 A = g (x0 ). x0 , (. . x0 ) g (x) - g (x0 ) 0; x - x0 0.

2. . , , (..) :
t

x(t) = x(t0 ) +
t
0

x( )d

t.

x L1 , . . . x(t) : 1 [t0 , t1 ] Rn W1 ([t0 , t1 ], Rn ) ( n ) (absolutely continuous functions). ), AC ([t0 , t1 ], R x(t) . , x
t AC

= |x(t0 )| + x 1 ,

(1.15)

u 1 = t01 |u(t)|dt (1.15), . , , u(t) : [t0 , t1 ] Rm . u

L1 . AC, , , L ([t0 , t1 ], Rm ) = vraimax |u(t)|.
t[t0 ,t1 ]



(1.16)

, vrai max ( , ess sup ) (t) : [t0 , t1 ] Rm . : vraimax (t) = min { C | (t) C .. [t0 , t1 ] } .
[t0 ,t1 ]

(, min .) C ([t0 , t1 ], Rn ) x(t) [t0 , t1 ] Rn . , x
C

= max |x(t)|.
t[t0 ,t1 ]

Rn .


1.3.

19

3. . . 2. = [t0 , t1 ] x = f (x, u), x(t0 ) = x0 (1.17)

f . x(t) ^ u L (, Rm ) x(t0 ) = x0 . ^ ^ ^ x0 , x0 , ^ u(t) , u(t) , > 0 , ^ |x0 - x0 | < , ^ u-u ^


< , x-x ^ 0.

(1.17) x(t) . , |x0 - x0 | 0, u - u 0 , ^ ^

C

, Rn ½ L () U (x0 ) ½ V (u) ^ ^ (x0 , u) ^^ F : (x0 , u) U (x0 ) ½ V (u) x(§) C (), ^ ^ (x0 , u) x(t) (1.17), (x0 , u) . (, , , ^^ .) , F . 3. F (x0 , u) , ^^ F (x0 , u) : Rn ½ L () C (), ^^ (x0 , u) x(t) , ïï ï x = fx (x, u)x + fu (x, u)u, ï ^^ï ^^ï x(t0 ) = x0 . ï ï (1.18)

. x0 U (x0 ) , u V (u) x (1.17). ^ ^ x0 = x0 - x0 , u = u - u, x = x - x, ï ^ ï ^ ^ x0 = x0 + x0 , ^ ï d (x + x) = f (x + x, u + u), ^ ^ ^ï dt (x + x) |t0 = x0 + x0 , ^ ^ ï u = u + u, ^ï x = x + x. ^

dx ^ = f (x, u), ^^ x(t0 ) = x0 . ^ ^ dt , f , x = fx (x, u) x + fu (x, u)u + (t, x, u), ^^ ^^ï ï
C

x(t0 ) = x0 , ï

(1.19)

= o( x

C

+u ï



)

x

C

0,

u ï



0.


20

1.

x(t) , (1.18), ï r(t) = x(t) - x(t) . (1.19) (1.18) ï r = fx (x, u)r + (t, x, u), ^^ ï r(t0 ) = 0

(., , [15, 9]) , r
C

const (t, x, u) 1 , ï r
C

r
C

C

o( x
C

C

+u ï




).

x = r + x , ï

o( r

) + o( x ï

+u ï
C

).

(1.18) , r
C

x ï
C

const(|x0 | + u 1 ), ï ï


(1 - o(1)) o(|x0 | + u ï ï



),

r

o(|x0 | + u ï ï

),

, x(t) x(t) ( ï x ), x0 u . ï ï (1.18) (1.17). [0 , 1 ] r (, ) 1 , . . . , r . k ( ) k , k = 1, . . . , r . dx = f (x, u( )) ^ d x(0 ) = x0 , zk , k = 1, . . . , r , x0 . , z v ( ) = dx = d (1.21) Rn í , u( ) í ^ = (z1 , . . . , zr ). f fx (x, u) . zk k ( ) , (1.20),(1.21) v f (x, u( )), x(0 ) = x0 , ^ (1.22)
r

zk k ( ),
k=1

(1.20)

4. (x0 , ^ x( ) , [0 , 1 ^ (x0 , z ) (1.20),(1.21) ^^ : (x0 , z ) W

z ) Rn ½ Rr (1.20),(1.21) ^ ] . W x(§) AC ([0 , 1 ], Rn ), (1.23)

(x0 , z ) , ^^ n ½ Rr x(§) AC ([ , ], Rn ) , (x0 , z ) R ïï ï 01 dx ï = fx (x( ), u( ))x ^ ^ ï d
r r

zk k ( ) + f (x( ), u( )) ^ ^ ^
k=1 k=1

zk k ( ), ï

(1.24) (1.25)

x(0 ) = x0 . ï ï


1.4. B . (1.24), (1.25) dx ï = v fx x + v f , ^^ ï ï^ d x(0 ) = x0 , ï ï

21

(1.26)

^ v = zk k ( ), v = zk k ( ), fx = fx (x, u), f = f (x, u). (1.24) ^ ^ ï ï ^^ ^ ^^ (1.20). x1 = P (x0 , z ) , P : (x0 , z ) W - x1 = x(1 ) Rn , (1.27)

x(§) = (x0 , z ) - (1.20),(1.21), W Rn ½ Rr - ^^ ( , z ) , 4. . 4 . 5. (1.27) W (x0 , z ). ^^ P : (x0 , z ) Rn ½ Rr - x1 = x(1 )) Rn , ïï ï ï x(§) (1.24), (1.25). ï

1.4

B .

1. B . B : J = F0 (x0 , x1 ) min, F (x0 , x1 ) 0, K (x0 , x1 ) = 0, (1.28) x = f (x, u), u U, (1.29) , x0 = x(t0 ), x1 = x(t1 ) , t0 , t1 - . (1.28) (1.29) .

l = l(x0 , x1 , 0 , , ) = 0 F0 (x0 , x1 ) + F (x0 , x1 ) + K (x0 , x1 ), 0 , - - , F K , .. (Rk ) , (Rq ) . , H = H (x, u, ) = f (x, u), (Rn ) . = (x(t), u(t) | t [t0 , t1 ]) - B . , , 0 , (Rk ) , (Rq ) (t) : [t0 , t1 ] (Rn ) , , (i) 0 0, 0, (ii) 0 + || + | | > 0,


22

1. (iii) i Fi (x(t0 ), x(t1 )) = 0, i = 1, . . . , k , (iv) - (t) = Hx (t, x(t), u(t), (t)) .. [t0 , t1 ], (v) (t0 ) = lx0 (x(t0 ), x(t1 ), 0 , , ), (vi) H (x(t), u, (t)) 0 - (t1 ) = lx1 (x(t0 ), x(t1 ), 0 , , ), t [t0 , t1 ] u U ,

(vii) H (x(t), u(t), (t)) = 0 .. [t0 , t1 ]. (vi) (vii) , t [t0 , t1 ] u H (x(t), u, (t)) u U u = u(t). (vi)í(vii) . H(x, ) = max H (x, u, ) (1.30)
uU

, . , H H : H(x(t), (t)) = H (x(t), u(t), (t)) .. [t0 , t1 ].

4.0. , H (x, u, ) H(x, ) ; . (1.30), "" . - ( ), . : (i) , (ii) , (iii) , (iv) ( ), (v) ( ). , , 0 , , . = (0 , , , (§)) B . 6. = (x(t), u(t) | t [t0 , t1 ]) B . . . ^ ^^ 2. . = (x(t), u(t) | t [t0 , t1 ]) ^ ^ B . B (x , z ), . 0 = {(t1 , u1 ), . . . , (ts , us )} ^ , t0 .
1

t

1

...

t

s

^ t1 ,

uk U,

k = 1, . . . , s. s = s()

^^ [0 , 1 ] : [t0 , t1 ], s , t . ^0 t ,...,t


1.4. B . ^ [0 , 1 ] 0 = t0 , 1 = [t1 , t1 + 1], M0 =
1

23 ^ 1 = t1 + s,

2 = [t2 + 1, t2 + 2], . . . ,
s

s = [ts + (s - 1), ts + s].

k , M+ = [0 , 1 ]\M0 .

v ( ) = dt ( ) = v ( ), d ^ t (0 ) = t0 , ^ t (1 ) = t1 .


0, 1,

M0 , M+ ,



^ t ( ) = t0 +
0

v (s)ds,

[0 , 1 ].

, t ( ) - , ^^ [0 , 1 ] [t0 , t1 ], k t (k ) = tk , k = 1, . . . , s. u(t ( )), M+ , ^ u ( ) = x ( ) = x(t ( )). ^ uk , k , k = 1, . . . , s; u ( ) : [0 , 1 ] Rm u ( ) U .. [0 , 1 ] (?), x ( ) : [0 , 1 ] Rn . 3. (a) x (0 ) = x(t0 ), x (1 ) = x(t1 ); ^^ ^^ (b) .. [0 , 1 ] dx ( ) = v ( )f (x ( ), u ( )). d (1.32) (1.31)

^ ^ . (a) t (0 ) = t0 , t (1 ) = t1 . k - M . t ( ) = const (b) 0 k , dx ( ) = 0 k . v ( ) = 0 k . d k , , , .. M . (1.32) 0 + - M+ . + : dx ( ) dx(t ( )) dt ( ) ^ = § = d dt d = f (x(t ( )), u(t ( )))v ( ) = f (x ( ), u ( ))v ( ) ^ ^ (1.32) + , .. M+ . M0 M+ = [0 , 1 ]. , (1.32) .. [0 , 1 ].


24

1.

M0 k , k = 1, . . . , s. M+ . , M0 M+ , k , k = 1, . . . , r. , [0 , 1 ] = r k , 1 k . k ( ) k , k = 1, . . . , r. zk ^ v ( ) k , k = 1, . . . , r, ..
r

v ( ) =
k=1

zk k ( ) ... ^

z = (z1 , . . . , zr ), x0 = x(t0 ). , zk = 0 , k M0 , zk = 1 , ^ ^ ^^ ^^ ^ ^ k M+ . 3. B . , ^ ^^ = (x(t), u(t) | t [t0 , t1 ]) B = {(t1 , u1 ), . . . , (ts , us )} ^ ^ [0 , 1 ] v , t , u , x . = (t ( ), x ( ), u ( ), v ( ) | [0 , 1 ]) ~ B , : F0 (p) min, F (p) 0 , K (p) = 0, (1.33)

dx dt = v f (x, u), = v , u U, v 0, d d p = (x0 , x1 ) = (x(0 ), x(1 )) , t( ), x( ) , u( ) , v ( ) , 0 1 . ~ 7. B . 3, (b), dx = v f (x , u ) d , dt = v , d v


...

0,

u U

...

v , t , u . , (a) ^ 3 x (0 ) = x(t0 ), x (1 ) = x(t1 ). ^ ^^ ~ B , 1 ( B B ) ~. B [0 , 1 ] 1 , . . . , r ( k M0 , M+ ). P : (x0 , z ) x(1 ),


1.4. B . x( ) : [0 , 1 ] R
n

25

zk k ( ), x(0 ) = x0 (1.34)

dx = f (x, u ( )) d

( k k ). x(§) = (x0 , z ) . v ( ) := zk k ( ), ^ dx = v f (x , u ), d x0 := x(t0 ) = x (0 ), ^ ^^

x = x0 , z = z (1.34) x ( ) , [0 , 1 ] , ^ ^ 5 P W (x0 , z ) ^^ Rn ½ Rr . B . (x0 , z ) Rn ½ Rr . B . F0 (x0 , P (x0 , z )) min, -z F (x0 , P (x0 , z )) 0, 0, K (x0 , P (x0 , z )) = 0 ,

(x0 , z ) W .

x0 = x(t0 ) , x1 = x(t1 ) , p = (x0 , x1 ) . , z ^ ^^ ^ ^^ ^ ^^ ^ , .. z = z () . ^^ 8. (x0 , z ) B . ^^ . : B (x0 , z ) , F0 (p) < F0 (p), ^ (1.35)

p = (x0 , P (x0 , z )) . x(§) = (x0 , z ) , v (§) = zk k (§) , t(§) dt = v , t(0 ) = t0 , d t0 . , ~ = (t( ), x( ), u ( ), v ( ) | [0 , 1 ]) ~ B , (x(0 ), x(1 )) = p. (1.33) B . (1.35) ~ ~ 7 . F0 , . B . .


26

1.

4. . í . 2 í . X = Rn , U X - , fi : U R, i = 0, . . . , k , g : U Rm . , fi g U . : f0 (x) min, fi (x) = (0 , . . . , k ) (Rk (1.36) :
+1 )

0, i = 1, . . . , k , g (x) = 0,

x U.

(1.36)

, (IRm ) .
k

L(x0 , , ) =
i=0

i fi (x) + g (x).

0 , . . . , k , , = (0 , . . . , k , ) = (, ) . , L = L(x, ). 9 ( ). x0 (1.36). = (0 , . . . , k , ) , 0 0, 1 0, . . . , k 0, (1.37)
k

i + | | > 0,
0

(1.38) (1.39) (1.40)

i fi (x0 ) = 0, i = 1, . . . , k , Lx (x0 , ) = 0.

, (1.37) , (1.38) , (1.39) , (1.40) x . . x0 ( ), , .

5. B . . 8, (x0 , z ) ^^ B . , . . B L = 0 F0 + F + K - ²z ,
0

, (Rk ) , ² (Rr ) , (Rn )

-.

l(x0 , x1 ) = 0 F0 (x0 , x1 ) + F (x0 , x1 ) + K (x0 , x1 )


1.4. B . L(x0 , z ) = l(x0 , P (x0 , z )) - ²z .

27

l L 0 , , ² . x0 , z (x0 , z ) ^^ ^ L
x0

= ^x0 + ^x1 Px0 , l l ^

^ Lz = ^x1 Pz - ² . l ^

( "", ). , (x0 , z ) : ^^ 0 0, 0, ² 0, (1.41) 0 + || + | | + |²| > 0, F (p) = 0, ²z = 0, ^ ^ ^x + ^x P = 0, l 0 l 1 ^x0 ^x P - ² = 0. l 1 ^z (1.42) (1.43) (1.44) (1.45)

0 + || + | | = 0 , lp (p) = 0 , (1.45) ² = 0 . , ^ (1.42) 0 + || + | | > 0 . , , 0 + || + | | = 1 . : 0 0, 0, ² 0, 0 + || + | | = 1, F (p) = 0, ²z = 0, ^ ^ (1.46) ^ + ^ P = 0, ^ P - ² = 0. lx0 lx1 ^x0 lx1 ^z , L(x0 , z ) (x0 , z ) , ^^ L (x0 , z )(x0 , z ) = ^x0 x0 + ^x1 (P ) (x0 , z )(x0 , z ) - ²z = 0 ^^ïï lï l ^^ïï ï (1.47)

x0 , z . , , , dL(x0 , z ) = 0 . ïï ^^ , 5 P (x0 , z ) ^^ ^ (P ) : (x0 , z ) x(1 ), ïï ï x(§) ï dx ï = v fx x + ï d
fx ( ) = fx (x ( ), u ( )), f ( ) = f (x ( ), u ( )).

zk k f , x(0 ) = x0 . ï ï ï

(1.48)

(1.49)

^ , Px0 x0 x(1 ) , x(§) ï ï ï dx ï = v fx x, ï d x(0 ) = x ï ï
0

(1.50)


28

1.

^ ( (1.48) z = 0 ). Pz ï z x(1 ) , x(§) ï ï ï dx ï = v fx x + ï d ( (1.48) x0 = 0 ). ï zk k f , ï x(0 ) = 0 ï (1.51)

6. . (1.47) ^x1 P (x0 , z )(x0 , z ) = ^x1 x1 . l ^^ïï lï x0 , z . ï ï . = [t0 , t1 ] x = A(t)x + v (t), ï ïï (1.52)

A(t) í . x(t) x0 Rn v (t) L , ï ï ï , , . l1 Rn , l1 x1 ( x1 = x(t1 ) ) x0 , u . ï ï ï ï ï (t) ( ) = - A (1.53)

(t1 ) = l1 . (t0 ) = 0 , (t1 ) = 1 . 2. (1.52) (t1 ) = l1
t
1

l1 x1 = -0 x0 - ï ï

v dt. ï
t
0

(1.54)

. (1.52) (1.53) d ( x) = - Ax + (Ax + v ) = v , ï ï ïï ï dt 1 x1 - 0 x0 = ï ï 1 = -l
1 t t1

v dt, ï
0

.

.
A = v ( )fx ( );

v= ï

zk k f ( ), ï

l1 = ^x1 . l





- d = v fx d (1.55)


1.4. B .

29

(1 ) = -^x1 . l

(1.56)

2 , ^x x0 + ^x dP (x0 , z ) - ²z = 0 l 0ï l1 ^^ ï ^x x0 - (0 )x0 - l 0ï ï
k

zk ï

k

f d =
k

²k zk . ï

(1.57)

x0 Rn zk k = 1, . . . , s. , ï ï , ²k 0 , k , k M0 , ²k = 0 , k M+ . (1.57) zk = 0 , ï ^x = (0 ). l0 (1.56) . x0 = 0 (1.57). k M+ ï f d = 0,

k

k M0 f d

k

0.

10. = (0 , , , ) , 0 0, 0, F (p) = 0, ^ 0 + i + | | = 1, - d = v f , ( ) = ^ , ( ) = -^ , lx0 lx1 0 1 x d (1.58) f d = 0, k M+ , k 0, k M0 , k = 1, . . . , r. k f d ( ) : [0 , 1 ] Rn í . 7. B (t) . . . k [ , ], [0 , 1 ] , k = 1, . . . , s , 01 M0 =
s

k ,

M+ = [0 , 1 ]\M0 . k M0

() M+ 1 , . . . , r . v ( ) M0 M+ . v ( ) = zk k ( ), ^

k=1


30

1. k , zk ^ zk = 0 ; k M+ , zk = 1 . , t ( ) ^ ^ v , ^ t (0 ) = t0 , ^ t (1 ) = t1 .

k 0 1. k M0 , , dt = d ^^ (t) : [t0 , t1 ] .

[0 , 1 ] t ( ) , ^ ^^ t = [t0 , t1 ]. (1.59)

t ( (t)) = t = (0 , , , )

, (1.58) 10. (t) = ( (t)) ^ t . (1.60)

2, - , - d (t) = (t)fx (x ( (t)), u ( (t))). dt d = v fx , d
fx = fx (x , u )

^^ N+ = [t0 , t1 ]\{t1 , . . . , ts }. u ( ) = u(t ( )) M+ (t) ^ N+ M+ , N+ : u ( (t)) = u(t ( (t))) = u(t). ^ ^ ^^ , .. [t0 , t1 ] u ( (t)) = u(t). , ^ x ( (t)) = x(t ( (t))) = x(t) ^ ^ , - d ^ ^^ = fx (t) .. [t0 , t1 ], dt ^ fx = fx (x, u). ^^
def (1.59) def (1.59)

^^ t [t0 , t1 ].

2: ^ ^ (t0 ) = (0 ), (t1 ) = (1 ). , ^ ^ (t0 ) = ^x0 , - (t1 ) = ^x1 , l l ^x = lx (p), l0 0^ p = (x0 , x1 ), ^ ^^ ( )f (x ( ), u ( )) d = 0,

k

^x = lx (p), l1 1^ x0 = x(t0 ). ^ ^^

x0 = x(t0 ), ^ ^^

(1.61)


1.4. B .

31

k M+ . k M+ . t (k ) í (), N+ . + . (1.61) = (t) . d (t) def ^ = 1 N+ , ( (t)) = (t) , dt ^ x ( (t)) = x(t) , u ( (t)) = u(t) N+ , , t (k ) , (1.61) : (t)f (x(t), u(t))dt = 0. ^ ^

+

+ N+ . ( )f (x ( ), u ( ))d

k

0,
k (tk ) = 0 .

(1.62)

kk k M0 . , k = [0 , 1 ], ( ) = tk k , t def

x ( ) = x(t ( )) = x(tk ) k . ^ ^ , - , k d = fx = 0 k d
def

k ( ) = (0 ) = ( (tk )) = (tk ).

, u u ( ) uk k . , (1.62) (tk )f (x(tk ), uk ) 0, k = 1, . . . , s. ^ , 10 : 11 ( ). = (0 , , , ) , (a) 0 0, 0; 0 + || + | | = 1; (b) (c) (d) (e) F (p) = 0; ^ ^ - = fx ; ^ (t0 ) = ^x0 ; l ^ - (t1 ) = ^x1 ; l 0, k = 1, . . . , s;
§

(tk )f (x(tk ), uk ) ^

(f ) + , N+ , ^ f dt = 0.
+


32

1. , , (x, u) . ^^ , (a)í(f ), M .

4. M



, = (0 , , , ) (0 , , ) (1.63)

M , (t) 0 , , . . , (0 , , ) l = 0 F0 + F + K , ^ - = fx , ^ (t0 ) = ^x0 . l

, (1.63) M . M 0 +||+| | = 1 . . , M M . 8. . B . , ^^ (tk , uk ), k = 1, . . . , s , tk , tk [t0 , t1 ], uk U . : 1 < 2 ( 2 1 ), ( tk , uk ) . , . 1 2 , , . k , uk ) (tk , uk ) . (t 3 . 3 = 1 2 . , 1 < 3 2 < 3 . , . M . , "" M : 1 < 2 M 2 M 1 . , , , M . , M 1 , . . . , M r , = 1 2 . . . r , M , M k ( k < ), , . M= M ,


. M . = (0 , , , ) M . (a)í(f ) ^^ . t = tk [t0 , t1 ], u = uk U k , uk ) . () : (t H (x(t), u, (t)) = (t)f (x(t), u) ^ ^ 0.

^^ , t [t0 , t1 ] u U , (vi). .


1.5. A .

33

3. h(t) : [t0 , t1 ] R . [t0 , t1 ] : h(t)dt = 0. h(t) = 0 .. [t0 , t1 ].
t

. y (t) =
t
0

h( ) d . -

. , y (t) = h(t) , y (t) 0 , h(t) = 0 .. ^^ [t0 , t1 ] , N+ . , (f ), ^ f dt = 0


^^ [t0 , t1 ].

3 : ^^ H (x(t), u(t), (t)) = (t)f (x(t), u(t)) = 0 .. [t0 , t1 ]. ^ ^ ^ ^ (vii). (, , , ) . B . , 6.

1.5

A .
F0 (p) min, F (p) 0, K (p) = 0, x = f (t, x, u), u U,

A :

p = (t0 , x0 , t1 , x1 ) , x0 = x(t0 ) , x1 = x(t1 ) , t0 , t1 , F Rk , K Rq , x Rn , u Rm , f Rn . , f t , F0 , F K t0 , t1 . = (x(t), u(t) | t [t0 , t1 ]) A . . l(p) = 0 F0 + F + K, 0 R, (Rk ) ,


(Rq ) , t R

H (t, x, u) = x f + t , x (Rn ) ,

( l 0 , , H ). , x t - , x t . ( x x , , fx ; x t ; , . , , .) 1.1. , A , 0 , (Rk ) , (Rq ) x (t) : [t0 , t1 ] (Rn ) , t (t) : [t0 , t1 ] R , :


34

1.

(I) (II) 0 + || + | | > 0; (III) i Fi (p) = 0, (IV) -x (t) = Hx (t, x(t), u(t)), -t (t) = Ht (t, x(t), u(t)); (V) t (t0 ) = lt0 (p), x (t0 ) = lx0 (p), (VI) H (t, x(t), u) 0 t [t0 , t1 ], u U; -t (t1 ) = lt1 (p), -x (t1 ) = lx1 (p); i = 1, . . . , k , p = (t0 , x(t0 ), t1 , x(t1 ));
0

0,



0;

(VII) H (t, x(t), u(t)) = 0 12. .. [t0 , t1 ].

A , .

. = (x(t), u(t) | t [t0 , t1 ]) A . 0 = t0 , 1 = t1 , t( ) = . = (t( ), x( ), u( ) | [0 , 1 ]) A : F0 (p) min, F (p) 0, K (p) = 0

p = (t(0 ), x , B

dt dx = f (t, x, u), = 1, d d (0 ), t(1 ), x(1 )). 1, 4. A B . : 0 , , x ( ) : [0 , 1 ] (Rn ) , t ( ) : [0 , 1 ] R

, 0 0, - 0, 0 + || + | | > 0, F (p) = 0, dx dt = Hx , - = Ht , d d x (0 ) = lx0 , -x (1 ) = lx1 , t (0 ) = lt0 , -t (1 ) = lt1 , (1.64)


1.6. H (t( ), x( ), u) 0, [0 , 1 ] u U ; .. [0 , 1 ],

35

H (t( ), x( ), u( )) = 0,

l = 0 F0 + F + K, H = x f + t , l (t(0 ), x(0 ), t(1 ), x(1 )) , H (t( ), x( ), u( )), [0 , 1 ]. (0 , , , t (t), x (t)). , A . .

1.6

.

1. . J = F0 (p) min, F (p) 0, K (p) = 0, x = f (t, x, u), u U, (1.65)

p = (x(t0 ), x(t1 )) , = [t0 , t1 ] ^ t0 - t0 = 0,

. A ^ t1 - t1 = 0,

: ï = 0 (t0 - t0 ) + 1 (t1 - t1 ) + l, ^ ^ l l = 0 F0 + F + K. t t (t0 ) = ït0 = 0 , l , 0 + || + | | > 0 (1.66) . , 0 + || + | | = 0. 0 = 0, = 0, = 0 , , ïx0 = lx0 = 0. x (t0 ) = ïx0 = 0 -x = x fx , l l x 0 , x f + t = 0 .. t = 0 , , 0 = 1 = 0 . , (1.66). t 0 1 . -t = Ht , , , . (1.65) , . (x(t), u(t) | t ) . 0 , , t ,
0

-t (t1 ) = ït1 = 1 . l

0,



0,

0 + || + | | > 0, .. ,

F (p) = 0 ,

(1.67) (1.68)

- = Hx ,


36

1. (t0 ) = lx0 ,
ï uU

-x (t1 ) = l

x1

,

(1.69) (1.70)

max H (t, x(t), u, (t)) = H (t, x(t), u(t), (t)) .. ,

t . H (t, x, u, ) = f (t, x, u) ( x ). (1.70) (-t (t)) , , , , t -t (t) = Ht (t, x(t), u(t), (t)) .. . 2. . x = f (x, u), u U.

f t . . H (x, u, x ) = x f (x, u) t. t -t = Ht = 0, , t = const . , max H (x(t), u, x (t)) = const = H (x(t), u(t), x (t)).
ï uU

,

.

3. .
t1

J=
t
0

F (x, u)dt min,

x = f (x, u),

u U,

x(t0 ) = a,

x(t1 ) = b,

(1.71)

= [t0 , t1 ] . , F, Fx , f , fx x, u. : J = y1 - y0 min, x = f (x, u), x0 - a = 0, y = F (x, u), x1 - b = 0, uU.

l = 0 (y1 - y0 ) + 0 (x0 - a) + 1 (x1 - a),

H = x f (x, y ) + y F .

t0 , t1 , : 0 -y = 0, -x = Hx ,
ï uU

0,

0 + |0 | + |1 | > 0, y (t1 ) = 0 , -x (t1 ) = 1 ,

(1.72) (1.73) (1.74) (1.75)

y (t0 ) = -0 , x (t0 ) = 0 ,

max H (x(t), u, x (t), y (t)) = const = H (x(t), u(t), x (t), y (t)), (1.75) ,

.


1.6.

37

(1.73) , y -0 . H = x f (x, y ) - 0 F (x, u). , 0 = 0 , 0 = 0 . x (t0 ) = 0, -x = x fx , x 0 , x (t1 ) = -1 = 0 . 0 + |0 | > 0, 0 + |x (t0 )| > 0. x = H = f (x, u) - 0 F (x, u) = H (x, u, , 0 ). (1.76)

(x(t), u(t) | t ) (1.71) : 0 (t) , 0 0, 0 + | (t0 )| > 0, (1.77) (1.78) (1.79) .

- = Hx (x(t), u(t), (t), 0 ) .. , max H (x(t), u, (t), 0 ) = const = H (x(t), u(t), (t), 0 ),
ï uU

(1.79) ,

4. . : T min, x(0) = a, x(T ) = b, x = f (x, u), u U. : t1 min, t0 = 0, x0 - a = 0, u U. x1 - b = 0, (1.80)

x = f (x, u),

l = 0 t1 + t t0 + 0 (x0 - a) + 1 (x1 - b), H = x f (x, y ) = H (x, u, x ). :
0

0,

0 + |t | + |0 | + |1 | > 0, x (t0 ) = 0 , t (t0 ) = t , -x (t1 ) = 1 , -t (t1 ) = 0 , t , .. .

(1.81) (1.82) (1.83) (1.84) (1.85)

-x = Hx , -t = 0,
ï uU

max H (x(t), u, x (t)) + t (t) = 0, H (x(t), u(t), x (t)) + t (t) = 0,

(1.83) , t -0 . , x (t0 ) x = x fx (x, u) , x 0. 0 = 1 = 0 - (1.85) , t = 0 . 0 = 0 (1.81). , |x (t0 )| .

= 0 . x f = 0 . t = 0 (1.83). > 0


38

1.

, (x(t), u(t) | t [0, T ]) (1.80) : = [0, T ] (t) , | (0)| > 0, - = Hx (x(t), u(t), (t)), 0
ï uU

(1.86) .. , .. . (1.87) (1.88)

max H (x(t), u, x (t)) = H (x(t), u(t), x (t)),

, max H (x(t), u, x (t)) const
ï uU

.

(1.89)

(1.88) t . , (1.71) (1.80), [1], .

1.7



B ( , B ). . : ? , . , : , , . , , , . , , A . , , , . . , , . , , , , , . , , .. .. . ( , "" , -, , , -, "" .)


1.7.

39

A. , [t0 , t1 ]. Q ( ) . 1.1. , x(t), u(t) ^ ^ , xk (t), uk (t), k = 1, 2, . . . , k ||xk - x||C 0, ^ ||uk - u||1 0, ^ (1.90)

||uk - u|| O(1), ^ k J (xk , uk ) < J (x, u). ^^

, , (x, u) (1.90). ^^ ( U , (1.90) , ; .. x C + u 1 .) . 1.2. (x(t), u(t)) ^ ^ , N > 0, (x, u), ||x - x||C < , ^ ||u - u||1 < , ^ ||u - u|| N , ^ (1.91)

J (x, u) J (x, u). ^^ , N (x, u) ^^ x C + u 1 ||u - u|| N . ^ , . (1.91) ||x - x||C < , ^ ||u - u|| < , ^

(x, u) , (1.91), (x, u) ^^ . (1.91) ||x - x||C < , ^ (x, u) , (1.91), (x, u) ^^ . .


40

1.

(x, u) , ^^ xk (t), uk (t), k = 1, 2, . . . , ||xk - x||C 0, ^ k J (xk , uk ) < J (x, u). ^^ ||xk - x||C 0. ^ ||uk - u|| 0, ^ (1.92)

(1.92) : (1.93)

(1.90), (1.92) (1.93), , : = = . , , .. (x, u) ^^ , , , . ( !) , A , . , (xk , uk ) ^ ^^ k = [tk , tk ], = [t0 , t1 ], 01 k (.. ^ (1.90) k , uk ) (x, u) ). (x ^^ , " " , Q, (1.90) : Q ( ), k (t, xk (t), uk (t)) k .

, Q , (1.90). 12, , . 13. (x, u) ^^ , . , ( , ). , , . : , , - . . , (xk , uk ) (x, u), (1.90). ( , , ^^ u; x .)


1.7.

41

4. (x, u) , . , , , ( M , ), , , .. : M = M (. [10, 11]). M, M = M. .

, , , . , , , , . . , ( ) , ( , í) .


2

. í
2.1 .

1. . X , Y , A : X Y . , A . BX , BY X Y . 14 ( ). A : X - Y . a > 0 , A(BX ) aBY ,


(2.1)

. . X a Y ( ). ., , [10, 11].


1 ( ). A : X - Y -1 : Y X , A .

A

(2.1) a > 0 . , a > 0 . : A(BX (x, r)) BY (Ax, ar), x X, r > 0,

BX (x, r) X r x (!). . 2. . (X, dX ) , (Y , dY ) , F : X Y , BX (x, r) r x X , BY (y , r) r y Y . U X . 42


2.1. 2.1. , F U a > 0, BX (x, r) U F (BX (x, r)) BY (F (x), ar).

43

(2.2)

a > 0, a < a. a, , F U . 2.2. S : X Y ( ) U b > 0, BX (x, r) U F (BX (x, r)) BY (F (x), br). (2.3)

, S : X Y U b , dY (S (x1 ), S (x2 )) bdX (x1 , x2 ) x1 , x2 U. (2.4) (, b 0.) , U , U . , , U b , U c . ( ?) , "" , . , .. ( ). 15 ( ). (X, d) , (Y , § ) , U X . T : X Y U a > 0 U , S : X Y U b > 0. a > b . F =T +S :X Y U a - b . . BX (x0 , R) R > 0 x0 X. BX (x0 , R) U. - X , Y . 5 ( ). T : X Y BX (x0 , R) a > 0 , S : X Y BX (x0 , R) b > 0 , a > b . F = T + S F (BX (x0 , R)) BY (F (x0 ), (a - b)R). (2.5) (F (x) := T (x) + S (x) x X)


44

2.

. a = 1, b < 1. (1 , Y § = a § , a .) X Y , X Y d . F (x0 ) = y0 . y B (y0 , (1 - b)), .. d(y , y0 ) ^ ^ (1 - b). , x B (x0 , ), F (x) = y . ^ ^ ^ x {xn }, ^ . r = (1 - b). T (x0 ) + S (x0 ) = y0 , (2.6)

y . T (x0 ) = y0 - S (x0 ) ^ 1- T . d (y - S (x0 ), y0 - S (x0 )) = d (y , y0 ) r, ^ ^ x1 B (x0 , r), T (x1 ) = y - S (x0 ), .. ^ T (x1 ) + S (x0 ) = y . ^ (2.7)

S (x0 ) S (x1 ). S b B (x0 , r), d(S (x1 ), S (x0 )) br, T (x1 ) + S (x1 ) = y1 , (2.8)

d(y , y1 ) br. ^ , (2.6) "" x0 (2.8) "" x1 , d(x0 , x1 ) r, d(y , y1 ) br. ^

(2.8) y1 y . ^ 1 = , r + br < r (1 + b + b2 + . . .) = r 1-b B (x1 , br) B (x0 , ); 1- T b- S, x2 B (x1 , br), T (x2 ) + S (x2 ) = y2 , d(y , y2 ) b2 r. ^

, xn , yn , F (xn ) = T (xn ) + S (xn ) = yn , (2.9) d(xn d(x0 , xn ) + bn r d(x0 , x1 ) + d(x1 , x2 ) + . . . + d(x
n-1 -1

, xn ) bn

-1

r,

d(y , yn ) bn r. ^

(2.10)

, xn ) + bn r


2.1. r + br + . . . + bn-1 r + bn r < r 1 = , 1-b

45 (2.11)

.. B (xn , bn r) B (x0 , ), . (, , (2.10), n .) xn . (2.10) , , X . x. (2.11) , d(x0 , x) , .. x B (x0 , ). ^ ^ ^ (2.10) , yn y , (2.9) ^ F F (x) = y , . ^ ^ . T (x) + ^ S (x) = y , (2.6). , ^ ^ : T , í S. T "" (2.7) x1 , S "" ; (2.8), , , .. .. . , T , , -, .. (2.7) , , . , U . 3. . (X, d) (Y ( , d , , , F : X Y . O (x) > 0 x , B (x, r) - X r 0 x . , d) ), X

2.3. F : X Y k > 0 X ½ Y , d x, F
-1

(y )

k d F (x), y

x X,

y Y .

(2.12)

1. X a > 0 1 X ½ Y k = a . . F : X Y X a > 0 . x X , y Y . r = d F (x), y : r F B (x, ) a B F (x), r .


46

2.
r a

y B F (x), r , xy B x, d(x, xy ) xy F d x, F
-1 -1

, F (xy ) = y .

r 1 = d F (x), y . a a
-1

(y ) , d x, F (y ) d(x, xy ), 1 d F (x), y . a y . -1 (y ) = + (-

(y )

, x F F (x) y y = 0 , . (2.12) F y . F -1 (y ) = , d x, F -1 (y ) + ).

"" " " " ". x0 X , y0 = F (x0 ) . 2.4. F a > 0 x0 , U x0 , F a , . . F (B (x, r)) B (F (x), ar) B (x, r) U. 2.5. F k > 0 x0 , U x0 X V y0 = F (x0 ) Y , , d(x, F
-1

(y ))

k d(F (x), y ) x U, y V .

. " (x0 , y0 ) X ½ Y " . 16. F x0 1 a > 0, k = a x0 y0 = F (x0 ). , x0 , : F k > 0 x0 y0 = F (x0 ), x0 1 a < k . . (=) a = 1 ( Y d(§, §) d(§, §)/a ). F 1 - x0 . = /3 , - x0 y0 = F (x0 ) 1 . x B (x0 , ), y B (y0 , ). F (x) = y , d(y , y ) = r. , d(x, F -1 (y )) r . , , x B (x, r) , F (x ) = y . (2.13)


2.1. : ) r 2 ) r < 2 (y y ""). (y y "" ),

47

) , 1- F B (x0 , ) B (y0 , ), x B (x0 , ) , F (x ) = y . d(x , x) d(x , x0 ) + d(x0 , x) + r, (2.13) . ) B (x, r) - x0 ( x x0 , r + d(x, x0 ) r + < 3 = ), 1- F (B (x, r)) B (y , r). y B (y , r) (. r ), (2.14) (2.13), ... (=) k = 1. , F x0 > 0, > 0 1 O (x0 ) O (y0 ) x0 y0 = F (x0 ) . , , , 0 < < /3 F (O (x0 )) O/3 (y0 ). (2.15) (2.14)

, F O (x0 ) a < 1. B (x, r) O (x0 ). , r 2 (.. x d(x , x) d(x , x0 ) + d(x0 , x) < 2 ). F (x) = y r < r. , F (B (x, r)) B (y , r ), a- F ( a < 1 ). y B (y , r ). (2.15) d(y0 , y ) d(y0 , y ) + r < + 2 < , 3 (2.16)

y O (y0 ). x O (x0 ), x, y d(x, F -1 (y )) d(y , y ) r < r,

x F -1 (y ) , d(x, x ) < r. , x B (x, r) F (x ) = y , (2.16) . . 4. . 1 . . X , Y , U x0 X, g : U Y ().


48

2.

2.6. g x0 , A : X Y , g (x0 + h) = g (x0 ) + Ah + r(x0 , h), r(x0 , h) = o( h ).

A g x0 . A = g (x0 ). , Y = R1 , x0 . g x0 , g (x) x0 g (x) - g (x0 ) 0 x - x g A : X Y , x1 , x2 , x1 - x0 g (x2 ) - g (x1 ) - A(x2 - x1 )
0

0.

x0 , > 0 > 0 , < , x2 - x0 < x2 - x1 . (2.17)

, . , - : > 0 > 0 , x - x0 < , g (x) - g (x0 ) - A(x - x0 ) x - x0 .

( (2.17) x1 , x2 x0 .) , , A = g (x0 ) . , . . 17 ( ). X , Y , U X , [a, b] U , f : U Y , x [a, b]. f (b) - f (a) sup
x[a,b]

f (x)

b-a .

. , [a, b] f , (. . ). . [16], . 148, . , , . g (x) x0 . (x) = g (x) - g (x0 )x. x x0 : (x) = g (x) - g (x0 ), , (x) 0 x - x0 0. g (x2 ) - g (x1 ) - g (x0 )(x2 - x1 ) = (x2 ) - (x1 )


2.1. sup
x[x1 ,x2 ]

49 (2.18)
0

(x)

x2 - x1 .

supx[x1 ,x2 ] (x) 0 x1 - x0 + x2 - x x0 . ,

0 , g

18. . 5. . g : U Y x0 U . g (x) = g (x0 + (x - x0 )) = g (x0 ) + g (x0 )(x - x0 ) + S (x), S (x) = g (x) - g (x0 ) - g (x0 )(x - x0 ). x1 , x2 U S (x2 ) - S (x1 ) = g (x2 ) - g (x1 ) - g (x0 )(x2 - x1 ). g x0 , > 0 > 0 , , S (x2 ) - S (x1 ) x2 - x1 x1 , x2 B (x0 , ), (2.20) (2.19)

. . S (, , ) > 0 B (x0 , ) . . x0 g : Y Y . 19 ( ). g x0 g (x0 )X = Y ( ). a > 0 x0 , g a ( g x0 1/a ). . g (x) = T (x) + S (x), T (x) = g (x0 ) + g (x0 )(x - x0 ) , S (x) (2.20). g (x0 )X = Y , T X a0 > 0 . , S (x) B (x0 , ) > 0, > 0. a0 - > 0. g = T + S a0 - x0 . . , a < a0 , a g (x0 ) .
0




50

2.

6. . X,Y , U X , g : U Y , M = {x U | g (x) = 0} = g -1 (0) . 20 ( ). x0 U , x0 g (x0 )X = Y O(x0 ) x0 k > 0 , x O(x0 ) x ï k g (x) , , , d(x, M ) k g (x) g (x0 ) = 0 (. . x0 M ), g . x X , g (x + x) = 0 ; ï ï .

. g O(x0 ) x0 . , k0 > 0 , d(x, M ) k0 g (x) x O(x0 ) (2.21)

( y = 0 ). k > k0 , k . g (x) = 0 , x = 0 , . ï g (x) > 0 . (2.21) , x1 M , x1 - x k g (x) , x = x1 - x . ï

7. . X , M , M X , x0 M M. 2.7. x () M ï x0 , d(x0 + x, M ) = o() 0+ . ï Tx0 (M ) x0 . , K X , x K , > 0 , x K . K , x, y K , x + y K. (). x Tx0 (M ) > 0 , x Tx0 (M ). , Tx0 (M ) ï ï ( ), , . . 10 . x0 int M , Tx0 (M ) = X . 20 . M = {x Rn | x 0} . M . 30 . M = {(x, y ) R2 | x2 + y {(x, y ) R2 | x 0}
2

1} . (1, 0)

40 . M = {(x, y ) R2 | x2 + y 2 = 1} . (1, 0) {(x, y ) R2 | x = 0} . 21 (. . ). X , Y , U X , g : U Y , x0 U , g (x0 ) = 0 . M = {x U | g (x) = 0}.


2.2.

51

g x0 g (x0 )X = Y ( ). Tx0 (M ) = {x X | g (x0 )x = 0}, ï ï . . , . 5. Tx0 (M ) K er g (x0 ). ~ . x Tx0 (M ) . 0 > 0 x() : ï (0, 0 ) X , g (x0 + x + x()) = 0 ï~ (0, 0 ), x() = o(). ~

g x0 , g (x0 ) + g (x0 )(x + x()) + r() = 0, ï~ r() = o() . g (x0 ) = 0 , , g (x0 )(x) + o() = 0 . ï +0 , , g (x0 )x = 0 , ï x K er g (x0 ) . ï . . K er g (x0 ) Tx0 (M ). g (x0 )x = 0 . ï g (x0 + x) = g (x0 ) + g (x0 )x + r() = r(), ï ï d(x0 + x, M ) ï r() = o() . k r() = o(). , x Tx0 (M ) . ï

, "", " " .

2.2



1. X , x1 , x2 X . [x1 , x2 ] X x1 , x2 , . . , x = x1 + x2 , 0 , 0 , + = 1 . , M X , . X X , x : X R . x


= sup x , x
x 1

X . X A , B X .

.

2.8. x X , x = 0 A B , sup x , x
xA xB

inf x , x

(2.22)


52

2.

c , (2.22). {x | x , x = c} A B ( A B ) , A {x | x , x c}, B , x {x | x c}. , . 22 ( ). A , B . A B = . , A B . í " ". , , [11]. M X , x X


.

2.9. x M , inf x (M ) = inf x , x > -.
xM

, M {x | x, x a} a . M M . , M . . 1. , M . 2. , K , x K x , x 0 x K.

K K . 3. , K , x K . 4. , , c , x -x . 5. L X í . , x , L , "" L , .. x , x = 0 x L . , , L L . ( ). , . 6. A B X . , , x y , x A , y B , x + y = 0, inf x (A) + inf x (B ) 0.

, , . , .


2.2.

53

2. í . 23 ( ). 1 , . . . , k , í X , 1 , . . . , k . 1 . . . k = , x , 1 1 , x + . . . + x + x = 0 . 1 k (2.24) ..., x , k k x , (2.23)

(2.24) 1 , . . . , k , . , , , í , ( , ) . ( ) í. . í . X1 , X2 , . . . , Xn , X = X1 ½ . . . ½ Xn . x X x Xk , x = (x1 , . . . , xn ) X ^ ^ k (x , x) = x , x1 + . . . + x , xn ^^ ( ). 1 n 23. . 1 . . . k = . ^ X = X ½ . . . ½ X = X k : k K0 = 1 ½ . . . ½ k , K = {x = (x1 , . . . , xk ) | x1 = . . . = xk = x }. ^ , K "" , K0 K = . (2.25) , x K0 K. x K0 , x = (x1 , . . . , xk ) , ^ ^ ^ x1 1 , . . . , xk k . x K , x1 = . . . = xk = x . , ^ x 1 . . . k , . K0 K , K0 . (2.25) x (X k ) , ^ x , K ^
0

½ . . . ½ = k . k

0,

x , K 0 , ^

x = 0 . x = (x , . . . , x ) , x X , i = 1, . . . , k , ^ ^ 1 i k , K x 0 0 , x1 , x1 + . . . + xk , xk 0 x1 1 , . . . , xk k .


54

2.

( ), x , . . . , x . 1 1 k k x , . . . , x , x = 0. ^ 1 k x , x , K 0 , x + . . . + x , x 0 x . ^ 1 k = -(x + . . . + x ) . x x + . . . + x + x = 0 . 1 1 k k

. x , . . . , x , x , (2.23) (2.24). 1 k 1 , . . . , k , , x ~ . x , . . . , x x0 . x0 , x > 0 , 1 i i~ k x int i0 . x : x , x 0 . , x , x 0 . ~ ~ ~ i i k + x , x > 0 , (2.24). C ~ i=1 xi , () , . 24 (í). M1 , . . . , Mk , M - X, M1 , . . . , Mk . M1 . . . Mk M = (x , . . . , x , x ) X 1 k , x + . . . + x + x = 0 , (2.26) 1 k inf x , M1 + . . . + inf x , M 1 k
k

+ inf x , M 0 .

(2.27)

, . X k A x = (x1 - x, . . . , xk - x) , x1 M1 , . . . , xk Mk , x M . ~ M1 , . . . , Mk M , A . , A . , x = (x , . . . , x ) , ~ 1 k A : x , x > 0 x A, ~ ~ x , x 1
1

+ . . . + x , x k

k

+ -x - . . . - x , x > 0 1 k

x1 M1 , . . . , xk Mk , x M . x = -x - . . . - x . , 1 k (x , . . . , x , x ) (2.26), (2.27). 1 k .

3. . L X . L = {x X | x , x = 0 x L} L. , x L L , (!). , L = L . , L X , .. . 6. L X . , X. L -


2.2.

55

. ([16], . 127) x X L. B (x, ) X L > 0. B (x, ) L . x L , x = 0 ( !). , L . (.)

4. . 7. X , Y , Z , A : X Y , B : X Z . AX = Y B (Ker A) ( A B ) Z. "" T : X Y ½ Z, x (Ax, B x) , . . {xn } X , A xn = yn , B xn = zn , (yn , zn ) (y0 , z0 ) .

, x X , Ax = y0 , B x = z0 . AX = Y , x0 X , Ax0 = y0 . xn = xn - x0 , A xn = yn - y0 0, B xn = zn - B x0 z0 - B x0 .

z0 - B x0 = z0 . , , A xn 0 , B xn z0

(., y0 = 0 ), x X , Ax = 0, B x = z0 ( A(x + x0 ) = y0 B (x + x0 ) = z0 ). A "", , xn 0 , A xn = Axn , .. A( xn - ï ï xn ) = 0. B ( xn - xn ) z0 ( z0 ). , ï ï xn = xn -xn Ker A, B xn z0 . ï B Ker A , , x Ker A , B x = z0 , . . , AX = Y , Y1 := AX Y . A : X - Y1 , , T = (A, B ) . , . 2. A : X Y "", B : X Z (dim Z < ) . T (x) = (Ax, B x) Y ½ Z . , B (Ker A) .


56

2. 5.

8. A : X Y

"". (K er A) = A Y ,

A . , x , Ker A, x , x = y , Ax , y - X Y (.. y Y ) . , Ker A ( A ). . ) , A Y (Ker A) . , x = A y , . . x , x = y , Ax x X. x , x = y , Ax = y , 0 = 0 x Ker A,

. . x (Ker A) . ( AX = Y .) ) , (Ker A) A Y . x Ker A. T : X Y ½ R, x (Ax, x , x ) ,

X Y ½ R . , Y ½ R. Y ½ R , (0, 1) , Ax = 0 , x , x = 0. 6 T X , . . y Y c R , y , Ax + c x , x = 0 x X,

y + |c| > 0 . , c = 0 . c = 0 , y , Ax = 0 x X . y , y = 0 y Y ( AX = Y ), .. y = 0 , . , c = 0 . 1 x X , x , x = - y , Ax c
.. x , x = y1 , Ax , y1 = - 1 y Y , . c

2.3



1. . . ([16], . 137) X, Y - , U X , x0 U , F : U Y , h X (). 2.10. lim F (x0 + h) - F (x0 ) ,

+0


2.3.

57

Y . F x0 h F (x0 ; h) . 2.11. F x0 F (x0 , h) h X, h F (x0 , h) . F (x0 ). F x0 . x0 , , h , F (x0 , h) = F (x0 )h. , x0 F (x0 ) , F (x0 ) , : F (x0 ) = F (x0 ). ; , R2 (. [16], . 141), , , . , x0 : . , g x0 ( , ) . , , g (x) = g (x0 )x + (g (x) - g (x0 )x), (g (x) - g (x0 )x) , x0 . , g x0 g (x0 ) , .

2. . X , f : X R , M X , x0 M . , f U x0 . 2.12. x0 f M , > 0 , x M , x - x0 < , f (x) f (x0 ) . f M . , f x0 .


58

2.

25. f x0 M x Tx0 (M ) x0 . ï x0 f M . f (x0 , x) ï 0 x Tx0 (M ). ï

. x Tx0 (M ). ï x() : (0, 0 ) X (0 > 0) , (0, 0 ) ~ x0 + x + x() M , ï~ x() = o(). ~

x0 M , f (x0 ) f (x0 + x + x()) ï~ > 0 . f x0 L > 0 , f (x0 + x + x()) ï~ , f (x0 ) f (x0 + x) + L x() = f (x0 + x) + o(). ï ~ ï

f (x0 + x) + o(). , ï f (x0 , x) = lim ï
+0

f (x0 + x) - f (x0 ) ï

0.

3. f x0 M , Tx0 (M ) . f M x0 , f (x0 )x = 0 ï x Tx0 (M ). ï

. x Tx0 (M ) ï . Tx0 (M ) , (-x) Tx0 (M ) . 25 ï f (x0 )x ï , f (x0 )x = 0 . ï 3. . X, Y - , U X - , f : U R - , g : U Y . : f (x) min, g (x) = 0. (2.28) g , U | g (x) = 0} 0 f (x0 )(-x) ï 0.

x0 U , .. g (x0 ) = 0. , f x0 (, , f g U , .. f C 1 (U ) g C 1 (U, Y )). , g (x0 )X = Y , M = {x x0 Ker g (x0 ) .


2.3.

59

x0 . 3, , f (x0 ), x = 0 ï x Ker g (x0 ). ï (2.29)

y Y , f (x0 ) + y g (x0 ) = 0, (2.30) y g (x0 ) = [g (x0 )] y


(.. y g (x0 ), x = y , g (x0 )x x X ).

, (2.29) (2.30). "" , . , , g (x0 )X . , , . , dim Y < +. g (x0 )X , , g (x0 )X = Y , , .. , : R y Y , , , f (x0 ) + y g (x0 ) = 0. (2.31)

, (2.31) = 1 , .. (2.30). , g (x0 )X Y . y = 0 , y , g (x0 )X = 0. (2.31) = 0. , , g (x0 )X = Y , . , . , . , , , (2.31) 0 , (-1) . L(x, , y ) = f (x) + y , g (x) , (2.31) : Lx (x0 , , y ) = 0, Lx ( ). .. 1934 . , : 26 (..). f , g (x0 ) = 0, g ( (2.28). y Y , + y g x0 U x0 )X Y . x0 R 0, > 0, (2.32) (2.33) (2.34)



Lx (x0 , , y ) = 0, L(x, , y ) = f (x) + y , g (x) .


60

2.

(2.32) , (2.33) (, y ) , (2.34) x . , , x0 , , . . 26 , Y = Rm , , g = (g1 , . . . gm ) - X . g (x)X . 4. . X , U X , fi : U R, i = 0, . . . , k , g : U Y , Y . , fi g U , , g (x)X Y x U. : f0 (x) min, fi (x) 0, i = 1, . . . , k , g (x) = 0. (2.35) ( ). = (0 , . . . , k ) R
k+1

, y Y . (2.35):
k

L(x, , y ) =
i=0

i fi (x) + y , g (x) .

0 , . . . , k , y , = (0 , . . . , k , y ) = (, y ) . , L = L(x, ). 27 ( ). x0 U (2.35). = (0 , . . . , k , y ) , 0 0, 1
k

0, . . . , k


0,

(2.36) (2.37) (2.38) (2.39)

i + y
0

> 0,

i fi (x0 ) = 0, i = 1, . . . , k , Lx (x0 , ) = 0.

: (2.36) , (2.37) , (2.38) , (2.39) x . , . . x0 U , .


2.3.

61

" ", .. x , f0 (x) < f0 (x0 ) . x0 I = {i {1, . . . , k } | fi (x0 ) = 0} {0}. (2.40)

9 ( ). x0 U (2.35). g (x0 )X = Y , .. g x0 . x X , : ï fi (x0 ), x < 0, i I , ï g (x0 )x = 0. ï (2.41) (2.42)

. , f0 (x0 ) = 0 , , , f0 (x) f0 (x) - f0 (x0 ) . x (2.41) (2.42). ï x ( (2.42)) g (x) = 0 x0 . ï , x() : (0, 0 ) X (0 > 0) , ~ g (x0 + x + x()) = 0, ï~ x() = o(). ~ (2.43)

x = x0 + x + x() . , {x } "" x0 . ï~ , g (x ) = 0 (0, 0 ), x - x0 0 ( +0). (2.44)

i I , i {1, . . . , k }, .. fi (x0 ) < 0 . fi x0 (2.44) , fi (x ) < 0 > 0 , i - . i I . fi (x0 ) = 0 fi (x0 ), x < 0 (2.41), ï fi (x ) = fi (x0 + x + x()) = fi (x0 ) + fi (x0 ), x + x() + o() = fi (x0 ), x + o(). ï~ ï~ ï > 0 , fi (x0 ), x < 0, fi (x ) < 0 > 0 , ï i = 0 i - , i = 0 f0 (x ) < 0 = f0 (x0 ) > 0. {x } , x0 . . . 6 ( ). l : X R . K = {x X | l, x > 0} . m K , .. m, x 0 x K. 0 , m = l. , , , . . , m, x = 0 x Ker l. , x0 Ker l , .. l, x0 = 0 x1 , l, x1 > 0 . x1 + x0 K R. , m, x1 + m, x0 0 R. m, x0 = 0 . , l, x = 0 m, x = 0. , , R , m = l. l, x > 0 , m, x 0 , , , 0.


62

2.

27. x0 (2.35). fi (x0 ) , i I , i 1, ( y ) , , (2.36)-(2.39) . , fi (x0 ) = 0 i I , (2.41) . . ) : g (x0 )X = Y . 9 (2.41) (2.42) . i = {x | fi (x0 ), x < 0}, i I , ï ï = {x | g (x0 ) = 0}. ï

i , i I . - x i , i I , x , , i I xi + x = 0. 6 x = -i fi (x0 ), i 0, i I , i x = [g (x0 )] (-y ) = -y g (x0 ), y Y ( ). , -
I

i fi (x0 ) - y g (x0 ) = 0.

i = 0 i I . (2.36)-(2.39) / . ( i , y , x , x i .) ) : g (x0 )X = Y . L = g (x0 )X Y , Y. y = 0, L. , y g (x0 ), x = 0 x X. i = 0 i = 0, . . . , k . = (0, . . . , 0, y ) (2.36)-(2.39) . ( , g (x0 )X = Y , .) . , . , , x0 fi , g , , , g (x0 )X Y .

2.4

.

1. . X , U X , f : U R , x0 U , x X ï (). ï f (x0 , x) f x0 x : ï ï
+0

lim

f (x0 + x) - f (x0 ) ï ï = f (x0 , x). ï


2.4. .

63

U fi : U IR, i = 0, 1, . . . , k , ï L, fi (x0 , x) ï x0 U x. ï M X, x0 M . Tx0 (M ) x0 M. Z : f0 (x) min, fi (x) 0, i = 1, . . . , k , xM. (2.45)

x0 U . , f0 (x0 ) = 0 ( f0 (x) - f0 (x0 ) ), x0 : I = {i {0, 1, . . . , k } | fi (x0 ) = 0}. i = 0 I , .. , , . 28. x x X , ï
0

Z , ï fi (x0 , x) < 0, ï x Tx0 (M ) . ï i I, (2.46) (2.47)

Z . x. , , (2.46), ï . {x | f0 (x0 , x) < 0} ïï ï x0 f0 , {x | fi (x0 , x) < 0} ïï ï i I , i = 0 x0 fi (x) 0. ( i I / X ). , Tx0 (M ) x0 x M . x0 . , , f0 (x) - f0 (x0 ) 0 , . , x0 . . , . , . 28. x , ï (2.46) (2.47). , x0 Z . (2.46) , i I fi (x0 , x) -a ï a > 0. x Tx0 (M ), x() : (0, 0 ) X , ï ~ x0 + x + x() M ; ï~ x() = o(). ~


64

2.

x = x0 + x + x(). x x0 ( +0) . i I ( ï~ / ). fi (x0 ) < 0 , , fi (x ) < 0 > 0 . , (0, 0 ) fi (x ) < 0 i I , / x U, x M . (2.48)

i I . fi U L , |fi (x0 + x + x()) - fi (x0 + x)| L x() = o(). ï~ ï ~ , , ï fi (x0 + x) - fi (x0 ) fi (x0 , x) + o() -a + o(). ï ï (2.49), (2.50) (2.46) : fi (x0 + x + x()) - fi (x0 ) -a + o() < 0 ï~ > 0 . , (0, 0 ). , (0, 0 ) fi (x ) < fi (x0 ), i I. (2.51) (2.50) (2.49)

(2.48) (2.51) x - x0 x0 Z . . , x, (2.46), (2.47), ï x0 , , s- . 2.13. , Z x0 s- , xn x0 , n xn M , fi (xn ) < 0 i = 0, 1, . . . , k . (, , fi (xn ) < 0 , .) s- x0 , x0 x M , fi (x) < 0, i = 0, 1, . . . , k . , x0 , s- . , s- . f0 fi , . , , s- , . , , fi , i = 0, 1, . . . , k ,


2.4. .

65

s- , x0 . s- , .. , s- , ( x0 ), . , 28 . 29. x0 s- Z , , .. x X, (2.46), ï (2.47). Z . . , () fi U (i = 0, . . . , k ) ; () fi x
0

x (i I ) ; ï

() M g (x) = 0, g : U Y - , x0 ; () g x
0

: g (x0 )X = Y .

Tx0 (M ) = {x | g (x0 )x = 0}. ï ï , Z1 : f0 (x) min, fi (x) 0, i = 1, . . . , k , g (x) = 0. (2.52)

, . 30. Z1 x0 ( s- ). : x = 0, ï fi (x0 , x) < 0, ï g (x0 ) x = 0. ï i I, (2.53) (2.54)

. , .. , . , fi (x0 , x) , x . ï ï , (2.53), , (2.53) (2.54) . . , , , . 40 , .


66

2.

2. . X . : X R , : (a) (x) = (x) x X, > 0, (b) (x + y ) (x) + (y ) x, y X . . (b) ; (). ( ), C > 0 , (c) |(x)| C x x X. X . , . . 10. (x) C x x X, |(x)| C x X C. . (-x) C ||x||, 0 = x + (-x), 0 = (0) = (x + (-x)) (x) + (-x), -(x) (-x) C ||x||. (x) C ||x|| |(x)| C ||x|| x X. , x, y X : (y ) = (x + (y - x)) (x) + (y - x) (x) + C y - x , , (y ) - (x) C y - x . (x) - (y ) C x - y , |(y ) - (x)| C y - x . l X , l(x) (x) x X. ( ), l (.. ) í ( ). , C > 0 , (x) C x x X, BC (0). , l , l, x (x) C x x . l, x C x x , | l, x | C x x , l C . , . , . , , -* . (!). , -* -* . , -* X . , . . x X,


2.4. .

67

31. : X R - . (x) = max l, x
l

x X ,

(2.55)

x X (, , -* ). . l , l, x (x) x , sup l, x (x) x.
l

, , x0 X l , l, x0 = (x0 ) . (2.55). . X ½ R K = {(x, t) | (x) < t }. , . . x0 X, t0 = (x0 ). (x0 , t0 ) K. , í (l, ) X ½ R , K : l, x + t l, x0 + t
0

(x, t) K .

(2.56)

. x t > (x) , 0 . = 0 , l, x - x0 0 x , l = 0 , (l, ) . < 0 , = -1 . (2.56) : l, x - l, x0 t - t0 t > (x),

t = (x), .. x X l, x - l, x0 (x) - (x0 ). (2.57)

x X - . x = N x, N ï ï . 1 1 l, x0 (x) - (x0 ), ï l, x - ï N N N , l, x (x) . x , ï ï ï l , , . , (2.57) x = 0 , l, x0 (x0 ). l , . , l, x0 = (x0 ). . . , = l í X , l . í ( ).


68

2.

32. ( ). X - , X í , : X - R í , : - R í . l ~ , ~(x) = l(x) x , .. ~ l l l l (, , ~ l l X. ) . 33. ( ). A : X - Y í , : Y - R í . f (x) = (Ax) . f = A , .. p f p, x = , Ax , . (2.58)

. p f , .. (p, x) (Ax) x X. X ½ Y (x, y ) = (y ), = {(x, y ) | y = Ax} ( A), l(x, y ) = p(x). l(x, y ) (x, y ) ( (p, x) (Ax) x X.) 32 l X ½ Y l . l X ½ Y l(x, y ) = (², x) + (, y ), ² X , Y . , (², x) + (, y ) (x, y ) = (y ), (², x) + (, y ) = l(x, y ) = (p, x) x, y , (x, y ) . (2.59) (2.60)

, ² = 0 (, y ) (y ) y , .. , , y = Ax , (p, x) = (, Ax), ... , , p = A . (p, x) = (, Ax) (Ax) = f (x) x , , p f . , (2.58).

3. . X í . 34. : X R í , K = {x X | (x) < 0} . ² K 0 l , ² = -l . . ² = 0 , = 0 , l í . ² = 0 . : (x) < 0 = ²(x) 0 . X ½ R : 0 = {(x, t) | (x) < t}, = {(x, t) | ²(x) < 0, t = 0}. (2.61)


2.4. . , , 0 . . , (x, 0) í , (x) < 0 (2.61). , 0 0 , , (x) < t = l(x) + t < 0 , ²(x) < 0 = l(x) 0 .

69 , 0 = ²(x) < 0, , l X

(2.62) (2.63)

(2.62). x = 0 , t = 1 . (2.62) , < 0 , , = -1 . (2.62) : (x) < t = l(x) < t x, t,

l(x) (x) x, .. l . (2.63) , l = -², 0 . = 0, l = 0, 0 , , (x) 0 x, K. > 0, 1 ² = - l, . : l 0 , ² = -l K . , x K, l(x) (x) < 0 = -l(x) 0 = -l K . , 4. K = - con( ), con M =
a0

M

,

M . 4. . X, Y í . 35. i (x) : X R í , i = 1, ..., k , G : X Y í . i (x) < 0, i = 1, ..., k ; Gx = 0, (2.64)

, 1 , ..., k x , ..., x X , y Y , 1 k
k

1 0, . . . k 0,
i=1

i > 0,

(2.65) (2.66) (2.67)

x i , i
k

i = 1, ..., k ,

i x + y G = 0. i
i=1


70

2.

. a) . (2.64) . , i = {x | i (x) < 0}, i = 1, ..., k , , 1 . 1 (x) 0 x . , 0 1 . x = 0 , 1 x . 1 = 1 , i i i = 0 . y = 0 . (1 , ..., k , x , ..., x , y ) 1 k (2.65)-(2.67). , i , i = 1, ..., k . , , = - con i , i L = {x | Gx = 0}. GX = Y : L = G Y
k

i = 1, ..., k .

(2.68)

(2.69)

( ). (
k i=1

i )

L = , - i = 1, ..., k , q L , ,

pi , i ,
i=1

pi + q = 0 .

(2.68), pi = -i x , i 0, x i , i = 1, ..., k , (2.69), i i q = -y G, y Y ( ). i = 0 i , pi , q = 0 , p1 , ..., pk , q . .
k i=1

i > 0 . -

) . 1 , ..., k , x , ..., x , y , 1 k (2.65)-(2.67), (2.64) , x í . ^ x , x i (x) < 0 i, Gx = 0. ^ ^ i^
k i=1

i > 0 ,

k i=1

i x , x + y , Gx < 0, ^ i^

(2.67). .

5. . . Z1 : f0 (x) - min; fi (x) 0, i = 1, ..., k ; g (x) = 0.

, . U X x0 U . , (a) fi : U - R, U X . i = 0, ..., k

(b) fi , i = 0, ..., k x0 fi (x0 , x) ï x , x X - fi (x0 , x) ï ï ï X .


2.4. . (c) g : U - Y x0 . (d) g (x0 )X Y .

71

(b) , (d) : , g (x0 )X = Y . Z1 , ZC . ( C ), . fi (x0 , §) () fi (x0 , §) . 36. x0 U í ZC . (0 , ..., k , x , ..., x , y ) ( i í , 0 k x X y Y í ) , : i (i) : i 0, i = 0, ..., k ; (ii) :
k i=0

i + y



>0;

(iii) : i fi (x0 ) = 0, i = 1, ..., k ; (iv) : x fi (x0 , §), i = 0, 1, ..., k ; i (v)
k i=0

i x + y g (x0 ) = 0 ( ). i

. ) g (x0 )X = Y ( ). g (x0 )X Y , y Y , y = 0 , y g (x0 ) = 0 . i = 0 i , x fi (x0 , §) . i ) g (x0 )X = Y . 30 fi (x0 , x) < 0, ï i I; g (x0 )x = 0 ï

(2.52) x . 35 ï . i I i = 0 , / f (x , §) . xi i0 30 35 : ( 30), ( 35). 36 ; .



[1] .. , .. , .. , .. . . ., , 1969. [2] .. , .. . . . , 1965, . 5, 3, . 395í453. [3] .. , .. . . , ( ), 1971. [4] .. , .. . . . " " (. .. ), ., , , 1981, . 6í47. [5] .. . . . , 1970. . 25, .5, . 110í116. [6] .. . . . " ", ., , 1990. [7] .. . . , 1970. [8] .. . . ., , 1950. [9] .. , .. . . ., , 1961. [10] . , .. . . . ., , 1962. [11] .. , .. . . ., , 1976. [12] .. , .. . . ., , 1982. [13] .. . . ., , 1974. [14] .. , .. . , . , 1972, . 27, N 3, c. 21í77. [15] .. , .. . . ., , 1974. [16] .. , .. , .. . . ., , 1979. 166


[17] .. . . ., , 1980. [18] .. , .. . . , - - , 1994.

167

[19] .. , .. , .. . . . , 1980, . 35, N 6, . 11í46. [20] .. , .. . . ., , 1989. [21] .. . . . " ", ., , 1990, . 14, . 26í42. [22] A.A. Milyutin, N.P. Osmolovskii. Calculus of Variations and Optimal Control. American Mathematical Society, 1998. [23] .. . . ., , 2001. [24] .. . . , 2007, . 256, . 102í114.