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Оптимизация управляющих функций и параметров в нелинейных системах на основе задач о неподвижной точке.

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1.
517.977
©
.
1
,
,
.
-
.
:
,
,
© A.S. Buldaev
OPTIMIZATION OF THE MANAGING FUNCTIONS AND PARAMETERS
IN NONLINEAR SYSTEMS ON BASIS OF THE FIXED POINT PROBLEMS
A new approach to the class of nonlinear optimal control problems containing both managing functions and parameters, is proposed on basis of the solution of special fixed point problems for operators
constructed in the space of controls. The fixed point problem makes it possible to obtain new conditions
for optimal control in the class of optimization problems and build improving controls.
Keywords: controlled system, fixed point problem, conditions of optimality
[1]
,
.
,
.
[2]
.
,
[3-5].
,
:
( )
( x(t1 ), )
F ( x (t ), u (t ), , t ) dt
inf ,
(1)
T
x(t ) f ( x(t ), u (t ), , t ) , x(t0 ) a , u (t ) U ,
W , a A, t T
x(t ) ( x1 (t ),..., xn (t )) –
, u (t ) (u1 (t ),..., um (t )) –
( 1 ,...,
,
Rl , A
W
l
)
a ( a1 ,..., an ) -
Rn
.
.
U.
Rm ,
-
T
(u , , a ) -
A.
:
Rn W ,
( x, )
1)
x, u,
R
1
U
.
T
V
V W
[t0 , t1 ] , (2)
n
F ( x, u , , t ) , f ( x, u , , t )
( x, u , , t )
U W T;
,
01-92200-
3
12-01-00914- , 12-01-98011-
, 13-
9(3)/2014
L 0:
f ( x, u , , t )
Rn U W T
x
f ( x, u , , t )
2)
f ( y , u, , t )
y .
L x
x(t , ) , t T
(2)
.
:
R
n
H ( , x, u , , t )
(t )
, f ( x, u , , t )
F ( x, u , , t )
H x ( (t ), x(t ), u (t ), , t ) , t T ,
(t1 )
x
( x(t1 ), ) ,
(3)
-
[1,2].
x(t )
(3)
(t , ) , t T u, ,
x (t , )
-
.
,
(
II
I
II
)
I
(
-
) 0.
.
,
,
.
1.
g ( y1 ,..., yl )
-
ys , ys
1
2
ys
1
ys g ( y1 ,..., yl )
2
ys , ys
1
2
g ( y1 ,..., ys
ys ,..., yl ) g ( y1 ,..., yl ) .
ys ,..., ys
1
1
2
(1)
2
,
I
:
(
I
)
( x(t1 ,
x ( t1 , ),
I
I
),
)
x ( t , ),u ( t ),
F ( x (t ,
I
), u I (t ),
I
, t ) dt .
(4)
T
x(t )
x ( t1 , ),
( x (t1 ,
x(t , ) x (t ,
I
),
I
)
I
I
),
( x (t1 , ),
I
)
, a a aI .
(4)
I
),
x ( t1 , ) ( x (t1 ,
:
I
).
p (t ) ( p1 (t ),..., pn (t )) ,
:
t T
p (t1 )
x
( x(t1 ,
I
),
I
) q,
(5)
q
x
( x(t1 ,
I
),
I
) q, x (t1 )
x ( t1 , )
( x(t1 ,
I
),
q 0
x(t1 , )
x (t1 ,
I
I
).
(6)
x,
).
(4)
4
:
.
.
I
( x (t1 ,
x ( t1 , ),
I
),
I
( x (t1 , ),
p(t0 ), a
)
T
I
( x(t1 , ),
p(t ), x (t )
I
( x (t1 , ),
)
p (t ),
p(t1 ), x(t1 )
)
d
p(t ), x (t ) dt
dt
p(t0 ), a
)
I
f ( x(t ,
x ( t , ),u ( t ),
), u I (t ),
I
, t ) dt.
T
(4)
:
(
I
I
( x(t1 , ),
)
p(t ), x (t )
p(t0 ), a
)
H ( p(t ), x (t ,
x ( t , ),u ( t ),
I
), u I (t ),
I
, t ) dt
T
I
( x(t1 , ),
p(t ), x (t )
(7)
p(t0 ), a
)
I
H ( p (t ), x(t , ), u (t ),
, t)
T
u (t )
H ( p(t ), x (t , ), u I (t ),
I
, t)
x (t , )
I
H ( p(t ), x(t ,
), u I (t ),
I
, t ) dt.
p (t )
(5), (6)
:
p (t )
H x ( p (t ), x(t , I ), u I (t ),
r (t ) ( r1 (t ),..., rn (t )) , t T
I
, t ) r (t ) ,
(8)
-
t T
I
), u I (t ),
I
, t ) r (t ), x (t )
H ( p(t ), x (t ,
r (t ) 0
I
), u I (t ),
H x ( p(t ), x (t ,
x(t, )
x(t , )
I
x (t ,
I
(9)
, t ).
F, f
).
p (t )
(8), (9)
(5), (6)
(
I
x,
(7)
{
)
-
:
( x(t1 , ),
I
)
H ( p(t ), x(t , ), u (t ),
I
, t )dt}
T
p(t0 ), a
u (t )
H ( p (t ), x (t , ), u I (t ),
I
(10)
, t )dt.
T
p (t )
p (t )
:
H x ( p (t ), x (t ), u (t ), , t ) r (t ) ,
H x ( p(t ), x(t ), u (t ), , t ) r (t ), y (t ) x (t )
p (t1 )
x
( x(t1 ), ) q ,
x
( x(t1 ), ) q, y (t1 ) x (t1 )
(11)
y (t )
(12)
H ( p (t ), x(t ), u (t ), , t )
(13)
y ( t1 )
( x(t1 ), ) ,
(14)
q 0 , r (t ) 0
, F, f
(1), (2)),
x(t , )
x (t ,
I
x
)
t T.
(1), (2)
,
(11)-(14)
(11)-(14)
(3).
p (t , I , ) , t T -
I
I
I
x(t ) x (t , ) , y (t ) x(t , ) , u (t ) u (t ) ,
(t , ) , t T .
p (t , , )
5
I
.
9(3)/2014
(10)
,
-
,
I
(
)
{
I
( x(t1 , ),
)
I
H ( p (t ,
, ), x(t , ), u (t ),
I
, t ) dt}
T
I
p(t0 ,
, ), a
u (t )
I
H ( p (t ,
, ), x(t , ), u I (t ),
I
(15)
, t ) dt.
T
(10),
-
.
«
»
.
,
-
,
,
,
[3 - 5].
,
-
,
.
2.
I
A :
:
u , u (t ) arg max H ( p(t ,
u
I
arg max{
,
( x (t1 , ), )
W
a
I
, ), x(t , ), u,
u U
I
H ( p (t ,
, t) , t T ,
, ), x(t , ), u (t ), , t )dt} ,
T
a , a
I
arg max p(t0 ,
a A
, ), a .
A
-
, t) , t T ,
(16)
A( )
:
I
u (t ) arg max H ( p(t ,
u U
arg max{
( x(t1 , ), )
W
, ), x(t , ), u ,
I
H ( p (t ,
, ), x(t , ), u(t ), , t ) dt} ,
I
a A
)
A ,
II
(16)-(18)
u
II
II
{
I
H ( p(t ,
( x(t1 ,
II
),
I
II
,
II
), x (t ,
I
H ( p(t ,
)
II
II
(18)
, a II ) (
,
-
,
), u I (t ),
,
, ), a .
(u II ,
.
u II ( t )
(17)
T
a arg max p (t0 ,
,
I
I
, t) 0 , t T ,
II
), x(t ,
), u II (t ),
I
, t )dt} 0 ,
T
I
p ( t0 ,
(15)
II
(
I
II
,
), a II
aI
0.
) 0.
u ( p, x, t ) arg max H ( p, x, u ,
I
u U
, t ) , p Rn , x R n , t T .
,
(16)-(18),
x(t )
p (t )
f ( x(t ), u ( p (t ), x (t ), t ),
I
I
H x ( p (t ), x (t ), u (t ),
H x ( p(t ), x I (t ), u I (t ),
I
I
, t ) r (t ) ,
, t ) r (t ), x(t ) x I (t )
I
I
x ( t ) H ( p (t ), x (t ), u (t ),
p (t1 )
x
( x I (t1 ),
I
) q,
6
, t ) , x(t0 ) a ,
I
(19)
(20)
(21)
, t)
(22)
.
.
x
( x I (t1 ),
I
arg max{
( x(t1 ), )
W
a
) q, x (t1 ) x I (t1 )
I
),
(23)
H ( p(t ), x(t ), u ( p(t ), x(t ), t ), , t ) dt} ,
(24)
T
arg max p (t0 ), a .
(25)
a A
( x(t ), p(t )) , t T
(
,
,
(19) – (25)
( x I (t1 ),
x ( t1 )
),
,
-
u II (t ) u ( p(t ), x(t ), t ) , t T ,
.
arg max{
II
T
II
a
x(t )
x (t ,
II
) , p (t )
I
p(t ,
,
u II ( t )
II
{
( x(t1 ,
II
II
arg max p(t0 ), a ,
a A
) ,
I
H ( p(t ,
),
H ( p(t ), x(t ), u II (t ), , t )dt} ,
( x(t1 ), )
W
I
II
,
), x (t ,
I
H ( p(t ,
)
II
II
,
I
), u (t ),
II
), x(t ,
II
,
I
, t) 0 ,
), u II (t ),
I
, t )dt} 0 ,
T
p ( t0 ,
(15)
,
I
II
,
I
(
II
), a II
aI
0.
) 0.
(19)-(25)
( x(t ), p(t )) , t T ,
.
(16)-(18)
(19)-(25).
II
,
(16)-(18).
( x(t ,
(16)-(18).
,
II
,
II
I
), p(t ,
,
II
-
)) , t T
(19)-(25).
I
,
(16)-(18)
(19)-(25).
.
(1), (2)
,
[1],
(15)
-
:
u (t ) arg max H ( (t , ), x (t , ), u , , t ) , t T ,
(26)
u U
arg max
W
( x(t1 , ), )
a arg max
a A
(26)-(28)
(
I
),
I
)
, (27)
(28)
[1]
a
,
x
(
(t0 , ), a .
,
(1), (2) (
I
H ( (t , ), x(t , ), u (t ), , t ) dt ,
T
t0 ).
-
(16)-(18).
.
:
I
I
u (t ) arg max H ( (t ,
u U
I
arg max{
W
( x(t1 ,
H ( (t ,
), )
I
), u ,
I
I
, t) , t T ,
), x(t ,
I
(29)
), u I (t ), , t ) dt} , (30)
T
aI
(1), (2)
I
), x(t ,
arg max
a A
(29)-(31)
7
(t0 ,
I
), a .
(31)
-
9(3)/2014
I
(26)-(28)
.
,
-
.
.
(1), (2)
,
(26)-(28)
-
( ).
).
( ).
(1), (2).
(
(1), (2)).
(16)-(18), ,
1.
,
(19)-(25),
.
(16)-(18)
,
2.
,
.
(1), (2).
-
[3 - 5]
3.
.
,
(26)-(28)
A( )
-
A:
:
u , u (t ) arg max H ( (t , ), x(t , ), u, , t ) , t T ,
u
u U
,
arg max
a, a
a
,
( x (t1 , ), )
W
H ( (t , ), x(t , ), u (t ), , t ) dt ,
,
T
arg max
a A
(t0 , ), a .
«
»(
)
,
u (t ) arg max H ( (t , ), x (t , ), u , , t ) , t T ,
A( ) :
u U
arg max
( x(t1 , ), )
W
H ( (t , ), x(t , ), u (t ), , t ) dt ,
,
T
a arg max
(t0 , ), a ,
a A
:
x(t ) f ( x(t ), u ( (t ), x(t ), , t ), , t ) , x(t0 ) a ,
(t )
H x ( (t ), x(t ), u ( (t ), x(t ), , t ), , t ) ,
(t1 )
x ( x(t1 ), ) ,
u ( , x, , t ) arg max H ( , x, u , , t ) ,
u U
arg max
W
( x(t1 ), )
Rn , x Rn ,
W, t T,
H ( (t ), x(t ), u ( (t ), x(t ), , t ), , t ) dt ,
,
T
a
,
arg max
a A
(t0 ), a .
(19)-(25)
,
.
-
(1), (2),
(11)-(14)
(3).
(16)-(18)
I
u (t ) arg max H ( (t ,
u U
8
), x(t , ), u,
I
, t) , t T ,
-
.
.
( x(t1 , ), )
arg max{
W
I
H ( (t ,
), x(t , ), u(t ), , t ) dt} ,
T
a arg max
I
(t0 ,
a A
), a .
a
.
(19)-(25)
-
f ( x(t ), u ( p(t ,
I
), x(t ), t ),
u ( p, x, t ) arg max H ( p, x, u ,
I
, t ) , p Rn , x R n , t T ,
x(t )
u U
arg max{
( x(t1 ), )
W
, t ) , x(t0 ) a ,
H ( (t ,
I
), x(t ), u ( (t ,
I
), x(t ), t ), , t ) dt} ,
T
arg max
a
(t0 ,
a A
I
), a .
,
.
3.
A ( ),
[6].
k
0,
,
(32)
-
,
:
k 1
A(
k
0
),
.
(32)
0:
A ( )) ,
(
k 1
k
(
k
A(
k
,
0
)) ,
» [6]
(33)
:
.
.
,
-
0,
.
(33)
I
-
II
.
.
,
II
(
0 -
)
(
I
)
(
I
),
.
.
A( )
.
.
,
-
.
9
9(3)/2014
1.
.,
.–
2.
.,
, 1976. – 392 .
.:
.,
3.
.
.
.–
–
.,
,
:
, 1997. – 175 .
.
:
4.
.
.
.
, 2008. – 260 .
.
,
.
//
.2, 1. – . 94-106.
5.
.
.
, ..
6.
.,
.
«
/
.
». – 2009. –
/
//
.
.–
.:
. – 2013. – 12. – . 5-14.
, 1989. – 432 .
,
,
,
. (301-2) 217733, E-mail:
buldaev@mail.ru.
Buldaev Alexander Sergeevich, doctor of physical and mathematical science, professor of applied
mathematics department of Buryat State University.
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