Iíäèâiäóàëüíå çàâäàííÿ 2
Ìàêñèì Ùåðáà
20 ëèñòîïàäà 2022 ð.
1
Ôîðìóëþâàííÿ çàäà÷i îïòèìàëüíîãî êîíòðîëþ
Z
Z
2
(y(x) − yd ) dx + γu
ψ0 (u) = γy
u2 (x)dx → min
Ω
Ω2
, ïîâèííi çàäîâîëüíÿòè ðiâíÿííÿ
u y
−y 00 + s(x, y, y 0 , u) = f (x, u),
x ∈ Ω = (x0 , xe )
s(x, y, y 0 , u) = r(x)y 0 + s1 (x)y + s3 (x)y 3 ,
f (x, u) = f0 (x) + fu (x),
êðàéîâi óìîâè Íåéìàíà òà Äiðiõëå
α0 y 0 (x0 ) = d0 y (0) ,
βe y(xe ) = de y (e) ,
äå u(x) = s1(x), u− ≤ u(x) ≤ u+ òà îáìåæåííÿ
Z
ψ1 (u) =
2
(y(x) − y + ) + |y(x) − y + | dx ≤ 0,
Ω1
äå Ω1 = [x0 + p1(xe − x0), xe − p2(xe − x0)], Ω2 = Ω \ Ω1.
2
Çàäà÷à íåëiíiéíîãî ïðîãðàìóâàííÿ
Z
xe
g(x, y, b)dx → min
ψ0 (b) =
, ïîâèííi çàäîâîëüíÿòè ðiâíÿííÿ
x0
b y
−y 00 + s(x, y, b) = f (x, b),
êðàéîâi óìîâè Íåéìàíà òà Äiðiõëå
x ∈ Ω = (x0 , xe )
y 0 (x0 ) = y01 ,
y(xe ) = ye0 ,
òà îáìåæåííÿ
b− ≤ b ≤ b+ ,
b = [b1 , . . . , bnb ]T ,
ψ1 (b) ≤ 0.
1
3
Ðîçâ'ÿçóâàííÿ êðàéîâî¨ çàäà÷i äëÿ ïî÷àòêîâèõ çíà÷åíü ïàðàìåòðiâ îïòèìiçàöi¨
Äëÿ ïàðàìåòðiâ x0 = 0, xe = 1, f0 = 20, fu = 3, r = 1, s1 = 3, s3 = 2 òà êðàéîâèõ óìîâ Íåéìàíà
y 0 (x0 ) = 1 òà Äiðiõëå: y(xe ) = 0 ðîçâ'ÿæåìî çàäà÷ó ç âèêîðèñòàííÿì ôóíêöi¨ bvp4c ó Matlab.
Ãðàôiê ðîçâ'ÿçêó y(x) íàâåäåíî íà ðèñóíêó.
4
ψ0 , ψ1
Îá÷èñëåííÿ
äëÿ ïî÷àòêîâèõ çíà÷åíü ïàðàìåòðiâ êåðóâà-
ííÿ
Äëÿ γy = 1, γu = 10−4, p1 = 0.25, p2 = 0.25 îäåðæàíî çíà÷åííÿ ψ0(b(0)) = 0.4670, ψ1(b(0)) = 0.0348.
5
Àíàëiç ÷óòëèâîñòi
Íåõàé (xi, xi+1) ïðîìiæîê, ÿêîìó âiäïîâiä๠ïàðàìåòð bi. Òîäi ∂b∂g = 2γubi íà öüîìó ïðîìiæêó
òà ∂b∂gR = 0 ïîçà Ríèì.
x ∂g
x
2γu bi dx = 2γu bi (xi+1 − xi ) = 2γu bi x −x
n .
x ∂b dx = x
∂g
∂s
∂s
2
∂y äîðiâíþ¹ 2γy (y − yd ) íà Ω2 i äîðiâíþ¹ 0 íà Ω1 . ∂y = u + 3y s3 , ∂b = y íà ïðîìiæêó (xi , xi+1 )
òà ∂b∂s = 0 ïîçà íèì, ∂b∂f = 0.
i
i
e
0
i+1
i
e
0
i
i
i
i
2
5.1
DDM
dψ0
=
dbi
Z
x0
äå zi ðîçâ'ÿçîê êðàéîâî¨ çàäà÷i
−zi00 +
xe
∂g
∂g
+
zi dx,
∂bi ∂y
∂s
∂s
∂f
−
,
zi =
∂y
∂bi ∂bi
i = 1..n,
x ∈ Ω = (x0 , xe ),
zi0 (x0 ) = 0,
zi (xe ) = 0.
Ïåðåïèøåìî
5.2
dψ0
xe − x0
= 2γu bi
+ 2γy
dbi
n
Z
(y − yd )zi dx,
i = 1..n.
Ω2
AM
dψ0
=
dbi
Z
xe
x0
äå µ ðîçâ'ÿçîê êðàéîâî¨ çàäà÷i
∂g
−µ
∂bi
−µ00 + µ
∂s
∂f
−
∂bi ∂bi
∂s
∂g
=
,
∂y
∂y
dx,
i = 1..n,
x ∈ Ω = (x0 , xe ),
µ0 (x0 ) = 0,
µ(xe ) = 0.
Ïåðåïèøåìî
dψ0
xe − x0
= 2γu bi
−
dbi
n
Z
3
xi+1
µydx,
xi
i = 1..n.
n=4
FDM (a) (0.0073, 0.0048, 0.0089, 0.0059)
FDM (b) (0.0073, 0.0048, 0.0090, 0.0060)
DDM (0.0070, 0.0048, 0.0073, 0.0063)
AM (0.0070, 0.0048, 0.0073, 0.0063)
Òàáë. 1: dψdb ó ïî÷àòêîâié òî÷öi
0
2
4
8
FDM (a) 0.2873 0.3060 0.3009
FDM (b) 0.2872 0.2812 0.2859
DDM 0.2872 0.2815 0.2803
AM 0.2872 0.2815 0.2803
Òàáë. 2: ψ0 ó òî÷öi ìiíiìóìó áåç îáìåæåíü
2
4
8
FDM (a) 0.7609 1.0306 1.3296
FDM (b) 0.7752 1.4441 0.8456
DDM 0.7601 1.3063 1.1853
AM 0.7601 1.3061 1.1853
Òàáë. 3: ψ1 ó òî÷öi ìiíiìóìó áåç îáìåæåíü
FDM (a)
FDM (b)
DDM
AM
2
(-0.04, -9.98)
(-0.16, -10.0)
(-0.01, -10.0)
(-0.01, -10.0)
2
FDM (a) (-2.21, -42.56)
FDM (b) (0.00, 0.01)
DDM
(0.00, 0.01)
AM
(0.00, 0.01)
4
8
(-0.09, -4.51, -9.97, -5.77) (-2.24, 2.27, -5.96, -9.42, -9.55, -9.83, -9.56, -0.60)
(3.88, -9.96, -10.0, -10.0) (-1.55, 1.41, 1.07, -3.43, -9.87, -9.91, -9.95, -9.83)
(3.33, -8.32, -10.0, -10.0) (-0.26, 5.57, -2.96, -10.0, -10.0, -10.0, -10.0, -10.0)
(3.33, -8.32, -10.0, -10.0) (-0.26, 5.57, -2.96, -10.0, -10.0, -10.0, -10.0, -10.0)
Òàáë. 4: b áåç îáìåæåíü
4
8
(-0.04, -0.01, 0.00, -1.60) (-0.19, 0.19, -0.08, 0.00, -0.05, -0.04, -0.04, -0.44)
(0.02, 0.02, 0.03, 0.00)
(0.08, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00)
(0.00, 0.00, 0.00, 0.00)
(0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00)
(0.00, 0.00, 0.00, 0.00)
(0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00)
Òàáë. 5: dψdb ó òî÷öi ìiíiìóìó áåç îáìåæåíü
0
2 4 8
FDM (a) 16 11 109
FDM (b) 17 63 481
DDM 16 40 179
AM
9 16 42
Òàáë. 6: ÷àñ CPU áåç îáìåæåíü
4
2
4
8
FDM (a) 0.4978 0.4965 0.4579
FDM (b) 0.5168 0.5051 0.4863
DDM 0.5191 0.5010 0.4550
AM 0.5167 0.4722 0.4720
Òàáë. 7: ψ0 ó òî÷öi ìiíiìóìó ç îáìåæåííÿìè
2
0.0044
0
0
4
8
0.0124
FDM (a)
6.1971 ·
FDM (b)
3.5671 · 10−8 2.4263 · 10−5
DDM
3.6172 · 10−4
0.0026
−22
AM 3.0569 · 10
1.7756 · 10−4 7.7861 · 10−4
Òàáë. 8: ψ1 ó òî÷öi ìiíiìóìó ç îáìåæåííÿìè
2
FDM (a) (4.94, 3.08)
FDM (b) (6.71, 2.08)
DDM (6.29, 2.80)
AM (6.63, 2.18)
10−9
4
8
(2.95, 9.93, 3.53, 0.67) (2.67, 3.75, 5.65, 4.23, 4.44, 0.48, 0.65, 1.06)
(4.52, 7.51, 4.05, 1.05) (3.52, 4.55, 8.54, 5.48, 6.07, 1.15, 0.26, 1.27)
(4.61, 6.26, 4.36, 1.15) (3.30, 3.75, 7.68, 4.90, 6.02, -0.76, -1.45, -1.16)
(2.60, 9.06, 3.06, -1.05) (3.33, 4.30, 7.95, 4.72, 6.65, 0.67, -0.82, -0.15)
Òàáë. 9: b ç îáìåæåííÿìè
2
4
8
FDM (a) (0.24, 0.01) (0.03, 0.01, 0.01, 0.01) (0.01, 0.00, 0.01, 0.00, 0.00, 0.03, 0.01, 0.00)
FDM (b) (0.02, 0.02) (0.01, 0.01, 0.01, 0.01) (0.01, 0.01, 0.00, 0.00, 0.00, 0.00, 0.01, 0.00)
DDM (0.02, 0.01) (0.01, 0.01, 0.01, 0.01) (0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00)
AM (0.02, 0.01) (0.01, 0.01, 0.01, 0.01) (0.01, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00)
Òàáë. 10: dψdb ó òî÷öi ìiíiìóìó ç îáìåæåííÿìè
0
2 4 8
FDM (a) 8 45 32
FDM (b) 57 57 187
DDM 117 94 201
AM 103 42 62
Òàáë. 11: ÷àñ CPU ç îáìåæåííÿìè
5
6
Ðîçâ'ÿçóâàííÿ êðàéîâî¨ çàäà÷i äëÿ îïòèìàëüíèõ çíà÷åíü ïàðàìåòðiâ êåðóâàííÿ
6
7
Íåîáõiäíi óìîâè îïòèìàëüíîñòi
Z
(y(x) − yd )2 dx + γu
L(u, y, µ) = γy
Z
u2 (x)dx +
Ω
ZΩ2
+
µ(−y 00 + ry 0 + uy + s3 y 3 − f )dx + µ(x0 )(y 0 (x0 ) − 1) + µ(xe )y(xe ) +
Ω
Z
Z
Z
+
+ 2
−
µ2 (u − u)dx +
µ3 (u − u+ )dx
+
µ1 ((y − y ) + |y − y |) dx +
Ω1
Ω
7
Ω