最適化問題と汎関数の極値問題

九州大学学術情報リポジトリ

Kyushu University Institutional Repository

最適化問題と汎関数の極値問題

倉橋, 貴彦

長岡技術科学大学機械系機械情報・制御工学大講座

https://doi.org/10.15017/1470160

出版情報：九州大学情報基盤研究開発センター全国共同利用システム広報. 2 (1), pp.17-28, 2008-08.

九州大学情報統括本部広報委員会 バージョン：

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− d dt

∂J ^∗

∂ ^{d x} _dt ^(t)

+ ∂J ^∗

∂x(t) = − dp(t)

dt + A ^T p(t) + Q ^T x(t)

= − d(P (t)x(t))

dt + A ^T P (t)x(t) + Q ^T x(t)

= − dP (t)

dt x(t) − P(t) dx(t)

dt + A ^T P (t)x(t) + Q ^T x(t) = 0 (94)

c,2A

(91)

^{)R 245A}

(95)

^)*:>Y

u(t) = (R ^T ) ⁻¹ B ^T p(t) = (R ^T ) ⁻¹ B ^T P (t)x(t) (95)

2 A

(94)

^{667 >A}

(89)

^:FG=>2A

(96)

^)*:>Y

dx(t)

dt + Ax(t) − Bu(t) = dx(t)

dt + Ax(t) − B(R ^T ) ⁻¹ B ^T P (t)x(t) = 0 (96)

A

(96)

^{6=>212345-U@A}

(97)

^{)*:*+ 9>Y}

dx(t)

dt = − (A − B(R ^T ) ^{− 1} B ^T P (t))x(t) (97)

c,2A

(97)

^6A

(94)

^{:FG => ` :4 )B}

P (t)

^:)>&5[2A

(98)

^6X>Y

− dP (t)

dt x(t) − P (t) dx(t)

dt + A ^T P (t)x(t) + Q ^T x(t)

= − dP (t)

dt x(t) + P(t)(A − B(R ^T ) ⁻¹ B ^T P (t))x(t) + A ^T P(t)x(t) + Q ^T x(t)

= − dP (t)

dt + P (t)(A − B(R ^T ) ⁻¹ B ^T P(t)) + A ^T P (t) + Q ^T x(t) = 0 (98)

2

x(t) 6 = 0

^)R2A

(98)

^3B5)B>Y)<<2JK:496=

>)B

P (t)

^::=>[A

(99)

^)*:>Y

− dP (t)

dt + P (t)(A − B(R ^T ) ⁻¹ B ^T P (t)) + A ^T P(t) + Q ^T = 0 (99)

A

(99)

^{[9>Y 2F\F]^_` %a4:D=>/067}

p(t f ) = 0

*+ 9<"R2

x(t f )

^{#$9 > 2 A}

(93)

^)RA

(99)

^{:D =>678<2}

P(t _f )

^{)B>67)R'(6)*YA}

(93)

:";<2`:4)B

P(t)

^:)>

>Ac9<"R2T:[:D=>T6> 89<;>YA

(99)

⁶

,:%Z6b82X 9,2 9,

P (t)

^{:)R 2}:"S >45

u(t)

^A

(100)

^)*:'(89>Y

u(t) = (R ^T ) ⁻¹ B ^T P (t)x(t) (100)

7

!"#$%&'()*+,-./01*+

-."#)*:23 4&5[6678,&:4L9# 2[:" ;

<`:4) B`:4 )BC 89> A6&5[X 92T6

:> 89<;>Y 2&:4L9>67[6,=)*:2 '

(:)R2&:46=>:46> %T: 9>Y

=>

[3]

^{Y -.c [6R>2JK:4L9#:D8<2 67[F\}

F]^ _`a4C6JK:4:+,HIJK:4L 96>#dZ:R+,

267[>[,2J K:4:+<;>AJK:49:6*+

; JK:4L9#6HIJK:4L 9# 8<#;, >YD

+6 :2-."#:D8<2 :S >:4

u(t)

²

l

l + 1

U >c )+,2

u(t) ^l+1 − u(t) ^l

^{::=>A6HIJK:4:+<L9}

#6; , 8<2JK:4L9#dZ:R >Y:2

l

^@4

92 :;<TO!O"R8#$89>9 >Y 2HIJK:4:

u(t) ^l+1 − u(t) ^l

^{:%R89,A6&8,JK:4}

K

^6A

(101)

^{% :/ =>Y}

K (p(t) ^l , x(t) ^l , u(t) ^l , u(t) ^l+1 ) = J ^∗ (p(t) ^l , x(t) ^l , u(t) ^l )

+ 1

2 Z t _f

t 0

(u(t) ^l+1 − u(t) ^l ) ^T W ^l (u(t) ^l+1 − u(t) ^l )dt (101)

:

W ^l

C 'F()*:%<<89,DE)B R2)B :; <6 +,8>,:-.8<*+ 9>'F()* >Yc,2/8,&:4

K

²

J ^∗

^{6H IJ}

K:4 :D8<./JK041>Y 2./2304

K

^{:D =>49#6}

,:2./23045 6@56 =>A

(102)

^{% :>7}

δK ^l =

Z t _f t 0

(δp(t) ^l ) ^T ∂J ^{∗ l}

∂p(t) ^l

+ ∂J ^{∗ l}

∂ ^{d x} _dt ^{(t) l} T

δ dx(t) dt

l

+ ∂J ^∗l

∂x(t) ^l T

δx(t) ^l + ∂J ^∗l

∂u(t) ^l T

δu(t) ^l dt = 0 (102)

-.de2A

(102)

^:D8<289:;:<>=>

x(t ₀ )

^{?@1<;> @5}

δx(t 0 )

^{>7A A}

(102)

^{A5 BC:D8}<D5E 5A6=>289:;:

<>=>A@56FG8<H6IJ =>KL2 A

(103)

^{A% :>7}

Z t _f

t 0

∂J ^{∗ l}

∂ ^{d x} _dt ^{(t) l} T

δ dx(t) ^l dt

dt = Z t _f

t 0

d dt

∂J ^{∗ l}

∂ ^{d x} _dt ^{(t) l} T

δx(t) ^l dt − Z t _f

t 0

d dt

∂J ^{∗ l}

∂ ^{d x} _dt ^{(t) l} T

δx(t) ^l dt

= ∂J ^{∗ l}

∂ ^{d x (t} _dt ^{f ) l} T

δx(t f ) ^l − ∂J ^{∗ l}

∂ ^{d x (t} _dt ^{0 ) l} T

δx(t ₀ ) ^l

− Z t f

t 0

d dt

∂J ^{∗ l}

∂ ^{d x} _dt ^{(t) l} T

δx(t) ^l dt

= ∂J ^{∗ l}

∂ ^{d x (t} _dt ^{f ) l} T

δx(t _f ) ^l − Z t _f

t ₀

d dt

∂J ^{∗ l}

∂ ^{d x} _dt ^{(t) l} T

δx(t) ^l dt (103)

%M<2 A

(103)

^6A

(102)

^:NO=>A

(104)

^{A% :>7}

δJ ^∗ =

Z t _f t ₀

∂J ^{∗ l}

∂p(t) ^l δp(t) ^l + − d dt

∂J ^{∗ l}

∂ ^{d x} _dt ^{(t) l}

+ ∂J ^{∗ l}

∂x(t) ^l

δx(t) ^l + ∂J ^{∗ l}

∂u(t) ^l δu(t) ^l dt + ∂J ^{∗ l}

∂ ^{d x (t} _dt ^{f ) l}

δx(t f ) ^l = 0 (104)

A

(104)

^{%PQR2304A56 @5}>S:2QR2304AT04:D=>U5 >=>A

(105)

²

(106)

²

(107)

²

(108)

^{6V>KL >7}

∂J ^{∗ l}

∂p(t) ^l = dx(t) ^l

dt + Ax(t) ^l − Bu(t) ^l = 0 (105)