517.977 22.193
1
. .2, . .3
( , -)
. , , .
: , , .
1.
[1, 2, 6] , .
1 ( 0701-90101, 08-01-00945).
2 , - , (buldaev@mail.ru).
3 , (hint@rambler.ru).
, , .
. , . , , .
2.
:
(1) x = A( x, t)u + b(x, t), x(t0) = x0, u = u(t) e U, t e T = [t0, t1 ],
(2) F(u) = {c, x(t1)) min,
(3) xi(ti) = xi1,
x = (x1(t), x2(t), ..., xn(t)) - ; u = (u1(t), u2(t), ..., um(t)) - ; T , x0 e Rn, x11 e R, c = (c1, c2, ..., cn) , c1 = 0. A(x, t) - b(x, t) x l > 1 t Rn T.
V - U Rm:
V = {u e PCm(T): u(t) eU, t e T j .
v e V (1) u = v(t) x(t, v), t e T.

W = {u e V: x1(t1,u) = x11 j.
-, , , Fo(u) min, Fi(u) = 0, i = 1,...,5, 5 > 1, (u) = j (x(O) + JT (di (x,t) + (gi (x,t),u))dt, i = 0,...,5,
(1)-(3) .
, (p(x), i = 0, ., 5 l1 > 1 Rn, di(x, t), gi(x, t),
i = 0, ., 5, x l1 > 1 t Rn T.
(1)-(3) p e R
H(p,x,u,t) = H0(p,x,t) + (H1(p,x,t),u),
H0(p,x,t) = (p, b(x,t)), H1(p,x,t) = A(x,t)Tp .
X e R:
L(u, l) = (c, x(t^) +x^) - x11) .
[1, 2]
DvL(u,1) = L(v,1) - L(u, l) u0, v (4) DvL(u0,1) = -J^H1(p(t,u0,v,1),x(t,v),t), v(t)-u0(t^dt.
p(t, u0, v, X), t e T -
<5> = - -!(:) - ...-!(...,*1,*1 ,
(6) = -,
(7) ( = -,-, * = 2, ,
= (, 0), = ∞(0, : = (, V) - (, 0).
, I = 1
= ~ (, , , t).
^, V, ), t - (6), (7) = (, V), = v(t>. ,
(^ V, V, 1) = , V, 1), t .
0 > 0 [2, 6] -
(, , t> = (> + 1(, , t>), , ,
- .
(, , 0 (, ) - t , [2, 6]:
(8) (, , 0, (, , 0 - ∞(0) > ||(, , 0 - ∞(0|| .
' '11 11
0
(9) ∞(0 = ((^0,1>,(^0>,t), t , > 0.
(9) > 0.
3.
0 : V ^> < (0).
.
1. > 0 (^>, (0), t
= (,{>(,, t> + (, t>, t , (^> = 0, x1(t1> = 1,
(10) = - - . *)-..-(- , ^ :) *)
(t1> = -!, * = 2, , = (^ 0), = u0(t>, : = x(t> - (^ 0).
2. v(t > = ( (0, x(t>, t >, t .
, ((0, (t>>, t
(10) (, ) . , (0 = (^ V), t V .
. , ^>, t (5) = (^ 0), = ∞(0, : = (^ V) - x(t, 0) (7). 1 = -1 (^ >, (t> = (t, 0, V, 1 >, t .
(4) (8) , V
^, 1) - 0,1) <- [ ^> - ї .
11 11
, 0, V,
(11) ^> - ( 0> < - - ||v(t > - ∞(0||2 .

:
(0) = {V : v(t > = ( ^, 0, v,>, ^, V >, t >, t }.

v(t> = (^,0,v>,^,V),t>, t .
.
. ( > , 0 (9).
, (10) , ,
^> = (^ 0), (0 = ^, 0,>, t , 1 = -1^1) .
() .
. , 0 (0).
(10) .
. 0 , ((, 0), 0, 1 >>, t -
(10).
1. (10) , V (0) 0 (11>. 0
2. , (10) V (0), V 0 0 (11).
, .
. ,
(0)={0}.
(10) .
. 0 , ((, 0), ^, 0, 1 >>, t (10).
, .
(10), .
1. (10) (I - 1) .
2. (10) (, )

.
. . : .
, , .
, , .
4.
(10) , [1].
(I = 2) (1)-(3).
= (,{>(,, t> + (, t>, t , (^> = 0, x1(t1> = 1,
(12) = - (, x(t, 0), 0(t >, t > -
- 2 (, ^, 0>, 0(t >, t > ( - ^, 0>) ,
(tl> =-! , * = 2, .
[0, 1]: = (,^(,, t> + (, t>, t ,
x(to> = 0, Xl(tl> = 1,
(13) = - (, x(t, 0), 0(t >, t > -
- ^ (, x(t, 0), 0 (t>, t > ( - x(t, 0>),
(t1> = -, * = 2, .
(12) = 1. = 0
= (,{>(,, t> + (, t>, t
(^> = 0, Xl(tl> = 1,
(14) = ~ (, x(t, 0>, 0(t>, 0,
(0 = , * = 2, .
.
(14) .
, (, ), t ,
= ~ (, ^, 0), 0 ^>, t>,
(0 = , , (tl> = -, , = 2, .
: = (, t>u(^,), ,0 + (, t>, t , (^>=0.
(^ ), ( - . (( ), (^ >>, t (14) ,
(15) (^, ) = 1 .
, (15) . (15) :
( > = 0.
[5]. , .
(13) (0, 1] [1] > 0:
+1 = (+1, t>(+1,+1, t> + (+1, t>, t ,
+1(to> = 0, +1(tl> = 1,
(16) + =- (+1, ^ 0), 0(t>, 0 -
-1 ( ^ >, x(t, 0), 0 (0,0 ( (t > - ^, 0 > ) ,
+1(t1> = -, * = 2, .
(∞(0, ∞(0), t , = 0 (14).
(16) , (14).
(16) [0, 1] > 0 [1].
, = 1. (16) , , (16) > 0. (16) . > 0.
(16) ( > <(0), (0 = ( (0, (0,0, t , > 0.
.
, , .
5.
1. , .

x = u, x2 = Xj2, |u(t)| < 1, t T = [0, p],
1
2
x,(0) = 0, x2(0) = 0,
(u) = x2(p) min, xj(p) = 0.

H = Piu - 2P2xi2, H0 = - 2P2xi2, Hi = Pj.
u0(t) = 0, x,(t, u0) = 0, x2(t, u0) = 0, t T (^) = 0. ua > 0
1, , > 1,
ua(p,x,t) = i-1, , <-1,
ap,, -1 < ap1 < 1.
= 1 . ,
x1 = ua(p, x, t), x2 =-2 x12, x1(0) = 0, x2(0) = 0, x1(p) = 0,
p1 = p2 X1, p 2 = 0, p2(p) =-1,
:
x1 = ua( p, x, t), x1(0) = 0, x1(p) = 0,
p1 = -x1.
, (p1(t) = 0, x1(t) = x1(t, u0) = 0) . , u0 l = 0. u0 .
, , . , |p1(t)| < 1, t T.
X = P1, X1(0) = 0, x(p) = ,
p1 = -X1
, x1 (t) = C sin t, p1 (t) = C cos t, t T, C - , |C| < 1.
, , v(t) = cos t
x1(t, v) = sin t, x2(t, v) = 1 (sin 2t - 2t), t T,
S
u0: ^) = - < (u0) = 0.
2. [7] :
X = au(1 - x) - bx, x(0) = x0, u(t) [o, u+], t T = [0, t1],
JT e~rt (cx - u )dt max,
x(t1 ) = x1.
x = x(t) - ( ); a, b, c, r, x1 - . u(t) [0, u+], t T .
T .

XI = au (1 - x1) - bx1, X2 = e~rt (u - cx1), t [0, t1 ], x1(0) = x0, x2(0) = 0, u(t) [0, u+] ,
0^ ) = x2 (t1 ) min ,
1(u ) = x1(t1) - x1 = 0 .
:
62
= 2, = 0,5, = 2, = 1, 0 = 0,75, 1 = 0,75, + = 1, ?! = 1.
() ,
() = () + /12(),
> 0 - .
:
1) () [3] ;
2) () [6], .
∞(0 = 1.
- (5-6) . 10-10. , , 0,001, = [0, 1]. .
106 .
.
10-6.

|(+1) -()| < ()|,
> 0 - , = 10-6.
. 1.
1 : 0 - ; 1 - ; N - . ( ) ( ).
- ? = 0,5 .
1
^ * ^ * N
-0,5883189 7,937110"5 3639 100
-0,5885496 4,319710"5 1218 100
-0,5882504 1,6391 10-5 64 10
, , , ( , ).
3.
[4] ( ):
= hlx1 - h2x1x2 - ux1, x2 = h4 (x3 - x2) - h8 x1x2, x3 = h3 x1 x2 - h5 (x3 -1), x4 = h6 x1 - h x4 ,
x1 (0) = xj > 0, x2(0) = 1, x3(0) = 1, x4(0) = 0, t e T = [0, t1], F0 (u) = x1 (t1) min,
JT x4 (t)dt - m = 0, m > 0 .
x1 = x1(t) (), x2 = x2(t), x3 = x3(t) - (, ), x4 = x4(t) - , h > 0, i = 1, ..., 8 - . u(t) e [0, u ], t e T = [0, t1] , .
x10 t0 = 0. u(t) = 0 .
[4] h1 = 2, h2 = 0,8, h3 = 104, h4 = 0,17, h5 = 0,5, h6 = 10, h7 = 0,12, h8 = 8, m = 0,1 x10 10-6.
. u = 0,5. T 20 : t1 = 20.

.

5 = 4, 5(0) = 0
1() = 5(^) - = 0 .
, , .
(0 = 0.
- (5-6) 10-10. , , 0,001.

10-5 .
, [5] , .

|(+1) - ()| < |()|, () = 0 () + 12 (), > 0 - , = 10-5.
. 2, 0 - ; 1 - ; N - . ( ) ( ).
2
0 ^ * 1 N
2,68669810"19 1,8548610-5 464 -10
1,142279 10'2∞ 5,8447210-5 167 -10
1,172261 10-20 1,5347910-5 88 1000
. , = 1.
- ^ = 5 ^ = 14.
, .
6.
.
, . .
, .
1. .
2. .
3. .
4. , .
.

1. . .
. - .: . - . 10. - : - , 2004. - 52 .
2. . .
// -
. . - 2004. - 1. - . 18-24.
3. . . . -: - , 1994. - 340 .
4. . . . . - .: , 1991.
- 304 .
5. . ., . . . -.: , 1989. - 432 .
6. . . . - .: , 2000. - 160 .
7. SETHI S. P., THOMSON G. L. Optimal control theory. Applications to management science. - USA, Boston, 1981. -370 p.
METHOD OF NON-LOCAL IMPROVEMENT IN POLYNOMIAL OPTIMAL CONTROL PROBLEMS WITH TERMINAL CONSTRAINTS
Alexander Buldaev, Buryat State University, Ulan-Ude, Doctor of Science, professor (buldaev@mail.ru).
Dmitry Trunin, Buryat State University, Ulan-Ude, assistant (hint@rambler.ru).
Abstract: A new approach is designed to solve polynomial in state optimal control problems with terminal constraints. The proposed method provides the non-local control improvement with all terminal constraints satisfied on each iteration without needle or weak variation and has an opportunity to improve extreme controls.
Keywords: optimal control problem, terminal constraints, nonlocal improvement.
..