;
Resh3 <E
,
D ,x13 0 0,
D 1@,x43 @ D @ D @ D D @ D
;
Resh4 <E
,
@ D+ Sin t ∗@ ,x14D 0 0, @ D @ D 0,x44 1 @ D <E @ D t , t,0,1 ;D<Dê @tD .Resh1,x31@ D< 8D<<@tDê.Resh1,
Dê tD<<
Dê=t .Resh2,
, x43 t =
D<<@tDê.Resh3,
8 @ DêDê t< @ Dê tD<<@ Dê=
.Resh4,x24 t .Resh4,
x44@@tDêDê.Resh4<; @ Dê
x11 t x12 t x13 t x14 t
@ D x41x31 t x42 t tX@tsD@ == InverseDx21@@@@@ttDDDD@HDx22x32@ @@Dê@DD
@@@DDD< @@DD{D; <L @ DY k X t ..Y s ; h1 s_ ,h2 s_ ,h3 s_ s_ = MK s
Построение и решение линейных уравнений a11,a12,a13,a14 =
8 |
NIntegrate h1, NIntegrate h1 τ .h4 τ , τ,0,1
=
,
8 |
NIntegrate@@@ @@ DD @ D 8 <D
NIntegrate
τ .h4 τ , τ,0,1
=
,
@@@τDDDD.h3@@@@τDDDD,8888τ,0,1<<<<<DDDD<,1.51071,0.764328,3.3364,1.89574@ τ τ τ
8 |
a41,a42,a43,a44 =
NIntegrate h4 τ <.h1@@τDD,88τ,0,1<<DD,
8NIntegrate@@@h4@@@@τDD.h2 τ , τ,0,1
NIntegrate h4,
0.779061,1.12012,1.89574,2.6037
V= @8 < ∗ v1 + a12 ∗ v2 + a13 ∗ v3 + a14 ∗ v4, c2 a21 ∗ v1 + a22 ∗ v2 + a23 ∗ v3 + a24 ∗ v4, c3 a31 ∗ v1 + a32 ∗ v2 + a33 ∗ v3 + a34 ∗ v4,
∗ ; v1 , v2 <DПостроение оптимального управления
u10 t_ ,u20 t_ =v1∗h1 t +v2∗h2 t +v3∗h3 t +v4∗h4 t ;
@ВычислениеD @ оптимальногоD< @ значенияD функционала@ D @ D @ D HNIntegrate@8u10@tD,u20@tD<.8u10@tD,u20@tD<,8t,0,1<DL^0.999712
Построение оптимального закона движения
Resh0 =
NDSolve t x30 t ,x20' t x40 t , x30' @@DDDD @@DD @@DD @ D @ D @tDD ,@ Dx40' + 1 ∗ + ∗ + u20 t , x100,x30 @D D== 0 ,
x10t ,x20 t ,x40@ t ,
8 @ @ x30D t_@ D<,8 @@DêD<<<E Dê.Resh0,x30@ D< t .Resh0,
правильности вычислений
−1.68561 1.60071 × 10−6, −2.40392 × 10−6
<
Пример 4.5
Ввод начального и конечного положений фазового вектора
X0;XT 80.7746,
8−80.7746, 147.179, < матрицы Коши
Resh1 =
DSolve x11'x31 t , x21'@8@t@ Dx31'D tDD @@DD < @8D @ D@ D @ DD@0D @1,D<
x21 0@D 0,x31 0 t ,x31 t ,
t ; @
Resh2 =
DSolve x12' tx32 t , x22'@8@ D x32'+ t 0,x22 1,x32 0@DD== 0<,@8x12D @tD,x22@ D @ D,x32@ D @tD<, tD; @ D @
Resh3 =
DSolve t x33 t , x23' @8@@ DDDDê @D@DD <@ 8@@@DDDDê@@ @ D @@D<D x33'+ 0 0, x23 0t ,x23t , t ; x21 t, x31 888888ReD@@x11@@x12x32@@t@D<DDêtDDê8 @@ <<D<D<êê@x21 888 @ @DDê<<.Resh1,D<<D<< D<<Re x31 . ;
x12 t, τ , x22 t, τ , x32 t, = Re .Resh2,Re.Resh2, Re.Resh2 . − τ ;
x13 t, τ x23 t, τ , x33 t, τ = Re .Resh3,Re.Resh3, 8 Re@@ @DDê .
X t, τ =
; @ D@ @ {Построение
MK τ = τ 1;
JRo@l2_,l3_D=NIntegrate@Pod@τ,l2,l3D,8τ,0,1<D
NIntegrateMinel=FindMinimum@Pod@τ,l2,l3D, <D0.731198, l2→ 0.038468,l3 <D
Minel,1D