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114
??????? ?????. 2015. ? 10(132)
?????????????? ?????????????
??? 517.997
?.?. ??????? 1
?? ??????????? ?????????? N -???????
????????????2
??????????????? ?????? ???????????? ?????????? ??? n-???????? ??????????? ? ????????????? ?????????? ????????? ? ??? ???????????? ???? ????? ? ????????????? Lq [t0 , tf ], q = 1, 2, ?. ??-??????, ??? ?????? ??????????? ??????? ???????? ??????????, ??????? ???????? ? L? -????????
????????; ??-??????, ?????? ?? ??????? ???????????? ???????? ???????????? ????????? (??? L 1 -???????? ????????) ?, ???????, ?????? ?? ??????? ??????????? ?????? ??????????? (??? L 2 -???????? ????????). ??????? ????? ???????? ? ??????? ???????? ????????? ?.?. ???????????
(????? ????????). ????????, ??? ??????????? ?????????? ??? ?????? ?????? ???????????????? ?-?????????? ???????????. ??????? ????? ???????
????????????? ?????????? ? ??????????? ??????? ? ???? ?????? ? ??????????? ?? ????????? ???????. ???????? ????? ??????? ?????? ??????, ???
??????? ???? ??????????? ??????? ????????????? ?????????? ? ??????????? ??????? ? ?? ????????????????? ? ???????? ??????? ?? ??????????????.
????????? ??????? ??????? ????????????? ?????. ??? ?????? ??????????
? ???????????? ???????????? ???? ???????? ????? ???????????, ??????????? ??? ?????????? ????????? ???????????? ??????????.
???????? ?????: n-??????? ??????????, ??????????? ??????????, ???????? ????????, ??????? ????????? ?.?. ???????????, ?????????? ????????.
1.
?????????? ??????
??????????????? ?????? ?????????? ???????? ??????????????? ???????????
???????????? ??? n-??????? ????????????
dx1 (t)
dxn?1 (t)
dxn (t)
= x2 (t); . . . ;
= xn (t);
= u(t),
(1.1)
dt
dt
dt
??? x1 , x2 , . . . , xn ? ?????????? ?????????, ??????-??????? ??? ??????? x =
= col(x1 , x2 , ..., xn ) ? Rn , ? u ? R1 ? ??????????? ????????.
1?
c ??????? ?.?., 2015
??????? ???? ?????????? (yungor07@mail.ru), ???????? ??????? ????????????? ? ?????????? ????????-??????????????? ?????????? ?????????? ???????????????? ???????????????? ???????????? ????? ????????? ?.?. ???????? (????????????? ?????????????????? ????????????),
443086, ?????????? ?????????, ?. ??????, ?????????? ?????, 34.
2 ???????????? ????????? ??? ????????? ????, ?????? ? 13-08-97019 ?_????????_?,
? 13-08-97002 ?_????????_?.
115
?? ??????????? ?????????? n-??????? ????????????
????????? ??????? ??? ??????? ?????????? (1.1) ? ????? ?????? ????? ????????? ???:
x1 (t0 ) = x10 ; x2 (t0 ) = x20 ; . . . ; xn (t0 ) = xn0 ;
(1.2)
x1 (tf ) = x1f ; x2 (tf ) = x2f ; . . . ; xn (tf ) = xn f ,
(1.3)
??? x10 , x20 , . . . , xn0 , x1f , x2f , . . . , xnf ? ????????? ?????????, t0 ? tf ? ?????????
? ???????? ??????? ??????? ????????? ??????????, ?????, ??? ??? ????????????
T = tf ? t0 ???????????.
??????? (1.1) ????? ?????????? ? ????????-????????? ????
?
dx(t)
= Ax(t) + en u(t),
dt
?
(1.4)
x(t0 ) = x0 ; x(tf ) = xf .
(1.5)
0 иии 0
1 иии 0 ?
?
.. . .
.. ? ? RnОn , e = col (0, ..., 0, 1) ? Rn , ? ?????????
n
. . ?
.
?
0 0 0 иии 1 ?
0 0 0 иии 0
??????? (1.2), (1.3), ??????????????, ? ????? ?? ????
?
?
?
??? A = ?
?
?
0
0
..
.
1
0
..
.
?????????? ??????????? ??? ??????? (1.5) ???????? ???????? ???? [1]:
?t
x(t) = ?(t, t0 )x0 +
?(t, ?)en u(?)d?,
(1.6)
t0
??? ? ? ?????????? ???????, ??? ??????? ????? ?????: ?(t, ?) = ?(t ? ?) ?
?(t ? ?)en = h(t ? ?) = col [h1 (t ? ?), h2 (t ? ?), ..., hn (t ? ?)],
(1.7)
1
??? hk (t ? ?) = (n?k)!
(t ? ?)n?k , k = 1, 2, ..., n (hn?1 (t ? ?) = t ? ?,hn (t ? ?) = 1).
??? t = tf ??????? (1.6) ????? ?????????? ? ????
?tf
?(tf ? ?)en u(?)d? = c,
(1.8)
t0
??? c = xf ? ?(tf ? t0 )x0 = col (c1 , c2 , ..., cn ), ? ?????????? ??????? c (???? ???????) ??????????? ?? ????????:
ck = xkf ?
n
?
m=k
1
(tf ? t0 )m?k xm0 , k = 1, 2, ..., n.
(m ? k)!
????????? (1.8) ????? ?????????? ? ???? ??????? ????????? ????????:
?tf
hk (tf ? ?)u(?)d? = ck ,
k = 1, 2, ..., n.
(1.9)
t0
????? ???????, ???????????? ????????? ?????? (1.4), (1.5) ???????? ? ??????? ????????? (1.9) ???????????? ?????????? u(и) = u[t0 , tf ] ???, ??? ?? ?? ?????, ? ???????? ???????? [2; 3]. ???? ????????????? ???????????, ????? ???????
?????????? ?????????? ??????? ??????-???? ?????????? ??? ????????, ????????,
116
?.?. ???????
???? ????? ??? u(и) = u[t0 , tf ], ????? (1.4), (1.5) ? ?????? ???????????? ??????????. ??????????? ???? ????? ? Lq [t0 , tf ], q = 1, 2, ?, ????? ????????? ??? [2]:
?tf
J1 (u) = || u(и) ||L1 =
| u(t) | dt;
(1.10)
t0
J? (u) = || u(и) ||L? = max
t? [t0 , tf ]
| u(t) |;
(1.11)
?tf
J2 (u) = || u(и) ||L2 =
u2 (t)dt.
(1.12)
t0
?????????????? ?????? ???????????? ?????????? (1.4), (1.5), (1.10) ?? ??????? ?????????? ?????????? ???????? ? L? -???????? ???????? ??? (1.9), ??????
?????????? (1.4), (1.5), (1.11) ?? ??????? ???????????? ?? ?????? ????????
?????????? ???????? ? L 1 -???????? ????????, ? ?????? (1.4), (1.5), (1.12) ?? ??????? ??????????? ??????? ?????????? ? ? L 2 -???????? ????????. ? ????? ?
?????????????? ??????? ???? ????? ?????????? ???????, ??? ??????? (1.4) ???????? ?????? ???????????, ????????? ??????? ????????????? ??????? [3; 4] ???
??? ? ????????????? ???????.
????????????? ???????, ??? ? ?????? ??????? ????????? ??????? (1.5), ???
??????? ??????????? ????????? ???????:
xf ? ?(tf , t0 )x0 = 0,
(1.13)
? ?????? (1.6) ? (1.8) c = 0. ??? ?????????? ??????? (1.13) ??????? ????????????? ???? ????? ?????????? ?? ?????????, ??? ??? ? ???? ?????? ?????????
??????? (1.5) ????????????? ??????????? ??? ???????????? ??????? ??????????:
u(t) ? 0, ?t ? [t0 , tf ]. ????? ????, ??????? (1.13) ????? ????? ??????????? ? ?
?????? ??? ?????????? ??????????????? ??? ????-???????? ?????????? [5; 6],
?tf
??????????????? ??????? ?(tf ? t)en u(t)dt = 0.
t0
?????? ???????????? ?? ?????????????? ?????????? n-??????? ????????????
(1.1) ???????? ? [7; 8]. ??? ???????????? ?????? ???????????? ?? ?????? ????????????? ???????, ?? ? ?????? ?????????? ???????? [9]. ?? ?? ????? ????????? ?
? ??????? ??????????????? ????? ?????????? n-??????? ???????????? ? ????????????? (1.10)?(1.12), ??? ??????? ? ???????????? ?????????? ??????????????
? [10?13].
2.
?????? ???????????? ?????????? ??? n-????????
??????????? ? ????????????? ???? ?????.
??????? ?????? ?? ??????? ???????????
??????? ??????????
??????? ????????? ?.?. ??????????? [3]. ????? h0 (?) = l0T ?(tf ? ?)en ? ??????? ????????? ?????? (?? l = l0 ):
min || lT ?(tf ? ?) en || Lp = min || lT h(tf ? ?) || Lp =
lT c=1
lT c=1
= || lT0 h(tf ? ?) || Lp = || h0 (?) ||Lp = ?0 (p = 1, 2, ?),
(2.1)
117
?? ??????????? ?????????? n-??????? ????????????
??? l = col (l1 , l2 , ... , ln ), ?0 ? ????? ???????????? ????????. ????? ???? ??????:
(
?tf
max
||u(и)||Lq =
h0 (t)u(t)dt = 1
1
?0
)
1 1
+ =1 ,
p q
(2.2)
t0
?
????? ???????????? ??????? u (и), ?? u? (t), ?t ? [t0 , tf ], ? ??????????? ??????????. 2.1. ? ???????????? ? ????????? ????????? ??????? ?????? ????????????
?????????? (1.4), (1.5), (1.10) ? ?????? (1.8) ???????? ? ????????????????? ??????? ????????? ????? [2; 3].
??-??????, ??????????? ????? ???????????? ????????, ? ??????:
(
)
T
T
min || l h(tf , и) ||L? = min
max | l h(tf ? t) | = || h0 (и) ||L? = ?0 , (2.3)
lT c=1
lT c=1
??? h0 (tf ? t) = lT0 h(tf ? t) =
t? [t0 ,tf ]
n
?
k=1
lk0 (tf ?t)n?k
,
(n?k)!
? lT0 c =
?
n
?
lk0 ck = 1.
k=1
??-??????, ?????? ???????????? ?????????? u (и) ?? ??????? (2.2):
?tf
max
||u(и)||L =
1
???
?tf
1
?0
(2.4)
h0 (t)u(t)dt = 1,
t0
?tf
?
h0 (t)u (t)dt = 1, || u(и) ||L 1 =
t0
| u(t) |dt =
1
.
?0
(2.5)
t0
?? ??????? ?????? (2.3) ?????, ??? ??????????? ??????? ? ????????? ???????? ??????? ???? ? ????????? ?? ????????? ?0 , ?? ???? ?????????, ???????? ???????????? ?? ???? ?? [ t0 , tf ] [14]. ????? ????, ??????????? ??????????
????? ? ??????? ???????? ?-?????????? ??????????? [10], ??????? ???????????
?-?????????? ? ?????? ??????????? ???????????? ???????? ? ? ????????? ?????? ????????? ??????????. ??? ???? ? ??????? ???????? ??? ?????? ???????
?????????? ? ?????? (2.5) ????? ????? ????????, ???????? ????? ????????????
?1
????????: J inf
1 = ?0 .
2.2. ??????? ?????? ???????????? ?????????? (1.4), (1.5), (1.11) ????? ???????? ? ????????????????? ??????? ????????? ?????.
??-??????, ? ??????????? ????? ???????????? ????????:
?tf
min || l h(tf , и) ||L1 = min
| lT h(tf ? t) | dt = || h0 (и) ||L1 = ?0 .
T
lT c=1
lT c=1
(2.6)
t0
??-??????, ? ??????? ???????????? ?????????? u? (и) ?? ???????:
?tf
max
||u(и)||L? =
???
?tf
t0
1
?0
(2.7)
h0 (t)u(t)dt,
t0
h0 (t)u? (t)dt = 1, || u(и) ||L ? =
max
t? [t0 , tf ]
| u(t) | =
1
.
?0
(2.8)
118
?.?. ???????
????????, ??? ?? ??????? ?????? (2.6) ??????? ??????????? ??????? ? ????
?????????? ???????? ??????? ???? ? ????????? ?? ?????????? ????????? [14].
?????????????? ? ?????? (2.8) ??????????? ?????????? ? ???? ?????? ????? ?????
????????? ???:
1
u? (t) =
sign h0 (t), ?t ? [t0 , tf ],
(2.9)
?0
min
??? h0 (t) = lT0 h(tf ? t), ? lT0 c = 1. ??? ???? ??????? J?
= ??1
0 .
2.3. ??? ??????? ?????? (1.4), (1.5), (1.12) ?? ??????? ??????????? ???????
?????????? ??????? ????? ??????? ????? ??????????? ??????? ? ??? ????? ?
L 2 [t0 , tf ], ? ??????:
? tf
? 12
?
min || lT h(tf , и) ||L2 = min ? [lT h(tf ? t)] 2 dt? = || h0 (и) ||L2 = ?0 .
(2.10)
lT c=1
lT c=1
t0
?tf
??????? ?????, ???
[lT h(tf ? t)] 2 dt = lT Gl, ??? G =
t0
?tf
h(tf ? t)h(tf ? t)T dt ?
t0
??????? ?????????????:
?tf
(tf ? t)2n?k?j
G = [gk j (tf ? t)]nОn dt, gk j (tf ? t) =
, k, j = 1, 2, ..., n,
(n ? k)! (n ? j)!
t0
???????? ???????? ??????????? ?? ????????:
?tf
?tf
(tf ? t)2n?k?j
(tf ? t0 )2n?k?j+1
gk j (tf ? t)dt =
dt =
.
(n ? k)! (n ? j)!
(n ? k)! (n ? j)!(2n ? k ? j + 1)
t0
t0
? ???? ?????? ????????????? ??????? (1.4) ?? ??????? ????????????? G ???????? ????????????? ????????. ???? ??????? t0 = ?1 ? tf = +1, ?? ??? ????????
????? ??????????? ?? ????????:
?tf
22n?k?j+1
gk j (tf ? t)dt =
, k, j = 1, 2, ..., n.
(n ? k)! (n ? j)!(2n ? k ? j + 1)
t0
??? ????????? ???????????? ???????? ? ??? ????? ? ??????????????? ??????
??????? ????? ??????? ???????????? ????? lT Gl ??? ???????, ??? lT c = 1.
????? ??? ?????? ??????? ?????????? ????????, ??????
G?1 c
l 0 = T ?1
c G c
?, ??????????????, ???????
1
cT G?1 h(tf ? t)
, ?0 = (cT G?1 c)? 2
h0 (t) =
(2.11)
cT G?1 c
????? ?? ??????? ????????? ????????? ? (2.2) ????????? ??????????? ??????????, ????????? ???????? ????? ????? ????????? ???:
u? (t) = ? h0 (t), ?> 0(?t ? [t0 , tf ]).
????????? ??? ????????? ? (2.2), ???????, ??? ? = ??2
0 , ?? ???? ???????????
?????????? ??? ?????? (1.4), (1.5), (1.12) ? ?????? (2.11) ????? ???
1
u? (t) = 2 h0 (t) = cT G?1 h(tf ? t), ?t ? [t0 , tf ],
(2.12)
?0
? ??????????? ???????? ??????????? (1.12) ????? ????? J2min =
1
?20
= cT G?1 c.
119
?? ??????????? ?????????? n-??????? ????????????
3.
??????????? ?????????? n-???????
???????????? ?? ??????? ????????? ??? ???????
???????? ??????????
??????? ?????? (2.3) ?????????? ? ?????? (1.7) ?? ??????????? ???????
h0 (t) =
n
?
lk0 hk (tf ? t) =
k=1
n
?
k=1
?
lk0
(tf ? t)n?k ,
lk0 ck = 1.
(n ? k)!
???????? ??????? ?????? (tf ? t)n?k =
n
(3.1)
k=1
n?k
?
m=0
m
(?1)m Cn?k
tfn?k?m t m , ?????????
????????? (3.1) ? ????
h0 (t) =
n
?
k=1
n?k
n?1
?
?
lk0
m
tfn?k?m t m =
(?1)m Cn?k
ap t p ,
(n ? k)! m=0
p=0
(3.2)
r!
??? Crq = q!(r?q)!
, ? ap , p = 0, 1, ..., n ? 1, ? ???????????? ?????????? ? ???????????? ???????? ?????? (2.3), ??????????? ?? ????????:
a p = (?1)p
n?p
?
k=1
p
lk0 Cn?k
tn?p?k , p = 0, 1, ..., n ? 1.
(n ? k)! f
(3.3)
????? a = col [(?1)n?1 an?1 , (?1)n?2 an?2 , ..., ?a 1 , a 0 )] ? l0 = col (l10 , l20 , ..., ln0 ),
??????????? (3.3) ????? ?????????? ? ????????-????????? ????:
(3.4)
M l0 = a,
??? ??????? M ? RnОn ????? ????????? ??????????:
?
1
0
0
иии
0
(n?1)!
n?2
? Cn?1 tf
1
? (n?1)!
0
иии
0
(n?2)!
?
n?3 2
n?3
? Cn?1
tf
Cn?2
tf
1
? (n?1)!
иии
0
(n?2)!
(n?3)!
?
?
?
?
..
..
..
..
..
.
M = ?
.
.
.
.
?
?
? 2 n?3
2
2
? Cn?1 tf
Cn?2
tn?4
Cn?3
tn?5
C 32 tf
f
f
?
иии
(n?2)!
(n?3)!
3!
? (n?1)!
1
1
1
? Cn?1
tn?2
Cn?2
tn?3
Cn?3
tn?4
C 31 t2f
f
f
f
?
иии
(n?2)!
(n?3)!
3!
? (n?1)!
tn?2
tn?3
tn?1
t3f
f
f
f
иии
(n?1)!
(n?2)!
(n?3)!
3!
0
0
0
0
0
0
..
.
..
.
1
2!
C21 tf
2!
t2f
2!
0
1
1!
tf
1!
0
?
?
0 ?
?
?
0 ?
?
?
?
.. ?
. ?
?.
?
?
?
0 ?
?
?
0 ?
?
(3.5)
1
????????? ??????? M ?????????????, ?? ?? (3.4) ?????? ????? ???????? l0 =
= M ?1 a. ??? ???? ????????? ??????? (3.5), ???????? ?? ??????????, ?????? ????????? ??????? ?????????? [15]. ????? ????, ??????????? ????????? ??????? M
????????? ?????? ??????? (3.4), ??????????????? ????????? lk0 , k = 1, 2, ..., n-1,
??????? ? l10 = (?1)n?1 (n ? 1)! a n?1 ? ?. ?.
??? ??? l0 ? ??????? ?????? (2.3) ??? ?????????? c ?= 0, ?? ????? ? ?????
??????, ????? l10 ?= 0, ????? ???????? ??????????, ??? ? ?????? (3.4) ? (3.5)
???????? ??????? an?1 ?= 0 ???, ??? ?? ?????, ????? deg h0 (t) = n ? 1. ?????,
???? deg h0 (t) < n ? 1, ?? ??????? ?????? (2.3) ????? ???????????, ??????? ????????????? ???????? ????? ??????????? ????. ? ?????? (3.4) ? cT l 0 = 1, c ?= 0
120
?.?. ???????
? ?????????? ?????? ?????? ????? ?????: cT M ?1 a = 1 ?, ??? ?????????, an?1 ?= 0
?, ????? ????, l10 ?= 0. ????????, ??? ??? ??????? ?? ??????????? ? ??? ??????,
????? ??????? c ? M ?1 a (??? M ?T c ? a) ????????????.
??? ??? ??????? (1.4) ????????????, ?? ??? ??????? ?????? ????????? ?????????? [t 0 , tf ] ??????????? ???????????? ????????, ???????? ???? ????????????
????????? ?????????? ? T = tf ?t 0 , ??????????? ???????????? ?????????? ??????????????? ????? ???????????? ??????????. ????????, ??? ??????????? ???????? ???????? ????????? ???????? ??????? ????????? [t 0 , tf ]: ??-??????, t0 = 0;
tf = T ; ??-??????, t0 = ?T ; tf = 0 ?, ?-???????, t0 = ?T /2; tf = +T /2. ????
tf = 0, ?? ???? ?? ?????? ??????, ?? ??????? (3.5) ????? ????????????.
????????? ??????? ??????????????? ????? ?????????? ????? ??????? ? ???????????? ????????, ?? ???????? ?????????? [t 0 , tf ] ?????? ????????????? ? ????????? [?1, +1], ???????? ?? ?????????? t ? (3.2) ? ?????????? ? :
1
[(tf ? t 0 )? + (tf + t 0 )], ? ? [?1, +1].
(3.6)
2
?????? ??????????? ?????????? ???????? (3.6) ? ????????? ??? ?????????? (3.2),
??? ???????????? ???????? ?????? (2.3), ???????? ??? ? ?????? ????:
t=
g0 (? ) =
n?1
?
bp ? p , ? ? [?1, +1],
(3.7)
p=0
??? bp , p = 0, 1, ..., n?1 ? ????????????, ??????????? ?? ??????????????? ???????? ? ??????????? ?? ???????? ??????? ?????? ????????? ?????????? [t 0 , tf ]
? ?????????? ??? ?????? ?????????? (3.6), ? ?????? ??? ?????? ?? ?????????
????????? ?????? ?????????? t ? [t 0 , tf ] ?? ? ? [?1, +1]:
T
T
T
(? + 1); 2) t = (? ? 1); 3) t = ?.
2
2
2
????????, ??? ??????? ???????? (3.8) ?? (3.2) ???????
n?1
n?1
n?1
?
? ( T )p
?
ap t p =
(? + 1) p =
b k ? k,
ap
2
p=0
p=0
(3.8)
1) t =
k=0
??? b k =
n?1
?
am
m= k
( T )m
2
m?k
Cm
, k = 0, 1, ..., n?1, ?, ? ?????????, b n?1 = an?1
( T )n?1
2
.
??????????????, ??? ????????? ????????? (3.8) ?? (3.2) ???????
( )m
( )k
n?1
?
T
T
m?k m?k
bk =
am
(?1)
Cm
? b k = ak
, k = 0, 1, ..., n ? 1.
2
2
m= k
????????? ???????????? ?????????? (3.7) ??????????? ?? ????????????? (3.3),
?????????, ????? ? ???????????? ??????-??????? b = col (bn?1 , bn?2 , ..., b 0 ), ?????
????? ????????? b ? a ????? ??????????? ? ????? ????:
Qa = b,
(3.9)
??? ??????? Q ? R
????? ??????????????? ?????????? ? ??????????? ?? ?????? ???????? ?????? (3.8). ??????? ? ?????? (3.9) ?? (3.4) ????? ???????
nОn
l 0 = M ?1 Q ?1 b.
(3.10)
?????? ??????? ?????? (2.3), ????? ?????????? ????????? ?0 ? ?????? ????,
??? ?? ???????? ???? ?????? ????????????? ???????? ?????????? (3.2) ?, ?????
121
?? ??????????? ?????????? n-??????? ????????????
????, (3.7) ?????? ? ?????????? ??????? ?????????? ?? ???? ???, ??? ?? ??
?????, ? ????????? ?? ????????? ?0 ? ?????????? ?????? ?????? ????? ?????
????? ???????????:
g0 (? ) = s0 ?0 Tn?1 (? ), ? ? [?1, +1],
(3.11)
??? s0 = ▒1, Tn?1 (? ) = 2n?2 ? n?1 +
n?2
?
?p ? p ? ????????? ???????? ??????? ????
p=0
(n?1)-? ??????? [14]. ?????????? ???????? ??????? ???? ?????? ???????, ?? ????
??? n = 2k (k = 0, 1, 2, ...), ????? ????????? ?????? ?????? ??????? ? , ? ?????????
???? ??? ??? ????? ????? (?1)k , ? ?????????? ???????? ??????? ? ??? n = 2k +1
(k = 0, 1, 2, ...) ? ?????? ???????? ??????? ? . ???????? [14]: T0 (? ) = 1; T1 (? ) = ? ;
T2 (? ) = 2? 2 ? 1; T3 (? ) = 4? 3 ? 3? ; T4 (? ) = 8? 4 ? 8? 2 + 1; T5 (? ) = 16? 5 ? 20? 3 + 5? , ?
?. ?. ? ???????????? ? ???????????? ????????????: Tn+1 (? ) = 2? Tn (? ) ? Tn?1 (? )
(n > 0). ?? ???? ? ????? ?????? ??? ????????????? ?????????? Tn?1 (? ) ?????
?????: ?n?2 = ?n?4 = ... = 0 ? ?n?1 = 2n?2 .
??????????? ???????????? ?? (3.7) ? ?????????????? ?????????? ?? s0 ?0
??????????????? ???????????? ?????????? Tn?1 (? ), ?? ????, ???????? b = s0 ?0 ?,
??? ? = col (?n?1 , ?n?2 , ..., ? 0 ), ?? (3.10) ???????
l 0 = s0 ?0 M ?1 Q ?1 ?.
(3.12)
??????? ????? (3.12) ???????? ?? c ? ????????, ??? ?0 > 0 ? cT l 0 = 1, ??????
???????? ????? ???????????? ???????? (3.2) ?0 ? s0 :
?0 = | cT M ?1 Q ?1 ? |?1 ; s0 = sign (cT M ?1 Q ?1 ?).
(3.13)
??????????? ?????????? ??? ??????????? ?????? ? ??????????????? ??????
????? ?????????? ?? ??????? (2.4), (2.5). ? [10] ???? ????????, ??? ???????????
?????????? ??? ???????? ??????????? ? ??????? ???????? ?-?????????? ???????????. ?? ?? ????? ????? ????? ????? ? ? ?????? n > 3, ? ?????? ? ???? ??????
??????????? ?????????? ????? ????? ????????????? ?-?????????? ? ???????????
????????? ????? ????????? ?????????? ? ????? ??????????? ???????????? ???????? ???, ??? ?? ?? ?????, ? ??????????? ????? ??????????? ?????????? Tn?1 (? )
?? ????????? ?????????? [?1, +1]. ??? ???? ???????? ??? ?????? ??????? ???????????? ?????????? ? ?????? (2.5) ? (3.13) ????? ?????
J min
1
?tf
?
= || u (и) ||L 1 =
T
| u (t) |dt =
2
?
t0
?+1
1
| u? (? ) |d? =
= | cT M ?1 Q ?1 ? |. (3.14)
?0
?1
????????? ???????????? ?????????? ? ?????? (2.4) ? (3.7) ????? ????? ???
u? (? ) = sign g0 (? )
n?1
?
u?k ?(? ? ?k ), ? ? [?1, +1],
(3.15)
k=0
??? ?(? ? ?k ) ? ?-???????, u?k > 0 (k = 0, 1, ..., n ? 1) ? ????????? ????????? ?????????? (???????? ?????????), ? ?k ? ????? ?????????? ?-?????????,
(? )
= 0. ??? ???
??????? ?? ????????? (?1, +1) ???????????? ?? ??????? dTn?1
d?
1 dTn?1 (? )
= Un?2 (? ), ?? ????????? ????? ???? ????????? ????? ?????????
n?1
d?
?????????? [?1, +1] ? ???? ??????????? ???????? ??????? ???? [14]:
?k = cos
(n ? k ? 1)?
, k = 0, 1, ..., n ? 1(?0 = ?1, ?n?1 = +1).
n?1
(3.16)
122
?.?. ???????
? ?????? ????????? ???????? (1.8), (1.9), ? ????? ???????, ?, ? ?????? ???????, ? ?????? ??????????? ??????????? (3.2) ? (3.7), ????????????? ??????? ?????????? (3.8), ?? (1.8) ??? ??????????? ? ???? (3.15) ???????:
?tf
T
h(tf ? t)u (t)dt =
2
?
t0
=
T
2
n?1
?
k=0
[
]
?+1 [
T
(1 ? ? ) u? (? )d? =
h
2
?1
]
[
]
n?1
T ? ? T
T
u?k h
(1 ? ?k ) sign g 0 (?k ) =
uk h
(1 ? ?k ) sign [s 0 Tn?1 (?k )] = c.
2
2
2
k=0
?????? ??????? ??????? ????????? ???????????? ?????????? ????????? ? (3.15),
??????? ? ????????-????????? ?????? ????? ???
Hu? = c,
?
??? u = col
(u?0 ,
u?1 , ...,
(3.17)
u?n?1 ),
H = [g0 |g1 | и и и |gn?1 ] ?, ??????????????,
]
T
gk = h
(1 ? ?k ) sign [s 0 Tn?1 (?k )] , k = 0, 1, ..., n ? 1.
2
[
???????? ??????????, ??? ??????? (3.17) ????? ????????????? ??????? ?????????? ? u? , ? ??????: (?1)k s0 u?k?1 ? u??k?1 , k = 1, 2, ..., n, ?? ???? u? ? u?? , ? ???
?????? ????? c ? c?, ??? (n ? k)! ck ? c?k , k = 1, 2, ..., n, ??? ??? (3.17) ????? ?????
???: W u?? = c?, ??? W ? ??????? ???????????.
????? ????????? (3.17), ??????? ?????? ?????????? ????????? u? ??? ???????????? ?????????? (3.15): u? = H ?1 c ? ??? ????? ??????? ? ????? ???? ??????? ?????? ???????????? ?????????? (1.4), (1.5), (1.10) ??? n-???????? ??????????? ?? ??????? ????????? ??? ??????? ???????? ?????????? ??? ???????????
??????. ???????, ??? ? ?????? (3.15) ?? (2.5) ????? ???????
n?1
?
u?k =
k=0
?????? ?????, ??? ??? ?0 ? ? ???????
2
.
?0 T
n?1
?
k=0
4.
(3.18)
u?k ? 0.
?? ???????? ?????????? ?????? ??????????
n-??????? ???????????? ?? ??????? ?????????
??? ??????? ????????
?????????? ??????? ????????????? ???????????? ??????? ? ?????? (2.3).
? ?????? ??????? cT l 0 = 1 ? (3.12) ??? ? ?????, ??? ? ? ?????????? ??????
?????? ????? ?????:
s0 ?0 cT M ?1 Q ?1 ? = 1.
(4.1)
????????? ????? ? = col (?n?1 , ?n?2 , ..., ? 0 ) ? ?????? ????????????? ??????????
Tn?1 (? ), ?? ??? ?????????? ??????? (4.1) ?????????? ? ??????????, ????? ??????? c ? qn = M ?1 Q ?1 ? ???? ?? ????????????, ?? ???? cT qn ?= 0. ?????, ?????
cT qn = 0, ??????? (4.1) ?? ??????????? ?, ????? ????, ????? ????? ?????????????
???????????? ??????? ?????? (2.3).
??? ??????????? ???????? ????????? ????? ?????? t0 = ?1 ? tf = +1. ?????
? (3.9) Q = In ? ????????? ??????? ? a = b = s0 ?0 ?, ?? ???? qn = M ?1 ?. ?????
?? ??????????? ?????????? n-??????? ????????????
123
????, ??????? (3.5) ????????? Mn ? RnОn ? ?????? ??? ??? ????????? ???????
?????????????:
?
?
1
T
]
[
0
0n?2
1
T
(n?1)!
0
n?1
1
1
T
? ? ?. ?.
0n?2
; Mn = ? (n?2)!
Mn = (n?1)!
(n?2)!
mn?1 Mn?1
?mn?1 mn?2 Mn?2
?????????????? ???????? ? ???? ???????? ????? ????? ????????? ???:
[
]
T
(n ? 1)!
0n?1
Mn?1 =
;
?1
?1
?(n ? 1)!Mn?1
mn?1 Mn?1
?
?
T
(n ? 1)!
0
0n?2
T
?
?(n ? 1)!
(n ? 2)!
0n?2
Mn?1 = ?
(4.2)
?1
?1
?1
(n ? 1)!Mn?2 (mn?2 ? ?mn?1 ) ?(n ? 2)!Mn?2 mn?2 Mn?2
? ?. ?. ?? ?????? ???????? ??????? ??? (3.5).
????? ????? c = col (c 1 , ?cn?1 ), a = col (a n?1 , ?an?1 ) ? l0 = col (l 10 , ?ln?1 ),
??????? cT l 0 = 1 ????? ?????????? ? ????
T
c 1 l 10 + ?cn?1
?l n?1 = 1,
(4.3)
? ??????? ????????? (3.4) ? ?????? (4.2) ??????????? ? ????:
?1
l 10 = (B ? 1)! a n?1 ; ?l n?1 = Mn?1
[?an?1 ? (n ? 1)! an?1 mn?1 ].
(4.4)
???? ?? cT l 0 = 0, ?? ??? ????????? c ?= 0 ?????????? ??????? ?????? (2.3) ?????????? ? ???? ??????????????? c ?= 0 ? qn = M ?1 ?. ?? ????? ?????, ?? ???? ???
cT qn = 0, ????? ????? a n?1 = b n?1 = 0 ? l 10 = 0. ????? ????, ????? ??? ?????????? ? (2.3) ??????? cT l 0 = 1 ? ???? (4.3) ? ?????? (4.4) ?????????? ? ??????????,
????? ??????????? ????????? ???????:
?1
T
?cn?1
Mn?1
?an?1 = 1,
(4.5)
(n?1)
(n?1)
??? ?an?1 = s0 ?0 ??n?1 , ? ??n?1 = col (2n?3 , 0, ?n?3 , ..., ? 0
) ? ?????? ????????????? ?????????? Tn?2 (? ). ??????? ????????, ??? ??????? ? = ?n ?
?n?1 = col (0, ??n?1 ) ???????????? ? ???? ??????? ????????????? ???????????
???????? ??????? ????. ???? (4.5) ???????????, ?? ?????? (2.3) ???????? ?????????? ???????????. ????????, ??? ??? ?????????? ????? ??????? ??????? ?????? (2.3) ????? ????????????? ??? ????????????, ?? ? ?????? ????, ??? ?????
deg h0 (t) = n ? 2, ? ??????????? ?????????? ?????? ????????? ?? ????? (n ? 1)-??
?1
T
?-????????. ???? ??????? (4.5) ?? ???????????, ?? ???? ?cn?1
Mn?1
?an?1 = 0 ???
l 10 = 0, ?? ?????? (2.3) ???????? ?? ????? ??? ????????? ???????????. ????????????? ????????? ???????????? ??????? ?????? (2.3) ??????????????? ????? ??
???????, ??? ? ? ?????? ???????????? ?????????? ???????. ??????? ?????? (2.3)
????? n-?????? ???????????, ????? ??????? (1.13) ??????????? ?????????????
??? ???????????? ??????? ??????????.
?????? 1. ????? ? (1.2), (1.3) x30 = . . . = xn0 = 0 ? x3f = . . . = xn f = 0,
? ????? t 0 = ?1 ? tf = +1. ????????? ????????, ??? ????? t ? ? ?, ????? ????,
?????????? (3.2) ? (3.7) ???????????? ? ? (3.9) Q = In ? ????????? ???????.
???????, ??? ???????? ??????? ??????? ????????? ??????? (1.2), (1.3) ????????
???????? ????-?????????? ???????????? ?????????? (3.15) [6] ? ????? ?????????? ??????? (1.1), ? ?????? ? ?????????? ??????? ?? n?2 ????????????, ???????
??????????????? ???????????? ? ???????? ??????????? ? ?????????? ??????????
?????????.
124
?.?. ???????
??? ????????? ???? ????????? ??????? ???????
c 1 = x1f ? x10 ? 2x20 ; c 2 = x2f ? x20 ; c3 = . . . = cn = 0.
(4.6)
?????? ?????, ??? (1.13) ???????????, ???? ?????? x1f ? x10 ? 2x20 = 0 ? x2f ?
? x20 = 0. ????? c 1 ?= 0 ?, ????????, c 2 ?= 0. ? ?????? (4.2) ? (4.6) ???????
cT M ?1 = [(n ? 1)!(c 1 ? c 2 )|(n ? 2)! c 2 |0| и и и |0] ??? cT M ?1 ? = 2n?2 (n ? 1)! (c 1 ? c 2 ).
????? ???????, ??? c 1 ?= c 2 , ?? ???? ????? c 1 ? c 2 = x1f ? x10 ? (x20 + x2f ) ?=
= 0, ????? ????? ?????????? ?????? ??????? ?????? (2.3) ? ?????????? ????
?????????? ?????????. ?????????? ?????? ????? ????? ? ??? x2f = x20 , ?????
c 2 = 0, ?? ??? ??????? x1f ? x10 ? 2x20 ?= 0. ??? ???? ???????
[
] ?1
?0 = 2n?2 (n ? 1)! | x1f ? x10 ? (x20 + x2f ) |
;
s0 = sign [x1f ? x10 ? (x20 + x2f )],
? ??????????? ?????????? (3.15) ????? ????? ????? ????????? ???:
u? (? ) = sign [s0 Tn?1 (? )]
n?1
?
u?k ?(? ? ?k ), ? ? [?1, +1],
k=0
??? ????????? ????????? ???????????? ?? ??????? ??????? (3.17).
???? ?? ????? c 1 ?c 2 = x1f ?x10 ?(x20 +x2f ) = 0, ?? ??????? ?????? (2.3) ? ????????? ?????????? ????????? ???????? ? ???????????? ???????????? ??????.
??? ????? ??????? ??????
?1
T
?cn?1
Mn?1
??n?1 = 2n?3 (n ? 2)! c 2 = 2n?3 (n ? 2)! (x2f ? x20 ).
?????? ?????, ??? ?????????? ??????????? ?????? (2.3) ????? ???????????? ??[
] ?1
?????, ???? c 2 = x2f ? x20 ?= 0. ??? ???? ?0 = 2n?3 (n ? 2)! | x2f ? x20 |
?
s0 = sign (x2f ? x20 ). ???? ? c 2 = x2f ? x20 = 0, ?? x1f ? x10 ? 2x20 = 0 ???, ???
?? ?? ?????, ????? ??????????? ??????? (1.13). ????????? ?????? ??????? ?????????? ??? ?????? (2.3), ????????, ??? ?????????? ?????? ??????? ?????? (2.3) ????? ?????, ????? ??????????? ???????
(4.1), ?? ???? ????? ??????? c ? Mn?1 ?n ?? ????????????. ????????, ??? ??????????? ??????? ?????? ??????????? ? ? ?????? ??????? ?????????? ??????????? ?????? (2.3), ??????? ??????? ????? ????????????? ??? ???????????? ?
?1
?????? ?????????? ??????? (4.5), ?? ???? ????? ??????? ?c n?1 ? Mn?1
?? n?1 ??
?1
T
????????????. ???? ?? ??? ??? ?cn?1 Mn?1 ?? n?1 = 0, ?? ??????????????? ?????? ????? ?? ????? ??? ????????? ???????????. ?????? ??????? ? ???????????
????????? ?????????? ?????? (2.3) ??????????????? ??????? ? ???????? ???????
???????????????, ???????? ????????????? (4.2), ? ?????????????????? ?????????
m ??? ????????:
?1
?1
(c, Mn?1 ?n ); (?c n?1 , Mn?1
?? n?1 ); ...; (?c n?m+1 , Mn?m+1
?? n?m+1 ).
(4.7)
? ?????? ?n?j = col (0, ..., 0, ??n?j ) ? R , j = 0, 1, ..., n ? 1, ??? ??n?j ? ??????
| {z }
j ???
????????????? ?????????? Tn?j?1 (? ), ??????????? (4.7) ? ?????? m-??????? ????????????? ?????? (2.3) ????? ?????????? ???:
n
cT Mn?1 ?n?j = 0, j = 0, 1, ..., m ? 1, cT Mn?1 ?n?m ?= 0,
???
(4.8)
125
?? ??????????? ?????????? n-??????? ????????????
cT qn?j = 0, j = 0, 1, ..., m ? 1, cT qn?m ?= 0,
(4.9)
Mn?1 ?n?j ,
j = 0, 1, ..., m ? 1.
??? qn?j =
?????????? ??????? (4.8) ????????, ??? ?????? Mn?T c ??????????? ?? ???? ???????? ?n ,?n?1 , ... ,?n?m+1 , ???????? ?????? ?n?m , ???, ??? ?? ?? ?????, ?????? c
??????????? ?? ???? ???????? qn ,qn?1 , ... ,qn?m+1 , ???????? ?????? qn?m . ? ?????
???? ????????, ??? ? ?????? (1.8) ? (1.13) c = 0, ?? ??????? ?????? ?? ??????????? ??????????? ?? ???? ???????? qn ,qn?1 , ..., q1 . ????? ????, ????? ???????
(1.13) ???????? n-??????? ??? ?????? ????????????? ?????? (2.3).
?????? ????? ? ???????????? ??????? ?n = [ ?n | ?n?1 | и и и | ? 1 ], ??????? ??
?????????? ???????? ?????? ??????????? ????????. ? ?????? ??????? ????????????? ??????????? ???????? ??????? ???? Tm (? ) (m = 0, 1, ..., n ? 1), ? ??????:
(m)
?m?j , j = 0, 1, ..., m, ??????? ?n ????? ????? ????????? ???:
?
? n?2
2
0
0
иии
0 0 0
?
0
2n?3
0
иии
0 0 0 ?
?
? (n?1)
n?4
? ?
0
2
и
и
и
0 0 0 ?
?
? n?3
(n?2)
?
?
0
? n?4
0
иии
0 0 0 ?
?
?
? (n?1)
(n?3)
? ? n?5
иии
0 0 0 ?
0
? n?5
?.
?n = [ ?n | ?n?1 | и и и | ? 1 ] = ?
(n?2)
?
0
иии
0 0 0 ?
0
? n?6
?
?
?
..
..
..
.. .. .. ?
..
?
.
.
.
.
. . . ?
?
?
?
2 0 0 ?
иии
?
?
?
?
?
?
0 1 0 ?
иии
?
?
?
?
?
?
иии
?1 0
1
?????? ?????, ??? ??????? ?n ? ????????????? ???????, ?, ??????????????,
???????????? ??????? ????? G = ?T
n ?n ??????? ?? ????. ?? ????? ??????? ???????? ?n?j , j = 0, 1, ..., n?1, ???????? ??????? ??????????? ????????, ? ??? ??????? ???????? ????? ???????????? Rn [15]. ??????? ????? ????? ????? ????????????
?????????? ??????? Mn?T c ?? ?????????? ??????. ?? ?? ????? ??????????? ? ???
??????? ???????? qn?j , j = 0, 1, ..., n ? 1. ???? ??? ??????? ??????????, ?? ????? ???????? ????? ???????????? Rn ? ??? ???? ????? ????? ????? ????????????
?????????? ??????? c.
??????????? ? ???????? (4.9), ???????, ??? ?? ?????????? ??? ??????? c ???????? ??? ??????????????? ? ???????? ???????? Lm ? Rn ??? ??????? m ????????
qn?j , j = 0, 1, ..., m ? 1, ??????? ???????? ???? ??????????????? Lm ?, ??????????????, (4.9) ???????? ????????? m-??????? ????????????? ?????? (2.3).
5.
??????????? ?????????? n-???????
???????????? ?? ??????? ????????????
?? ?????? ???????? ??????????. ?? ????????
?????????? ??????
????????? ? ??????? ?????? ???????????? ?????????? (1.4), (1.5), (1.11), ??????? ????????, ??? ??? ???????? ???????? ? ?????? ?? ??????????????, ? ???????
?
?????????, ????? J?
(u) = (tf ? t0 ) ? min, ? ?? ??????????? ???????? ???????-
126
?.?. ???????
?????? ???????????:
| u(t) | 6 umax , ?t ? [t0 , tf ],
(5.1)
??? umax ? ??????????? ?????????? ?? ?????? ???????? ??????????.
??????? ?????? (1.4), (1.5), (1.11) ?????????? ??????????? ???? ??????? ?????? (1.4), (1.5), (1.10), ?? ???? ????? ??? ??, ??? ? ??? ????? (2.3), (2.4), ???
???????? ? ??????? ????? (2.6), (2.7). ????? ????, ??????? ??????? ????? ??????????? ??????? ?? ??????? ?????? (2.6) ? ???? ?????????? h0 (t) (3.2), ???????
?? ???????? ?????? ? ????????? ?? ?????????? ????????? ?????? ???? ??????????? ???????? ??????? ???? (n ? 1)-? ??????? ? Un?1 (? ) [14]. ????? ?????
???????? ????????????? ????? ?????????? a ? ???????? l0 ????? ?????????? (3.4)
? ? ??? ?? ???????? M (3.5). ? ??????? ?????? (3.6) ????????? h0 (t) ???????
???????? ? ???????????? ???? (3.7), ?? ???? ? ?????????? g0 (? ), ? ? [?1, +1]. ???????? ????? ??? ??????????? ???????? ?????????: tf = 1; t0 = ?1, ???????????
??????? ?????? (2.6) ????? ????? ?????????? ? ????
g0 (? ) =
n
?
lk0 hk (1 ? ? ) =
k=1
???
n
?
n
?
k=1
n?1
?
lk0
(1 ? ? )n?k =
bp ? p ,
(n ? k)!
p=0
(5.2)
lk0 ck = 1, ? ???????????? bp , p = 0, 1, ..., n?1, ???????????? ?? ????????,
k=1
??????? ?????????? (3.3):
b p = (?1)
p
n?1
?
1
p
ln?m,0 Cm
, p = 0, 1, ..., n ? 1.
m!
m=p
(5.3)
???? ?????? b
=
col [(?1)n?1 bn?1 , (?1)n?2 bn?2 , ..., ?b 1 , b 0 )] ? l0
=
= col (l10 , l20 , ..., ln0 ), ?? ??????????? (5.3) ????? ?????????? ? ????????????????? ????:
M l0 = b,
(5.4)
??? ??????? M , ???????? ??????? (3.5), ?????? ??? tf = 1. ??????????????, l0 =
= M ?1 b ? ????? ??????? ??????????????? ??????, ????? l10 ?= 0. ??? ? ? ??????
(2.3), ???? ?????? ????? ???????? ?????????? ? ??? ????? ???????? bn?1 ?= 0
??? deg g0 (? ) = n?1. ??????? ????????????? ??????????? ??????? ?????? (2.6),
????? deg g0 (? ) < n ? 1, ????? ??????????? ????.
????????? ??? ???????????? ???????? ?????? || g0 (и) ||L1 [?1,+1] = ?0 , ?? ? ?????? || Um (и) ||L1 [?1,+1] = 2 (m = 0, 1, 2, ...) [14; 16] ????? ?????? ????? ?????:
g0 (? ) =
n?1
?
p=0
bp ? p =
1
s0 ?0 Un?1 (? ), ? ? [?1, +1],
2
(5.5)
?n?2
??? s0 = ▒1, Un?1 (? ) = 2n?1 ? n?1 + p=0 ?p ? p ? ????????? ???????? ??????? ????
(n ? 1)-? ???????, ?p , p = 0, 1, ..., n ? 2, ? ??? ????????????. ????????: U0 (? ) =
= 1; U1 (? ) = 2? ; U2 (? ) = 4? 2 ? 1; U3 (? ) = 8? 3 ? 4? ; U4 (? ) = 16? 4 ? 12? 2 + 1 ? ?. ?.
???????? ???????????? ???????: Un+1 (? ) = 2? Un (? ) ? Un?1 (? ) (n > 0). ???????
????????, ??? ? (5.5) ? ?????? ?????? (3.6) ?0 = T2 ??0 , ??? ??0 = || h0 (и) ||L1 [t0 , tf ] ?
????? ???????????? ???????? ??? ??????? ?????? (2.6).
??????????? ????? ? ?????? (5.5) ??????????????? ???????????? ? (5.2) ?
?????????? ?? s0 ?0 /2 ???????????? ?????????? Un?1 (? ), ??????? ????????????
?? ??????????? ?????????? n-??????? ????????????
127
?????????? lk0 , k = 1, 2, ..., n, ??????????? ??????? ????????? (5.4), ? ??????:
(?1)
m
n?m
?
k=1
m
Cn?k
1
lk0 = s0 ?m ?0 , m = 0, 1, ..., n ? 1.
(n ? k)!
2
(5.6)
????? ? ???????????? ??????-??????? ? = (?1)n?1 col (?n?1 , ?n?2 , ..., ? 1 , ?0 ) ?
l0 = col (l10 , l20 , ..., ln0 ), ????????? ??????? (5.6) ? ????????? ???? (5.4):
M l0 =
? ?????? ??????? lT0 c =
???????????? ????????:
?n
k=1 lk0 ck
1
s0 ?0 ?.
2
(5.7)
= 1 ?????? ??? ?????? (2.6) ???????? ?????
?0 = 2 |cT M ?1 ?|
?1
,
(5.8)
?1
T
? ????? s0 = sign (c M ?).
??????????? ?????????? ? ??????????????? ?????? ????? ?????????? ?? ??????? (2.7), (2.8), ? ?????? ? ?????? (5.8) ????? ????? ?????:
u? (? ) =
1
1
sign g0 (? ) = sign [s0 Un?1 (? )], ? ? [?1, +1],
?0
?0
(5.9)
min
= ?10 = |cT M ?1 ?|.
??? g0 (? ) = lT0 h(1 ? ? ), ? lT0 c = 1. ??? ???? ??????? J?
?????????? ?????? ??????? ????????????? ???????????? ??????? ??? ??????
(2.6). ????, ? ?????? (5.4), (5.7) ? cT l 0 = 1 ? ????? ?, ??????????????, ? ?????????? ?????? ?????? ????:
1
s0 ?0 cT M ?1 ? = 1.
2
(5.10)
???? ??????? c ? p = M ?1 ? ????????????, ?? cT p = 0 ?, ??????????????, ? ????
?????? ?????????? ??????????? ??????? ?????? (2.6).
??? m-??????? ????????????? ?????? (2.6) ??????????????? ????? ??????
??????? ? ?????? (4.2) ? ?? ???????? ? (4.7), (4.8) ????? ???????? ???:
cT Mn?1 ?n?j = 0, j = 0, 1, ..., m ? 1, cT Mn?1 ?n?m ?= 0,
??? ?n?j = col (0, ..., 0, ??n?j ) ? Rn , ? ??n?j ? ?????? ????????????? ??????????
| {z }
j ???
Un?j?1 (? ) (j = 0, 1, ..., n?1). ???? ?????? ??????????? pn?j = Mn?1 ?n?j , ?? ???????
m-??????? ????????????? ?????? (2.6) ????? ?????????? ? ????
cT pn?j = 0, j = 0, 1, ..., m ? 1, cT pn?m ?= 0.
(5.11)
? m-?????? ??????????? ?????? (2.6) ??????????? ?????????? (5.9) ?????
?????? (n ? m) ?????????????, ?? ??????? ??? ?????????. ????????? ????????
??????? ???????? pn?j ? (5.11) ?????????? ????????? qn?j ?? (4.9), ?? ???????
(5.11) ????? ????? ?????????? ? ????, ??????????? (4.10).
?????? 2. ?????????? ??????? ?????? (1.4), (1.5), (1.11) ??? ?????????
???????? ? (1.2), (1.3): t 0 = ? T2 ; tf = T2 ; x20 = . . . = xn0 = 0 ? x3f = . . . =
= xn f = 0, x 1f = ?T , 0 < ?T 6 ?. ????? ?? (1.8 c 1 = ?T , c2 = . . . = cn = 0, ?? ????
l 10 = ?1T ? lk0 ? R1 , k = 2, 3, ..., n. ???????? ??????????? ??????? ?????? (2.6)
1
h0 (t) =
?T (n ? 1)!
( )n?1 (
)n?1
T
2t
1?
+
2
T
128
?.?. ???????
+
l20
(n ? 2)!
( )n?2 (
)n?2
(
)
T
2t
ln?1,0 T
2t
1?
+ ... +
1?
+ ln0 .
2
T
2!
T
T
2
? ??????? g0 (? ) (5.2):
( )n?1
1
T
n?1
g0 (? ) =
(1 ? ? )
+
(n ? 1)! ?T 2
( )n?2
l20
T
ln?1,0 T
n?2
+
(1 ? ? )
+ ... +
(1 ? ? ) + ln0 =
(n ? 2)! 2
2!
( )n?1
s0
T
= n?1
(2n?1 ? n?1 + ?n?2 ? n?2 + ... + ?1 ? + ?0 ) =
2
(n ? 1)! ?T 2
( )n?1
1
T
T n?1
| Un?1 (? ) |,
= n?1
| Un?1 (? ) | = 2(n?1)
2
(n ? 1)! ?T 2
2
(n ? 1)! ?T
?????? ? ?????? ?????? t =
??? s0 = (?1)n?1 . ????? ? ?????? (5.5), ? ????? d? = 2dt
T ? || Um (и) ||L1 [?1,+1] = 2
??????? ????? ???????????? ???????? ??? ?????? (2.6):
?
+
T
2
| h0 (t) |dt =
?
T
2
Tn
= ??0 ,
22(n?1) (n ? k)!?T
?, ????? ????, ????? ????? ?????:
??0 =
1
umax
=
Tn
,
22(n?1) (n ? 1)!?T
? ?????? ? ??? ??????? ??????????? (5.1) ? ???????, ??? ??????? ?????? ? ?????? ??????? ?????????? ?????? ?????, ????? ???????????? ??????? ?????? ??
??????, ???
?
n 22(n?1) (n ? 1)!?T
Tmin =
.
(5.12)
umax
?
T
[17], [18], ? ??? n =
? ?????????, ??? n = 2 ?? (5.12) ???????: Tmin = u4?
max
3? 32?T
= 3: Tmin =
umax [11], ??? ???????? ????????? ?????????????? ? ???????
?????????? ? ????????? ??????? ?????????? ??????? ??? ????? (5.1). ?????
????, ?????? ??????? ??? ???????????? ?????????? ???????? (5.12) ????? ?????
22(n?1) (n?1)!?T
. ????????
T n?1
min
???????, ??????? J 1 = 12 umax T
umax T =
??? ????????? J min
(3.14) ??? ???????? ??????
1
2n?3
T
= 2 T(n?1)!?
. n?1
?????? 3. ?????????? ??? ??? ??????? ?????? (1.4), (1.5), (1.11), ?? ???
????????? ???????? ?????? [9]: n = 3; t 0 = ? T2 ; tf = T2 ; x 10 = ?0 , 0 < ?0 6 ?;
x20 = ? 0 ; x30 = ? 0 , ??? ? 0 , ? 0 , ? 0 ? ????????? ???????? ?????????, ? ?????
x1f = x2f = x3f = 0. ????? ?? (1.8) ??????? ????????? ???????? ????????:
1
c 1 = ? (?0 + T ?0 + T 2 ?0 ); c 2 = ? (?0 + T ?0 ); c 3 = ? ?0 .
2
??????????? ??????? ??????
(
)2
(
)
l10 T 2
2t
2t
l20 T
h0 (t) =
1?
1?
+ l30 ,
+
8
T
2
T
(5.13)
(5.14)
?? ??????????? ?????????? n-??????? ????????????
129
??? l10 , l20 , l30 ? ?????????, ??????????????? ??????? c 1 l10 + c 2 l20 + c 3 l30 = 1.
????? ????? ??????????? ?????????? ? = 2t
T , ????????? (5.14) ? ????
)
(
l20 T
l10 T 2
T?
(1 ? ? )2 +
(1 ? ? ) + l30 =
h0
= g0 (? ) =
2
8
2
=
l10 T
32
=
[
2
l10 T 2 2 T
l10 T 2
l20 T
? ? (T l10 + 2l20 )? +
+
+ l30 =
8
4
8
2
(
)]
8
32
l10 T 2
l20 T
4? 2 ?
(T l10 + 2l20 ) ? +
+
+
l
.
30
l10 T
l10 T 2
8
2
2
10 T
??? l10 ?= 0 ???? ????????? ? ????????? ?? ????????? s0 l32
(s0 = ▒1) ??????
????????? ? ??????????? ???????? ??????? ???? U2 (? ) = 4? 2 ? 1, ?? ???? ?????
2
10 T
????? ????? g0 (? ) = s0 l32
(4? 2 ?1) ?, ??????????????, ??????? ??????? ?????????
???????????? l10 , l20 ? l30 :
c 1 l10 + c 2 l20 + c 3 l30 = 1; T l10 + 2l20 = 0;
5T 2
T
l10 + l20 + l30 = 0.
32
2
????? ??? ??????? ?? ??????? ??????? [15], ???????
l10 =
??? D0 = 2c 1 ? T c 2 +
3T 2
16 c 3 ,
D1
D2
D3
; l20 =
; l30 =
,
D0
D0
D0
D 1 = 2, D 2 = ? T , D 3 =
3T 2
16 ,
?, ????? ????, ?????
?1
)?1
(
T 2 3T 2
3T 2 c3
, ?0 =
c
2c 1 ? T c 2 +
2c
?
T
c
+
,
3
1
2
16
16 16
(
)
2
16 3T 2 ? ????? s0 = sign 2c 1 ? T c 2 + 3T
c
,
u
=
c
?
c
T
+
c
2
3
max
1
2
3
16
T
16 . ????????,
l10 = 2
??? ????? ??? ?0?? ? umax ? 0, ??? ????? ????? ? ?????? (5.13) ??? T ? T? ,
64?02 ?96?0 ?0
0
(???? 2?02 ? 3?0 ?0 > 0 ? T? > 0).
??? T? = ? 8?
3?0 ▒
3?0
? ??????????? ??????, ????? l10 = 0, ????????? (5.14) ? ????????? ?? ????????? ????? ????? U1 (? ) = 2? . ????? ????? ????? ?????:
(
)
T?
l20 T
l20 T
l20 T
h0
= g0 (? ) =
(1 ? ? ) + l30 = ?
?+
+ l30 .
2
2
2
2
?????????????? ?????? ???????
l20 =
2
T
T
; l30 = ?
; s0 = sign (2c 2 ? T c 3 ); ?0 =
.
2c 2 ? T c 3
2c 2 ? T c 3
|2c 2 ? T c 3 |
?1
???? ?? l10 = 0 ? l20 = 0, ?? l30 = c?1
.
3 , s0 = sign c 3 ? ?0 = 2 | c 3 |
??????? ????????, ??? ? ?????? ???????, ??? ? ? ??????? 2, ? ????? ??????
?????? (1.4), (1.5), (1.11) ?? ???????????? ??????????????? ?????? ?? ??????????????, ??? ??????????? ???????????? ????????????? ?? ????????? ??????????
?????? ?????????? ???????? ? ??????????????? ?? ????????? ????????? ???????????? ??????????. ??????? ??? ???????? ???????????? ??????? T ? ??????
(1.4), (1.5), (1.11) ? ??????????? ??? ??? ???????? umax ?? ???????? ? ??? ?????? ?? ??????????????, ? ??????? ???????? ??????????? (5.1), ?????? ????? ?????
????????? ?????? ??? ??????????????: Tmin 6 T . 130
?.?. ???????
??????????
??????????? ?????? ???????????? ??????????, ??????? ?????????? ????????
??? ??????? ????????? ????? ?????????? ??????????? ??????????? ?????????,
? ??????: ?????? ???????????? ?????????? n-??????? ???????????? ? ????????????? ?????????? ????????? ? ? ????????????? ???? ????? ? Lq [t0 , tf ], q =
= 1, 2, ?, ??????? ???????? ? ??????? ???????? ????????? ?.?. ???????????
(??????? ????????). ??-??????, ???? ???????? ?????? ??????? ?????? ?? ??????? ??????????? ??????? ??????????, ??????? ???????? ? L 2 -???????? ????????.
??-??????, ?????? ?????? ?? ??????? ?????????? ??? ??????? ???????? ??????????, ??????? ???????? ? L? -???????? ???????? ? ??? ??????? ????????, ???
??????????? ?????????? ???????? ?-?????????? ???????????. ???????? ???????
????????????? ??????????? ??????? ?????? ? ??????????? ?? ????????? ???????,
? ????? ??????? ????????????? ?? ??????????? ???????. ?-???????, ????????
????? ??????? ?????? ?? ??????? ???????????? ?? ?????? ???????? ???????????? ????????? (??? L 1 -???????? ????????) ? ??????? ????????????? ??? ??????????, ??? ? ??????????? ??????? ???? ??????. ????????? ??????? ???????
?????, ??????? ???????? ? L? - ? L 1 -????????? ????????. ??????????? ????????????????? ?????? ?? ??????? ???????????? ?????????? ??????????????? ??????
?? ??????????????.
??????????
[1]
??????? ?.?. ????????? ?????????? ? ???????????? ? ?????? ???????????? ??????. ?.: ?????????, 1994. 544 ?.
[2] ?????????? ?.?. ?????? ?????????? ?????????: ???????? ???????. ?.: ?????,
1965. 476 ?.
[3] ????? ?.?. ???? ?????? ??????. ?.: ?????? ?????, 1987. 304 ?.
[4] ?????? ??????????????? ??????????: ? 2 ?. ?. II. ?????? ?????????? ? ??????????? ?????? ??????????????? ?????????? / ?.?. ???????, ?.?. ???, ?.?. ?????
[? ??.]; ??? ???. ?.?. ????????. 2-? ???., ???????. ? ???. ?.: ?????? ?????, 1986.
504 ?.
[5] ?????????? ?.?. ?????? ?????????? ????????? ? ??????????????? ???????????.
?.: ?????, 1975. 568 ?.
[6] ??????? ?.?. ??????? ?????????? ???????? ?????????? ?????????. ?.: ???-??
???, 1981. 200 ?.
[7] ??????? ?.?., ????? ?.?. ??????????? ?????????????? ? ????????? ???????? ???????? // ?????????????? ???????. 1987. ?. 134(176). ? 2(10). ?. 186?206.
[8] ??????? ?.?., ????? ?.?. ?????? ??????? ????? n-?????? ?????? ?????????????? // ??????? ?? ????. 1988. ?. 296. ? 6. ?. 1304?1308.
[9] ??????? ?.?. (Widnall W.S.) ??????????? ????? ?????????? ???????? ???? ? ?????????? ??????? ?????? ???????????? ??????? ????????? /?/ ?????????? ? ???????:
????? III ?????????????? ?????????? ???? ?? ??????????????? ?????????? ?
?????? ????????????? ???????????? ????????????. ?.: ?????, 1972. ?. 2. ?. 36?49.
[10] ??????? ?.?., ???????? ?.?. ??????????? ?? ???????? ???????? ??????????
??????? ???????????? // ??????? ?????????? ???????????????? ????????????.
2012. ? 9(100). ?. 118?129.
[11] ??????? ?.?., ???????? ?.?. ?????? ??????????? ?????????? ???????????? ?????????? ???????????? ???????? ??????? ???????? // ???????? ??? ???. 2012.
T. 14. ? 6. ?. 166?176.
?? ??????????? ?????????? n-??????? ????????????
131
[12] On Optimization of Attitude Control Programs for Earth Remote Sensing Satellite /
Yu.N. Gorelov [et al.] // Gyroscopy and Navigation. 2014. Vol. 5. ? 2. P. 90?97.
[13] ??????? ?.?. ? ??????? ?????? ??????? ???????????? ?????????? ??????????????? ???????????? ???????? ??? ??????????????? ?????????? ???????????? ?????
??????? ???????????????? ??????????? // ???????? ??? ???. 2014. T. 16. ? 4.
?. 127?131.
[14] ??????? ?.?. ?????????? ????????. ?.: ????????? ????, 2003. 160 ?.
[15] ???????? ?.?., ???????? ?.?. ??????? ? ??????????. ?.: ?????, 1984. 320 ?.
[16] ????????? ?.?., ???????? ?.?. ???????? ???????????? ?? ???? ????????? //
?????????????? ???????????? / ??. ???. ?.?. ??????????. ?.: ????????? ????????????, 1982. ?. 3. ?. 874?875.
[17] ???????? ?.?. ? ?????? ??????????? ?????????????? ???????????? ???????? //
??????????? ????????????. 1973. T. 11. ? 2. ?. 180?187.
[18] ??????? ?.?., ????? ?.?. ?? ??????????? ?????????????? ???????????? ???????????? ???????? // ??????????? ????????????. 1978. ?. 16. ? 2. ?. 157?162.
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[3]
Moroz A.I. Course of systems theory. M., Vysshaia shkola, 1987, 304 p. [in Russian].
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133
?? ??????????? ?????????? n-??????? ????????????
Yu.N. Gorelov 3
ON THE OPTIMAL CONTROL OF THE N -FOLD
INTEGRATOR4
The optimal control problem n-fold integrator with arbitrary boundary
conditions and functionals of type norms in spaces of Lq [t0 , tf ], q = 1, 2, ?
is considered. First, it is the problem of minimizing the total controling
impulse, which boils down to L? - problem of moments; secondly, the problem
of minimizing the maximum values of the control parameter (represented
as L 1 -problem of moments), and, ?nally, it is the problem of minimizing
?generalized work control? (as L 2 -problem of moments). Solving problems is
obtained by using the method of moments in the form of the maximum principle
by N.N. Krasovsky. It is shown that optimal control in the ?rst problem is
approximated by a ?-impulsive control. Conditions for the existence of regular
and singular solutions to this problem depending on the boundary conditions
are also speci?ed. The general solution of the second problem, which is the
conditions for existence of regular and singular solutions and not equivalence
with the mutual problem of time-optimal control is obtained. Examples of
solution for the considered control tasks are given. In case of a quadratic
functional general relations required for constructing a program optimal control
were obtained.
Key words: the n-fold integrator, optimal control, problem of moments,
maximum principle by N.N. Krasovsky, Chebyshev polynomials.
?????? ????????? ? ???????? 24/VIII/2015.
The article received 24/VIII/2015.
3 Gorelov Yury Nickolaevich (yungor07@mail.ruu), Institute of Modeling and Control Sciences,
Samara State Aerospace University, 34, Moskovskoye Shosse, Samara, 443086, Russian Federation.
4 The research is carried out with support from the Russian Foundation for Basic Research, project
? 13-08-97019 ?_????????_?.
?? ????????:
?tf
hk (tf ? ?)u(?)d? = ck ,
k = 1, 2, ..., n.
(1.9)
t0
????? ???????, ???????????? ????????? ?????? (1.4), (1.5) ???????? ? ??????? ????????? (1.9) ???????????? ?????????? u(и) = u[t0 , tf ] ???, ??? ?? ?? ?????, ? ???????? ???????? [2; 3]. ???? ????????????? ???????????, ????? ???????
?????????? ?????????? ??????? ??????-???? ?????????? ??? ????????, ????????,
116
?.?. ???????
???? ????? ??? u(и) = u[t0 , tf ], ????? (1.4), (1.5) ? ?????? ???????????? ??????????. ??????????? ???? ????? ? Lq [t0 , tf ], q = 1, 2, ?, ????? ????????? ??? [2]:
?tf
J1 (u) = || u(и) ||L1 =
| u(t) | dt;
(1.10)
t0
J? (u) = || u(и) ||L? = max
t? [t0 , tf ]
| u(t) |;
(1.11)
?tf
J2 (u) = || u(и) ||L2 =
u2 (t)dt.
(1.12)
t0
?????????????? ?????? ???????????? ?????????? (1.4), (1.5), (1.10) ?? ??????? ?????????? ?????????? ???????? ? L? -???????? ???????? ??? (1.9), ??????
?????????? (1.4), (1.5), (1.11) ?? ??????? ???????????? ?? ?????? ????????
?????????? ???????? ? L 1 -???????? ????????, ? ?????? (1.4), (1.5), (1.12) ?? ??????? ??????????? ??????? ?????????? ? ? L 2 -???????? ????????. ? ????? ?
?????????????? ??????? ???? ????? ?????????? ???????, ??? ??????? (1.4) ???????? ?????? ???????????, ????????? ??????? ????????????? ??????? [3; 4] ???
??? ? ????????????? ???????.
????????????? ???????, ??? ? ?????? ??????? ????????? ??????? (1.5), ???
??????? ??????????? ????????? ???????:
xf ? ?(tf , t0 )x0 = 0,
(1.13)
? ?????? (1.6) ? (1.8) c = 0. ??? ?????????? ??????? (1.13) ??????? ????????????? ???? ????? ?????????? ?? ?????????, ??? ??? ? ???? ?????? ?????????
??????? (1.5) ????????????? ??????????? ??? ???????????? ??????? ??????????:
u(t) ? 0, ?t ? [t0 , tf ]. ????? ????, ??????? (1.13) ????? ????? ??????????? ? ?
?????? ??? ?????????? ??????????????? ??? ????-???????? ?????????? [5; 6],
?tf
??????????????? ??????? ?(tf ? t)en u(t)dt = 0.
t0
?????? ???????????? ?? ?????????????? ?????????? n-??????? ????????????
(1.1) ???????? ? [7; 8]. ??? ???????????? ?????? ???????????? ?? ?????? ????????????? ???????, ?? ? ?????? ?????????? ???????? [9]. ?? ?? ????? ????????? ?
? ??????? ??????????????? ????? ?????????? n-??????? ???????????? ? ????????????? (1.10)?(1.12), ??? ??????? ? ???????????? ?????????? ??????????????
? [10?13].
2.
?????? ???????????? ?????????? ??? n-????????
??????????? ? ????????????? ???? ?????.
??????? ?????? ?? ??????? ???????????
??????? ??????????
??????? ????????? ?.?. ??????????? [3]. ????? h0 (?) = l0T ?(tf ? ?)en ? ??????? ????????? ?????? (?? l = l0 ):
min || lT ?(tf ? ?) en || Lp = min || lT h(tf ? ?) || Lp =
lT c=1
lT c=1
= || lT0 h(tf ? ?) || Lp = || h0 (?) ||Lp = ?0 (p = 1, 2, ?),
(2.1)
117
?? ??????????? ?????????? n-??????? ????????????
??? l = col (l1 , l2 , ... , ln ), ?0 ? ????? ???????????? ????????. ????? ???? ??????:
(
?tf
max
||u(и)||Lq =
h0 (t)u(t)dt = 1
1
?0
)
1 1
+ =1 ,
p q
(2.2)
t0
?
????? ???????????? ??????? u (и), ?? u? (t), ?t ? [t0 , tf ], ? ??????????? ??????????. 2.1. ? ???????????? ? ????????? ????????? ??????? ?????? ????????????
?????????? (1.4), (1.5), (1.10) ? ?????? (1.8) ???????? ? ????????????????? ??????? ????????? ????? [2; 3].
??-??????, ??????????? ????? ???????????? ????????, ? ??????:
(
)
T
T
min || l h(tf , и) ||L? = min
max | l h(tf ? t) | = || h0 (и) ||L? = ?0 , (2.3)
lT c=1
lT c=1
??? h0 (tf ? t) = lT0 h(tf ? t) =
t? [t0 ,tf ]
n
?
k=1
lk0 (tf ?t)n?k
,
(n?k)!
? lT0 c =
?
n
?
lk0 ck = 1.
k=1
??-??????, ?????? ???????????? ?????????? u (и) ?? ??????? (2.2):
?tf
max
||u(и)||L =
1
???
?tf
1
?0
(2.4)
h0 (t)u(t)dt = 1,
t0
?tf
?
h0 (t)u (t)dt = 1, || u(и) ||L 1 =
t0
| u(t) |dt =
1
.
?0
(2.5)
t0
?? ??????? ?????? (2.3) ?????, ??? ??????????? ??????? ? ????????? ???????? ??????? ???? ? ????????? ?? ????????? ?0 , ?? ???? ?????????, ???????? ???????????? ?? ???? ?? [ t0 , tf ] [14]. ????? ????, ??????????? ??????????
????? ? ??????? ???????? ?-?????????? ??????????? [10], ??????? ???????????
?-?????????? ? ?????? ??????????? ???????????? ???????? ? ? ????????? ?????? ????????? ??????????. ??? ???? ? ??????? ???????? ??? ?????? ???????
?????????? ? ?????? (2.5) ????? ????? ????????, ???????? ????? ????????????
?1
????????: J inf
1 = ?0 .
2.2. ??????? ?????? ???????????? ?????????? (1.4), (1.5), (1.11) ????? ???????? ? ????????????????? ??????? ????????? ?????.
??-??????, ? ??????????? ????? ???????????? ????????:
?tf
min || l h(tf , и) ||L1 = min
| lT h(tf ? t) | dt = || h0 (и) ||L1 = ?0 .
T
lT c=1
lT c=1
(2.6)
t0
??-??????, ? ??????? ???????????? ?????????? u? (и) ?? ???????:
?tf
max
||u(и)||L? =
???
?tf
t0
1
?0
(2.7)
h0 (t)u(t)dt,
t0
h0 (t)u? (t)dt = 1, || u(и) ||L ? =
max
t? [t0 , tf ]
| u(t) | =
1
.
?0
(2.8)
118
?.?. ???????
????????, ??? ?? ??????? ?????? (2.6) ??????? ??????????? ??????? ? ????
?????????? ???????? ??????? ???? ? ????????? ?? ?????????? ????????? [14].
?????????????? ? ?????? (2.8) ??????????? ?????????? ? ???? ?????? ????? ?????
????????? ???:
1
u? (t) =
sign h0 (t), ?t ? [t0 , tf ],
(2.9)
?0
min
??? h0 (t) = lT0 h(tf ? t), ? lT0 c = 1. ??? ???? ??????? J?
= ??1
0 .
2.3. ??? ??????? ?????? (1.4), (1.5), (1.12) ?? ??????? ??????????? ???????
?????????? ??????? ????? ??????? ????? ??????????? ??????? ? ??? ????? ?
L 2 [t0 , tf ], ? ??????:
? tf
? 12
?
min || lT h(tf , и) ||L2 = min ? [lT h(tf ? t)] 2 dt? = || h0 (и) ||L2 = ?0 .
(2.10)
lT c=1
lT c=1
t0
?tf
??????? ?????, ???
[lT h(tf ? t)] 2 dt = lT Gl, ??? G =
t0
?tf
h(tf ? t)h(tf ? t)T dt ?
t0
??????? ?????????????:
?tf
(tf ? t)2n?k?j
G = [gk j (tf ? t)]nОn dt, gk j (tf ? t) =
, k, j = 1, 2, ..., n,
(n ? k)! (n ? j)!
t0
???????? ???????? ??????????? ?? ????????:
?tf
?tf
(tf ? t)2n?k?j
(tf ? t0 )2n?k?j+1
gk j (tf ? t)dt =
dt =
.
(n ? k)! (n ? j)!
(n ? k)! (n ? j)!(2n ? k ? j + 1)
t0
t0
? ???? ?????? ????????????? ??????? (1.4) ?? ??????? ????????????? G ???????? ????????????? ????????. ???? ??????? t0 = ?1 ? tf = +1, ?? ??? ????????
????? ??????????? ?? ????????:
?tf
22n?k?j+1
gk j (tf ? t)dt =
, k, j = 1, 2, ..., n.
(n ? k)! (n ? j)!(2n ? k ? j + 1)
t0
??? ????????? ???????????? ???????? ? ??? ????? ? ??????????????? ??????
??????? ????? ??????? ???????????? ????? lT Gl ??? ???????, ??? lT c = 1.
????? ??? ?????? ??????? ?????????? ????????, ??????
G?1 c
l 0 = T ?1
c G c
?, ??????????????, ???????
1
cT G?1 h(tf ? t)
, ?0 = (cT G?1 c)? 2
h0 (t) =
(2.11)
cT G?1 c
????? ?? ??????? ????????? ????????? ? (2.2) ????????? ??????????? ??????????, ????????? ???????? ????? ????? ????????? ???:
u? (t) = ? h0 (t), ?> 0(?t ? [t0 , tf ]).
????????? ??? ????????? ? (2.2), ???????, ??? ? = ??2
0 , ?? ???? ???????????
?????????? ??? ?????? (1.4), (1.5), (1.12) ? ?????? (2.11) ????? ???
1
u? (t) = 2 h0 (t) = cT G?1 h(tf ? t), ?t ? [t0 , tf ],
(2.12)
?0
? ??????????? ???????? ??????????? (1.12) ????? ????? J2min =
1
?20
= cT G?1 c.
119
?? ??????????? ?????????? n-??????? ????????????
3.
??????????? ?????????? n-???????
???????????? ?? ??????? ????????? ??? ???????
???????? ??????????
??????? ?????? (2.3) ?????????? ? ?????? (1.7) ?? ??????????? ???????
h0 (t) =
n
?
lk0 hk (tf ? t) =
k=1
n
?
k=1
?
lk0
(tf ? t)n?k ,
lk0 ck = 1.
(n ? k)!
???????? ??????? ?????? (tf ? t)n?k =
n
(3.1)
k=1
n?k
?
m=0
m
(?1)m Cn?k
tfn?k?m t m , ?????????
????????? (3.1) ? ????
h0 (t) =
n
?
k=1
n?k
n?1
?
?
lk0
m
tfn?k?m t m =
(?1)m Cn?k
ap t p ,
(n ? k)! m=0
p=0
(3.2)
r!
??? Crq = q!(r?q)!
, ? ap , p = 0, 1, ..., n ? 1, ? ???????????? ?????????? ? ???????????? ???????? ?????? (2.3), ??????????? ?? ????????:
a p = (?1)p
n?p
?
k=1
p
lk0 Cn?k
tn?p?k , p = 0, 1, ..., n ? 1.
(n ? k)! f
(3.3)
????? a = col [(?1)n?1 an?1 , (?1)n?2 an?2 , ..., ?a 1 , a 0 )] ? l0 = col (l10 , l20 , ..., ln0 ),
??????????? (3.3) ????? ?????????? ? ????????-????????? ????:
(3.4)
M l0 = a,
??? ??????? M ? RnОn ????? ????????? ??????????:
?
1
0
0
иии
0
(n?1)!
n?2
? Cn?1 tf
1
? (n?1)!
0
иии
0
(n?2)!
?
n?3 2
n?3
? Cn?1
tf
Cn?2
tf
1
? (n?1)!
иии
0
(n?2)!
(n?3)!
?
?
?
?
..
..
..
..
..
.
M = ?
.
.
.
.
?
?
? 2 n?3
2
2
? Cn?1 tf
Cn?2
tn?4
Cn?3
tn?5
C 32 tf
f
f
?
иии
(n?2)!
(n?3)!
3!
? (n?1)!
1
1
1
? Cn?1
tn?2
Cn?2
tn?3
Cn?3
tn?4
C 31 t2f
f
f
f
?
иии
(n?2)!
(n?3)!
3!
? (n?1)!
tn?2
tn?3
tn?1
t3f
f
f
f
иии
(n?1)!
(n?2)!
(n?3)!
3!
0
0
0
0
0
0
..
.
..
.
1
2!
C21 tf
2!
t2f
2!
0
1
1!
tf
1!
0
?
?
0 ?
?
?
0 ?
?
?
?
.. ?
. ?
?.
?
?
?
0 ?
?
?
0 ?
?
(3.5)
1
????????? ??????? M ?????????????, ?? ?? (3.4) ?????? ????? ???????? l0 =
= M ?1 a. ??? ???? ????????? ??????? (3.5), ???????? ?? ??????????, ?????? ????????? ??????? ?????????? [15]. ????? ????, ??????????? ????????? ??????? M
????????? ?????? ??????? (3.4), ??????????????? ????????? lk0 , k = 1, 2, ..., n-1,
??????? ? l10 = (?1)n?1 (n ? 1)! a n?1 ? ?. ?.
??? ??? l0 ? ??????? ?????? (2.3) ??? ?????????? c ?= 0, ?? ????? ? ?????
??????, ????? l10 ?= 0, ????? ???????? ??????????, ??? ? ?????? (3.4) ? (3.5)
???????? ??????? an?1 ?= 0 ???, ??? ?? ?????, ????? deg h0 (t) = n ? 1. ?????,
???? deg h0 (t) < n ? 1, ?? ??????? ?????? (2.3) ????? ???????????, ??????? ????????????? ???????? ????? ??????????? ????. ? ?????? (3.4) ? cT l 0 = 1, c ?= 0
120
?.?. ???????
? ?????????? ?????? ?????? ????? ?????: cT M ?1 a = 1 ?, ??? ?????????, an?1 ?= 0
?, ????? ????, l10 ?= 0. ????????, ??? ??? ??????? ?? ??????????? ? ??? ??????,
????? ??????? c ? M ?1 a (??? M ?T c ? a) ????????????.
??? ??? ??????? (1.4) ????????????, ?? ??? ??????? ?????? ????????? ?????????? [t 0 , tf ] ??????????? ???????????? ????????, ???????? ???? ????????????
????????? ?????????? ? T = tf ?t 0 , ??????????? ???????????? ?????????? ??????????????? ????? ???????????? ??????????. ????????, ??? ??????????? ???????? ???????? ????????? ???????? ??????? ????????? [t 0 , tf ]: ??-??????, t0 = 0;
tf = T ; ??-??????, t0 = ?T ; tf = 0 ?, ?-???????, t0 = ?T /2; tf = +T /2. ????
tf = 0, ?? ???? ?? ?????? ??????, ?? ??????? (3.5) ????? ????????????.
????????? ??????? ??????????????? ????? ?????????? ????? ??????? ? ???????????? ????????, ?? ???????? ?????????? [t 0 , tf ] ?????? ????????????? ? ????????? [?1, +1], ???????? ?? ?????????? t ? (3.2) ? ?????????? ? :
1
[(tf ? t 0 )? + (tf + t 0 )], ? ? [?1, +1].
(3.6)
2
?????? ??????????? ?????????? ???????? (3.6) ? ????????? ??? ?????????? (3.2),
??? ???????????? ???????? ?????? (2.3), ???????? ??? ? ?????? ????:
t=
g0 (? ) =
n?1
?
bp ? p , ? ? [?1, +1],
(3.7)
p=0
??? bp , p = 0, 1, ..., n?1 ? ????????????, ??????????? ?? ??????????????? ???????? ? ??????????? ?? ???????? ??????? ?????? ????????? ?????????? [t 0 , tf ]
? ?????????? ??? ?????? ?????????? (3.6), ? ?????? ??? ?????? ?? ?????????
????????? ?????? ?????????? t ? [t 0 , tf ] ?? ? ? [?1, +1]:
T
T
T
(? + 1); 2) t = (? ? 1); 3) t = ?.
2
2
2
????????, ??? ??????? ???????? (3.8) ?? (3.2) ???????
n?1
n?1
n?1
?
? ( T )p
?
ap t p =
(? + 1) p =
b k ? k,
ap
2
p=0
p=0
(3.8)
1) t =
k=0
??? b k =
n?1
?
am
m= k
( T )m
2
m?k
Cm
, k = 0, 1, ..., n?1, ?, ? ?????????, b n?1 = an?1
( T )n?1
2
.
??????????????, ??? ????????? ????????? (3.8) ?? (3.2) ???????
( )m
( )k
n?1
?
T
T
m?k m?k
bk =
am
(?1)
Cm
? b k = ak
, k = 0, 1, ..., n ? 1.
2
2
m= k
????????? ???????????? ?????????? (3.7) ??????????? ?? ????????????? (3.3),
?????????, ????? ? ???????????? ??????-??????? b = col (bn?1 , bn?2 , ..., b 0 ), ?????
????? ????????? b ? a ????? ??????????? ? ????? ????:
Qa = b,
(3.9)
??? ??????? Q ? R
????? ??????????????? ?????????? ? ??????????? ?? ?????? ???????? ?????? (3.8). ??????? ? ?????? (3.9) ?? (3.4) ????? ???????
nОn
l 0 = M ?1 Q ?1 b.
(3.10)
?????? ??????? ?????? (2.3), ????? ?????????? ????????? ?0 ? ?????? ????,
??? ?? ???????? ???? ?????? ????????????? ???????? ?????????? (3.2) ?, ?????
121
?? ??????????? ?????????? n-??????? ????????????
????, (3.7) ?????? ? ?????????? ??????? ?????????? ?? ???? ???, ??? ?? ??
?????, ? ????????? ?? ????????? ?0 ? ?????????? ?????? ?????? ????? ?????
????? ???????????:
g0 (? ) = s0 ?0 Tn?1 (? ), ? ? [?1, +1],
(3.11)
??? s0 = ▒1, Tn?1 (? ) = 2n?2 ? n?1 +
n?2
?
?p ? p ? ????????? ???????? ??????? ????
p=0
(n?1)-? ??????? [14]. ?????????? ???????? ??????? ???? ?????? ???????, ?? ????
??? n = 2k (k = 0, 1, 2, ...), ????? ????????? ?????? ?????? ??????? ? , ? ?????????
???? ??? ??? ????? ????? (?1)k , ? ?????????? ???????? ??????? ? ??? n = 2k +1
(k = 0, 1, 2, ...) ? ?????? ???????? ??????? ? . ???????? [14]: T0 (? ) = 1; T1 (? ) = ? ;
T2 (? ) = 2? 2 ? 1; T3 (? ) = 4? 3 ? 3? ; T4 (? ) = 8? 4 ? 8? 2 + 1; T5 (? ) = 16? 5 ? 20? 3 + 5? , ?
?. ?. ? ???????????? ? ???????????? ????????????: Tn+1 (? ) = 2? Tn (? ) ? Tn?1 (? )
(n > 0). ?? ???? ? ????? ?????? ??? ????????????? ?????????? Tn?1 (? ) ?????
?????: ?n?2 = ?n?4 = ... = 0 ? ?n?1 = 2n?2 .
??????????? ???????????? ?? (3.7) ? ?????????????? ?????????? ?? s0 ?0
??????????????? ???????????? ?????????? Tn?1 (? ), ?? ????, ???????? b = s0 ?0 ?,
??? ? = col (?n?1 , ?n?2 , ..., ? 0 ), ?? (3.10) ???????
l 0 = s0 ?0 M ?1 Q ?1 ?.
(3.12)
??????? ????? (3.12) ???????? ?? c ? ????????, ??? ?0 > 0 ? cT l 0 = 1, ??????
???????? ????? ???????????? ???????? (3.2) ?0 ? s0 :
?0 = | cT M ?1 Q ?1 ? |?1 ; s0 = sign (cT M ?1 Q ?1 ?).
(3.13)
??????????? ?????????? ??? ??????????? ?????? ? ??????????????? ??????
????? ?????????? ?? ??????? (2.4), (2.5). ? [10] ???? ????????, ??? ???????????
?????????? ??? ???????? ??????????? ? ??????? ???????? ?-?????????? ???????????. ?? ?? ????? ????? ????? ????? ? ? ?????? n > 3, ? ?????? ? ???? ??????
??????????? ?????????? ????? ????? ????????????? ?-?????????? ? ???????????
????????? ????? ????????? ?????????? ? ????? ??????????? ???????????? ???????? ???, ??? ?? ?? ?????, ? ??????????? ????? ??????????? ?????????? Tn?1 (? )
?? ????????? ?????????? [?1, +1]. ??? ???? ???????? ??? ?????? ??????? ???????????? ?????????? ? ?????? (2.5) ? (3.13) ????? ?????
J min
1
?tf
?
= || u (и) ||L 1 =
T
| u (t) |dt =
2
?
t0
?+1
1
| u? (? ) |d? =
= | cT M ?1 Q ?1 ? |. (3.14)
?0
?1
????????? ???????????? ?????????? ? ?????? (2.4) ? (3.7) ????? ????? ???
u? (? ) = sign g0 (? )
n?1
?
u?k ?(? ? ?k ), ? ? [?1, +1],
(3.15)
k=0
??? ?(? ? ?k ) ? ?-???????, u?k > 0 (k = 0, 1, ..., n ? 1) ? ????????? ????????? ?????????? (???????? ?????????), ? ?k ? ????? ?????????? ?-?????????,
(? )
= 0. ??? ???
??????? ?? ????????? (?1, +1) ???????????? ?? ??????? dTn?1
d?
1 dTn?1 (? )
= Un?2 (? ), ?? ????????? ????? ???? ????????? ????? ?????????
n?1
d?
?????????? [?1, +1] ? ???? ??????????? ???????? ??????? ???? [14]:
?k = cos
(n ? k ? 1)?
, k = 0, 1, ..., n ? 1(?0 = ?1, ?n?1 = +1).
n?1
(3.16)
122
?.?. ???????
? ?????? ????????? ???????? (1.8), (1.9), ? ????? ???????, ?, ? ?????? ???????, ? ?????? ??????????? ??????????? (3.2) ? (3.7), ????????????? ??????? ?????????? (3.8), ?? (1.8) ??? ??????????? ? ???? (3.15) ???????:
?tf
T
h(tf ? t)u (t)dt =
2
?
t0
=
T
2
n?1
?
k=0
[
]
?+1 [
T
(1 ? ? ) u? (? )d? =
h
2
?1
]
[
]
n?1
T ? ? T
T
u?k h
(1 ? ?k ) sign g 0 (?k ) =
uk h
(1 ? ?k ) sign [s 0 Tn?1 (?k )] = c.
2
2
2
k=0
?????? ??????? ??????? ????????? ???????????? ?????????? ????????? ? (3.15),
??????? ? ????????-????????? ?????? ????? ???
Hu? = c,
?
??? u = col
(u?0 ,
u?1 , ...,
(3.17)
u?n?1 ),
H = [g0 |g1 | и и и |gn?1 ] ?, ??????????????,
]
T
gk = h
(1 ? ?k ) sign [s 0 Tn?1 (?k )] , k = 0, 1, ..., n ? 1.
2
[
???????? ??????????, ??? ??????? (3.17) ????? ????????????? ??????? ?????????? ? u? , ? ??????: (?1)k s0 u?k?1 ? u??k?1 , k = 1, 2, ..., n, ?? ???? u? ? u?? , ? ???
?????? ????? c ? c?, ??? (n ? k)! ck ? c?k , k = 1, 2, ..., n, ??? ??? (3.17) ????? ?????
???: W u?? = c?, ??? W ? ??????? ???????????.
????? ????????? (3.17), ??????? ?????? ?????????? ????????? u? ??? ???????????? ?????????? (3.15): u? = H ?1 c ? ??? ????? ??????? ? ????? ???? ??????? ?????? ???????????? ?????????? (1.4), (1.5), (1.10) ??? n-???????? ??????????? ?? ??????? ????????? ??? ??????? ???????? ?????????? ??? ???????????
??????. ???????, ??? ? ?????? (3.15) ?? (2.5) ????? ???????
n?1
?
u?k =
k=0
?????? ?????, ??? ??? ?0 ? ? ???????
2
.
?0 T
n?1
?
k=0
4.
(3.18)
u?k ? 0.
?? ???????? ?????????? ?????? ??????????
n-??????? ???????????? ?? ??????? ?????????
??? ??????? ????????
?????????? ??????? ????????????? ???????????? ??????? ? ?????? (2.3).
? ?????? ??????? cT l 0 = 1 ? (3.12) ??? ? ?????, ??? ? ? ?????????? ??????
?????? ????? ?????:
s0 ?0 cT M ?1 Q ?1 ? = 1.
(4.1)
????????? ????? ? = col (?n?1 , ?n?2 , ..., ? 0 ) ? ?????? ????????????? ??????????
Tn?1 (? ), ?? ??? ?????????? ??????? (4.1) ?????????? ? ??????????, ????? ??????? c ? qn = M ?1 Q ?1 ? ???? ?? ????????????, ?? ???? cT qn ?= 0. ?????, ?????
cT qn = 0, ??????? (4.1) ?? ??????????? ?, ????? ????, ????? ????? ?????????????
???????????? ??????? ?????? (2.3).
??? ??????????? ???????? ????????? ????? ?????? t0 = ?1 ? tf = +1. ?????
? (3.9) Q = In ? ????????? ??????? ? a = b = s0 ?0 ?, ?? ???? qn = M ?1 ?. ?????
?? ??????????? ?????????? n-??????? ????????????
123
????, ??????? (3.5) ????????? Mn ? RnОn ? ?????? ??? ??? ????????? ???????
?????????????:
?
?
1
T
]
[
0
0n?2
1
T
(n?1)!
0
n?1
1
1
T
? ? ?. ?.
0n?2
; Mn = ? (n?2)!
Mn = (n?1)!
(n?2)!
mn?1 Mn?1
?mn?1 mn?2 Mn?2
?????????????? ???????? ? ???? ???????? ????? ????? ????????? ???:
[
]
T
(n ? 1)!
0n?1
Mn?1 =
;
?1
?1
?(n ? 1)!Mn?1
mn?1 Mn?1
?
?
T
(n ? 1)!
0
0n?2
T
?
?(n ? 1)!
(n ? 2)!
0n?2
Mn?1 = ?
(4.2)
?1
?1
?1
(n ? 1)!Mn?2 (mn?2 ? ?mn?1 ) ?(n ? 2)!Mn?2 mn?2 Mn?2
? ?. ?. ?? ?????? ???????? ??????? ??? (3.5).
????? ????? c = col (c 1 , ?cn?1 ), a = col (a n?1 , ?an?1 ) ? l0 = col (l 10 , ?ln?1 ),
??????? cT l 0 = 1 ????? ?????????? ? ????
T
c 1 l 10 + ?cn?1
?l n?1 = 1,
(4.3)
? ??????? ????????? (3.4) ? ?????? (4.2) ??????????? ? ????:
?1
l 10 = (B ? 1)! a n?1 ; ?l n?1 = Mn?1
[?an?1 ? (n ? 1)! an?1 mn?1 ].
(4.4)
???? ?? cT l 0 = 0, ?? ??? ????????? c ?= 0 ?????????? ??????? ?????? (2.3) ?????????? ? ???? ??????????????? c ?= 0 ? qn = M ?1 ?. ?? ????? ?????, ?? ???? ???
cT qn = 0, ????? ????? a n?1 = b n?1 = 0 ? l 10 = 0. ????? ????, ????? ??? ?????????? ? (2.3) ??????? cT l 0 = 1 ? ???? (4.3) ? ?????? (4.4) ?????????? ? ??????????,
????? ??????????? ????????? ???????:
?1
T
?cn?1
Mn?1
?an?1 = 1,
(4.5)
(n?1)
(n?1)
??? ?an?1 = s0 ?0 ??n?1 , ? ??n?1 = col (2n?3 , 0, ?n?3 , ..., ? 0
) ? ?????? ????????????? ?????????? Tn?2 (? ). ??????? ????????, ??? ??????? ? = ?n ?
?n?1 = col (0, ??n?1 ) ???????????? ? ???? ??????? ????????????? ???????????
???????? ??????? ????. ???? (4.5) ???????????, ?? ?????? (2.3) ???????? ?????????? ???????????. ????????, ??? ??? ?????????? ????? ??????? ??????? ?????? (2.3) ????? ????????????? ??? ????????????, ?? ? ?????? ????, ??? ?????
deg h0 (t) = n ? 2, ? ??????????? ?????????? ?????? ????????? ?? ????? (n ? 1)-??
?1
T
?-????????. ???? ??????? (4.5) ?? ???????????, ?? ???? ?cn?1
Mn?1
?an?1 = 0 ???
l 10 = 0, ?? ?????? (2.3) ???????? ?? ????? ??? ????????? ???????????. ????????????? ????????? ???????????? ??????? ?????? (2.3) ??????????????? ????? ??
???????, ??? ? ? ?????? ???????????? ?????????? ???????. ??????? ?????? (2.3)
????? n-?????? ???????????, ????? ??????? (1.13) ??????????? ?????????????
??? ???????????? ??????? ??????????.
?????? 1. ????? ? (1.2), (1.3) x30 = . . . = xn0 = 0 ? x3f = . . . = xn f = 0,
? ????? t 0 = ?1 ? tf = +1. ????????? ????????, ??? ????? t ? ? ?, ????? ????,
?????????? (3.2) ? (3.7) ???????????? ? ? (3.9) Q = In ? ????????? ???????.
???????, ??? ???????? ??????? ??????? ????????? ??????? (1.2), (1.3) ????????
???????? ????-?????????? ???????????? ?????????? (3.15) [6] ? ????? ?????????? ??????? (1.1), ? ?????? ? ?????????? ??????? ?? n?2 ????????????, ???????
??????????????? ???????????? ? ???????? ??????????? ? ?????????? ??????????
?????????.
124
?.?. ???????
??? ????????? ???? ????????? ??????? ???????
c 1 = x1f ? x10 ? 2x20 ; c 2 = x2f ? x20 ; c3 = . . . = cn = 0.
(4.6)
?????? ?????, ??? (1.13) ???????????, ???? ?????? x1f ? x10 ? 2x20 = 0 ? x2f ?
? x20 = 0. ????? c 1 ?= 0 ?, ????????, c 2 ?= 0. ? ?????? (4.2) ? (4.6) ???????
cT M ?1 = [(n ? 1)!(c 1 ? c 2 )|(n ? 2)! c 2 |0| и и и |0] ??? cT M ?1 ? = 2n?2 (n ? 1)! (c 1 ? c 2 ).
????? ???????, ??? c 1 ?= c 2 , ?? ???? ????? c 1 ? c 2 = x1f ? x10 ? (x20 + x2f ) ?=
= 0, ????? ????? ?????????? ?????? ??????? ?????? (2.3) ? ?????????? ????
?????????? ?????????. ?????????? ?????? ????? ????? ? ??? x2f = x20 , ?????
c 2 = 0, ?? ??? ??????? x1f ? x10 ? 2x20 ?= 0. ??? ???? ???????
[
] ?1
?0 = 2n?2 (n ? 1)! | x1f ? x10 ? (x20 + x2f ) |
;
s0 = sign [x1f ? x10 ? (x20 + x2f )],
? ??????????? ?????????? (3.15) ????? ????? ????? ????????? ???:
u? (? ) = sign [s0 Tn?1 (? )]
n?1
?
u?k ?(? ? ?k ), ? ? [?1, +1],
k=0
??? ????????? ????????? ???????????? ?? ??????? ??????? (3.17).
???? ?? ????? c 1 ?c 2 = x1f ?x10 ?(x20 +x2f ) = 0, ?? ??????? ?????? (2.3) ? ????????? ?????????? ????????? ???????? ? ???????????? ???????????? ??????.
??? ????? ??????? ??????
?1
T
?cn?1
Mn?1
??n?1 = 2n?3 (n ? 2)! c 2 = 2n?3 (n ? 2)! (x2f ? x20 ).
?????? ?????, ??? ?????????? ??????????? ?????? (2.3) ????? ???????????? ??[
] ?1
?????, ???? c 2 = x2f ? x20 ?= 0. ??? ???? ?0 = 2n?3 (n ? 2)! | x2f ? x20 |
?
s0 = sign (x2f ? x20 ). ???? ? c 2 = x2f ? x20 = 0, ?? x1f ? x10 ? 2x20 = 0 ???, ???
?? ?? ?????, ????? ??????????? ??????? (1.13). ????????? ?????? ??????? ?????????? ??? ?????? (2.3), ????????, ??? ?????????? ?????? ??????? ?????? (2.3) ????? ?????, ????? ??????????? ???????
(4.1), ?? ???? ????? ??????? c ? Mn?1 ?n ?? ????????????. ????????, ??? ??????????? ??????? ?????? ??????????? ? ? ?????? ??????? ?????????? ??????????? ?????? (2.3), ??????? ??????? ????? ????????????? ??? ???????????? ?
?1
?????? ?????????? ??????? (4.5), ?? ???? ????? ??????? ?c n?1 ? Mn?1
?? n?1 ??
?1
T
????????????. ???? ?? ??? ??? ?cn?1 Mn?1 ?? n?1 = 0, ?? ??????????????? ?????? ????? ?? ????? ??? ????????? ???????????. ?????? ??????? ? ???????????
????????? ?????????? ?????? (2.3) ??????????????? ??????? ? ???????? ???????
???????????????, ???????? ????????????? (4.2), ? ?????????????????? ?????????
m ??? ????????:
?1
?1
(c, Mn?1 ?n ); (?c n?1 , Mn?1
?? n?1 ); ...; (?c n?m+1 , Mn?m+1
?? n?m+1 ).
(4.7)
? ?????? ?n?j = col (0, ..., 0, ??n?j ) ? R , j = 0, 1, ..., n ? 1, ??? ??n?j ? ??????
| {z }
j ???
????????????? ?????????? Tn?j?1 (? ), ??????????? (4.7) ? ?????? m-??????? ????????????? ?????? (2.3) ????? ?????????? ???:
n
cT Mn?1 ?n?j = 0, j = 0, 1, ..., m ? 1, cT Mn?1 ?n?m ?= 0,
???
(4.8)
125
?? ??????????? ?????????? n-??????? ????????????
cT qn?j = 0, j = 0, 1, ..., m ? 1, cT qn?m ?= 0,
(4.9)
Mn?1 ?n?j ,
j = 0, 1, ..., m ? 1.
??? qn?j =
?????????? ??????? (4.8) ????????, ??? ?????? Mn?T c ??????????? ?? ???? ???????? ?n ,?n?1 , ... ,?n?m+1 , ???????? ?????? ?n?m , ???, ??? ?? ?? ?????, ?????? c
??????????? ?? ???? ???????? qn ,qn?1 , ... ,qn?m+1 , ???????? ?????? qn?m . ? ?????
???? ????????, ??? ? ?????? (1.8) ? (1.13) c = 0, ?? ??????? ?????? ?? ??????????? ??????????? ?? ???? ???????? qn ,qn?1 , ..., q1 . ????? ????, ????? ???????
(1.13) ???????? n-??????? ??? ?????? ????????????? ?????? (2.3).
?????? ????? ? ???????????? ??????? ?n = [ ?n | ?n?1 | и и и | ? 1 ], ??????? ??
?????????? ???????? ?????? ??????????? ????????. ? ?????? ??????? ????????????? ??????????? ???????? ??????? ???? Tm (? ) (m = 0, 1, ..., n ? 1), ? ??????:
(m)
?m?j , j = 0, 1, ..., m, ??????? ?n ????? ????? ????????? ???:
?
? n?2
2
0
0
иии
0 0 0
?
0
2n?3
0
иии
0 0 0 ?
?
? (n?1)
n?4
? ?
0
2
и
и
и
0 0 0 ?
?
? n?3
(n?2)
?
?
0
? n?4
0
иии
0 0 0 ?
?
?
? (n?1)
(n?3)
? ? n?5
иии
0 0 0 ?
0
? n?5
?.
?n = [ ?n | ?n?1 | и и и | ? 1 ] = ?
(n?2)
?
0
иии
0 0 0 ?
0
? n?6
?
?
?
..
..
..
.. .. .. ?
..
?
.
.
.
.
. . . ?
?
?
?
2 0 0 ?
иии
?
?
?
?
?
?
0 1 0 ?
иии
?
?
?
?
?
?
иии
?1 0
1
?????? ?????, ??? ??????? ?n ? ????????????? ???????, ?, ??????????????,
???????????? ??????? ????? G = ?T
n ?n ??????? ?? ????. ?? ????? ??????? ???????? ?n?j , j = 0, 1, ..., n?1, ???????? ??????? ??????????? ????????, ? ??? ??????? ???????? ????? ???????????? Rn [15]. ??????? ????? ????? ????? ????????????
?????????? ??????? Mn?T c ?? ?????????? ??????. ?? ?? ????? ??????????? ? ???
??????? ???????? qn?j , j = 0, 1, ..., n ? 1. ???? ??? ??????? ??????????, ?? ????? ???????? ????? ???????????? Rn ? ??? ???? ????? ????? ????? ????????????
?????????? ??????? c.
??????????? ? ???????? (4.9), ???????, ??? ?? ?????????? ??? ??????? c ???????? ??? ??????????????? ? ???????? ???????? Lm ? Rn ??? ??????? m ????????
qn?j , j = 0, 1, ..., m ? 1, ??????? ???????? ???? ??????????????? Lm ?, ??????????????, (4.9) ???????? ????????? m-??????? ????????????? ?????? (2.3).
5.
??????????? ?????????? n-???????
???????????? ?? ??????? ????????????
?? ?????? ???????? ??????????. ?? ????????
?????????? ??????
????????? ? ??????? ?????? ???????????? ?????????? (1.4), (1.5), (1.11), ??????? ????????, ??? ??? ???????? ???????? ? ?????? ?? ??????????????, ? ???????
?
?????????, ????? J?
(u) = (tf ? t0 ) ? min, ? ?? ??????????? ???????? ???????-
126
?.?. ???????
?????? ???????????:
| u(t) | 6 umax , ?t ? [t0 , tf ],
(5.1)
??? umax ? ??????????? ?????????? ?? ?????? ???????? ??????????.
??????? ?????? (1.4), (1.5), (1.11) ?????????? ??????????? ???? ??????? ?????? (1.4), (1.5), (1.10), ?? ???? ????? ??? ??, ??? ? ??? ????? (2.3), (2.4), ???
???????? ? ??????? ????? (2.6), (2.7). ????? ????, ??????? ??????? ????? ??????????? ??????? ?? ??????? ?????? (2.6) ? ???? ?????????? h0 (t) (3.2), ???????
?? ???????? ?????? ? ????????? ?? ?????????? ????????? ?????? ???? ??????????? ???????? ??????? ???? (n ? 1)-? ??????? ? Un?1 (? ) [14]. ????? ?????
???????? ????????????? ????? ?????????? a ? ???????? l0 ????? ?????????? (3.4)
? ? ??? ?? ???????? M (3.5). ? ??????? ?????? (3.6) ????????? h0 (t) ???????
???????? ? ???????????? ???? (3.7), ?? ???? ? ?????????? g0 (? ), ? ? [?1, +1]. ???????? ????? ??? ??????????? ???????? ?????????: tf = 1; t0 = ?1, ???????????
??????? ?????? (2.6) ????? ????? ?????????? ? ????
g0 (? ) =
n
?
lk0 hk (1 ? ? ) =
k=1
???
n
?
n
?
k=1
n?1
?
lk0
(1 ? ? )n?k =
bp ? p ,
(n ? k)!
p=0
(5.2)
lk0 ck = 1, ? ???????????? bp , p = 0, 1, ..., n?1, ???????????? ?? ????????,
k=1
??????? ?????????? (3.3):
b p = (?1)
p
n?1
?
1
p
ln?m,0 Cm
, p = 0, 1, ..., n ? 1.
m!
m=p
(5.3)
???? ?????? b
=
col [(?1)n?1 bn?1 , (?1)n?2 bn?2 , ..., ?b 1 , b 0 )] ? l0
=
= col (l10 , l20 , ..., ln0 ), ?? ??????????? (5.3) ????? ?????????? ? ????????????????? ????:
M l0 = b,
(5.4)
??? ??????? M , ???????? ??????? (3.5), ?????? ??? tf = 1. ??????????????, l0 =
= M ?1 b ? ????? ??????? ??????????????? ??????, ????? l10 ?= 0. ??? ? ? ??????
(2.3), ???? ?????? ????? ???????? ?????????? ? ??? ????? ???????? bn?1 ?= 0
??? deg g0 (? ) = n?1. ??????? ????????????? ??????????? ??????? ?????? (2.6),
????? deg g0 (? ) < n ? 1, ????? ??????????? ????.
????????? ??? ???????????? ???????? ?????? || g0 (и) ||L1 [?1,+1] = ?0 , ?? ? ?????? || Um (и) ||L1 [?1,+1] = 2 (m = 0, 1, 2, ...) [14; 16] ????? ?????? ????? ?????:
g0 (? ) =
n?1
?
p=0
bp ? p =
1
s0 ?0 Un?1 (? ), ? ? [?1, +1],
2
(5.5)
?n?2
??? s0 = ▒1, Un?1 (? ) = 2n?1 ? n?1 + p=0 ?p ? p ? ????????? ???????? ??????? ????
(n ? 1)-? ???????, ?p , p = 0, 1, ..., n ? 2, ? ??? ????????????. ????????: U0 (? ) =
= 1; U1 (? ) = 2? ; U2 (? ) = 4? 2 ? 1; U3 (? ) = 8? 3 ? 4? ; U4 (? ) = 16? 4 ? 12? 2 + 1 ? ?. ?.
???????? ???????????? ???????: Un+1 (? ) = 2? Un (? ) ? Un?1 (? ) (n > 0). ???????
????????, ??? ? (5.5) ? ?????? ?????? (3.6) ?0 = T2 ??0 , ??? ??0 = || h0 (и) ||L1 [t0 , tf ] ?
????? ???????????? ???????? ??? ??????? ?????? (2.6).
??????????? ????? ? ?????? (5.5) ??????????????? ???????????? ? (5.2) ?
?????????? ?? s0 ?0 /2 ???????????? ?????????? Un?1 (? ), ??????? ????????????
?? ??????????? ?????????? n-??????? ????????????
127
?????????? lk0 , k = 1, 2, ..., n, ??????????? ??????? ????????? (5.4), ? ??????:
(?1)
m
n?m
?
k=1
m
Cn?k
1
lk0 = s0 ?m ?0 , m = 0, 1, ..., n ? 1.
(n ? k)!
2
(5.6)
????? ? ???????????? ??????-??????? ? = (?1)n?1 col (?n?1 , ?n?2 , ..., ? 1 , ?0 ) ?
l0 = col (l10 , l20 , ..., ln0 ), ????????? ??????? (5.6) ? ????????? ???? (5.4):
M l0 =
? ?????? ??????? lT0 c =
???????????? ????????:
?n
k=1 lk0 ck
1
s0 ?0 ?.
2
(5.7)
= 1 ?????? ??? ?????? (2.6) ???????? ?????
?0 = 2 |cT M ?1 ?|
?1
,
(5.8)
?1
T
? ????? s0 = sign (c M ?).
??????????? ?????????? ? ??????????????? ?????? ????? ?????????? ?? ??????? (2.7), (2.8), ? ?????? ? ?????? (5.8) ????? ????? ?????:
u? (? ) =
1
1
sign g0 (? ) = sign [s0 Un?1 (? )], ? ? [?1, +1],
?0
?0
(5.9)
min
= ?10 = |cT M ?1 ?|.
??? g0 (? ) = lT0 h(1 ? ? ), ? lT0 c = 1. ??? ???? ??????? J?
?????????? ?????? ??????? ????????????? ???????????? ??????? ??? ??????
(2.6). ????, ? ?????? (5.4), (5.7) ? cT l 0 = 1 ? ????? ?, ??????????????, ? ?????????? ?????? ?????? ????:
1
s0 ?0 cT M ?1 ? = 1.
2
(5.10)
???? ??????? c ? p = M ?1 ? ????????????, ?? cT p = 0 ?, ??????????????, ? ????
?????? ?????????? ??????????? ??????? ?????? (2.6).
??? m-??????? ????????????? ?????? (2.6) ??????????????? ????? ??????
??????? ? ?????? (4.2) ? ?? ???????? ? (4.7), (4.8) ????? ???????? ???:
cT Mn?1 ?n?j = 0, j = 0, 1, ..., m ? 1, cT Mn?1 ?n?m ?= 0,
??? ?n?j = col (0, ..., 0, ??n?j ) ? Rn , ? ??n?j ? ?????? ????????????? ??????????
| {z }
j ???
Un?j?1 (? ) (j = 0, 1, ..., n?1). ???? ?????? ??????????? pn?j = Mn?1 ?n?j , ?? ???????
m-??????? ????????????? ?????? (2.6) ????? ?????????? ? ????
cT pn?j = 0, j = 0, 1, ..., m ? 1, cT pn?m ?= 0.
(5.11)
? m-?????? ??????????? ?????? (2.6) ??????????? ?????????? (5.9) ?????
?????? (n ? m) ?????????????, ?? ??????? ??? ?????????. ????????? ????????
??????? ???????? pn?j ? (5.11) ?????????? ????????? qn?j ?? (4.9), ?? ???????
(5.11) ????? ????? ?????????? ? ????, ??????????? (4.10).
?????? 2. ?????????? ??????? ?????? (1.4), (1.5), (1.11) ??? ?????????
???????? ? (1.2), (1.3): t 0 = ? T2 ; tf = T2 ; x20 = . . . = xn0 = 0 ? x3f = . . . =
= xn f = 0, x 1f = ?T , 0 < ?T 6 ?. ????? ?? (1.8 c 1 = ?T , c2 = . . . = cn = 0, ?? ????
l 10 = ?1T ? lk0 ? R1 , k = 2, 3, ..., n. ???????? ??????????? ??????? ?????? (2.6)
1
h0 (t) =
?T (n ? 1)!
( )n?1 (
)n?1
T
2t
1?
+
2
T
128
?.?. ???????
+
l20
(n ? 2)!
( )n?2 (
)n?2
(
)
T
2t
ln?1,0 T
2t
1?
+ ... +
1?
+ ln0 .
2
T
2!
T
T
2
? ??????? g0 (? ) (5.2):
( )n?1
1
T
n?1
g0 (? ) =
(1 ? ? )
+
(n ? 1)! ?T 2
( )n?2
l20
T
ln?1,0 T
n?2
+
(1 ? ? )
+ ... +
(1 ? ? ) + ln0 =
(n ? 2)! 2
2!
( )n?1
s0
T
= n?1
(2n?1 ? n?1 + ?n?2 ? n?2 + ... + ?1 ? + ?0 ) =
2
(n ? 1)! ?T 2
( )n?1
1
T
T n?1
| Un?1 (? ) |,
= n?1
| Un?1 (? ) | = 2(n?1)
2
(n ? 1)! ?T 2
2
(n ? 1)! ?T
?????? ? ?????? ?????? t =
??? s0 = (?1)n?1 . ????? ? ?????? (5.5), ? ????? d? = 2dt
T ? || Um (и) ||L1 [?1,+1] = 2
??????? ????? ???????????? ???????? ??? ?????? (2.6):
?
+
T
2
| h0 (t) |dt =
?
T
2
Tn
= ??0 ,
22(n?1) (n ? k)!?T
?, ????? ????, ????? ????? ?????:
??0 =
1
umax
=
Tn
,
22(n?1) (n ? 1)!?T
? ?????? ? ??? ??????? ??????????? (5.1) ? ???????, ??? ??????? ?????? ? ?????? ??????? ?????????? ?????? ?????, ????? ???????????? ??????? ?????? ??
??????, ???
?
n 22(n?1) (n ? 1)!?T
Tmin =
.
(5.12)
umax
?
T
[17], [18], ? ??? n =
? ?????????, ??? n = 2 ?? (5.12) ???????: Tmin = u4?
max
3? 32?T
= 3: Tmin =
umax [11], ??? ???????? ????????? ?????????????? ? ???????
?????????? ? ????????? ??????? ?????????? ??????? ??? ????? (5.1). ?????
????, ?????? ??????? ??? ???????????? ?????????? ???????? (5.12) ????? ?????
22(n?1) (n?1)!?T
. ????????
T n?1
min
???????, ??????? J 1 = 12 umax T
umax T =
??? ????????? J min
(3.14) ??? ???????? ??????
1
2n?3
T
= 2 T(n?1)!?
. n?1
?????? 3. ?????????? ??? ??? ??????? ?????? (1.4), (1.5), (1.11), ?? ???
????????? ???????? ?????? [9]: n = 3; t 0 = ? T2 ; tf = T2 ; x 10 = ?0 , 0 < ?0 6 ?;
x20 = ? 0 ; x30 = ? 0 , ??? ? 0 , ? 0 , ? 0 ? ????????? ???????? ?????????, ? ?????
x1f = x2f = x3f = 0. ????? ?? (1.8) ??????? ????????? ???????? ????????:
1
c 1 = ? (?0 + T ?0 + T 2 ?0 ); c 2 = ? (?0 + T ?0 ); c 3 = ? ?0 .
2
??????????? ??????? ??????
(
)2
(
)
l10 T 2
2t
2t
l20 T
h0 (t) =
1?
1?
+ l30 ,
+
8
T
2
T
(5.13)
(5.14)
?? ??????????? ?????????? n-??????? ????????????
129
??? l10 , l20 , l30 ? ?????????, ??????????????? ??????? c 1 l10 + c 2 l20 + c 3 l30 = 1.
????? ????? ??????????? ?????????? ? = 2t
T , ????????? (5.14) ? ????
)
(
l20 T
l10 T 2
T?
(1 ? ? )2 +
(1 ? ? ) + l30 =
h0
= g0 (? ) =
2
8
2
=
l10 T
32
=
[
2
l10 T 2 2 T
l10 T 2
l20 T
? ? (T l10 + 2l20 )? +
+
+ l30 =
8
4
8
2
(
)]
8
32
l10 T 2
l20 T
4? 2 ?
(T l10 + 2l20 ) ? +
+
+
l
.
30
l10 T
l10 T 2
8
2
2
10 T
??? l10 ?= 0 ???? ????????? ? ????????? ?? ????????? s0 l32
(s0 = ▒1) ??????
????????? ? ??????????? ???????? ??????? ???? U2 (? ) = 4? 2 ? 1, ?? ???? ?????
2
10 T
????? ????? g0 (? ) = s0 l32
(4? 2 ?1) ?, ??????????????, ??????? ??????? ?????????
???????????? l10 , l20 ? l30 :
c 1 l10 + c 2 l20 + c 3 l30 = 1; T l10 + 2l20 = 0;
5T 2
T
l10 + l20 + l30 = 0.
32
2
????? ??? ??????? ?? ??????? ??????? [15], ???????
l10 =
??? D0 = 2c 1 ? T c 2 +
3T 2
16 c 3 ,
D1
D2
D3
; l20 =
; l30 =
,
D0
D0
D0
D 1 = 2, D 2 = ? T , D 3 =
3T 2
16 ,
?, ????? ????, ?????
?1
)?1
(
T 2 3T 2
3T 2 c3
, ?0 =
c
2c 1 ? T c 2 +
2c
?
T
c
+
,
3
1
2
16
16 16
(
)
2
16 3T 2 ? ????? s0 = sign 2c 1 ? T c 2 + 3T
c
,
u
=
c
?
c
T
+
c
2
3
max
1
2
3
16
T
16 . ????????,
l10 = 2
??? ????? ??? ?0?? ? umax ? 0, ??? ????? ????? ? ?????? (5.13) ??? T ? T? ,
64?02 ?96?0 ?0
0
(???? 2?02 ? 3?0 ?0 > 0 ? T? > 0).
??? T? = ? 8?
3?0 ▒
3?0
? ??????????? ??????, ????? l10 = 0, ????????? (5.14) ? ????????? ?? ????????? ????? ????? U1 (? ) = 2? . ????? ????? ????? ?????:
(
)
T?
l20 T
l20 T
l20 T
h0
= g0 (? ) =
(1 ? ? ) + l30 = ?
?+
+ l30 .
2
2
2
2
?????????????? ?????? ???????
l20 =
2
T
T
; l30 = ?
; s0 = sign (2c 2 ? T c 3 ); ?0 =
.
2c 2 ? T c 3
2c 2 ? T c 3
|2c 2 ? T c 3 |
?1
???? ?? l10 = 0 ? l20 = 0, ?? l30 = c?1
.
3 , s0 = sign c 3 ? ?0 = 2 | c 3 |
??????? ????????, ??? ? ?????? ???????, ??? ? ? ??????? 2, ? ????? ??????
?????? (1.4), (1.5), (1.11) ?? ???????????? ??????????????? ?????? ?? ??????????????, ??? ??????????? ???????????? ????????????? ?? ????????? ??????????
?????? ?????????? ???????? ? ??????????????? ?? ????????? ????????? ???????????? ??????????. ??????? ??? ???????? ???????????? ??????? T ? ??????
(1.4), (1.5), (1.11) ? ??????????? ??? ??? ???????? umax ?? ???????? ? ??? ?????? ?? ??????????????, ? ??????? ???????? ??????????? (5.1), ?????? ????? ?????
????????? ?????? ??? ??????????????: Tmin 6 T . 130
?.?. ???????
??????????
??????????? ?????? ???????????? ??????????, ??????? ?????????? ????????
??? ??????? ????????? ????? ?????????? ??????????? ??????????? ?????????,
? ??????: ?????? ???????????? ?????????? n-??????? ???????????? ? ????????????? ?????????? ????????? ? ? ????????????? ???? ????? ? Lq [t0 , tf ], q =
= 1, 2, ?, ??????? ???????? ? ??????? ???????? ????????? ?.?. ???????????
(??????? ????????). ??-??????, ???? ???????? ?????? ??????? ?????? ?? ??????? ??????????? ??????? ??????????, ??????? ???????? ? L 2 -???????? ????????.
??-??????, ?????? ?????? ?? ??????? ?????????? ??? ??????? ???????? ??????????, ??????? ???????? ? L? -???????? ???????? ? ??? ??????? ????????, ???
??????????? ?????????? ???????? ?-?????????? ???????????. ???????? ???????
????????????? ??????????? ??????? ?????? ? ??????????? ?? ????????? ???????,
? ????? ??????? ????????????? ?? ??????????? ???????. ?-???????, ????????
????? ??????? ?????? ?? ??????? ???????????? ?? ?????? ???????? ???????????? ????????? (??? L 1 -???????? ????????) ? ??????? ????????????? ??? ??????????, ??? ? ??????????? ??????? ???? ??????. ????????? ??????? ???????
?????, ??????? ???????? ? L? - ? L 1 -????????? ????????. ??????????? ????????????????? ?????? ?? ??????? ???????????? ?????????? ??????????????? ??????
?? ??????????????.
??????????
[1]
??????? ?.?. ????????? ?????????? ? ???????????? ? ?????? ???????????? ??????. ?.: ?????????, 1994. 544 ?.
[2] ?????????? ?.?. ?????? ?????????? ?????????: ???????? ???????. ?.: ?????,
1965. 476 ?.
[3] ????? ?.?. ???? ?????? ??????. ?.: ?????? ?????, 1987. 304 ?.
[4] ?????? ??????????????? ??????????: ? 2 ?. ?. II. ?????? ?????????? ? ??????????? ?????? ??????????????? ?????????? / ?.?. ???????, ?.?. ???, ?.?. ?????
[? ??.]; ??? ???. ?.?. ????????. 2-? ???., ???????. ? ???. ?.: ?????? ?????, 1986.
504 ?.
[5] ?????????? ?.?. ?????? ?????????? ????????? ? ??????????????? ???????????.
?.: ?????, 1975. 568 ?.
[6] ??????? ?.?. ??????? ?????????? ???????? ?????????? ?????????. ?.: ???-??
???, 1981. 200 ?.
[7] ??????? ?.?., ????? ?.?. ??????????? ?????????????? ? ????????? ???????? ???????? // ?????????????? ???????. 1987. ?. 134(176). ? 2(10). ?. 186?206.
[8] ??????? ?.?., ????? ?.?. ?????? ??????? ????? n-?????? ?????? ?????????????? // ??????? ?? ????. 1988. ?. 296. ? 6. ?. 1304?1308.
[9] ??????? ?.?. (Widnall W.S.) ??????????? ????? ?????????? ???????? ???? ? ?????????? ??????? ?????? ???????????? ??????? ????????? /?/ ?????????? ? ???????:
????? III ?????????????? ?????????? ???? ?? ??????????????? ?????????? ?
?????? ????????????? ???????????? ????????????. ?.: ?????, 1972. ?. 2. ?. 36?49.
[10] ??????? ?.?., ???????? ?.?. ??????????? ?? ???????? ???????? ??????????
??????? ???????????? // ??????? ?????????? ???????????????? ????????????.
2012. ? 9(100). ?. 118?129.
[11] ??????? ?.?., ???????? ?.?. ?????? ??????????? ?????????? ???????????? ?????????? ???????????? ???????? ??????? ???????? // ???????? ??? ???. 2012.
T. 14. ? 6. ?. 166?176.
?? ??????????? ?????????? n-??????? ????????????
131
[12] On Optimization of Attitude Control Programs for Earth Remote Sensing Satellite /
Yu.N. Gorelov [et al.] // Gyroscopy and Navigation. 2014. Vol. 5. ? 2. P. 90?97.
[13] ??????? ?.?. ? ??????? ?????? ??????? ???????????? ?????????? ??????????????? ???????????? ???????? ??? ??????????????? ?????????? ???????????? ?????
??????? ???????????????? ??????????? // ???????? ??? ???. 2014. T. 16. ? 4.
?. 127?131.
[14] ??????? ?.?. ?????????? ????????. ?.: ????????? ????, 2003. 160 ?.
[15] ???????? ?.?., ???????? ?.?. ??????? ? ??????????. ?.: ?????, 1984. 320 ?.
[16] ????????? ?.?., ???????? ?.?. ???????? ???????????? ?? ???? ??????
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