`nØ{²§ UÁ¯
¢ù¥
ó £ÚUEg,´ 0, b > 0© - y(·) Ǒëùü:^ C §= y(0) = 0, y(a) = b© âfÉå^± 0 ÷ y(·) l (0, 0...
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`nØ{²§ UÁ¯
¢ù¥
ó £ÚUEg,´ 0, b > 0© - y(·) Ǒëùü:^ C §= y(0) = 0, y(a) = b© âfÉå^± 0 ÷ y(·) l (0, 0) :w (a, b)©· '%¯K´XÛÀJ y(·) Tâfl (0, 0) w (a, b) má© bâfäkþ m©^ s(t) L«Tâfl: (0, 0) Ñu 3mã [0, t] S (÷X) rL´§©u´ s˙ = v Ò´âf Ý©dUþÅð½Æ§·k 1 mv 2 = mgy, 2
Ù¥ y ´âflm© R £©ù§v = √2gy, l § p 1 + (y ′ )2 dx √ . 2gy
DR A
ds dt = √ = 2gy
Ïd§âf£Ä (a, b) ¤ImǑ t=
Z
a
0
p 1 + y ′ (x)2 △ p dx = J(y(·)). 2gy(x)
u´§©¯KÒzǑ3^ y(0) = 0 Ú y(a) = b e§z ¼ J(y(·)) C©¯K© (0, 0)
a
x
y = y(x)
b
y
(a, b)
1Ù Ú ó
ã 1.2 `¯K
FT
6
~ 1.5. ÄÜ>F©b>FoþǑ m§l/ålǑ y©- F (t) ǑǑ t Ú>Få©d Newton 1½Æ§ ·k m¨ y (t) = F (t).
g,/§·Ib½>FÚõÇ´kµéu, F 0§
0
|F (t)| ≤ F0 .
>
(1.10)
DR A
b>F©G´·3/¡ (y = 0), = y(0) = 0,
y(0) ˙ = 0.
·F"£Ä>F3Ǒ t = T Ê3pÝ y = h©ù±L« Ǒ y(T ) = h,
y(T ˙ ) = 0.
ùp§t = T ´Ǒ©·8´3ámS pÝ h©5¿3vkÚõÇå¹e (= F = +∞ ), T > 0 U ?¿/. g,ù´vk¢S¿Â. Ïdå^ (1.10) ·¯KC¢S© 0
l²5w§8Z{AT´µ 0 ≤ y(t) ≤ h2 , F0 , F (t) = −F0 , h2 < y(t) < h, 0, y(t) = h.
1.
¼ê4!C©¯K9`
7
Z
Z
T
F (t) dt =
0
Ïd§
Z
mh
T
0
T
ds
0
Z
s
m¨ y (t) dt = m y(T ˙ ) − y(0) ˙ = 0,
F (t) dt =
0
=
Z
Z
dt
0
=
T
Z
T
my(s) ˙ ds = my(T ) = mh.
F (t) ds =
F (t) dt − Z
−mh =
Z
Z
T
(T − t)F (t) dt
0
T
tF (t) dt = −
0
T
(t + r)F (t) dt,
0
Z
T
tF (t) dt.
0
∀ r ∈ IR.
DR A
dd
T
T
t
0
l §
Z
0
T
FT
·y3Ò5y²þ¡üÑ´©U¤I^§·k
mh
≤
Z
=
F0
T
0
|t + r| |F (t)| dt ≤ F0
Z
r
Ïd,
T +r
|t| dt,
mh ≤ inf r∈IR F0
P ϕ(r) = R
T +r r
Z
T
0
|t + r| dt
∀ r ∈ IR.
Z
T +r
|t| dt.
r
©ǑÏé ϕ(·) , ·-
|t| dt
r = − ©u´,
0 = ϕ′ (r) = |T + r| − |r|,
T 2
mh T ≤ ϕ(− ) = F0 2
Z
T /2
−T /2
|t| dt =
T2 . 4
1Ù Ú ó
dd, T ≥2
r
FT
8
mh △ ∗ =t . F0
e¡§·y² t ´¤Iám©Ǒd§P ( ∗
t∗ F (t) = −F0 sgn t − ≡ 2
F0 ,
∗
KA y (·) ÷v: ∗
my˙ (t ) =
Z
t∗
∗
F (t) dt =
0
±9
Z
t∗ /2
F0 dt −
0
Z
my ∗ (t∗ ) =
Z
ds 0
Z
s
0 Ø´½§·k n
0
DR A
0
F,Γ
0
M
JM (T, y(·), u(·)) = h(T, y(T )),
Ù¥ h : [0, +∞) × IR
n
JL (T, y(·), u(·)) =
Ù¥ f
Z
→ IR
T
B
∀ (y(·), u(·)) ∈ D(JM ),
´½N;
f 0 (t, y(t), u(t))dt,
0
0
L
: [0, +∞) × IRn × U → IR;
JB (T, y(·), u(·)) = h(T, y(T )) +
Z
T
∀ (y(·), u(·)) ∈ D(JL ),
f 0 (t, y(t), u(t))dt,
0
∀ (y(·), u(·)) ∈ D(JB ),
1Ù Ú ó
20 n
FT
Ù¥ h : [0, +∞) × IR → IR, f : [0, +∞) × IR × U → IR©y3§X J·- f ≡ 1! J Ǒ5UI§¿± (2.2) ǑGå§ ·Ò ãm`¯K© 2. =¯K ÄkG§ (m ≥ 2): 0
0
n
L
△
y(t) ˙ = f (t, y(t), d),
t ∈ [0, T ], d ∈ D = {1, 2, · · · , m}.
(2.10)
z§L« XÚ«$1ª©3zǑ§ÑkÙ¥ ª3$1§ 0 n
|x| ≤ M,
∀ x ∈ E.
l §d·K 1.4 co E ´k.8© e¡§·5y² co E ´4© x ∈ co E, ·y x ∈ co E©d½n 1.5, ·k±eL«: m
xm =
n X i=0
m αm i xi ,
m ≥ 1,
©
lim xm = x
m→∞
1.
à8
33
n X
αm i ≥ 0,
FT
Ù¥
αm i = 1,
i=0
xm i ∈ E.
du {α } Ú {x } Ñ´k.§·±b α →α, x →x, m → ∞, i = 0, · · · , n. w,§ m i
m i
m i
m i
i
i
αi ≥ 0,
?Ú§
n X
x = lim xm = lim
m→∞
n X
xi ∈ E.
m αm i xi =
n X
αi xi ∈ co E.
DR A
m→∞
αi = 1,
i=0
i=0
i=0
ùÒy² co E ´4, l ·Ky©
2
I5¿´§3y² co E 45§=k·K 1.4 ´Ø
©ÖöÄǑo. ·K 1.7. E ⊆ IR ´à48, Ké?Û x ∈ IR , k x¯ ∈ E n
n
△
|x − x¯| = d(x, E) = inf |x − y|. y∈E
(1.2)
?Ú§x¯ d±eC©ØªǑx: hx − x ¯, y − x¯i ≤ 0,
∀ y ∈ E.
(1.3)
1Ù O£
FT
34
x
y
x¯
E
ã 2.1 y². Äk§·5y² x¯ 35©P △
Kk x
∈E
y∈E
lim |xk − x| = d.
(1.4)
DR A
k
d = inf |x − y| ≡ d(x, E) ≥ 0,
k→∞
d²1o>/{K
2
|xj − xk |
= 2|x − xj |2 + 2|x − xk |2 − |2x − (xj + xk )|2 2 xj + xk = 2|x − xj |2 + 2|x − xk |2 − 4 x − . 2
(1.5)
|xj − xk |2
(1.6)
du E ´à8§l x
j
+ xk ∈E 2
©u´
≤ 2|x − xj |2 + 2|x − xk |2 − 4d2 .
ù§·w {x } ¯¢þ´ IR ¥ Cauchy S©d E 45§·k k
n
xk → x ¯ ∈ E.
d (1.4) á= (1.2) ¤á©ù·Òy² x¯ 35©
1.
à8
35
FT
·Ñ§5¯¢þ®¹3þ¡y²¥©bk, y¯ ∈ E ÷v |x − y¯| = d,
K3 (1.6) ¥ x = x¯, x = y¯ |¯x − y¯| ≤ 0©5y© §é?Û y ∈ E ±9 α ∈ (0, 1), ·k k
2
j
x¯ + α(y − x¯) = αy + (1 − α)¯ x ∈ E.
l |x − x¯|2
= =
=
d2 ≤ |x − x ¯ − α(y − x ¯)|2
|x − x ¯|2 − 2αhx − x ¯, y − x ¯i + α2 |y − x ¯|2 . 2hx − x ¯, y − x ¯i ≤ α|y − x ¯|2 .
DR A
- α → 0 , ·Ò (1.3)©L5§XJ (1.3) ¤á, Ké?Û +
y ∈ E,
|x − y|2
l (1.2) ¤á©
= |x − x¯ − (y − x ¯)|2
= |x − x¯|2 − 2hx − x¯, y − x ¯i + |y − x ¯|2 ≥ |x − x¯|2 .
2
½n 1.8. E , E ⊂ IR ´üpØà48§E k., K3 c ∈ IR ±9 λ ∈ IR , |λ| = 1 ÷v 1
2
n
1
n
hλ, x1 i < c < hλ, x2 i,
∀ x1 ∈ E1 , x2 ∈ E2 .
(1.7)
þª`²²¡ hλ, xi = c î©lm 8Ü E Ú E © 1
2
1Ù O£
FT
36 hλ, xi = c
λ E2
E1
ã 2.2 y². Äk·b E = {0}©d§du 0 6∈ E E ´ à4§d·K 1.7, 3 x¯ ∈ E 1
2
2
2
0 < |¯ x| = inf |x2 |. x2 ∈E2
x ¯ , |¯ x|
c=
|¯ x| > 0, 2
DR A
λ=
K |λ| = 1© d (1.3), ·k l
hλ, x2 − x ¯i = −
Ïd,
1 h0 − x ¯, x2 − x ¯i ≥ 0, |¯ x|
∀ x2 ∈ E2 .
hλ, x2 i ≥ hλ, x¯i > c,
∀ x2 ∈ E2 .
hλ, 0i < c < hλ, x2 i,
∀ x2 ∈ E2 .
ù§·Òé E = {0} /y² (J© é/§ E = E − E , Kd·K 1.2, E ´à4© du E Ú E ا·k 0 6∈ E©d ¡¤y²(J§ 3 λ ∈ IR , |λ| = 1 ±9 c ∈ IR 1
2
1
1
2
n
1
0 = hλ, 0i < c1 < hλ, xi,
∀ x ∈ E.
2. Lebesgue
È©
37
hλ, x2 i > c1 + hλ, x1 i,
u´-
∀ x1 ∈ E1 , x2 ∈ E2 .
c1 + sup hλ, x1 i, 2 x1 ∈E1
c≡
= (1.7)©
FT
l
2
§2. Lebesgue
È©
ÿ8 Äk§·5£ Lebesgue ÿ8Ú Borel 8Vg©P 2 Ǒd IR ¥¤kf8¤8x© ½Â 2.1. ¡ F ⊆ 2 ´ σ- §XJ n
DR A
IRn
IRn
(i) ∀ Ei ∈ F , i = 1, 2, · · · ,
∞ [
i=1
Ei ∈ F ;
(ii) ∀E1 , E2 ∈ F , E1 \ E2 ∈ F ;
(iii) IRn ∈ F
©
é?Û G ⊆ 2 , ¹ G σ- ¡Ǒd G )¤ σ§PǑ σ(G )©d IR ¥m8N O )¤ σ- ¡Ǒ IR Borel σ- §PǑ B(IR ) ({PǑ B)©±y² B Ǒ´d IR ¥48N)¤ σ- ©B ¥¡Ǒ Borel 8© x = (x , x , · · · , x ) Ǒ¥%!>Ǒ 2δ > 0 éu IR ¥± Y N Q (x) = (x − δ, x + δ)©½Â IRn
n
n
n
n
1
n
δ
i
n
2
n
i
i=1
m(Qδ (x)) = (2δ)n .
(2.1)
1Ù O£
38
FT
/§é?Û E ⊆ IR , ½Â n
m(E) = inf{
∞ X i=1
∞ [ m(Qi ) E ⊆ Qi , Qi i=1
ǑN},
(2.2)
±y² E ´N§þã½ÂØ¬Ú (2.1) gñ© N = {E ⊆ IRn m(E) = 0}.
¡ N ¥Ǒ"ÿÝ8©´"ÿÝ8f8E´"ÿÝ8© P L = L (IR ) = σ(B S N ) Ǒd B S N )¤ σ- ©L ¥¡Ǒ Lebesgue ÿ8(k{¡Ǒÿ8)©5¿, L 6= 2 , = IR ¥3 Lebesgue Øÿ8©¯¢þ, ·kXe (J: n
IRn
n
½n 2.2. E ⊆ IR , K±e^d: (i) E ´ Lebesgue ÿ8; (ii) 3 Borel 8 B ∈ B Ú"ÿÝ8 N ∈ N ,
DR A
n
E=B
(iii)
é?Û Ee ⊆ IR ,
[
N;
n
e = m(E e m(E)
\
e \ E); E) + m(E
E ´k.8§ e E e ⊆ E Ǒ48} = inf{m(G) G ⊇ E Ǒm8}. sup{m(E)
(iv)
?Ú, 'u m, ·k
È©
39
½n 2.3. (i) m(φ) = 0; (ii) m(E) ≥ 0, ∀ E ∈ L ; (iii) m
∞ [
i=1
FT
2. Lebesgue
©
∞ X \ Ei = m(Ei ), ∀ Ei ∈ L , Ei Ej = φ, i 6= j i=1
8 §·¡ m Ǒ IR þ Lebesgue ÿݧ¡ m(E) Ǒ E ∈ L Lebesgue ÿÝ©± , ·Ǒ~^ |E| 5L« m(E)©3 þ¡½n¥, (iii) ¡Ǒ m \5©+Uì (2.2), m(·) é ¤k E ∈ 2 k½Â, \5=é L ¥8Ü⤠᩠ÿ¼ê y3§·5Ä IR þ¼ê© ½Â 2.4. E ⊆ IR ÿ© (i) ¼ê f : E → IR ¡Ǒ{ü¼ê§XJ3 E ∈ L , E ⊆ E (1 ≤ i ≤ k), n
IRn
DR A
n
n
i
f (x) =
Ù¥ α ∈ IR, χ
k X
αi χEi (x),
i=1
i
x ∈ E,
´ E AƼê: ( 1, XJ x ∈ E , χ (x) = 0, XJ x 6∈ E . (ii) ¼ê f : E → IR ¡Ǒ (Lebesgue) ÿ¼ê§XJé?Û i
Ei (·)
i
i
Ei
i
c ∈ IR
△ {f ≥ c} = f −1 [ c, +∞) = {x ∈ E f (x) ≥ c}
´Lebesgueÿ8©
1Ù O£
40
FT
¼ê f : E → IR ¡Ǒ Borel ÿ¼ê§XJé?Û c ∈ IR, {f ≥ c} ´ Borel 8© ±y²3þã½Â (ii) Ú (iii) ¥, {f ≥ c} ±¤ {f > c}, {f ≤ c} ½ {f < c}©?Ú§·±y² f ÿ
=é ?Û B ∈ B, f (B) ∈ L © ·K 2.5. ÿ¼êäkXeÄ5: (i) XJ f, g ÿ§K f + αg (α ∈ IR), f · g, f /g(XJé?Û x, g(x) 6= 0), min{f, g},max{f, g}, ±9 |f | Ñ´ÿ© (ii) XJ f ÿ§K lim f , lim f , inf f ±9 sup f þ ÿ© (iii) f ÿ
=3{ü¼ê f , 3ØK "ÿÝ8 , ::Âñu f§AO§?Û{ü¼ê´ÿ© ÿ¼êk±e5: ½n 2.6. E ⊂ IR Ǒk.ÿ8, f ´ E þÿ¼ê §A??Âñu f , K (i) f ½UÿÝÂñu f © (ii) (Egorov) é?Û ε > 0, 3 E ⊆ E, m(E \ E ) < ε,
f 3 E þÂñu f© (iii)
−1
3
k
k
k
k→∞
k→∞
k
k
k
k
DR A
k
n
k
4
5
k
ε
k
ε
ε
dd§f ÿ
=é?Ûm8 O ∈ O, f (O) ∈ L , =m8_´ÿ8©
âÿÀÆ*:§f ëY
=é?Ûm8 O ∈ O, f (O) ∈ O©ù´k ¿Â'© XJØ E "ÿÝf8 E §,«5 P 3 E \ E ¤á§·Ò¡ P 3 E þA??¤á§PǑ P a.e. E©ù f → f, a.e. E , = f 3 E þA? ?Âñu f ´3"ÿÝ8 E ⊆ E , f (x) → f (x), ∀ x ∈ E \ E © =é?Û ε > 0, lim m{x ∈ E |f (x) − f (x)| > ε} = 0© 3
−1
−1
4
0
0
k
0
5
k→∞
k
k
k
0
2. Lebesgue
È©
41 n
FT
½n 2.7. (Riesz) E ⊂ IR Ǒÿ8©f ´ E þÿ¼ ê§UÿÝÂñu f , K3 f f {f } 3 E þA?? Âñu f© k
k
kj
½n 2.8. (Luzin) E ⊂ IR ÿ, f : E → IR ÿ, Kéu ?¿ ε > 0, 348 E ⊆ E, m(E \ E ) < ε, f 3 E þ´ëY © n
ε
ε
6
ε
`5§ÿ¼êEܼêؽ´ÿ©±e~fÄ uØÿ835© ~ 2.1. E ⊂ IR ´ Lebesgue Øÿ8©½Â f : IR → IR Ǒ ( 1, XJ x = y ∈ E, f (x, y) = 0, Ù§, K´ f ´ IR ¥"ÿÝ8 {(x, y)|x = y ∈ E} þAƼê, l
§´ÿ©é x ∈ IR, - g(x) = x, K g(·) ´ IR þÿ¼ ê (¯¢þ´ëY¼ê)©- F (x) ≡ f (x, g(x))§K´ F (·) Ø ´ÿ¼ê© éuÿ¼ê§Eܼêÿ5´'E,¯K©ù p·Ñü(J©éù¯K(J§I^ Souslin mk'£©
DR A
2
2
½n 2.9. k¼ê f : IR × IR → IR ÷vµé?Û y ∈ IR , f (·, y) ´ÿ, é?Û x ∈ IR , f (x, ·) ´ëY , Ké?Û n
m
m
n
7
E Ǒ IR ¥8ܧ·¡¼ê f 3 E þëY§´µ ∀ x ∈ E , ∀ ε > 0, ∃δ > 0, |x − x | < δ
x ∈ E §k |f (x) − f (x )| < ε©d/§é IR ¥?Ûm8 U §f (U ) = {x ∈ E f (x) ∈ U } ´ E (é) m8© Ï~¡ù¼êǑ Carath´eodory ¼ê© 6
n
0
0
n
7
−1
0
1Ù O£
42 8
n
→ IRm ,
Eܼê x 7→ f (x, ξ(x)) ´
FT
þ ) ÿ¼ê ξ : IR ÿ©
(
y². IR ¥Èf8 {y , y , · · ·}©éu i, j ≥ 1, ½ m
Â
1
1 △ Eij = {x ∈ IRn |ξ(x) − yj | < }, i
2
△
Fij = Eij \
l é½ i, F ÿ§üüØ,
IR Â
n
ij
ξi (x) = yj ,
=
∞ [
j=1
j−1 [
Eik .
k=1
Fij
©u´½
x ∈ Fij , j = 1, 2, · · · .
´ ξ (·) ÿ§
ξ (·) 3 IR þ::Âñu ξ(·)©5¿ x 7→ f (x, ξ (x)) ÿ, f (x, ξ(x)) = lim f (x, ξ (x))©u´d½n 2.5 =(Ø© 2 ½n 2.10. f : IR × IR → IR ´ Borel ÿ¼ê©ξ : IR → IR ÿ, KEܼê x 7→ f (x, ξ(x)) ´ÿ© ù½ny²3Öög1¤© i
n
i
i
i
DR A
i→∞
n
m
n
m
È© 3Ö§XJvkAO`²§·^È©Ñò´ Lebesgue È©©e¡·£e Lebesgue È©Ä5© ·K 2.11. E ⊆ IR ÿ©f : E → IR ÿ, K (i) e m(E) < +∞, f : E → IR k.§K f È© Lebesgue
n
8
þ¼ê9Ùÿ½5½Âe!©
(ii) f
È©
43
È
= |f | È©?Úk
FT
2. Lebesgue
Z Z f (x) dx ≤ |f (x)| dx. E
(2.3)
E
·K 2.12. E ⊆ IR ÿ,
f, g : E → IR È, K (i) é?Û α ∈ IR§f + αg È© (ii) f ∨ g ≡ max(f, g) ±9 f ∧ g ≡ min(f, g) È© T S (iii) XJ E , E ∈ L §
E E = φ, E E = E, K f 3 E , E þþȧ
n
1
1
2
1
2
Z
Z
f (x) dx =
E
f (x) dx +
1
Z
2
f (x) dx.
E2
é?Û ε > 0, 3{ü¼ê ϕ
DR A
(iv)
E1
2
Z
E
|f (x) − ϕ(x)| dx < ε.
(2.4)
XJ E ´«, Ké?Û ε > 0, 3©¬~¼ê ϕ (2.4) ¤á© (vi) XJ E ´«, Ké?Û ε > 0, 3 E þ C ¼ ê ϕ (2.4) ¤á© (vii) (ýéëY5) é?Û ε > 0, 3 δ > 0, é?Û e ∈ L , e ⊆ E§e m(e) < δ, Bk (v)
∞
Z
e
|f (x)| dx < ε.
e¡´ Lebesgue È©nØ¥n~½n©
1Ù O£
44
FT
½n 2.13. (Lebesgue Âñ½n) E ⊆ IR ÿ§f , g : E → IR È (k ≥ 1)§÷v n
K f ȧ
|fk (x)| ≤ g(x),
a.e. E, ∀ k ≥ 1,
lim fk (x) = f (x), f (x) dx = lim
k→∞
E
(2.5)
a.e. E,
k→∞
Z
Z
fk (x) dx.
(2.6)
E
½n 2.14. (L´evy üNÂñ½n) E ⊆ IR ÿ§f IR ȧ
'u k üNO\: n
fk (x) ≤ fk+1 (x),
K
Z
fk (x) dx =
k
:E→
a.e. k ≥ 1,
Z
lim fk (x) dx,
DR A
lim
k
k→∞
E k→∞
Ù¥þªü>Ñ#N +∞© ½n 2.15. (Fatou Ún) E ⊆ IR ÿ§f , g : E → IR ȧ
E
n
K
fk (x) ≥ g(x),
Z
a.e. k ≥ 1,
lim fk (x) dx ≤ lim
E k→∞
§3.
k
k→∞
Z
fk (x) dx.
E
þ¼ê9 Liapounoff ½n
!¥§·ò#N¼êu IR §Ǒ{ü姷 ļê½ÂǑf8/© n
3.
þ¼ê9 Liapounoff ½n
45
FT
½Â 3.1. E ⊆ IR ÿ§f ≡ (f , · · · , f ) : E → IR , K f ¡Ǒþ¼ê © (i) ¡ f ´ (Lebesgue, Borel) ÿ§XJ¤k f : E → IR ´ (Lebesgue, Borel) ÿ© (ii) ¡ f ´ (Lebesgue) ȧXJ¤k f : E → IR ´È ©d§·P Z 1
9
n
⊤
n
k
k
Z
f1 (t) dt
E f (t) dt = Z E
.. .
fn (t) dt
E
Z
.
5¿´ n > 1 § f (t) dt ∈ IR , Ø´ IR© þ¼êê±aq/½Â. ·Ñ·K 2.11—2.13(Ø 2.12(ii)) éþ¼êǑ¤á© y3§·Qã'uþ¼êÈ©~¤ (J§ 3± Ù§ù(Jò~k^© ½n 3.2. (P. K. Liapounoff ½n) y(·) ∈ L (a, b; IR ), λ(·) : [a, b] → [0, 1] ÿ, K3ÿ8 E ⊆ [a, b], n
DR A
E
1
Z
b
λ(t)y(t) dt =
a
Z
n
y(t) dt.
E
íØ 3.3. y(·) ∈ L (a, b; IR ), λ ∈ (0, 1) ´~ê, K 3ÿ8 E ⊆ [a, b], m(E) = λ(b − a), 1
λ
9
Z
n
b
y(t) dt =
a
Z
y(t) dt.
E
Ó:L«aq§31©¥§·Ǒ²~r f ¤ (f , · · · , f 1
n ).
1Ù O£
46
j
K3ÿ8 E
ℓ X
λj (t) = 1,
j=1
FT
d Liapounoff ½n§N´8B/y²±e Liapounoff .½n: ½n 3.4. E ⊆ [a, b] ÿ§y (·) ∈ L (E; IR )§λ (·) : [a, b] → [0, 1] ÿ (1 ≤ j ≤ ℓ), ÷v 1
⊆E ℓ
j
i
j=1
ℓ Z X
E
j
= φ (i 6= j),
ℓ Z X j=1
yj (t) dt.
Ej
DR A
j=1
λj (t)yj (t) dt =
j
a.e. E,
§ [ \ E= E ,
E E j
n
§4.
¼©Û¥ (J
3C©nØ!`nØ¥§Ùö$^¼©Û¥«
Ǒ(J§´4Ǒ©3ù!§·XÚ^§~^ (J© ÄVg ½Â 4.1. 5m X ¡Ǒ´D5m§XJ3 ¡ǑêN k · k : X → [0, +∞) ÷v±e^: (i) kxk ≥ 0, ∀ x ∈ X,
kxk = 0
= x = 0; (ii) kαxk = |α| kxk, ∀ α ∈ IR, x ∈ X;
4.
¼©Û¥ (J
47
FT
nÆت) kx + yk ≤ kxk + kyk, ∀ x, y ∈ X© XJé?Û÷v±e^S x ∈ X (¡Ǒ Cauchy S): (iii) (
k
lim kxk − xℓ k = 0,
k,ℓ→∞
ok x ∈ X
10
lim kxk − xk = 0,
k→∞
K¡Ǒ X ´D5m (Ï~¡Ǒ Banach m)© ~^ Banach mk (i) Euclidean m IR , éuÙ¥ x = (x , x , · · · , x )§D ê n
1
△
n X
|xi |p
1 p
,
n
(4.1)
DR A
kxkp =
2
i=1
Ù¥ p ∈ [1, +∞]© p = 2 §(4.1) ÒÑ Ï~ Euclidean ê k · k = k · k © p = +∞ §kxk = max |x |©±y²?Û 4 IR ¤ǑD5mê ||| · ||| Ñ´ k · kd§=3 ~ê C , C > 0 △
2
∞
i
1≤i≤n
n
1
2
C1 kxk ≤ |||x||| ≤ C2 kxk,
∀ x ∈ IRn .
(4.2)
m
(ii) lp : n o X k k k p {a } ∈ I R, |a | < +∞ , p ∈ [1, +∞), a k≥1 p △ k≥1 n o l = k k k {a }k≥1 a ∈ IR, sup |a | < +∞ , p = +∞, k≥1
ù x ¡Ǒ x (r) 4©éuD5m, N´y²S4 3§4o´© 10
k
1Ù O£
Dê △
k{ak }klp =
(iii) Lp
m:
FT
48
X 1 k p p |a | , p ∈ [1, +∞), k≥1
sup |ak |, k≥1
△
p = +∞.
Lp (a, b; IRn ) = Z b {f : (a, b) → IRn |f (t)|p dt < +∞}, p ∈ [1, +∞), a n {f : (a, b) → IR esssup |f (t)| < +∞}, p = +∞, t∈(a,b)
Dê
b
|f (t)|p dt
p1
, p ∈ [1, +∞),
DR A
△
Z
kf kLp(a,b;IRn ) =
a
esssup t∈(a,b)
|f (t)|,
p = +∞.
Ï~ L (a, b; IR) PǑ L (a, b)© (iv) C m: p
Dê
p
△ C([a, b]; IRn ) = {f : [a, b] → IRn f (·)
ëY},
kf kC([a,b];IRn ) = max |f (t)|. t∈[a,b]
5¿´3þ¡~f¥§k IR ´k©? " ξ ∈ IR , K {t ξ k ≥ 1} ⊆ C([a, b]; IR ) ⊆ L (a, b; IR ) ´5 ' 8©ùL² L (a, b; IR ) Ú C([a, b]; IR ) Ñ ´ ©Ó±y² l Ǒ´ ©8 ·ò¬w§ mkmmkXNõ«O©ù:§3? n
n
k
n
p
p
n
p
n
n
4.
¼©Û¥ (J
49
FT
n m¯K½Úåv À© éukD 5m§þ§ÑÓuÓê Euclid m© ±e§·ò± X L« Banach m (k½ )© ½Â 4.2. N f : X → IR ¡Ǒ´k.5¼, XJ: f (αx + βy) = αf (x) + βf (y),
3 M ≥ 0§
∀ α, β ∈ IR, x, y ∈ X,
|f (x)| ≤ M kxk,
∀ x ∈ X.
(4.3)
(4.4)
DR A
l (4.4) §k.5¼ f ò X ¥k.8N¤k.8© ,¡§±y²e f ÷v (4.3)(¡Ǒ5¼)§
ò X ¥ ?Ûk.8N¤k.8§K f ½´k.5¼©ùÒ´Ǒ o·rù«¼¡Ǒ´k.©?Ú§d (4.3)—(4.4), |f (x) − f (y)| ≤ M kx − yk,
∀ x, y ∈ X.
Ïd§k.5¼½´ Lipschitz ëY©,¡§ ±y²5¼XJ´ëY§K§½´k.©Ïd§é u X þ5¼§8 òØ2«Ok.5ÚëY5© éu?Ûk.5¼ f : X → IR§·½Â △
|f (x)| ≡ sup |f (x)|, kxk6=0 kxk kxk=1
kf k = sup
(4.5)
¡Ǒ f ê©ù´ÏǑXJ·P X Ǒ X þk.5¼ N, ¿3Ù¥Ú\5(: ∗
(αf + βg)(x) = αf (x) + βg(x),
∀ x ∈ X, f, g ∈ X ∗ ,
1Ù O£
50 ∗
FT
K X Ò¤Ǒ5m§ (4.5) ¤Ǒ X þê©? Ú±y²3ùêe§X ´ Banach m©·¡ Ǒ X éó (éóm)© ±ea.(J¡Ǒ Riesz L«½n§§Ñ éóm «Lª© ½n 4.3. p ∈ [1, +∞), - q( ~PǑ p ) Ǒ p éó ê: p , XJ p 6= 1, p−1 q=p = +∞, XJ p = 1, K ∗
∗
′
′ △
(4.6)
(4.7)
DR A
´,
(lp )∗ = lq , n o∗ Lp (a, b; IRn ) = Lq (a, b; IRn ).
(l∞ )∗ 6= l1 ,
±þ¡ äkL«
n o∗ L∞ (a, b; IRn ) 6= L1 (a, b; IRn ).
Ǒ~ ٹ´: 35N n o é?Û F ∈ L (a, b; IR ) ,
(4.7) , n o∗ p n q R : L (a, b; IR ) → L (a, b; IRn )
p
n
∗
Z b F (g) = hR(F )(t), g(t)i dt, ∀ g ∈ Lp (a, b; IRn ), a kF k = kR(F )k q L (a,b;IRn ) . n o∗ R Lq (a, b; IRn ) Lp (a, b; IRn )
ǑÒ´`§ ´ åÓ© ·¡Ǒ Riesz N©Ó§(4.6) ¹Â´aq© du X ´ Banach m§·±?Ú½ÂÙéó (X ) , ¡Ǒ X géó§PǑ X ©5¿§é?Û x ∈ X, ÏL½Â ∗
∗ ∗
∗∗
△
x∗∗ (f ) = f (x),
∀ f ∈ X ∗,
¼©Û¥ (J
4.
51
∗∗
∗∗
FT
± x ∈ X
kxk = kx k ©Ïd§x 7→ x ½Â
X X N©ù§·Ò±r X X f8Óå5: X ⊆ X ©·½Â: ∗∗
X
∗∗
∗∗
X ∗∗
∗∗
∗∗
½Â 4.4. Banach m X ¡Ǒ´g§XJ X
∗∗
=X
©
w,§é?Û p ∈ (1, +∞)§l ±9 L (a, b; IR ) Ñ´g© Ó§IR Ǒ´g©´ l , l , L (a, b; IR ), L (a, b; IR ) ± 9 C([a, b]; IR ) ÑØ´g© Banach mÿÀ5 Äk§·5w Banach m¥Âñ5© p
n
1
∞
p
n
n
∞
½Â 4.5. X ´ Banach m© (i) ¡S x ∈ X (r)Âñu x ∈ X( PǑ
k→∞
n
§X
DR A
n
1
J
k
lim xk = x)
lim kxk − xk = 0;
k→∞
XJ
(ii)
¡S x
∈X
k
fÂñu x ∈ X( PǑ w- lim x
lim f (xk − x) = 0,
k→∞
¡S f §XJ (iii)
k
f )12
∈ X∗
∗
lim fk (x) = f (x),
12
Ǒ~PǑ x Ǒ~PǑ x
w
k
−→ x ∗ w
k
−→ x
½x
k
⇀x
§
= x)11
∀ f ∈ X ∗;
f Âñu f ∈ X ( PǑ w - lim f
k→∞
11
k
k→∞
∗
∗
k→∞
∀ x ∈ X.
k
=
1Ù O£
52
FT
3 Banach m¥§SrÂñ½fÂñ© éu X é óm§3Ù¥Q±!fÂñ, q±!f Âñ©d§f ñruf Âñ© X ´gm§3 X ¥fÂñÚf Âñ´d© e¡§·5wA8 ~f© p ∈ [1, +∞), x (·), x(·) ∈ L (a, b; IR ), K3 L (a, b; IR ) ¥ x (·) Âñu x(·)
= ∗
∗
∗
p
k
k
lim
k→∞
Z
b
a
n
p
k
k→∞
Z
n
|xk (t) − x(t)|p dt = 0;
x (·) fÂñu x(·)
=é?Û g(·) ∈ L lim
∗
b
p′
(a, b; IRn )
hg(t), xk (t) − x(t)i dt = 0.
a
(4.8)
éu p ∈ (1, +∞], 3 L (a, b; IR ) ¥ x (·) f Âñu x(·)
=é?Û g(·) ∈ L (a, b; IR ), (4.8) ¤á© du p ∈ (1, +∞) §L (a, b; IR ) ´gm§Ï 3Ù¥ fÂñÚf Âñ´©`5§éu p = 1§·Ï~ Øùf Âñ§ éu p = +∞, ·Ï~ØùfÂñ©3A^ ¥§p = 1, 2, +∞ /´^õ©
¡®²Ñ§rÂñ%ºXfÂñ§ fÂñq%ºXf Âñ©L5(Øؽ¤á© ~ 4.1. Ä x (t)Z= sin kt, K x (·) ∈ L (0, π)§
n
∗
k
DR A
p
′
p
n
p
n
∗
∗
∗
k
2
k
kxk k2 =
π
sin2 kt dt =
0
π , 2
∀ k ≥ 1.
,¡§d Riemann-Lebesgue Ún§é?Û y(·) ∈ L (0, π), k Z 2
π
lim
k→∞
y(t) sin kt dt = 0,
0
4.
¼©Û¥ (J
53
2
k
FT
= x (·) 3 L (0, π) ¥fÂñu 0©´§w, x (·) Ø´rÂñ (Ǒo?)© k
~ 4.2. ½Â
△ c0 = {xi }i≥1 lim xi = 0 , i→∞
Dê k · k , K c ´ l 4fm©±y² l∞
0
∞
c∗0 = l1 .
y3 x
k
K kx k
k l1
∈ l1 :
xk = {δik }i≥1 .
©w,§é?Û y ≡ (y ) i
=1
∈ c0 ,
k → ∞.
DR A
xk (y) = y k → 0,
i≥1
ùÒ´`S x 3 l ¥f Âñu 0©´XJ k
1
∗
y¯ ≡ {(−1)i }i≥1 ∈ (l1 )∗ = l∞ ,
·k
y¯(xk ) = (−1)k ,
Ïd x 3 l ¥Ø´fÂñ© k
∀ k ≥ 1.
1
y3·= Qã Banach mA Ǒ5©Äk ·£eÿÀmÄVg© ½Â 4.6. Y ´8©T ⊆ 2 ¡ǑÿÀ§XJ (i) Y, φ ∈ T © T (ii) e A, B ∈ T , K A B ∈ T © Y
1Ù O£
54
FT
K [ A∈ T© ¡óé (Y, T ) ǑÿÀm( T ²(§{PǑ Y )©¡ T ¥ǑÿÀm (Y, T ) m8©XJ Y þküÿÀ T , T ÷v T ⊆ T §K¡ T ´' T rÿÀ© 3 Y ¤kÿÀ¥,{Y, φ} ±9 2 ©O´fÚrÿÀ©
ö¡Ǒ²TÿÀ§ ö¡ǑlÑÿÀ© ½Â 4.7. (Y, T ) ǑÿÀm©y, y ∈ Y (k = 1, 2, · · ·), X Jé?Û¹ y U ∈ T , 3 N ∈ IN, k ≥ N §ok y ∈ U , K¡ y ´ y 4©PǑ lim y = y© (iii) T1 ⊆ T ,
A∈T1
1
1
2
2
1
2
Y
k
k
k
k
k→∞
`5§ÿÀm¥§S4±Ø©éu Banach m X 9Ùéóm X , ·±Ú\XeÿÀ: ½Â 4.8. (i) X þrÿÀ: 8Ü G ⊆ X ´m8
=é ?Û x ∈ G, 3 ε > 0,
DR A
∗
0
{x ∈ X kx − x0 k < ε} ⊆ G;
þfÿÀ: 8Ü G ⊆ X ´m8
=é?Û x G, 3 f , · · · , f ∈ X ±9 ε , · · · , ε > 0, (ii) X
1
∗
m
1
∈
m
{x ∈ X |fi (x − x0 )| < εi , 1 ≤ i ≤ m} ⊆ G;
þf ÿÀ: 8Ü G ⊆ X ´m8
=é?Û ∈ G, 3 x , · · · , x ∈ X ±9 ε , · · · , ε > 0, (iii) X ∗
f0
0
∗
∗
1
m
1
m
{f ∈ X ∗ |f (xi ) − f0 (xi ))| < εi , 1 ≤ i ≤ m} ⊆ G;
4.
¼©Û¥ (J
55
FT
u´§ ¡0rÂñ!fÂñ9f ÂñÒ©OéAu UrÿÀ!fÿÀ9f ÿÀÂñ© ±e(J3A^¥´4Ǒ© ∗
∗
½n 4.9. {x } ´ Banach m¥fÂñS, K§½ ´k.§=3~ê K > 0 k
kxk k ≤ K,
∀ k ≥ 1.
½n 4.10. X ´© Banach m, K X ¥k. 7kf Âñf© ∗
∗
DR A
½n 4.11. (Eberlein-Shmulyan) X ´ Banach m, K X ´gm
= X ¥k.ÑkfÂñf© ½n 4.12. (Banach-Alaoglu) X ´D5m, K X ¥ü 4¥´f ;©
∗
∗
½n 4.13. (Mazur) X ´ Banach m§x, x ∈ X, w- lim x = x, K3 α ∈ [0, 1], j = 1, 2, · · · , N , k
k→∞
k
kj
Nk X
k
αkj = 1,
j=1
lim
k→∞
Nk X j=1
∀ k = 1, 2, · · · ,
αkj xk+j = x.
ó§éu Banach m¥fÂñS§±é§à|Ü rÂñ54©
1Ù O£
56
FT
½n 4.9 ´Í¶ (Banach-Steinhaus) ´½nA~© ½n 4.10 ´'N´y²½n§½n 4.13 ´ Hahn-Banach òÿ½níا ½n 4.11 ±9 4.12 K´éǑ(J© d½n 4.11§·é?Û p ∈ (1, +∞), L (a, b; IR ) ¥k. 7kfÂñf©du©5§ù:Ǒ±d½n 4.10 ©d½n 4.10 ·± L (a, b; IR ) ¥k.7½k f Âñf©ǑÒ´`§XJ f ∈ L (a, b; IR ) ÷v p
∞
∗
n
∞
k
kfk kL∞ ≤ M < +∞,
K3 f f f 9 f ∈ L k
kj
lim
j→∞
Z
∞
b
n
∀ k = 1, 2, · · · ,
§
(a, b; IRn )
fkj (t)g(t) dt =
a
n
Z
b
f (t)g(t) dt
a
é?Û g ∈ L (a, b; IR ) ¤á§´§½n 4.11 w·§(¢ 3X L (a, b; IR ) ¥k.§§vkfÂñf© X QØ´ gmqØ´©m§X ¥k.ÒØUykfÂñ ½f Âñf§+Xd§d½n 4.12§·%±Xe (J: n
DR A
1
∞
n
∗
∗
íØ 4.14. X ´D5m©f ∈ X ´k., K 3 f ∈ X é?Û x ∈ X, 3 f f f ¤áX: k
∗
k
∗
kj
lim fkj (x) = f (x).
j→∞
I5¿´3þ¡íØ¥§fÀ´6u x © 3A^¥§ ù(Ø®²v
© §5.
~©§
5.
~©§
57
(
FT
·y3{/£~©§)Vg9Ù5©Ä± e~©§: y(t) ˙ = f (t, y(t)), t ∈ [0, T ], y(0) = y0 .
·b: (L) N f : [0, T ] × IR 3~ê M > 0
n
→ IRn
|f (t, x) − f (t, y)| ≤ M |x − y|,
(5.1)
ÿ§f (· , 0) ∈ L (0, T ; IR ),
1
∀ t ∈ [0, T ], x, y ∈ IRn .
n
(5.2)
±e´'u)35Ľn© ½n 5.1. ^ (L) ¤á, Ké?Û y ∈ IR , 3 y(·) ∈ C([0, T ]; IR ) ÷v 0
DR A
n
n
y(t) = y0 +
Z
t
f (s, y(s)) ds,
0
∀ t ∈ [0, T ].
(5.3)
§ (5.3) ¡Ǒ (5.1) È©/ª©A/§(5.1) ¡Ǒ (5.3) ©/ª©?Û÷v (5.3) ¼ê y(·) ∈ C([0, T ]; IR ) ¡Ǒ (5.1) )©d (5.3)§· (5.1) )½´ýéëY§ù¿ X y(·) ˙ A??3
Newton-Leibniz úª¤á: n
y(t) = y(0) +
y² E
Z
t
y(s) ˙ ds,
0
t ∈ [0, T ].
SS
. Picard : y0 (t) = y0 , t ∈ [0, T ], Z t y (t) = y + f (s, yk (s)) ds, t ∈ [0, T ], k ≥ 0. k+1 0 0
(5.4)
1Ù O£
58
|yk+1 (t) − yk (t)| ≤ Z
FT
·58B/y² N (M t)k , k!
∀ t ∈ [0, T ], k ≥ 0,
Ù¥ N = |f (s, y )| ds© éu k = 0, ·k T
0
0
|y1 (t) − y0 (t)| ≤
Z
0
t
|f (s, y0 )| ds ≤ N,
l (5.5) éu k = 0 ¤á© b (5.5) é, k ¤á§Kd (5.2),
(5.5)
t ∈ [0, T ].
DR A
Z t |yk+2 (t) − yk+1 (t)| ≤ M |yk+1 (s) − yk (s)| ds 0 Z Z t N M k+1 t k N (M s)k ds = s ds ≤ M k! k! 0 0 N (M t)k+1 = . (k + 1)!
ùÒy² (5.5)©du?ê
X N (M t)k k!
k≥0
3 t ∈ [0, T ] þÂñ, Ïd
yk (t) = y0 (t) +
k h i X yj (t) − yj−1 (t) j=1
3 t ∈ [0, T ] þÂñ, y(·) ∈ C([0, T ]; IR )©u´§3 (5.4) ¥4á= y(·) ÷v (5.3)©ùÒy² (5.1) )35© e¡·5y²5©b x(·) ´,), K n
|x(t) − y(t)| ≤ M
Z
0
t
|x(s) − y(s)| ds.
5.
~©§
59
FT
|^e¡ Gronwall ت§á= x(t) = y(t),
∀ t ∈ [0, T ].
ùÒy² 5©
2
Ún 5.2. (Gronwall ت) α ∈ IR, ϕ(·), ψ(·), β(·) ´ [0, T ] þëY¼ê§ψ(·) K©XJ Z th i ϕ(t) ≤ α + ψ(s)ϕ(s) + β(s) ds, t ∈ [0, T ], 0
K
ϕ(t) ≤ αe
0
ψ(s)ds
+
Z
t
e 0
Rt s
ψ(r)dr
β(s) ds, t ∈ [0, T ].
(5.7)
DR A
y². -
Rt
(5.6)
θ(t) = α +
K
Z th 0
i ψ(s)ϕ(s) + β(s) ds, t ∈ [0, T ],
˙ = ψ(t)ϕ(t) + β(t) ≤ ψ(t)θ(t) + β(t), t ∈ [0, T ]. θ(t) Rt − ψ(r)dr e 0 Rt Rt d − ψ(r)dr − ψ(r)dr 0 e θ(t) ≤ e 0 β(t), t ∈ [0, T ]. dt
ü>Ó±
(5.8)
(5.9)
é (5.9) È© e
−
Rt 0
ψ(r)dr
θ(t) − α ≤
Z
(Ü (5.6),(5.8) = (5.7)©
0
t
e
−
Rs 0
ψ(r)dr
β(s) ds, t ∈ [0, T ].
2
1Ù O£
60
(
FT
e¡§·5we)éëêëY65©ÄXe¹ëê ~©§: y(t) ˙ = f (t, y(t), λ), t ∈ [0, T ], y(0) = y0λ .
(5.10)
Ù¥ f : [0, T ] × IR × (0, 1] → IR , y ∈ IR ©·b: (L) N f : [0, T ] × IR → IR ÿ§3 λ ∈ (0, 1] ' β(·) ∈ L (0, T ; IR ) Ú~ê M > 0, n
n
′
n
1
(
n
n
|f (t, 0, λ)| ≤ β(t),
λ 0
n
∀ t ∈ [0, T ],
|f (t, x, λ) − f (t, y, λ)| ≤ M |x − y|, ∀ t ∈ [0, T ], x, y ∈ IRn . (5.11)
w,§d½n 5.1, 3^ (L) e§é?Û y ∈ IR ,(5.10) ok )§PǑ y (·) ≡ y(·; y , λ)©e¡½nÑ y (·) § (5.1) ) y(·) m'X© ′
λ 0
n
λ
DR A
λ
λ 0
½n 5.3. ^ (L) Ú (L) ¤á©?Ú§b ′
lim
λ→0+
Z
t
f (s, y, λ) ds =
0
Z
t
f (s, y) ds,
0
∀ (t, y) ∈ [0, T ] × IRn .
P y(·), y (·) ©OǑ (5.1) Ú (5.10) )©e
(5.12)
λ
K
lim |y0λ − y0 | = 0,
λ→0+
lim max |y λ (t) − y(t)| = 0.
λ→0+ t∈[0,T ]
Ǒ y²±þ½n§·k5Qã±eÚn©
(5.13)
5.
~©§
61
FT
Ún 5.4. (Arzel`a-Ascoli) F ⊆ C([a, b]; IR ) k.: 3 K > 0, n
|η(t)| ≤ K,
∀ t ∈ [a, b], η ∈ F ,
ÝëY, =é?Û ε > 0, 3 δ > 0, ÷v
|η(t) − η(s)| < ε, ∀ s, t ∈ [a, b], |t − s| < δ, η ∈ F ,
K3 {η (·)} ⊆ F 3 C[0, T ] ¥Âñu, η(·): k
lim max |ηk (t) − η(t)| = 0.
k→∞ t∈[0,T ]
±þÚny²3¼©Ûá¥é© Ún 5.5. (L) Ú(L) ±9 (5.12) ¤á© ζ(·), ϕ(· , λ) : [0, T ] → IR ëY§
′
DR A
n
lim max |ϕ(t, λ) − ζ(t)| = 0,
(5.14)
λ→0+ t∈[0,T ]
K
lim
λ→0+
Z
t
f (s, ϕ(s, λ), λ) ds =
'u t ∈ [0, T ] Âñ© y². Äk§é?Û 0 ≤ t 0
lim
λ→0+
=
=
Z
Z
t
f (s, ζ(s)) ds,
0
1
< t2 ≤ T
±9 y ∈ IR , d (5.12) n
t2
f (s, y, λ) ds
t1
Z t1 Z t2 lim f (s, y, λ) − f (s, y, λ) ds λ→0+ 0 0 Z t1 Z t2 Z t2 f (s, y) − f (s, y) ds = f (s, y) ds. 0
0
t1
1Ù O£
l yéu?ÛF¼ê ϕ(·) k lim+
λ→0
Z
t
FT
62
f (s, ϕ(s), λ) ds =
0
Z
t
f (s, ϕ(s)) ds.
(5.15)
0
e¡§·y² (5.15) 'u t ∈ [0, T ] ´© d (5.11), |f (s, ϕ(s), λ)| ≤ β(s) + M |ϕ(s)|,
s ∈ [0, T ].
ù§d Lebesgue È©ýéëY5, é?Û ε > 0§3 σ > 0 |t − t | < σ ¤áX 1
|
Z
2
t2
t1
À 0 = τ
Z
< τ1 < · · · < τℓ = T ,
t2
f (s, ϕ(s)) ds|
0 é?Û 0 < λ < δ, k |
Z
τi
f (s, ϕ(s), λ) ds−
0
Z
τi
f (s, ϕ(s)) ds|
t ©XÚ (1.1) ¡Ǒ´3 [t , t ] þU§X Jé?Û y , y ∈ IR , Ñk u(·) ∈ U [t , t ], µ (iii)
0
n
p
1
0
1
y(t1 ; t0 , y0 , u(·)) = y1 .
XÚ (1.1) ¡Ǒ´U§XJé?Û 0 ≤ t +∞§XÚÑ´3 [t , t ] þU© (v) XÚ (1.1) ¡Ǒ´3 [t , t ] þK§XJ (iv)
0
1
0
1
0
< t1
0, 3 δ = δ(ε, t ) > 0, |t − t | < δ , IRm
0
0
0
△ Q(t) ⊆ Q(t0 ) + Bε (0) = {u ∈ IRm d(u, Q(t0 )) ≤ ε}.
f : [0, T ] × IR
m
→ IRn
ëY, y(·) : [0, T ] → IR È,
(2.2)
n
△ y(t) ∈ f (t, Q(t)) = {f (t, u) u ∈ Q(t)},
K3ÿ¼ê u(·) ∈ [0, T ] → IR
a.e. t ∈ [0, T ],
m
u(t) ∈ Q(t), f (t, u(t)) = y(t),
a.e. t ∈ [0, T ].
DR A
XJ Q(t) ≡ Q Ø6u t, K§g,´þëY, = (2.2) g,¤á© ½n 2.3 y². du U ´k.8§U [0, T ] = U [0, T ]© V [0, T ] = {v : [0, T ] → co U v(·) ÿ}. w,§U [0, T ] ⊆ V [0, T ]©P 2
△
S(T ) = {y(T ; 0, y0, v(·)) v(·) ∈ V [0, T ]}.
dÚn 2.4, S(T ) ´à;8©g,/§
R(T ) ⊆ S(T ).
(2.3)
·y35y²ª¤á©é?Û ξ ∈ S(T ), k v(·) ∈ V [0, T ] ÷v Z T
ξ = y(T ; 0, y0, v(·)) ≡ Φ(T, 0)y0 +
Φ(T, s)B(s)v(s) ds.
0
1nÙ 5XÚm`
P
FT
86
m n o X Σm+1 = λ = (λ0 , λ1 , · · · , λm ) ∈ IRm+1 λi ≥ 0, λi = 1 ,
¿½Â f : Σ
m+1
Xe:
× U m+1 → IRm
f (λ, u) =
m X
i=0
∀ (λ, u) ∈ Σm+1 × U m+1 .
λi ui ,
i=0
w,§f ´ëY© d Carath´eodory ½n (1Ù§½n 1.5), v(s) ∈ co U = f (Σm+1 × U m+1 ),
a.e. s ∈ [0, T ].
Ïd§dÚn 2.5, 3ÿ¼ê u (·), λ (·) (j = 0, 1, · · · , m) j
uj (s) ∈ U,
λj (s) = 1, v(s) =
j=0
m X
λj (s)uj (s).
j=0
DR A
ù§
m X
j
ξ = Φ(T, 0)y0 +
m Z X j=0
T
λj (s)Φ(T, s)B(s)uj (s) ds.
0
u´§d1Ù½n 3.4, 3ÿ8Ü E
j
⊆ [0, T ]
m [ \ [0, T ] = Ej , Ej Ek = φ, (j 6= k), j=0 m Z T X λj (s)Φ(T, s)B(s)uj (s) ds 0 j=0 m Z X = Φ(T, s)B(s)uj (s) ds. j=0
u(·) =
m X j=0
Ej
χEj (·)uj (·) ∈ U [0, T ],
U8
87
K ξ = Φ(T, 0)y0 +
Z
T
0
Φ(T, s)B(s)u(s) ds ∈ R(T ).
dd= (2.3) ¥ª¤á© Â
FT
2.
2
y3ÄU8ëY5©éu IR ?Ûf8 P, Q, ½ n
△
ρ(P, Q) =
o 1n sup d(p, Q) + sup d(q, P ) , 2 p∈P q∈Q
(2.4)
Ù¥ d(p, Q) = inf |p − q|© ´XJ P ½ Q .§K ρ(P, Q) kU © ·K 2.6. éu IR ¥8 P, Q, R, ¤á q∈Q
n
DR A
(i)
ρ(P, Q) = ρ(Q, P ),
(ii)
ρ(P, Q) = 0, ⇐⇒ P = Q,
(iii)
ρ(P, R) ≤ ρ(P, Q) + ρ(Q, R).
y². (i) ´w,© (ii) d½Â§
ρ(P, Q) = 0
⇐⇒ d(p, Q) = d(q, P ) = 0, ∀ p ∈ P, q ∈ Q, ⇐⇒ P ⊆ Q, ⇐⇒ P = Q.
Q⊆P
(2.5)
1nÙ 5XÚm` (iii)
·k
FT
88
d(p, R) = inf |p − r| r∈R n o ≤ inf |p − q˜| + |˜ q − r| r∈R
= |p − q˜| + d(˜ q , R)
≤ |p − q˜| + sup d(q, R), q∈Q
éþª'u q˜ ∈ Q e(.,
∀ p ∈ IRn , q˜ ∈ Q.
∀ p ∈ IRn .
d(p, R) ≤ d(p, Q) + sup d(q, R), q∈Q
2éþª'u p ∈ P þ(., =k p∈P
(2.6)
q∈Q
DR A
aq/k
sup d(p, R) ≤ sup d(p, Q) + sup d(q, R).
p∈P
sup d(r, P ) ≤ sup d(r, Q) + sup d(q, P ).
r∈R
r∈R
(2.7)
q∈Q
r (2.6) Ú (2.6) \2ر 2 = (2.5).
2
P K Ǒd IR ¥¤k;8¤8x, Kd±þ·K ρ(· , ·) ´ K þÝþ©ùÝþ¡Ǒ Hausdorff Ýþ©, ¡§5¿´ n
△
d(P, Q) =
inf
p∈P,q∈Q
|p − q|,
Ø´Ýþ© ½n 2.7. U ⊂ IR k., K
∀ P, Q ∈ K
m
lim ρ(R(t), R(t¯)) = 0. t→t¯
(2.8)
m`3ÚǑx
89
y². du U k.©Ï k M > 0, é?Û u(·) ∈ U
k
Z t¯ Φ(t, 0) − Φ(t¯, 0) |y0 | + M Φ(t, s) − Φ(t¯, s) |B(s)| ds 0 Z t Φ(t, s) |B(s)| ds +M t¯
△
γ(t, t¯).
DR A
=
[0, T ]
|y(t; 0, y0 , u(·)) − y(t¯; 0, y0 , u(·))| Z t¯ Φ(t, 0) − Φ(t¯, 0) |y0 | + Φ(t, s) − Φ(t¯, s) |B(s)| |u(s)| ds 0 Z t Φ(t, s) |B(s)| |u(s)| ds + t¯
≤
2
ku(·)kL∞ (0,T ;IRm ) ≤ M .
·k ≤
§
FT
3.
ù, é?Û p ∈ R(t), 9 q ∈ R(t¯) Ñk
d(p, R(t¯)) ≤ γ(t, t¯), d(q, R(t)) ≤ γ(t, t¯).
l t → t¯ ,
ùÒy² (2.8)©
ρ(R(t), R(t¯)) ≤ γ(t, t¯) → 0.
§3.
½:
2
m`3ÚǑx
!¥§·ïÄm`¯K©ÄXÚ (1.1), b
1nÙ 5XÚm`
90
FT
Ǒ;8©M : [0, +∞) → 2 ´'u Hausdorff Ýþ ρ ëYõ¼ê§
éz t ∈ [0, +∞), M (t) ´ à;8© du U ´;§¤±8 U [0, +∞) Ñ´Ó§{P Ǒ U [0, +∞)©,¡§dþ!(J§é?Û (t , y ) ∈ [0, +∞) × IR , ±9 t ≥ t , U8 R(t) ≡ R(t; t , y ) ´à;©y 3½ (t , y ) ∈ [0, +∞) × IR , ·b IRn
(L2) U ⊆ IRm
p
0
n
0
0
0
n
0
o [ n \ M (t) R(t; t0 , y0 ) 6= φ.
t≥t0
0
0
(3.1)
þªL«XÚ´l (t , y ) 8I M (·) U©½Â 0
0
J(u(·)) ≡ J(u(·); t0 , y0 ) = inf{t ≥ t0 y(t; t0 , y0 , u(·)) ∈ M (t)}.
= J(u(·); t , y ) ´; y(· ; t , y , u(·)) Äg8I M (·) m©·½ inf φ = +∞©w,§ØÓ u(·) U¬kØÓÄ gÂ¥mJ(t , y ; u(·))©·¯K´: ¯K (TC). éu½ (t , y ) ∈ [0, +∞) × IR , bU5 ^ (3.1) ¤á©Ïé u¯(·) ∈ U [t , +∞) 0
0
0
DR A
0
0
0
0
n
0
0
J(¯ u(·); t0 , y0 ) =
inf
u(·)∈U [t0 ,+∞)
J(u(·); t0 , y0 ).
(3.2)
þã¯K¡Ǒm`¯K© △ t¯ =
inf
u(·)∈U [t0 ,+∞)
J(u(·); t0 , y0 )
¡Ǒ`m. ?Û÷vª (3.2) u¯(·) ∈ U [t , +∞) ¡Ǒ m`©du·'%´ u¯(·) 3 [t , t¯] þ§Ï Ï ~·¤`´ u˜(·) = u¯(·)| ©e¡?Ø (t , y ) ∈ [0, +∞) × IR ½/©Ø5§ t = 0©·òÑ m`ǑxÚ35(J©Äk, kXeÚn: 0
0
[t0 ,t¯ ]
n
0
0
0
3.
m`3ÚǑx
91
FT
Ún 3.1. (L1)±9 (3.1) ¤á, K`m \ t¯ = inf{t ≥ 0 M (t) R(t) 6= φ}.
(3.3)
y². ù(Ø*þ´~²w©§Ù¢´é`m t¯ ,«£ã. P \ t∗ = inf{t ≥ 0 M (t) R(t) 6= φ}.
Äk, e
M (t)
\
R(t) 6= φ,
K`²é, u(·) ∈ U [t , +∞) k 0
J(u(·)) ≤ t.
dd
t¯ ≤ t.
DR A
l
t¯ ≤ t∗ .
,¡, e
0 ≤ t < t∗ ,
Kk u´ l
ùqk
y(t; u(·)) 6∈ M (t),
J(u(·)) ≥ t,
∀ u(·) ∈ U [0, +∞).
∀ u(·) ∈ U [0, +∞), 0 ≤ t < t∗ .
t¯ ≥ t,
∀ 0 ≤ t < t∗ .
t¯ ≥ t∗ .
2
1nÙ 5XÚm`
92
FT
½n 3.2. (L1)—(L2) ±9 (3.1) ¤á, K¯K (TC) 3m`©
`m t¯ ÷v \ t¯ = min{t ≥ 0 M (t) R(t) 6= φ}.
y². dÚn 3.1, 3 t M (tk )
\
k
↓ t¯
(3.4)
÷v
R(tk ) 6= φ.
zk ∈ M (tk )
\
R(tk ),
Kdu M (·) Ǒ;8§
'u Hausdorff ÝþëY§ØJw z ´k.. u´Ø z Âñ. d M (·) 'u Hausdorff ÝþëY5 k
DR A
k
d zk , M (t¯) ≤ 2ρ M (tk ), M (t¯) → 0.
d½n 2.7§R(·) 'u Hausdorff ÝþëY§l qk ùÒ
d zk , R(t¯) ≤ 2ρ R(tk ), R(t¯) → 0, z¯ ≡ lim zk ∈ M (t¯) k→∞
\
R(t¯).
þª(Ü (3.3) = (3.4). Ó, §L²3 u¯(·) ∈ U [0, +∞) u´
y(t¯; u ¯(·)) ∈ M (t¯).
= u¯(·) ´m`©
J(¯ u(·)) ≤ t¯.
2
3.
m`3ÚǑx
93
FT
½n 3.2 L²§3b (L1)—(L2) Ú (3.1) e§`mÚ `½Â¥e(.¯¢þÑ´±. y3§·5Ǒx`mÚ`© ½n 3.3. (L1)—(L2) ±9 (3.1) ¤á§y K(TC) `m, K
0
§ ´¯
6∈ M (0) t¯
h i\h i \ ∂M (t¯) ∂R(t¯) = M (t¯) R(t¯) 6= φ.
(3.5)
AO§ M (·) = {z(·)} ´üëY¼ê§(3.5) ÒC¤ z(t¯) ∈ ∂R(t¯).
DR A
y². ½n3*þ´~²w. t¯ ´ M (t) R(t) Äg Ǒ. du M (·) Ú R(·) Ñ´U Hausdorff ÝþëY! Ǒ488¼ê, g,§Äk¬3>.þ. e¡, ·5 éd\±îØy. Äk, d (3.4) M (t¯)
\
R(t¯) 6= φ.
du M (t) R(t) Ñ´48, Ï
h i\h i \ ∂M (t¯) ∂R(t¯) ⊆ M (t¯) R(t¯).
ù, XJ (3.5) ؤá§K±é: ½
i o R (t¯) ,
(3.6)
i\ o R(t¯). M (t¯)
(3.7)
z¯ ∈ M (t¯)
z¯ ∈
h
\h
1nÙ 5XÚm`
94
M (t˜)
\
FT
Ø (3.6) ¤á. d R(·) 9 M (·) ëY5!à45, */k δ > 0 ±9 t˜ ∈ (0, t¯) Bδ (¯ z ) 6= φ,
(3.8)
Bδ (¯ z ) ⊆ R(t˜).
(3.9)
B2δ (¯ z ) ⊆ R(t¯).
(3.10)
e¡·5y²ù:©d (3.6), 3 δ > 0
d R(·) 9 M (·) ëY5, t˜ ∈ (0, t¯), δ ρ R(t˜), R(t¯) < , 2 δ ρ M (t˜), M (t¯) < . 2
d(¯ z , M (t˜)) ≤ 2ρ M (t˜), M (t¯) < δ.
DR A
u´§
l 3 z ∈ M (t˜) |¯z − z | < δ, = (3.8) ¤á© y3§e (3.9) ؤá, K3 ζ ∈ B (¯z) t˜
t˜
δ
ζ 6∈ R(t˜).
d R(t˜) à45±91Ù·K 1.7§3 y¯ ∈ R(t˜)
(
|ζ − y¯| = d(ζ, R(t˜)),
hζ − y¯, y − y¯i ≤ 0, ∀ y ∈ R(t˜). z =ζ+δ
K
ζ − y¯ , |ζ − y¯|
|z − z¯| ≤ |z − ζ| + |ζ − z¯| < δ + δ = 2δ,
(3.11)
(3.12)
m`3ÚǑx
95
FT
3.
δ |z − y¯| = |ζ − y¯| 1 + > δ, |ζ − y¯| hz − y¯, y − y¯i ≤ 0, ∀ y ∈ R(t˜).
l d1Ù·K 1.7,
d(z, R(t˜)) = |z − y¯| > δ.
(3.13)
,¡, d (3.10) ±9 (3.12) , z ∈ B
z) 2δ (¯
©u´
⊆ R(t¯)
d(z, R(t˜)) ≤ 2ρ(R(t¯), R(t˜)) < δ.
ù (3.13) gñ©Ïd§(3.9) ¤á©d (3.8) 9 (3.9) M (t˜)
\
R(t˜) 6= φ.
DR A
5¿ 0 < t˜ < t¯, Ï t¯ Ø´`m©bgñ©Ïd (3.5) ¤á© 2 ½n 3.4. (L1)—(L2) 9 (3.1) ¤á§y m ´±e¼ê3 [0, +∞) þ":© t¯
F (t) = inf
|λ|=1
n
max hλ, Φ(t, 0)y0 −zi+
z∈M(t)
?Ú§XJ |λ | = 1 ÷v
Z
t
0
6∈ M (0),
K`
o maxhλ, Φ(t, s)B(s)ui ds .
0 u∈U
(3.14)
0
max hλ0 , Φ(t¯, 0)y0 − zi +
z∈M(t¯ )
Z
t¯
maxhλ0 , Φ(t¯, s)B(s)ui ds = 0, (3.15)
0 u∈U
K` u¯(·) ÷v±e^:
maxhλ0 , Φ(t¯, s)B(s)ui = hλ0 , Φ(t¯, s)B(s)¯ u(s)i, u∈U
a.e. s ∈ [0, t¯],
(3.16)
1nÙ 5XÚm`
y¯ ≡ y(t¯; u¯(·)) ÷vXeî^: hλ0 , z − y¯i ≥ 0,
FT
96
∀ z ∈ M (t¯).
(3.17)
y². Äk·5w (3.14) mà¹Â©|^ Fillipov Ún§ ± uˆ(·) ∈ U [0, +∞) ÷v maxhλ, Φ(t, s)B(s)ui = hλ, Φ(t, s)B(s)ˆ u(s)i, a.e. s ∈ [0, t]. u∈U
dd Z
t
maxhλ, Φ(t, s)B(s)ui ds =
0 u∈U
u(·)∈U [0,t]
u´
Z
t
0
hλ, Φ(t, s)B(s)u(s)i ds.
DR A
F (t)
max
=
=
= =
inf
|λ|=1
n
max hλ, Φ(t, 0)y0 − zi
z∈M(t)
Z t o + max hλ, Φ(t, s)B(s)u(s)i ds u(·)∈U [0,t] 0 n D inf max λ, Φ(t, 0)y0
|λ|=1
z∈M(t),u(·)∈U [0,t]
+ n
Z
t
Φ(t, s)B(s)u(s) ds − z 0 o inf max hλ, y − zi |λ|=1 z∈M(t),y∈R(t) n o inf max hλ, yi . |λ|=1
Eo
y∈R(t)−M(t)
ù§ 0 6∈ R(t) − M (t) §d1Ù½n 1.8§3 λ ∈ IR §|λ| = 1§±9 c = c < 0 n
t
0 > c ≥ hλ, yi,
∀ y ∈ R(t) − M (t).
3.
m`3ÚǑx
97
FT
l d F (t) ≤ c < 0©
0 ∈ R(t) − M (t) §é?Û λ ∈ IR , |λ| = 1§ n
max
hλ, yi ≥ 0.
y∈R(t)−M(t)
l d F (t) ≥ 0© d½n 3.2 `3©éu`m t¯, ·k R(t¯)
\
M (t¯) 6= φ.
½= 0 ∈ R(t¯) − M (t¯)©u´þ¡?ØL² (
F (t) < 0, F (t¯) ≥ 0.
∀ t ∈ [0, t¯),
(3.18)
DR A
,¡§´¼ê F (·) ´ëY©ù§d (3.18) F (t¯) = 0.
= t¯ ´ F (·) 3 [0, +∞) þ":© y3 λ ∈ IR , |λ | = 1 ÷v (3.15)©P`Ǒ u¯(·), K 0
n
0
y¯ ≡ y(t¯; u ¯(·)) = Φ(t¯, 0)y0 +
u´|^ (3.15)
Z
t¯
Φ(t¯, s)B(s)¯ u(s) ds ∈ M (t¯)
0
0 = max hλ0 , Φ(t¯, 0)y0 − zi + z∈M(t¯ )
≥ hλ0 , Φ(t¯, 0)y0 − y¯i + = 0.
Z
0
t¯
Z
\
R(t¯).
t¯
maxhλ0 , Φ(t¯, s)B(s)ui ds
0 u∈U
hλ0 , Φ(t¯, s)B(s)¯ u(s)i ds
1nÙ 5XÚm`
ùL²
FT
98
max hλ0 , Φ(t¯, 0)y0 − zi = hλ0 , Φ(t¯, 0)y0 − y¯i,
z∈M(t¯ )
Z t¯ n 0
(3.19)
o maxhλ0 , Φ(t¯, s)B(s)ui − hλ0 , Φ(t¯, s)B(s)¯ u(s)i ds = 0. (3.20) u∈U
du (3.20) ¥È¼ê´K§·Ò^ (3.16)©
d (3.19) î^ (3.17) ¤á© 2 XJ·P ψ(t) = Φ(t¯, t)⊤ λ0 ,
(
DR A
K ψ(·) ÷v
t ∈ [0, ¯t ],
˙ ψ(t) = −A(t)⊤ ψ(t), ψ(t¯) = λ0 .
t ∈ [0, t¯],
(3.21)
ùXÚ¡Ǒ (1.1) ݧ©|^¼ê ψ(·), ·±ò ^ (3.16) ¤ maxhψ(t), B(t)ui = hψ(t), B(t)¯ u(t)i, a.e. t ∈ [0, ¯t ]. u∈U
(3.22)
î^ (3.17) zǑ
hψ(t¯), z − y¯i ≥ 0,
∀ z ∈ M (t¯).
(3.23)
ù§·Ò: ½n 3.5. (n) (L2) ¤á, u¯(·) ´¯K (TC) `§t¯ > 0 ´`m, K3 (3.21) ") ψ(·) ^ (3.22) Úî^ (3.23) ¤á©
3.
m`3ÚǑx
99
u ¯(t) ∈ ∂U,
FT
½n 3.6. (bang-bang n) (L2) Ú (3.1) ¤á, K3 ` u¯(·) ∈ U [0, ¯t ] a.e. t ∈ [0, t¯].
(3.24)
U = [0, 1] , (3.24) ¿X u ¯(t) = 0 ½ 1, a.e. t ∈ [0, t¯]. ùÒ´`§u¯(t) 0 ½ 1©Ônþ§ùÏ~L«ì±´ {üm';/m0½/'0¤ 1Ǒ©ùÒ´Ǒo·¡ ½n 3.6 Ǒ bang-bang nÏ©8 §·¡÷v (3.24) Ǒ bang-bang © y². P m {w(·) ∈ L∞ loc [0, +∞; IR ) w(t) ∈ ∂U, a.e. }, m {v(·) ∈ L∞ loc [0, +∞; IR ) v(t) ∈ co(∂U ), a.e. }.
DR A
W [0, +∞) = V [0, +∞) =
du
∂U ⊆ U ⊆ co(∂U ) = co U,
·k
△
Q(t) =
⊆
n o y(t; w(·)) w(·) ∈ W [0, +∞) ⊆ R(t) n o △ S(t) = y(t; v(·)) v(·) ∈ V [0, +∞) .
l½n 2.3 y² l
Q(t) = S(t),
Q(t) = R(t),
t ≥ 0.
∀ t ≥ 0.
1nÙ 5XÚm`
100
FT
ùéu` u¯(·) ∈ U [0, ¯t ], 3 w(·) ¯ ∈ W [0, t¯], y(t¯; u ¯(·)) = y(t¯; w(·)). ¯
w,§w(·) ¯ ´ bang-bang §§´`©
2
3þ¡?Ø¥§©G´½©©G#N3 k.à48¥Cħ·±ïáA(J©
½n 3.7. (L2) ¤á, Q, M (t) (t ∈ [0, +∞)) ´ IR ¥ à48©b t¯ > 0 ´òGl Q =£ M (·) `m§= n
t¯ =
inf{t ≥ 0 y(t; y0 , u(·)) ∈ M (t), y0 ∈ Q, u(·) ∈ U [0, +∞) }.
q u¯(·) ´`§y¯ ∈ Q Ú y¯ ∈ M (t¯) ©O´A`; 3 0 ǑÚ t¯ ǑG, K3 (3.21) ") ψ(·) ^ (3.22)§'uªàî^ (3.23) Ú±e'u©G î^¤á©
DR A
0
hψ(0), y0 − y¯0 i ≤ 0,
∀ y0 ∈ Q.
(3.25)
y². r
y˜(t) = y(t; y0 ) − Φ(t, 0)y0 ,
t>0
w#GCþ§
△ f(t) = M M (t) − Φ(t, 0)Q = {z − Φ(t, 0)y0 z ∈ M (t), y0 ∈ Q}
w#8I, K´ t¯ Ú u¯(·) Ò´A©G½Ǒ 0 m`¯K`mÚ`© AG;3 t¯ ªàGǑ y¯ − Φ(t¯, 0)¯y ©N´y½n 3.5 ^÷v, l 0
3.
m`3ÚǑx
101
FT
k (3.21) ") ψ(·) ^ (3.22) ÷v§ 'uªà î^CǑ hψ(t¯), z − Φ(t¯, 0)y0 − (¯ y − Φ(t¯, 0)¯ y0 )i ≥ 0,
3þª¥ y l
0
= y¯0
∀ z ∈ M (t¯), y0 ∈ Q.
= (3.23)© z = y¯ ∈ M (t¯) K
hψ(t¯), Φ(t¯, 0)y0 − Φ(t¯, 0)¯ y0 i ≤ 0,
∀ y0 ∈ Q.
hψ(0), y0 − y¯0 i = h(Φ(t¯, 0))⊤ ψ(t¯), y0 − y¯0 i
= hψ(t¯), Φ(t¯, 0)y0 − Φ(t¯, 0)¯ y0 i ≤ 0,
∀ y0 ∈ Q.
ù§Ò'u`©Gî^ (3.25)©
DR A
~ 3.1. H > 0, ?ØrXÚ
2
d2 x = u(t), dt2
|u(t)| ≤ 1,
(3.26)
lG (−H, 0) ¯/=£G (0, 0) m`©
·^n5©Ûù¯K©P y = dxdt , ·k x˙ y˙
(3.27)
)
=
ݧ´
ϕ˙ ψ˙
0 1 0 0
=−
ϕ(t) = C1 ,
x y
0 0 1 0
+
0
1
ϕ
ψ
u(t).
.
ψ(t) = C2 − C1 t,
(3.27)
1nÙ 5XÚm`
102
1
FT
Ù¥ C , C Ǒ~ê© u¯(·) Ǒ¤m`©d n§3ØǑ"~ê C , C 2
1
2
h i max (C2 − C1 t)u = (C2 − C1 t)¯ u(t),
u∈[−1,1]
l
a.e. [0, ¯t ].
a.e. [0, t¯].
u ¯(t) = sgn (C2 − C1 t),
du (C − C t) õk":§Ïd§3A??¿Âe§ ` u¯(·) Ǒ 1 Ú −1§
§õUCgÎÒ© XJ`3; (0, 0) ãǑ 1, G ;Ǒ √ 2
1
L1 :
y = − 2x,
x ≥ 0,
DR A
r´ y O\© XJ`3; (0, 0) ãǑ −1, KG ;Ǒ √ L2 :
y=
−2x,
x ≤ 0,
r´ y ~©Ïd§`;;7½ L ½ L ©bX u¯(·) 3m©Ǒ −1§KG3ùã;Ǒ 1
L3 :
y=−
2
p −2(x + H)
ã©5¿ L ØU L ½ L §`7,3m ©Ǒ 1©d§ 3
1
L4 :
y=
2
p 2(x + H).
L L §u¯(·) UCǑ −1©dd±Ñ: ( √ 1, 0 ≤ t < H, √ √ u ¯(t) = −1, H ≤ t ≤ 2 H, 4
2
(3.28)
3.
m`3ÚǑx
103
FT
ùp·b½©ǑǑ 0© `m´ √ t¯ = 2 H.
ù´·31Ù¥(J© y
L2
x
(0, 0)
L4
−H L3
L1
ã 3.1
DR A
/§-
v(x, y) =
KXÚ
(
1,
−1,
(x, y)3 L , L e½ L þ, (x, y)3 L , L þ½ L þ, x˙ y˙
=
1
2
1
1
2
2
y
v(x, y)
(3.29)
;Ñ´XÚ (3.26) ¯ (0, 0) `;©ùpÖö 5¿´§+3êÆþ§(3.29) (3.28) ÓÑ´` ©´ (3.29) äkG"/ª, 3¢SA^¥òǑk^©A O§|^ (3.29) O`¬äk|Z65©3ù~K¥§ ·|^ém¯K (`¯K) ïÄ(J 4¯K ("`¯K) )§Ǒ´~k¿Â© §4.
m`5
1nÙ 5XÚm`
104
FT
3ù!¥§·Äm`5© ±eØ CXÚ: ( y(t) ˙ = Ay(t) + Bu(t),
y(0) = y0 ,
Ù¥ A ∈ IR õ¡N:
n×n
, B ∈ IRn×m
t ≥ 0,
(4.1)
©?Ú§b U ´ IR ¥à m
U = {u ∈ IRm hλi , ui ≤ ci , 1 ≤ i ≤ k},
Ù¥ λ ∈ IR , c ∈ IR (1 ≤ i ≤ k) ´½© ½Â 4.1. U ´ IR ¥k.àõ¡N, A ∈ IR , B ∈ IR ©XJéu²1u U ,^ "þ w ∈ IR § þ| i
m
i
m
n×n
n×m
m
Bw, ABw, · · · , An−1 Bw
DR A
o´5 '§Ò¡ U A, B ?u2 © m = 1, U = [−1, 1], B ∈ IR §U A, B ?u2
= n
B, AB, · · · , An−1 B
´5 '§=XÚ (4.1) ´U©e¡½nL²§3
ã^e§ ψ(·) (½§^ (3.16) /(½ : ½n 4.2. U A, B ?u2 ©ψ(·) 6= 0 ´XÚ (4.1) ݧ ˙ ψ(t) = −A⊤ ψ(t),
t ∈ [0, T ]
), K3 [0, T ] þØ k: §^
hψ(t), Bu(t)i = maxhψ(t), Bvi, t ∈ [0, T ] v∈U
(4.2)
4.
m`5
105
1
k
FT
/(½ u(t)§¿
u(·) 3 [0, T ] þ´u U º: Åã~¼ê© y². éu½ t ∈ [0, T ], hψ(t), Bvi ´ v ∈ U 5¼ê© XJ§3 U þØ´~§K=3 U >.þ©Ó n§±?Úy² hψ(t), Bvi ½ö=3 U ,º: §½ö3 U ,^ þ~© ·Äk5y²§Øk:± §d (4.2) (½ u(t) ´ © ÄK§3 ØÓ t , t , · · · ∈ [0, T ] t = t (k = 1, 2, · · ·), d (4.2) (½ u(t ) Ø´©d§hψ(t ), Bvi 73 U ,^ þ~© u , u ´T üº:©P w =u −u , K 2
k
k1
k
k2
k1
k
k2
DR A
hψ(tk ), Bwk i = hψ(tk ), Buk2 i − hψ(tk ), Buk1 i = 0.
5¿ U kk^ §Ï k k éA Ó þ w ≡ w ©=: j
kj
hψ(tkj ), Bwi = 0.
(4.3)
du ψ(·) ´~Xê5©§|)§¤± hψ(t), Bwi ´ Cþ t )Û¼ê©d)Û¼ê":á5½n9 (4.3) : hψ(t), Bwi ≡ 0,
∀ t ∈ [0, T ].
éþª'u t n − 1 ê§= hψ(t), Bwi ≡ 0,
hψ(t), ABwi ≡ 0, ············
hψ(t), An−1 Bwi ≡ 0,
∀ t ∈ [0, T ].
1nÙ 5XÚm`
106 n−1
Bw
5 '§l
ψ(t) ≡ 0,
FT
db§Bw, ABw, · · · , A
∀ t ∈ [0, 1].
ù ψ(·) 6= 0 gñ©ùÒy² Øk:± §(4.2) / (½ u(t)©
§ù (½ u(t) ½´ U º:© e¡§·5y² u(·) ´Åã~© tˆ < tˆ < · · · < tˆ ∈ [0, T ] ´ ØUd (4.2) /(½ u(t) t©? J = (tˆ , tˆ ) (1 ≤ k ≤ N − 1)© e , e , · · · , e Ǒ U ܺ:©P 1
2
N
k
k+1
1
2
Ej = {t ∈ J|hψ(t), Bej i > hψ(t), Bek i, ∀ k 6= j}.
dué t ∈ J §(4.2) /(½ u(t) ∈ {e , e , · · · , e }§l 1
j=1
2
(4.4)
q
Ej .
DR A
J=
q [
q
,¡§é?Û k, j,
hψ(t), Bej i − hψ(t), Bek i
'u t ´ëY©Ïd§éz j = 1, 2, · · · , q, E ´m8©5¿ E üüاd (4.4) ±9 J ´ëÏm8§ E ¥k
k 2 ´©ùÒ´`§3 J þ§u(·) ~© j
j
j
þã½nL²§ U Ú A, B ?u2 §÷v^ §AO´`§½´Åã~, ÙǑk.à õ¡Nº:§
UCgêǑkg© I5¿´§3þã½n¥§¤¢ (4.2) /(½ u(t)(Ø k:) ´3 ψ(·) ½ Je©Ï ØUdd `5©e¡½nL² 3½^e§m` ´©
4.
m`5
107 m
0
0
1
1
Z t¯ ¯ ¯ e(t−s)A B u ¯(s) ds, y1 = etA y0 + Z0 t¯ ¯ y1 = et¯A y0 + e(t−s)A B¯ v (s) ds. 0
Ïd§
Z
FT
½n 4.3. U ´ IR ¥k.àõ¡N§U A, B ? u2 , KXÚ (4.1) lG y ¯£G y m` ´© y². u¯(·), v¯(·) ∈ U [0, t¯] Ñ´^`m t¯ > 0 òXÚ (4.1) lG y =£ y `, l
t¯
¯
e(t−s)A B u ¯(s) ds =
Z
t¯
¯
e(t−s)A B¯ v (s) ds.
du u¯(·) ´`§Ï §d½n 3.5, 3 λ (4.1) ݧ 0
0
6= 0
±9XÚ
DR A
0
(4.5)
(
) ψ(·) 5¿
˙ ψ(t) = −A⊤ ψ(t), ψ(t¯) = λ0 ,
t ∈ [0, t¯],
hψ(t), B u ¯(t)i = maxhψ(t), Bvi, a.e. t ∈ [0, t¯]. v∈U
¯
(Ü (4.5) Z
0
=
Z
0
=
Z
⊤
ψ(t) = e(t−t)A λ0 ,
t¯
hψ(t), B¯ v (t)i dt =
t¯
¯
t ∈ [0, ¯t ],
Z
0
t¯
hλ0 , e(t−t)A B u ¯(t)i dt =
t¯
maxhψ(t), Bvi dt.
0 v∈U
¯
hλ0 , e(t−t)A B¯ v (t)i dt Z
0
t¯
hψ(t), B u ¯(t)i dt
(4.6)
1nÙ 5XÚm`
Ïd
FT
108
hψ(t), B¯ v (t)i = maxhψ(t), Bvi v∈U
= hψ(t), B u ¯(t)i,
u´§d (4.7) ±9½n 4.2
a.e. t ∈ [0, t¯].
(4.7)
a.e. t ∈ [0, t¯].
u¯(t) = v¯(t),
ùÒy² m`5©
2
½Â 4.4. ψ(·) ´§ (4.1) ݧ (4.6) ")©¡ ÷v^ hψ(t), Bu(t)i = maxhψ(t), Bvi v∈U
(4.8)
DR A
u(·) ∈ U Ǒ4© ·®²`´4©3r^e§·Ǒ ±y²4´`© ½n 4.5. U ´ IR ¥k.àõ¡N§U A, B ? u2 ,
0 ∈U , KXÚ (4.1) lG y =£G 0 4 ´©AO§ù47,´`© y². u(·), u˜(·) ´òXÚl y £ 0 ü4, K 3 T, S ∈ [0, +∞)(Ø T ≥ S) : m
o
0
0
0 = 0 =
eT A y 0 + eSA y0 +
Z
Z
T
e(T −t)A Bu(t) dt,
0 S
0
e(S−t)A B u ˜(t) dt.
m`5
l −y0 =
Z
T
109
FT
4.
Z
e−tA Bu(t) dt =
S
e−tA B u ˜(t) dt.
(4.9)
ψ(·) ´§ (4.6) ²
)§§ u(·) 3 [0, T ] þ÷v ^ (4.8), Kd (4.9), y: Z
Z
Z
Z
hψ(t), B u ˜(t)i dt = T
hψ(0), e
0
=
0
S
0
=
0
−tA
0
du 0 ∈ U , Ï
Z
T
0
maxhψ(t), Bvi dt.
hψ(t), Bu(t)i dt
(4.10)
v∈U
maxhψ(t), Bvi ≥ 0, v∈U
ddd (4.10) 9 T ≥ S
Z
DR A
Z
hψ(0), e−tA B u ˜(t)i dt
Bu(t)i dt =
T
0
S
u´
0
S
hψ(t), B u ˜(t)i dt ≥
S
0
maxhψ(t), Bvi dt. v∈U
hψ(t), B u ˜(t)i = maxhψ(t), Bvi, v∈U
d½n 4.2
u(t) = u ˜(t),
a.e. t ∈ [0, S].
a.e. t ∈ [0, S].
(4.11)
þª(Ü (4.10, ¿5¿ u(·) Ú ψ(·) ÷v^ (4.8), hψ(t), Bu(t)i = maxhψ(t), Bvi = 0, v∈U
a.e. t ∈ [S, T ].
(4.12)
·äó T = S. ÄK T > S. d½n 4.2, (4.12) ªA?? (½ u(·) 3 [S, T ] þ,
ÙǑ U >.:©d5 u(t) = 0,
a.e. t ∈ [S, T ].
1nÙ 5XÚm`
110
FT
0 ∈U , ù u(·) Ǒ U >.:gñ©Ï (4.12) %º
T = S ©d§·y² T = S ±9 u(·) = u ˜(·)© 2 o
5P 1.
3=©©z¥§U8'kü µ/attainable set0Ú/reachable set0©Ï~, öÙ¥0Vg§ öǑ [
△
ep (T ; t0 , y0 ) = R
Rp (t; t0 , y0 ).
t∈[t0 ,T ]
éu Re (T ; t , y ), Au·K 2.1!½n 2.2 (JØý© 2. d¤k;|¤8Ü p
0
0
{y (·; t0 , y0 , u(·))|u(·) ∈ U p }
DR A
¿Ø½´à©~XÄ U = {−1, 1} ±9XÚ y˙ (t) = u(t),
t ∈ [0, T ],
K y(t) ≡ −t Ú y(t) ≡ t ´Au u ≡ −1 Ú u ≡ 1 ü^G;© ´§à|Ü y(t) ≡ 0 ¿Ø´XÚ^G;© 3. XJ« U ´k.8§Kéu?Û p ∈ [1, +∞], U [t , T ] Ñ´ Ó§l U8 R (T ) Ǒ´Ó© U .§¹q¬XÛ? d §=éuØC5XÚ§R (T ) Ǒ(¢±X p Cz C z©we~µ T = π/4, 1 ≤ r ≤ +∞©ÄXÚ p
p
p
Ù¥
y˙ (t) = Ay (t) + u(t),
0 ≤ t ≤ T,
y (0) = 0,
A=
U=
ϕ(s) s
0
1
−1
0
,
s ∈ [1, +∞) ,
0
5P
ùp ϕ(s) =
ù y (T ; u(·))
=
1/ ln(2s), 2/[(r − 1)sr−1 ], 3 exp(−s),
Z
π/4
0
Z
=
π/4
0
FT
111
XJ r = 1, XJ 1 < r < +∞, XJ r = +∞.
cos(π/4 − t)
sin(π/4 − t)
− sin(π/4 − t) cos t
sin t
− sin t
cos t
éu½ t ∈ (0, π/4], ØJy
cos(π/4 − t)
u(t) dt
u(π/4 − t) dt.
ϕ′ (1) cos t + sin t ≤ [ϕ′ (1) + 1] cos t < 0,
±9
lim
ϕ(s) cos t + s sin t
= +∞.
ù ϕ(s) cos t + s sin t 3 [1, +∞) þ½3 (1, +∞) S,: s §
DR A
s→+∞
ϕ′ (s) = − tan t.
du ϕ (s) > 0, þã§k) s = s(t)© ′′
u ¯(t) =
K ?Ú§ØJy²
ϕ(s(π/4 − t))
u ¯(t) ∈ U,
u ¯(·) ∈ U q ⇐⇒
±9 u¯(·) ´§
0
=
min
u∈U
∀ t ∈ (0, π/4].
XJ r = 1, XJ 1 < r ≤ +∞,
q = 1,
1 ≤ q < r,
1
∀ t ∈ (0, π/4],
,
s(π/4 − t)
,
cos t
sin t
− sin t
cos t
1 0
,
u ¯(π/4 − t)
cos t
sin t
− sin t
cos t
u
,
1nÙ 5XÚm`
)§½d/§ u ¯(·) ´ 1
, y (T ; u ¯(·))
0
=
)©ùL²éu u(·) ∈ U
FT
112
1
min
u(·)∈U 1 1,
0
, y (T ; u(·))
y (T ; u(·)) = y (T ; u ¯(·)) ⇐⇒ u(·) = u ¯(·),
l ,¡§´
,
a.e. [0, T ].
y (T ; u ¯(·)) ∈ Rq ⇐⇒ u ¯(·) ∈ U q .
XJ r = 1, XJ 1 < r ≤ +∞. Ïd r = 1 §XJ q = 1 < p ≤ +∞§½ 1 < r ≤ +∞ §XJ 1 ≤ q < r ≤ p ≤ +∞§·B½k R 6= R © 4. XJéu q ∈ [1, +∞], 3 q NC R u)Cz§(/ù§XJ R 6= R , ∀ p 6= q, XJ q = 1, +∞, XJ q ∈ (1, +∞), R 6= R , ∀ 1 ≤ r < q < s ≤ +∞, K¡ q ´.ê© ¡~fL²éu5½~XÚ§[1, +∞] ¥ ?ÛÑU¤Ǒ.꧴,¡§3 [33] ¥§·y ² éu5½~XÚ (A, B þǑ 2 × 2 Ý §U Ǒ IR f8) u ¯(·) ∈ U q ⇐⇒
q = 1,
1 ≤ q < r,
q
p
p
p
q s
DR A
r
2
y˙ (t) = Ay (t) + Bu(t),
0 ≤ t ≤ T,
y (0) = y0 ,
.êêجL 12 ©AO B ´ÛÉ (= B u 2) §.êêǑ"§ǑÒ´`d Ø U ´o§¤k R Ñ ´Ó© A äkéE a ± bi (b > 0)§T > §.ê êǑǑ"© p
π b
SK
1.
0
1
0
0 . A = .. 0
0 .. .
1 .. .
0
0
0
0
0
©y² (1.5) ¤á©
Bn 6= 0
··· ··· .. . ··· ···
0
0 .. .
1 0
B 1
,
B2 B= . .. Bn
n×n
n×m
,
SK
Ä IR ¥XÚ: 2
x˙ (t) = y (t)
y˙ (t) = −x(t) + u(t).
TXÚ´ÄUºǑoº 3. éu±eXÚ:
FT
2.
113
x˙ (t) = y˙ (t) = u(t), x(0) = y (0) = 0,
Áé t ∈ [0, 4] (½U8 R(t)© 4. Ä
U = {−1, 0, 1}.
y˙ (t) = A(t)y (t) + b(t, u(t)),
y (t0 ) = y0 ,
t ∈ [t0 , T ],
Ù¥ U ´ IR ¥k.48§A(·) : [t , T ] → IR È©éu?Û t ∈ [t , T ], b(t, ·) : IR → IR ëY©é?Û u ∈ U , b(·, u) : [t , T ] → IR È©
é?Ûk.8 K ⊂ IR , 3È µ(·) : [t , T ] → IR m
0
m
0
n
n×n
0
k
n
0
DR A
|b(t, u)| ≤ µ(t).
ÁéþãXÚy²AU8´à;8© 5. ÁE~f½Â 1.2 ¥§(i) =⇒ (ii) =⇒ (iv) Øý© 6. y²·K 1.4© 7. ¡XÚ (1.1) 3 [t , +∞) þU§XJé?Û y , y t ≥ t ±9 u(·) ∈ U [t , +∞) 0
1
0
p
0
1
∈ IRn ,
3
0
y (t1 ; t0 , y0 , u(·)) = y1 .
ÁÞ~`²éu½~XÚ§Uþã½ÂU5ØUíÑÏ ~¿ÂeU5© (ii) XJüpØ'XÚU½Â 1.2 ´U§Kòùü XÚÜå5wXÚ§TXÚE´U©Þ~`² ù(Øé·ff½ÂU5Øý© 8. ?ØrXÚ d y (i)
2
+ y = u(t), dt2 |u(t)| ≤ 1,
l (y , y˙ ) =£ (0, 0) m`© 0
0
9.
éuXÚ
?د¥8I8 10. éuXÚ
d2 y = u(t), dt2 y=0
FT
1nÙ 5XÚm`
114
|u(t)| ≤ 1,
m`©
dx = y, dt
?د¥8I8 m`¯K© 11. éuXÚ
dy = −x + u(t), dt |u(t)| ≤ 1,
x2 + y 2 ≤ 1
x˙ = y + u(t),
y˙ = −x + v(t),
|u(t)| ≤ 1,
|v(t)| ≤ 1.
DR A
?ØòG¯=£ (0, 0) m`¯K©
§1.
FT
1oÙ 5XÚ`35 ¼êz
·Äk5w¼êz¯K©éù¯K?n¹
g©w±e~K© ~ 1.1. J : [0, 1] → IR ëY, K¯¤±3 u¯ ∈ [0, 1] J(¯ u) = inf J(u).
(1.1)
u∈[0,1]
k
∈ [0, 1],
DR A
éd§±y²Xe: de(.½Â§3 u ¡Ǒ4zS, lim J(uk ) = inf J(u).
k→∞
u∈[0,1]
du [0, 1] ´;§Ï 3 {u } f {u } ±9, u¯ ∈ [0, 1]§ k
ki
lim uki = u ¯.
i→∞
?Ú§d J(·) ëY5
J(¯ u) = lim J(uki ) = lim J(uk ) = inf J(u), i→∞
ùÒy² (1.1)©
k→∞
u∈[0,1]
(1.2) 2
[* þãy²§·±uy§(ؤá§3 (1.2) ¥1ª´±eت¤á=: J(¯ u) ≤ lim J(uki ). i→∞
1oÙ 5XÚ`35
116
1
FT
Ïd§·±òþã(Jí2Ǒ/©Ǒd§·Ú \XeVg© ½Â 1.1. U ´Ýþm © (i) ¡N J : U → IR ≡ [−∞, +∞] ´eëY({PǑ l.s.c.)§XJé?Û r ∈ IR, 8Ü {u ∈ U J(u) ≤ r} ´4©·½  J ½ÂXe: △ D(J) = {u ∈ U |J(u)| < +∞}.
(1.3)
DR A
D(J) 6= φ , ·¡ J(·) ´~© (ii) ¡N J : U → IR ≡ [−∞, +∞] ´þëY({PǑ u.s.c.)§XJ −J ´eëY§½d/§é?Û r ∈ IR, 8Ü {u ∈ U J(u) ≥ r} ´4©éuþëY¼ê J §Ù½ÂÓ^ (1.3) ½Â©aq/§ D(J) 6= φ , ¡ J(·) ´~© N´y²e¡{ü¯¢©·ïÆÖög1y²© ·K 1.2. U ǑÝþm, KN J : U → IR ëY
=§Q´þëYq´eëY© ±þ·K`²ëY¼ê7½´þëYÚeëY©e~ L²Ø,©T~¥¼ê9Ùãéu«©þëYÚeë YVgǑ´~kÏ© ~ 1.2. U = [−1,(1]©½Â △
J(u) =
0,
u ∈ [−1, 1] \ {0},
−1, u = 0.
¯¢þ§3ÿÀm¥Ò±½Â¼êþëY5½eëY5©Ö ¥§·ÌÄ´Ýþm§Ï ·½Â´9 Ýþm/©XJÖ öéÝþm£ØÙG§±r U w¤ IR f8© 1
m
¼êz
117
△ ˆ J(u) =
(
FT
1.
0, u ∈ [−1, 1] \ {0}, 1, u = 0.
ˆ ´þëY§§ÑØ´ N´y J(·) ´eëY§J(·) ëY©
du?Û Banach m, ) IR , Ñ´Ýþm§Ï ½Â 1.1 ®ºX ·a,/©5¿´3½Â 1.1 ¥§· #NeëY¼ê ±∞ © n
2
~ 1.3. U ´Ýþm§U IU0 (u) =
(
0
⊆U
0,
´4f8©½Â
u ∈ U0 ,
(1.4)
+∞, u ∈ U \ U0 ,
DR A
K I (·) ´eëY©?Ú§XJ U ´~§= U , K I (·) ´~§
U0
0
0
6= φ, U ,
U0
·¡ I
U0 (·)
D(IU0 ) = U0 .
Ǒ8Ü U «5¼ê© 0
kÿ§±e'ueëYd½ÂǑB©
·K 1.3. U ´Ýþm, K J : U → IR ´eëY
=é?Û u → u¯, ¤áX k
J(¯ u) ≤ lim J(uk ).
(1.5)
k→∞
Ï~3A^¥§·ÄØ −∞ eëY¼ê (ÚØ +∞ þëY ¼ê)©kÿ§ lim J(uk ). k→∞
3þª¥à±´ +∞, mà±´ −∞, ´3 ℓ ∈ IR k→∞
DR A
u´§3 N
J(¯ u) > ℓ > lim J(uk ).
J(¯ u) > ℓ ≥ J(uk ),
∀ k ≥ N.
Ïd {u ∈ U J(u) ≤ l} Ø´4©ù J(·) eëY5gñ. l (1.5) ¤á© 2 e¡(JL²eëY53, $eE,±3© ·K 1.4. U ǑÝþm§J : U → IR ´xeëY¼ ê§α ∈ A©(i) ¼ê sup J (·) ´eëY© (ii) XJ A ´k8§K min J (·) ´eëY© X X (iii) XJ A ´k8§
J (·) k¿Â§K J (·) ´ eëY© α
α∈A
α
α∈A
α
α
α∈A
α
α∈A
¼êz
119
y². (i) é?Û r ∈ IR, ·k
FT
1.
\ {u ∈ U sup Jα (u) ≤ r} = {u ∈ U Jα (u) ≤ r}. α∈A
α∈A
deëY5§màz8ÜÑ´4§l §8Ǒ´4 8©ùÒ sup J (·) eëY5© (ii) é?Û r ∈ IR, ·k α∈A
α
[ {u ∈ U min Jα (u) ≤ r} = {u ∈ U Jα (u) ≤ r}. α∈A
dd´(Ø© (iii) d·K 1.3, é?Û u Jα (¯ u) ≤
X
k
→u ¯,
|^e45
lim Jα (uk ) ≤ lim
α∈A k→∞
(1.6)
X
Jα (uk ),
k→∞ α∈A
DR A
X
α∈A
α∈A
l (Ø©
2
xëY¼êþ(.7´ëY§þã·Kw· §½´eëY© ·ÑxeëY¼êe(.7´eëY© ~ 1.4. K´
Jα (u) = (1 − |u|)α ,
inf Jα (u) =
α∈A
u ∈ [−1, 1], α ∈ [0, +∞),
(
1, u = 0,
0, u 6= 0,
Ø´eëY© J (·) ¯¢þ´ëY© α
1oÙ 5XÚ`35
120
FT
y3§·5í2~ 1.1 ¥(J©
·K 1.5. U ´;Ýþm§J : U → IR eëY!~
ek., K3 u¯ ∈ U J(¯ u) = inf J(u). u∈U
y². 4zS u
k
∈U
(1.7)
§=
lim J(uk ) = inf J(u). u∈U
k→∞
du U ´;§Øb u ·
k
©u´§d J(·) eëY5,
→ u¯
J(¯ u) ≤ lim J(uk ) = inf J(u). k→∞
DR A
ùÒ (1.7)©
u∈U
2
þãy²¹ Ùïá`35nØÌg© §2.
`35 — Ú(J
!¥§·é`35Ú&?©Ä± e`¯K: y(t) ˙ = f (t, y(t), u(t)),
t ∈ [0, T ],
(2.1)
Ù¥ y(·) ∈ C([0, T ]; IR ) ǑG;, u(·) ∈ U [0, T ] = {u : [0, T ] → U u(·) ÿ } Ǒ§U ǑÝþm§f : [0, T ]×IR ×U → IR Ǒ ½N©·Ï~Ñ^òyé?Û (y , u(·)) ∈ IR × n
△
n 0
n
n
`35 — Ú(J
121
ok
©P
FT
2.
U [0, T ], y(·) ≡ y(· ; y0 , u(·)) △ Y[0, T ] = {y(· ; y0 , u(·)) y0 ∈ IRn , u(·) ∈ U [0, T ]}, △ P[0, T ] = {(y(· ; y0 , u(·)), u(·)) y0 ∈ IRn , u(·) ∈ U [0, T ]}.
?Û (y(·), u(·)) ∈ P[0, T ] ¡ǑG – é©Ǒ{ü姷 ÄGåǑXe/ªµ △
Ù¥ y
(y(0), y(T )) ∈ S = {y0 } × S0 ,
0
∈ IRn
½ S
0
⊆ IRn
(2.2)
Ǒ48©Ä Bolza .5UI:
J(y(·), u(·)) = h(y(T )) +
Z
T
f 0 (t, y(t), u(t))dt,
0
Ù¥ h : IR → IR§f : [0, T ] × IR × U → IR© P P [0, T ], Y [0, T ] ±9 U [0, T ] ©OǑ1é8§1; 8Ú18; P P [0, T ], Y [0, T ] Ú U [0, T ] ©OǑ#N é8§#N;8Ú#N8 (1Ù, §2)©·'% `¯KǑ: ¯K (B). (¯y(·), u¯(·)) ∈ P [0, T ], n
0
S
S
DR A
S
n
ad
ad
ad
ad
J(¯ y (·), u¯(·)) =
inf
(y(·),u(·))∈Pad [0,T ]
J(y(·), u(·)).
(2.3)
÷v (2.3) (¯y(·), u¯(·)) ∈ P [0, T ] ¡Ǒ`é; A y¯(·) Ú u ¯(·) ©O¡Ǒ`;Ú`© Äk§·ò¬±e¯K: (i) #Né8 P [0, T ] ´Ä? (ii) XJ P [0, T ] 6= φ, `é (¯ y (·), u¯(·)) ´Ä3?
¡·®²Ñ§¯K (i) U5¯K'©e¡´ #Né8Ǒ8~f© ad
ad
ad
~ 2.1. ÄXeG§: y(t) ˙ = u(t),
FT
1oÙ 5XÚ`35
122
t ∈ [0, 1],
(2.4)
(y(0), y(1)) ∈ S = {(0, 2)}.
(2.5)
ÚGå
«Ǒ U = [−1, 1], Ké?Û u(·) ∈ U [0, 1], |y(T ; 0, u(·))| ≤ 1.
´Ø3÷v (2.5) u(·) ∈ U [0, 1], l U [0, 1] = φ, P [0, 1] = φ© e¡~fw·= U [0, T ] , `EkU Ø3© ~ 2.2. G§ÚGåE´ (2.4) Ú (2.5)©-« Ǒ U = IR§5UI½ÂǑ ad
DR A
ad
ad
J(y(·), u(·)) =
1
0
éuù¯K§´ U
ad [0, 1]
Au u (·),
Z
|y(t) − 2|2 dt.
©'X§±
uk (·) = 2kχ[0, k1 ] (·) ∈ Uad [0, 1], ∀ k ≥ 1.
k
yk (t) ≡ y(t; 0, uk (·)) =
l u (·) ∈ U ,
k
(
2ktχ[0, k1 ] (t), 0 ≤ t ≤ k1 , 2,
1 k
≤ t ≤ 1.
ad
J(yk (·), uk (·)) =
Z
0
1 k
|2kt − 2|2 dt =
4 → 0, 3k
(k → ∞).
`35 — Ú(J
123
Ïd§ inf
(y(·),u(·))∈Pad [0,1]
FT
2.
J(y(·), u(·)) = 0.
´§´e(. 0 ´ØU©ùÒ´`ù¯K `Ø3© þ¡~f¥§« U ´;§Ï Ø´~ −∞.
1oÙ 5XÚ`35
126
Z
T
0
|uk (t)|2 dt ≤ C,
FT
d (L1) U ´;§l 3~ê C > 0 ∀ k ≥ 1.
Ïd§d1Ù½n 4.11, Øb3 L (0, T ; IR ) ¥, 2
m
w uk (·) −→ u ¯(·).
d Mazur ½n§·k u (·) à|Ü, 3 L (0, T ; IR 2
k
△
u˜k (·) =
X i≥1
Ù¥
m
)
αik ui+k (·) → u ¯(·),
αik ≥ 0,
X i≥1
du U ⊆ IR à;§Ïd u¯(·) ∈ U ´5, XJP m
αik = 1.
©,¡§du (2.10)
DR A
ad [0, T ]
y˜k (·) = y(· ; y0 , u ˜k (·)), yk (·) = y(· ; y0 , uk (·)),
K
y˜k (·) =
X
αik yi+k (·),
i≥1
N´y²3 C([0, T ], IR ), n
y˜k (·) → y¯(·) ≡ y(· ; y0 , u ¯(·)).
u´|^ f (·) Ú h(·) à5!eëY5±9 Fatou Ún 0
=
J(¯ y (·), u¯(·)) Z h(¯ y (T )) +
0
T
f 0 (¯ y (t), u¯(t)) dt
G;8;5 ≤
lim k→∞
=
127 Z n h(˜ yk (T )) +
0
FT
3.
T
f 0 (˜ yk (t), u˜k (t)) dt
lim J(˜ yk (·), u˜k (·)) k→∞ X X lim J( αik yi+k (·), αik ui+k (·))
=
k→∞
≤
lim
i≥1
X
i≥1
αik J(yi+k (·), ui+k (·))
k→∞ i≥1
=
J.
o
Ïd u¯(·) ´`©
2
þ¡y²î/6uG§5Ú«5 ( (U à5±9 L (0, T ; IR ) f;5)©ù 5éu /¿Ø¤á§Ï Ǒïá35nا·I ?Úó© m
DR A
2
§3.
G;8;5
[* þ!½n 2.1 y²§·±uy3Ï~¿Â e§#Né8 P [0, T ] ¿Ø´;©?Ú* ±uy§ 5`§éu`¯K§#N8 U [0, T ] Ï~Ø´ ;©´é/§±wÑ3½^e§#NG;8ò¬ ´;© `35qǑõ/6uG;8 ;5§éu#N8;5f ©AO§5UI¥ È©¼ê f Ø6u u §#NG;8 Y [0, T ] ½1 ;8 Y [0, T ] 3 C([0, T ]; IR ) ¥;5òy`3 5© Äk§·5w1;8 Y [0, T ] 5© ad
ad
0
S
ad
n
S
1oÙ 5XÚ`35
128
FT
1;85 ·b (B1) T > 0, U ´;Ýþm§S ⊆ IR ´48© (B2) ¼ê f : [0, T ] × IR × U → IR 'u t ∈ [0, T ] ÿ, ' u u ∈ U ëY§
3~ê L > 0 ±9 ϕ(·) ∈ L (0, T ; IR) (p > 1), n
0
n
n
p
|f (t, x, u) − f (t, y, u)| ≤ L|x − y|, ∀ t ∈ [0, T ], x, y, ∈ IRn , u ∈ U, |f (t, 0, u)| ≤ ϕ(t), ∀ t ∈ [0, T ], u ∈ U .
·k ½n 3.1. ½ y ∈ IR , ¿ (B1)—(B2) ¤á, Ké?Û u(·) ∈ U [0, T ], XÚ (2.1) 3÷v^ y(0) = y ) y(·) ≡ y(· ; u(·))©?Ú§ÄGåǑ (2.2)§K1;8 Y [0, T ] 3 C([0, T ]; IR ) ¥ © y². 3^ (B1)—(B2) e§§ (2.1) )35d 1Ù½n 5.1 ©y3·y²8 Y [0, T ] 5©d (B2), n
0
DR A
0
n
S
4
S
≤
≤
|y(t; u(·))| Z t |y0 | + |f (τ, y(τ ), u(τ ))| dτ |y0 | +
Z
0
T
ϕ(τ ) dτ + L
0
Z
0
t
|y(τ )| dτ .
u´d Gronwall ت§ u(·) ∈ U [0, T ] '~ê C > 0 |y(t; u(·))| ≤ C(1 + |y0 |),
∀ t ∈ [0, T ], u(·) ∈ U [0, T ].
¡8Ü A 3ålm X ¥, XJ§4 A ´ X ¥;8. d/, A ¥ ?Û:ÑkÂñf (Âñu X ¥Ø½´ A ¥:)© 4
G;8;5
129
?Ú§´
FT
3.
|y(t; u(·)) − y(s; u(·))| Z t ≤ |f (τ, y(τ ), u(τ ))| dτ s Z t Z t ≤ ϕ(τ ) dτ + L |y(τ )| dτ s s Z t ≤ ϕ(τ ) dτ + LC(1 + |y0 |)|t − s|, s
∀ s, t ∈ [0, T ], u(·) ∈ U [0, T ].
l Y [0, T ] k.
ÝëY©d Arzel`a-Ascoli ½n§§3 C([0, T ]; IR ) ¥© 2 1;8;5 Ǒ Y [0, T ] 3 C([0, T ]; IR ) ¥;5§·Ú\Xe^ (B3) N f : [0, T ] × IR × U → IR ÷vXe Filippov– Roxin ^: éA¤k t ∈ [0, T ], ±e8Üé?Û y ∈ IR Ñ ´à4: S
n
n
DR A
S
n
n
n
△
f (t, y, U ) = {f (t, y, u) u ∈ U }.
?Ú·I±e Filippov Ún, ù´1nÙÚn 2.5 í2© Ún 3.2. U Ǒ;Ýþm©
g : [0, T ] × U → IR 'u t ∈ [0, T ] ÿ§'u u ∈ U ëY§ n
0 ∈ g(t, U ),
K3 u(·) ∈ U [0, T ],
g(t, u(t)) = 0,
a.e. t ∈ [0, T ],
a.e. t ∈ [0, T ].
(3.1)
(3.2)
y². d ´ U þÝþ§½Â d(u, v) , 1 + d(u, v)
△ d¯(u, v) =
FT
1oÙ 5XÚ`35
130
∀u, v ∈ U,
K d¯ E´ U þÝþ©3dÝþe§U E´;§g 'u u ∈ U ǑE´ëY ©e5, ½Â △
Γ(t) = {u ∈ U
g(t, u) = 0},
t ∈ [0, T ].
Ø5, d (3.1) é?Û t ∈ [0, T ], Γ(t) 6= φ©- U = {v k ≥ 1} ´ U Èf8©·k5y²¼ê d¯(u, Γ(·)) ÿ5©½ u ∈ U , · äóé?Û c ∈ IR, =
k
{t ∈ [0, T ] d¯(u, Γ(t)) ≤ c} ∞ ∞ \ [
1 1 {t ∈ [0, T ] d¯(u, vj ) ≤ c + , |g (t, vj )| ≤ }, i i
i=1 j=i
ùp
△
0
△ d¯(u, Γ(t)) =
(3.3)
inf d¯(u, v).
v∈Γ(t)
¯¢þ§d¯(u, Γ(t)) ≤ c
=3f {v
jk }
⊆ U0
÷v
DR A (
lim d¯(u, vjk ) ≤ c,
k→∞
lim d¯(vjk , Γ(t)) = 0.
(3.4)
k→∞
du U ´;§ g(t, ·) ëY§l (3.4) ¥1ªfdu lim g (t, vjk ) = 0.
k→∞
u´ (3.4) qdu: é?Û i ≥ 1, 3 j ≥ i,
d¯(u, vj ) ≤ c + 1i ,
|g (t, vj )| ≤
1 . i
ùÒy² (3.3)©du (3.3) mà´ÿ§Ï àǑ´ÿ©ùÒ´ `§é?Û u ∈ U , d¯(u, Γ(·)) ½Â [0, T ] þÿ¼ê© y3½Â △
u0 (t) ≡ u1 (t) = v1 ,
´ u (·), u (·) ´ÿ§ 0
1
d¯(u1 (t), Γ(t)) < 1, d¯(u1 (t), u0 (t)) < 2,
∀ t ∈ [0, T ].
∀ t ∈ [0, T ].
3.
G;8;5
131
0
1
FT
b·®²½Â u (·), u (·), · · · , u (·) ÷v k
d¯(ui (t), Γ(t)) < 2 , d¯(ui (t), ui−1 (t)) < 22−i , 1−i
∀ t ∈ [0, T ], i = 1, 2, · · · , k,
Ké?Û t ∈ [0, T ], 3 u ∈ Γ(t)
(3.5)
d¯(uk (t), u) < 21−k .
qdu U ´ U Èf8, 3 v ∈ U 0
0
d¯(v, u) < min(2−k , 21−k − d¯(uk (t), u)),
d7k
y3½Â
d¯(v, Γ(t)) ≤ d¯(v, u) < 2−k , d¯(v, uk (t)) ≤ d¯(v, u) + d¯(u, uk (t)) < 21−k .
(3.6)
△ Fik = {t ∈ [0, T ] d¯(vi , uk−1 (t)) < 21−k }, △
Eik = {t ∈ [0, T ] d¯(vi , Γ(t)) < 2−k },
Aki = Eik ∩ Fik , k, i ≥ 1.
d (3.6), éu?Û t ∈ [0, T ], 3 i t ∈ A ©ùÒk
DR A
k i
[0, T ] =
∞ [
Aki .
i=1
,¡, du t 7→ d(v , Γ(t)) ÿ, l E ÿ©q F ´ÿ§Ï A ´ÿ8© y3½Â u (·) : [0, T ] → U ⊆ U Xe: i
k i
k i
k i
0
k+1
i−1
∀ t ∈ Aki \
uk+1 (t) = vi ,
d E 9 F ½Â±9 (3.7) ´ k i
k i
[
Akj .
(3.7)
j=1
d¯(uk+1 (t), Γ(t)) < 2−k , d¯(uk+1 (t), uk (t)) < 21−k .
(3.8)
ù§·Ò8B/½Â {u (·)}, k = 0, 1, · · ·
(3.5) é?Û k ≥ 1 ¤á© dd±é?Û t ∈ [0, T ], {u (t)} ´ U ¥ Cauchy ©d U 5§· k
k
△
u(t) = lim uk (t) ∈ U, k→∞
t ∈ [0, T ].
1oÙ 5XÚ`35
132
k
u(t) ∈ Γ(t),
l (3.2) ¤á©
FT
du u (·) ´ÿ§l Ù4 u(·) Ǒ´ÿ©= u(·) ∈ U [0, T ]©? Ú§d (3.8) 1ª±9 Γ(t) 45, ∀ t ∈ [0, T ].
2
k ±þÚn§·±ïá±e(J© ½n 3.3. (B1)—(B3) ¤á§K Y [0, T ] ´ C([0, T ]; IR ) ¥ ;8© y². ?S {y (·)} ⊆ Y [0, T ]©- u (·) ∈ U [0, T ] Ǒ A: y (·) = y(· ; y , u (·))©Ø5§·±b3 C([0, T ]; IR ) ¥, n
S
k
k
k≥0
0
k
n
±93 L (0, T ; IR ) ¥,
k
yk (·) → y¯(·),
n
(3.9)
DR A
p
S
△ w fk (·) = f (·, yk (·), uk (·)) −→ f¯(·).
(3.10)
ù§d (3.10) ±9 Mazur ½n, k f (· , y (·), u (·)) à|ÜS k
△ f˜k (·) =
X i≥1
αik f (· , yi+k (·), ui+k (·)), αik ≥ 0,
3 L (0, T ; IR ) ¥, p
k
X
αik = 1, (3.11)
i≥1
m
f˜k (·) → f¯(·).
(3.12)
,¡§d (B2) 9 (3.9) k → ∞ , |f˜k (t) −
≤ L
X i≥1
X
αik f (t, y¯(t), ui+k (t))|
i≥1
αik |yi+k (t) − y¯(t)| → 0
(3.13)
G;8;5
133
'u t ∈ [0, T ] ¤á©d,
FT
3.
y¯(t) = lim yk (t) k→∞ Z t = lim y0 + fk (s) ds k→∞ 0 Z t f¯(s) ds, ∀ t ∈ [0, T ]. = y0 + 0
e¡, ·ky y¯(·) ∈ Y[0, T ]©Ǒd§y²3 u¯(·) ∈ U [0, T ], f¯(t) = f¯(t, y¯(t), u¯(t)),
a.e. [0, T ].
(3.14)
(Ü (3.12) Ú (3.13)§¿|^ (B3), ·
DR A
=
f¯(t) = lim f˜(t) k→∞ X αik f (t, y¯(t), ui+k (t)) lim
∈
k→∞
i≥1
co f (t, y¯(t), U ) = f (t, y¯(t), U ).
´N (t, u) 7→ f (t, y¯(t), u) ÷vÚn 3.2 ^©u´3 u ¯(·) ∈ U [0, T ], (3.14) ¤á©= y¯(·) ∈ Y[0, T ]© §d y (·) ÷vå (2.2) ±9 (3.9), á= y¯(·) Ǒ÷v (2.2)©Ïd y¯(·) ∈ Y [0, T ], ùÒy² Y [0, T ] ;5© 2 k
S
S
IÑ, +?ÛS {(y (·), u (·))} ⊆ P [0, T ], Ñk f {y (·)} Âñ, y¯(·) = y(· ; y , u¯(·)) ∈ Y [0, T ], ù¿ ØL² {u (·)} ½¬kfÂñ u¯(·)©·$ØUäó ´Ä½k {u (·)} fÂñ© k △
ki
i≥1
k
k
k≥1
0
k≥1
k
k≥1
§4.
`35
S
S
1oÙ 5XÚ`35
134
FT
þ¡·®²w3^ (B1)—(B3) e, 1;8 Y [0, T ] ´ C([0, T ]; IR ) ¥;8©Uì §1 ¥g, ·I5U IeëY5©Ǒd·Ú\±eb© (B4) f : [0, T ] × IR → IR ´ Borel ÿ¼ê§
'u y ∈ IR eëY; h : IR → IR eëY©?Ú§3~ê L > 0, S
n
0
n
n
n
f 0 (t, y) ≥ −L,
∀ (t, y) ∈ [0, T ] × IRn .
h(y) ≥ −L,
þ¡Ä f Ø6u u ∈ U ©ù·±ÄuG; ;5'u`3Ú(J© ½n 4.1. (B1)—(B4) ¤á§P [0, T ] 6= φ, K¯K (B) k`é© y². ?4zS {(y (·), u (·))} ⊆ P [0, T ]©U½ n 3.3§3 y¯(·) ≡ y(· ; y , u¯(·)) ∈ Y [0, T ], (3.9) ±9 (3.11)—(3.12) ¤á©d (B4), ·k 0
DR A
ad
k
k
k≥1
0
ad
S
f 0 (t, y¯(t)) ≤ lim f 0 (t, yk (t)),
a.e. t ∈ [0, T ],
k→∞
y (T )) ≤ lim h(yk (T )). h(¯ k→∞
u´§d Fatou Ún, =
J(¯ y (·), u¯(·)) Z h(¯ y (T )) +
T
f 0 (t, y¯(t)) dt
0
≤
lim h(yk (T )) +
k→∞
≤
k→∞
Z
T
0
lim h(yk (T )) + lim
lim f 0 (t, yk (t)) dt
k→∞ Z T
k→∞
0
f 0 (t, yk (t)) dt
`35 ≤
135 lim J(yk (·), uk (·)) k→∞
=
inf
(y(·),u(·))∈Pad [0,T ]
FT
4.
J(y(·), u(·)).
ùÒy² (¯y(·), u¯(·)) ´`é©
2
ØJw3^ (B4) e, ¼ J(y(·), u(·)) 3±e¿Âe´ eëY: é?Û (y (·), u (·)) ∈ P[0, T ]§e3 C([0, T ]; IR ) ¥ y (·) → y¯(·), K k
n
k
k
J(¯ y (·), u¯(·)) ≤ lim J(yk (·), uk (·)). k→∞
ù, 8Ü P [0, T ] “;5” \þ J(y(·), u(·)) “eëY5” ÒÑ `é35© f (t, y) ≡ 0, ·±w¯K (B) ¤ǑA (M) ¯K© Ïdþã(JǑw·A (M) ¯Kk`é© ,¡§f u 'ùb´ØU-0
co E(t, Oδ (y)) = E(t, y),
Ù¥ O (y) ´± y Ǒ%»Ǒ δ > 0 m¥© ´XJ E(t, y) 3 y äk Cesari 5, K E(t, y) ´à48© e¡'u Cesari ^¤á¿©^L² Cesari 5þ ´'u E(t, y) Ǒà48^© δ
4.
`35
137
0
FT
·K 4.2. ±e^¤á: (B6) éA¤k t ∈ [0, T ], N f (t, ·, u) 'u u ∈ U ëY§f (t, ·, u) 'u u ∈ U eëY§=é?Û½ y ∈ IR ±9 ε > 0, 3 σ = σ(t, y) > 0, é?Û y ∈ O (y) ¤áX n
′
(
|f (t, y ′ , u) − f (t, y, u)| < ε,
f 0 (t, y ′ , u) > f 0 (t, y, u) − ε,
σ
∀ u ∈ U,
(4.1)
Kéu½ t ∈ [0, T ], E(t, ·) 3 y :äk Cesari 5
= E(t, y) Ǒà48© y². ·Iy²¿©5© (B6) 3: t ∈ [0, T ] ¤ á§y ∈ IR 8Ü E(t, y) ¤Ǒà48, Kd (B6), é?Û ε > 0, 3 σ = σ(t, y) > 0, y ∈ O (y) , (4.1) ¤á©ù, é 0 < δ < σ ±9?Û (z , z ) ∈ E(t, O (y)), 3 y ∈ O (y), u ∈ U , n
′
δ
σ
δ
δ
δ
δ
DR A
δ 0
z0δ ≥ f 0 (t, y δ , uδ ),
l d (4.1) ±9 (4.2)§·k (
z δ = f (t, y δ , uδ ).
z0δ ≥ f 0 (t, y δ , uδ ) > f 0 (t, y, uδ ) − ε,
|z δ − f (t, y, uδ )| = |f (t, y δ , uδ ) − f (t, y, uδ )| < ε.
ù¿X (z , z ) ∈ O (E(t, y))©u´ δ 0
l , du
δ
ε
E(t, Oδ (y)) ⊆ Oε E(t, y) .
\
δ>0
co E(t, Oδ (y)) ⊆
\
ε>0
E(t, y) ⊆
Oε E(t, y) = E(t, y) = E(t, y).
\
δ>0
co E(t, Oδ (y))
(4.2)
1oÙ 5XÚ`35
g,¤á, Ïd E(t, y) =
\
δ>0
FT
138
co E(t, Oδ (y)).
= E(t, y) 3 y :äk Cesari 5© éu~ 4.1, ·k
2
E(t, y) = {(z 0 , u) z 0 ≥ u2 , u ∈ IR},
∀ (t, y) ∈ IR.
DR A
ù E(t, y) ´à48©Ó´ (B6) ¤á©u´3~f 4.1 ¥§ éuA¯K (B)§Cesari ^÷v© 3 (B6) ¥éuëY5b´éf§ùÒ´Ǒo·` Cesari ^þ´'u E(t, y) à45^©e¡(J Kw« Cesari ^fu Filippov–Roxin ^© ·K 4.3. (B6) ¤á§f (t, y, u) ek.©XJ f (t, y, u) ÷ v Filippov–Roxin ^§K E(t, y) äk Cesari 5© y². d·K 4.2§·Iy² E(t, y) à45© Äk·5w45©é?ÛÂñS (z , z ) ∈ E(t, y), ·k u ∈ U , ( 0
0 k
k
k
△
zk0 ≥ f 0 (t, y, uk ) = ζk0 , zk = f (t, y, uk ),
∀ k ≥ 0.
(z , z ) → (¯z , z¯)©du f (t, y, u ) ek.§ Ùþk ñ§Ïd§´k.. l ·±b (7{§À f)ζ → ζ¯ ©du (ζ , z ) ∈ f (t, y, U ), f (t, y, U ) ´48§l 7k (ζ¯ , z¯) ∈ f (t, y, U()©ùÒ´`3 u¯ ∈ U , 0 k
0
k
0 k 0
0
0
0 k
k
k
f 0 (t, y, u ¯) = ζ¯0 ≤ z¯0 ,
f (t, y, u ¯) = z¯.
4.
`35
139
FT
Ïd, (¯z , z¯) ∈ E(t, y), = E(t, y) ´4© y35y²à5© (z , z ) ∈ E(t, x) (i = 1, 2)§λ ∈ (0, 1)©· k u , u ∈ U 0
0 i
1
i
2
(
△
zi0 ≥ f 0 (t, y, ui ) = ζi0 ,
zi = f (t, y, ui ),
i = 1, 2.
d f (t, y, U ) à53 u¯ ∈ U (
λz10 + (1 − λ)z20 ≥ λζ10 + (1 − λ)ζ20 = f 0 (t, y, u ¯),
λz1 + (1 − λ)z2 = f (t, y, u ¯),
ùÒ E(t, y) à5. l d·K 4.2 E(t, y) äk Cesari 5© 2
´XJ f (t, y, u) 'u u Ǒ5, f (t, y, u) 'u u Ǒà§
U Ǒà8, K E(t, y) ´à©·y3Ñ E(t, y) Ǒà8 ¿©^© ·K 4.4. - (t, y) ∈ [0, T ] × IR © f (t, y, U ) Ǒà8§
3 à¼ê ϕ(· ; t, y) : IR → IR
DR A
0
n
n
f 0 (t, y, u) = ϕ(f (t, y, u); t, y),
∀ u ∈ U,
K E(t, y) ´à8© y². (z , z ) ∈ E(t, y), i = 1, 2, Kk u , u 0 i
i
(
1
zi0 ≥ f 0 (t, y, ui ),
zi = f (t, y, ui ),
2
∈U
∈U
i = 1, 2.
d f (t, y, U ) à5é?Û λ ∈ (0, 1), 3 u λz1 + (1 − λ)z2
(4.3)
3
1oÙ 5XÚ`35
FT
140
= λf (t, y, u1 ) + (1 − λ)f (t, y, u2 ) = f (t, y, u3 ).
d (4.3) λz10 + (1 − λ)z20 ≥
λf 0 (t, y, u1 ) + (1 − λ)f 0 (t, y, u2 )
=
λϕ(f (t, y, u1 ); t, y) + (1 − λ)ϕ(f (t, y, u2 ); t, y) ϕ λf (t, y, u1 ) + (1 − λ)f (t, y, u2 ); t, y
≥ =
ϕ(f (t, y, u3 ); t, y) = f 0 (t, y, u3 ).
Ïd E(t, y) ´à©
2
Ǒ8Ü E(t, y) ´à8§¼ê f (t, y, ·)§f (t, y, ·) ±98Ü U AT÷v,«^©^ (4.3) Ò´ù^© e¡·Ñ¯K (B) `é35©Äk·ò^ (B4) UǑ (B4) f : [0, T ] × IR × U → IR ´ Borel ÿ¼ê§
'u u ∈ U ëY; h : IR → IR eëY©?Ú§3~ê L > 0,
DR A
0
′
0
n
n
f 0 (t, y, u), h(y) ≥ −L,
∀ (t, y, u) ∈ [0, T ] × IRn × U .
½n 4.5. (B1)—(B2), (B4) ±9(B5) ¤á§P [0, T ] 6= φ, K¯K(B) k`é© y². d½nb§4zS {(y (·), u (·))} ⊆ P [0, T ] ′
ad
k
k
ad
△
J(yk (·), uk (·)) → J =
inf
(y(·),u(·))∈Pad [0,T ]
J(y(·), u(·)).
k≥1
`35
141
d½n 3.1, 3 C([0, T ]; IR ) ¥, n
FT
4.
yk (·) → y¯(·).
(4.4)
?Ú (3.9) Ú (3.11)—(3.12) ¤á©fj0 (·) ≡
X
αij f 0 (· , yi+j (·), ui+j (·)),
i≥1
f¯0 (t) = lim fj0 (t) ≥ −L, j→∞
a.e. t ∈ [0, T ],
Kd (4.4) ±9 (B5) eªA??¤áµ (f¯0 (t), f¯(t)) ∈
δ>0
co E(t, Oδ (¯ y (t))) = E(t, y¯(t)).
h i+ g(t, u) = |f¯(t) − f (t, y¯(t), u)| + f 0 (t, y¯(t), u) − f¯0 (t) ,
(4.5)
DR A
-
\
K g(·, ·) 'u t ∈ [0, T ] ÿ§'u u ∈ U ëY© d (4.5), (3.1) ¤á©u´§d Filippov Ún (Ún 3.2) =3ÿ u¯(·) ∈ U [0, T ] (
f¯0 (t) ≥ f 0 (t, y¯(t), u¯(t)), f¯(t) = f (t, y¯(t), u¯(t)),
a.e. t ∈ [0, T ].
,¡§d Fatou Ún
J(¯ y (·), u¯(·)) Z = h(¯ y (T )) +
T
f 0 (t, y¯(t), u¯(t)) dt
0
≤
=
lim h(yk (T )) +
k→∞
lim h(yk (T )) +
k→∞
Z
T
f¯0 (t) dt
0
Z
0
T
lim fj0 (t) dt
j→∞
≤ ≤ ≤ =
lim h(yk (T )) + lim
j→∞
k→∞
lim h(yk (T )) + lim
k→∞
k→∞
lim
k→∞
h(yk (T )) +
Z
T
Z
T
0
Z
0
lim J(yk (·), uk (·))
k→∞
= J.
FT
1oÙ 5XÚ`35
142
T
fj0 (t) dt
f 0 (t, yk (t), uk (t)) dt
0
f 0 (t, yk (t), uk (t)) dt
(4.6)
?Ú, N´då8 S = {x } × S 45 (¯y(·), u¯(·)) ∈ P [0, T ], (Ü (4.6) = (¯ y (·), u¯(·)) ´`é© 2 0
ad
5P
0
DR A
31Ù§·®²Ñ§C©¯K35nØÑy´3 19 V" Weierstrass 5¿ek.C©¯K¿Øo´k) ©?Ú§' u35¯±JPص N ´ (g,) ê§K N ≥ N , l du N ´§q k N ≥ N ©ù N = N , N = 1© ùØw·§XJ·Ø/b g,ê3 5§Ò¬Ø(Ø©aq/§XJ·3~ 2.4 ¥b`é (¯ y (·), u ¯(·)) 3§·½± y¯(·) ≡ 0, ? u ¯(·) = 0, a.e. © ¯ ¢þ§ u(·) ≡ 0 éA J (u(·)) = 0§ýe(. −1 $ ©3Nõ¢S¯K?n¥§·Ù¢¿ØAO'%` O©· IÑCq`©´ (Cq) ` L§²~´±`¤÷v7^ǑÄ:©þ¡~fL ²XJvk35(J |^7^§U(J¬ýÏ »Ì©ù´35nØ3¢SA^¥5©,¡§35n ØéuÆu¤äknØ¿Âg,Ǒ´ T© 2. L©W©Neustadt 3Ù 1961 ©Ù [34] ¥Ä XeXÚµ 1.
2
2
2
y˙ (t) = A(t)y (t) + b(t, u(t)),
y (t0 ) = y0 ,
t ∈ [t0 , T ],
5P
Ú5UI: J (u(·)) =
Z
FT
143
T
t0
{ha0 (t), y (t, u(t)i + b0 (t, u(t))}dt.
Ù¥, A(·) : [t , T ] → IR ±9 a (·) : [t , T ] → IR ´È; U ´ IR ¥k.48; éu?Û t ∈ [0, T ], b(t, ·) : IR → IR ± 9 b (t, ·) : IR → IR ëY; é?Û u ∈ U , b(·, u) : [t , T ] → IR ±9 b (·, u) : [t , T ] → IR È;
é?Ûk.8 K ⊂ IR , 3È µ(·) : [t , T ] → IR n×n
0
0
m
n
0
m
0
m
0
n
n
0
k
0
0
|b(t, u)| + |b0 (t, u)| ≤ µ(t);
y0 , y1
Ǒ IR ¥½ü:; 18Ǒ: n
Uad = {u(·) : [t0 , T ] → U
u(·)
ÿ, }.
`¯K´Ïé uˆ(·) ∈ U y(T ; uˆ(·)) = y , ad
J (ˆ u(·)) =
Z
J (u(·)).
DR A
P
inf
u(·)∈Uad y(T ;u(·))=y1
1
y 0 (t) =
y¯ =
K
y0 y
t
t0
,
y¯˙ =
{ha0 (s), y (s, u(s)i + b0 (s, u(s))}ds,
¯b(t, u) =
b0 (t, u) b(t, u)
0
(a0 (t))⊤
0
A(t)
y¯(t0 ) = y¯0 ,
,
y¯0 =
0
y0
,
y¯ + ¯b(t, u(t)),
(1)
'u y¯ ´5©`¯KÒzǑ (^ y¯(t ) = y¯ ) Ïé u ˆ(·) ∈ U y (T ; u ˆ(·)) = y , ad
0
y 0 (T ; u ˆ(·)) =
P
0
1
inf
u(·)∈Uad y(T ;u(·))=y1
y 0 (T ; u(·)).
b(t) = {¯y |¯y = y¯(t; u(·), u(·) ∈ Uad }, R
K Rb(T ) ´XÚ (1) U8©§´ IR ¥k.à48©
ã `¯Kdu n+1
b(T )}, inf{y 0 |¯ y ∈L∩R
1oÙ 5XÚ`35
144
n+1
FT
Ù¥ L Ǒ IR ¥ y=y . u´§XJk u(·) ÷v y(T ; u(·)) = y , `¯Kk)©´§ 3 Neustadt ?Ø/¥ Cesari ^Ø7÷v©¯¢þ§ù(J´ 3vkà5^¹e'u`35¡(J (3m þ§§u Cesari ^/¤m)© 3. + Cesari ^´à5^g,í2©´§´K `w,3/§'X5UIäk/ª Z 1
1
T
f 0 (t, u(t))dt
d§|^ Filippov Ún§A å`¯K35BC¤ Ǒ¼ê4¯K§ f 'u u këY5§U ´k.48§
È©ke.§K`o´3©d Cesari ^7¤á© 4. 3/ªþ, 1nÙÚn 2.5 ¿Ø´Ún 3.2 AÏ/, ´·ØJd Ún 3.2 y²±e(Ø: U Ǒ;Ýþm©Q : [0, T ] → 2 ´Ǒ;8þëY8 ¼ê©g : [0, T ] × U → IR 'u t ∈ [0, T ] ÿ§'u u ∈ U ëY© 0
0
U
DR A
n
0 ∈ g (t, Q(t)),
K3 u(·) ∈ U [0, T ],
g (t, u(t)) = 0, u(t) ∈ Q(t),
a.e. t ∈ [0, T ],
a.e. t ∈ [0, T ],
a.e. t ∈ [0, T ].
Ǒd§·IòÚn¥ g ^ |g(t, u)| + |d(u, Q(t))| O©?Ú§ ±|^Ún 3.2 y²ò(Jí2Ǒ Q(·) Ǒÿ!U Ǒ©Ý þm/. 5. 3"yà5^§?Ø`35óä´tµn Ø©tµ´±VÇÿÝǑ¼êa©uÏ~gX2 ¼êuÏ~¼ê©|^tµy²`35Äg ´´k3tµa¥?Ø`tµ¯K§`tµ 35©, ÏL`²¤`tµǑÏ~§½±U EǑÏ~5`²¯K`35©tµ ´ Young 3Vn Ú\2©Gamrelidze!McShane Ú Warga étµnØ(áÚuå ^©édk, Öö±ëw Gamrelidze [23, 24]!McShane [35∼39]!Warga [45∼50] Ú Young [53∼56] ©Ǒ ?n«à/§Fattorini 3 1991 Ú? dk\ (VÇ) ÿݽÂtµVg (ë [20, 21])©
SK
SK
FT
145
y²·K 1.2© 2. 3·K 1.4(iii) ¥§A Ǒ8(ØXÛº 3. E~f`²eëY¼êe(.7´eëY© 4. ÁE`¯K~f§ Cesari ^ؤá§`é 3© 5. ÁéÙ§/ª`¯K?Ø`é35½n©~X§ G åØÓu (2.2) /½5UIØ´ Bolza ./©
DR A
1.
§1.
FT
1ÊÙ n Úó
Ù8´ïáǑx` Pontryagin n© ù´'u`é7^©Äk·Ñ¯K£ã Ú ~5b©ÄXeXÚ: (
y(t) ˙ = f (t, y(t), u(t)), y(0) = y0 ,
(1.1)
DR A
±95UI
a.e. t ∈ [0, T ],
J(u(·)) =
Z
T
f 0 (t, y(t), u(t))dt,
0
Ù¥ u(·) u8Ü U . ·Xeb: (C1) U ⊆ IR , T > 0© (C2) N f : [0, T ]×IR ×U → IR Ú f : [0, T ]×IR ×U → IR ´ÿ,
3~ê L > 0 ÚëY ω : [0, +∞) → [0, +∞), é ϕ(t, y, u) = f (t, y, u), f (t, y, u) ¤áX m
n
n
0
n
1
0
ˆ|), |ϕ(t, y, u) − ϕ(t, yˆ, uˆ)| ≤ L|y − yˆ| + ω(|u − u n ∀t ∈ [0, T ], y, yˆ ∈ IR ; u, u ˆ ∈ U, |ϕ(t, 0, u)| ≤ L, ∀ (t, u) ∈ [0, T ] × U.
(1.2)
·¡ ω : [0, +∞) → [0, +∞) ǑëY§´§3 [0, +∞) þëY§î üO§
3":Ǒ"© 1
§1.
Úó
147 0
FT
N f, f 'u y ´ C ,
3ëY ω : [0, +∞) → [0, +∞), é ϕ(t, y, u) = f (t, y, u), f (t, y, u) ¤áX (C3)
1
0
|ϕy (t, y, u) − ϕy (t, yˆ, u ˆ)| ≤ ω(|y − yˆ| + |u − u ˆ|),
∀ t ∈ [0, T ], y, yˆ ∈ IRn ; u, u ˆ ∈ U.
P U [0, T ] = {u(·) : [0, T ] → U u(·)ÿ}, Ké?Û u(·) ∈ U [0, T ], db(C1)—(C2) 91Ù½n 5.1, § (1.1) k ) y(·) ≡ y(· ; u(·))©éu f ±9 f = (f , f , · · · , f ), ·Ú\P Òµ 0
1
2
fy0
∂f 0 △ = ≡ .. ∂y .
∂f 0 ∂yn
n
,
(1.3)
DR A
±9
∂f 0 ∂y1 ∂f 0 ∂y2
2
∂f 1 ∂y1 ∂f 1 ∂y2
∂f 2 ∂y1 ∂f 2 ∂y2
∂f 1 ∂yn
∂f 2 ∂yn
∂f △ fy ≡ = .. ∂y .
.. .
··· ··· .. .
···
∂f n ∂y1 ∂f n ∂y2
.. . .
(1.4)
∂f n ∂yn
Ù§·òïá¯K`é¤÷v^ — n©énïÄ´`nØSN©31 !¥, ·ò?ØG å (©^½)
U = IR /©ù´N´?n/©éù/?رw¤´²;C ©g$^©, §·òé«Ǒ/?1 ?اù§·òÚ\GC©ù{©3Ù1n! ·òÄGäkªàå/©T!ÌgK´| ^ Ekeland C©nòkå¯KzǑ åCq¯K© m
2 (1.3)
±w´ (1.4) A~©
1ÊÙ n
148
ªà å¯K
FT
§2.
3ù!¥§·Ä±e¯Kµ ¯K (C).Ïé` u¯(·) ∈ U [0, T ] u¯(·) 3 U [0, T ] þz5UI J(·), = J(¯ u(·)) =
inf
u(·)∈U [0,T ]
J(u(·)).
3þ¡Qã`¯K¥, ©G´½, G3Ù Ǒvkå, ù´'N´?na¯K©·Äkѱe( J: ½n 2.1. U = IR , f , f 'u y Ú u këY §f , f , f , f k.. e (¯y(·), u¯(·)) ´¯K (C) ` ¯ : [0, T ] → IR ÷v é§K3 ψ(·) m
u
0 y
0 u
DR A
y
0
n
¯ dψ ¯ + f 0 (t, y¯(t), u = −fy (t, y¯(t), u¯(t))ψ(t) ¯(t)), y dt ¯ ψ(T ) = 0,
(2.1)
¯ − f 0 (t, y¯(t), u¯(t)) = 0 fu (t, y¯(t), u ¯(t))ψ(t) u
(2.2)
3 [0, T ] þA??¤á© y². ? u(·) ∈ L (0, T ; IR ) ⊂ U [0, T ], Ké?Û α ∈ IR, u (·) = u ¯(·) + αu(·) ∈ U [0, T ]. l J(u (·)) 3 α = 0 ©l XJ d ∞
α
m
△
α
dα
J(uα (·))|α=0
3{, §Òu 0. ½ny²L§Ò´uO J(u (·)) 3 α = 0 :êL§©Ä α > 0, P y (·) ≡ y(·; u (·)), · α
α
α
ªà å¯K
k
149
FT
2.
1 0≤ J(uα (·)) − J(¯ u(·)) α Z T 0 f (t, y α , uα ) − f 0 (t, y¯, u ¯) dt α 0 Z T n Z 1 hD y α − y¯ E fy0 (t, y¯ + θ(y α − y¯), u¯ + θαu), α 0 0 D Ei o + fu0 (t, y¯ + θ(y α − y¯), u¯ + θαu), u dθ dt. (2.3)
= =
,¡§eP
Y α (t) =
K
y α (t) − y¯(t) , α
Y α (0) = 0,
DR A
dY α dt Z 1 fy (t, y¯ + θ(y α − y¯), u¯ + θαu)⊤ dθ Y α
=
0
+
Z
0
1
fu (t, y¯ + θ(y α − y¯), u¯ + θαu)⊤ u dθ.
du f , f k., Ï k~ê M > 0, y
u
|Y α (t)| ≤ M
Z
t
0
|Y α (τ )| dτ + M
¤±d Gronwall ت§Y [0, T ], ·k
α
(·)
Z
T
0
|u(τ )| dτ .
3 [0, T ] þk.©ù§∀ t ∈
lim y α (t) = lim y¯(t) + αY α (t) = y¯(t).
α→0
α→0
u´§|^1Ù½n 5.3 ·k
kY α (·) − Y (·)kC[0,T ] → 0,
1ÊÙ n
Ù¥
FT
150
dY = f (t, y¯(t), u¯(t))⊤ Y (t) + f (t, y¯(t), u ¯(t))⊤ u(t), y u dt Y (0) = 0.
(2.4)
¯ d (2.1) ½Â§|^ (2.4) 9 f , f ëY5§3 (2.3) y3, ψ(·) ¥- α → 0+ 0 y
0
≤ =
Z
Z
h i hfy0 (t, y¯(t), u¯(t)), Y (t)i + hfu0 (t, y¯(t), u¯(t)), u(t)i dt
T 0 T 0
+
Z
¯ dψ(t) , Y (t)i dt + h dt T
0
Z
T
0
Z
Z
T
¯ hfy (t, y¯(t), u¯(t))ψ(t), Y (t)i dt
0
hfu0 (t, y¯(t), u¯(t)), u(t)i dt ¯ hψ(t),
dY (t) i dt + dt
Z
T
0
¯ hψ(t), fy (t, y¯(t), u¯(t))⊤ Y (t)i dt
DR A
= −
0 u
+
T
0
= −
+
=
Z
Z
T
0
Z
T
0
T
0
hfu0 (t, y¯(t), u¯(t)), u(t)i
dt
¯ hψ(t), fu (t, y¯(t), u¯(t))⊤ u(t)i dt hfu0 (t, y¯(t), u¯(t)), u(t)i dt
¯ hfu0 (t, y¯(t), u¯(t)) − fu (t, y¯(t), u¯(t))ψ(t), u(t)i dt
(2.5)
þ¡1ªd (2.4) §1ªd (2.1) §3í 1nªÿ^ ©ÜÈ©{§ 1oªK|^ Y (·) ¤÷v§ (2.4)© , d (2.5) ±9 u(·) ?¿5= (2.2)© 2
3¯K¥, `5§U L«´Uå, Ï Ï~ U Ø´m©XJ U ´m8§KEâ?n§·E,
2.
ªà å¯K
151
FT
±|^þ¡{5í`é¤÷v7^©?Ú§X J U ´à8§Kéu u(·) ∈ U [0, T ]§α ∈ (0, 1) ·k u ¯(·) + α(u(·) − u ¯(·)) ∈ U [0, T ].
l 0≤
J u¯(·) + α(u(·) − u¯(·)) − J(¯ u(·)) α
,
∀ α ∈ (0, 1).
|^ù'XªÓ±^ ¡{aq(J©´X J U ==´ IR ¥8ܧo`5§u¯ + αu(·) ½ u ¯(·) + α(u(·) − u ¯(·)) ÒؽE,3 U [0, T ] ¥©3ù«¹e§ ·ÒI¤¢GC©©e¡§·ò½n 2.1 (Jí2 «Ǒ/© m
DR A
½n 2.2. (C1)—(C3) ¤á§(¯y(·), u¯(·)) ´¯K (C) ¯ : [0, T ] → IR ÷v `é§K3 ψ(·) n
¯ dψ ¯ + f 0 (t, y¯(t), u = −fy (t, y¯(t), u¯(t))ψ(t) ¯(t)), y dt ¯ ψ(T ) = 0,
(2.6)
±9Xe^:
¯ hψ(t), f (t, y¯(t), u ¯(t))i − f 0 (t, y¯(t), u¯(t))
¯ = max{hψ(t), f (t, y¯(t), u)i − f 0 (t, y¯(t), u)} u∈U
(2.7)
3 [0, T ] þA??¤á© XJP
△
H(t, y, u, ψ) = hψ, f (t, y, u)i − f 0 (t, y, u),
(2.8)
1ÊÙ n
K`;÷v§¤
FT
152
d¯ y ∂H ¯ = (t, y¯(t), u¯(t), ψ(t)), dt ∂ψ
(2.9)
dψ¯ ∂H ¯ =− (t, y¯(t), u¯(t), ψ(t)), dt ∂y
(2.10)
d, § (2.6) ±¤
^ (2.7) K¤Ǒ
¯ ¯ H(t, y¯(t), u¯(t), ψ(t)) = max H(t, y¯(t), u, ψ(t)). u∈U
(2.11)
Ï~, ò/Ǒ (2.9)—(2.10) |§¡Ǒ Hamilton XÚ, H ¡Ǒ Hamilton ¼ê, § (2.6) ¡Ǒ§ (1.1) §. ½n 2.2 y². ? u(·) ∈ U [0, T ], ·Ä f 0 (·, y¯(·), u(·)) − f 0 (·, y¯(·), u¯(·))
DR A
f (·, y¯(·), u(·)) − f (·, y¯(·), u ¯(·))
d1Ù½n 2.9§§´ÿ§d1ÙSK 18 §é?Û ε ∈ (0, 1)§3 E ⊂ [0, T ] ÷v 3
ε
|Eε | = εT,
±9
ε
=
Z
Z t 0
Eε
+
T
f 0 (τ, y¯(τ ), u(τ )) − f 0 (τ, y¯(τ ), u¯(τ ))
dτ f (τ, y¯(τ ), u(τ )) − f (τ, y¯(τ ), u¯(τ )) 0 f (τ, y¯(τ ), u(τ )) − f 0 (τ, y¯(τ ), u¯(τ ))
[0,t]
rε0 (t) rε (t)
f (τ, y¯(τ ), u(τ )) − f (τ, y¯(τ ), u¯(τ ))
,
dτ
(2.12)
ùp·Ñ§y²nØI^Ǒ Liapounoff ½n§ I^ Liapounoff ½nCq(J (ë1ÙSK 15)© 3
2.
ªà å¯K
153
FT
Ù¥ |E| L«8Ü E Lebesgue ÿݧ
|rε0 (t)| + |rε (t)| ≤ ε2 .
-
(
ε
u (t) =
u ¯(t),
t ∈ [0, T ] \ Eε ,
u(t), t ∈ Eε ,
K u (·) ∈ U [0, T ]©·~¡ u (·) ´ u¯(·) GC©©P y (·) = y(·; u (·)), K ε
ε
ε
ε
0 ≤ =
DR A
=
J(uε (·)) − J(¯ u(·)) ε Z i 1 Th 0 f (t, y ε (t), uε (t)) − f 0 (t, y¯(t), uε (t)) dt ε 0 Z i 1 Th 0 + f (t, y¯(t), uε (t)) − f 0 (t, y¯(t), u¯(t)) dt ε 0 Z T DZ 1 y ε − y¯ E fy0 (t, y¯ + θ(y ε − y¯), uε ) dθ, dt ε 0 0 Z h i 1 + f 0 (t, y¯, u) − f 0 (t, y¯, u¯) dt ε Eε Z T DZ 1 y ε − y¯ E dt fy0 (t, y¯ + θ(y ε − y¯), uε ) dθ, ε 0 0 Z Th i r0 (T ) + f 0 (t, y¯, u) − f 0 (t, y¯, u ¯) dt + ε . ε 0
=
,¡§eP
Y ε (·) =
Kaqu (2.13), ·k ε
Y (s)
=
Z
nZ
(2.13)
y ε (·) − y¯(·) , ε
o fy (t, y¯ + θ(y ε − y¯), uε )⊤ dθ Y ε dt 0 Z s h0 Z s i rε (t) dt, + f (t, y¯, u) − f (t, y¯, u ¯) dt + ε 0 0 ∀ s ∈ [0, T ]. (2.14) s
1
1ÊÙ n
dþª9 (1.2), |Y ε (t)|
≤ L
Z
t
0
FT
154
|Y ε (τ )| dτ + 2LT (k¯ y(·)kC[0,t] + 1) + T ε.
u´d Gronwall ت,
h i kY ε (·)kC[0,T ] ≤ 2LT (k¯ y(·)kC[0,t] + 1) + T eLT .
= Y (·) 3 C[0, T ] ¥k.©AO§ y (·) → y¯(·), u C[0, T ]. u´§|^1Ù½n 5.3 ·k ε
ε
kY ε (·) − Y (·)kC[0,T ] → 0,
Ù¥
DR A
dY = f (t, y¯, u ¯)⊤ Y + f (t, y¯, u) − f (t, y¯, u ¯), y dt Y (0) = 0.
¯ d (2.6) ½Â§Kaqu (2.5) ψ(·) 0 ≤
Z
T
0
+
Z
¯ hf (t, y¯(t), u¯(t)) − f (t, y¯(t), u(t)), ψ(t)i dt
0
=
Z
T
0
T
h
i f 0 (t, y¯(t), u(t)) − f 0 (t, y¯(t), u¯(t)) dt.
h i ¯ ¯ H(t, y¯(t), u¯(t), ψ(t)) − H(t, y¯(t), u(t), ψ(t)) dt ≥ 0, ∀ u(·) ∈ U [0, T ].
(2.15)
ª (2.15) ¡ǑÈ©.^, e¡·|^§5í ^ (2.7)©du U ´©, ·± U Èf8 {v } ©P ¯ Q = {s ∈ (0, T ) s ´H(·, y¯(·), u¯(·), ψ(·)) Lebesgue :}, ∞ j j=1
0
2.
ªà å¯K
155
j
|Qj | = T,
-Q=
∞ \
Qj ,
j=0
FT
¯ ´H(·, y¯(·), v , ψ(·)) Lebesgue :}, j ≥ 1, Kd Lebesgue ȼê5 Qj = {s ∈ (0, T ) s
∀ j = 0, 1, 2, · · · .
KE,k
|Q| = T .
é?Û t ∈ Q, ? h ∈ (0, min(t, T − t)), ¿ u(s) =
Kd (2.15)
u ¯(s), |s − t| ≥ h,
vj ,
|s − t| < h,
h i ¯ ¯ H(s, y¯(s), u¯(s), ψ(s)) − H(s, y¯(s), vj , ψ(s)) ds ≥ 0.
DR A
Z
(
t+h
t−h
ü>Óر 2h, ¿- h → 0+, d Lebesgue :½Â ¯ ¯ H(t, y¯(t), u¯(t), ψ(t)) − H(t, y¯(t), vj , ψ(t)) ≥ 0.
d {v } È5±9 H(t, y, ·, ψ) ëY5þªé?Û v ∈ U ¤á§= ∞ j j=1
¯ ¯ H(t, y¯(t), u ¯(t), ψ(t)) ≥ H(t, y¯(t), v, ψ(t)),
ùÒy² (2.7)©
∀ v ∈ U, t ∈ Q.
2
·w§½n 2.1 ´½n 2.2 A~§|^^ (2.7)§ f 'u u §á= (2.2)©Ï GC©Ø= ·£; «vk5((J§Ǒ4·t é¼ 0
1ÊÙ n
156
FT
ê f, f 1w5©·5¿, þ¡?Ø´ Lagrange ¯K,
1Ù §2 (Jw·, 3½^e, Bolza ¯K!Lagrange ¯KÚ Mayer ¯K´d©Ïd§éu Bolza ¯KÚ Mayer ¯ KATkAn©·ò§îQãÚy²3 ÖöǑöS© 0
§3.
äkªàå¯K
DR A
!¥§·òĪàkå`¯K©£1n Ù¤Äm`¯K¥§·3ªǑ T §G y(T ) á3,½8I8Ü¥. /§·Ä ©G±38Ü¥Cz/©!¥§Ǒ{²å§· ==Ä©GÚªàGþǑ½/©·b¤Ä ¯KäkXeå^µ y(T ) = y1 .
P
Uad [0, T ] ≡ {u(·) ∈ U [0, T ] y(T ; u(·)) = y1 }.
Ǒ#N8©G§!5UIEd1!ѧ!Ä `¯KǑ e Ïé` u ¯K (C). ¯(·) ∈ U [0, T ] u ¯(·) 3 U [0, T ] þz5UI J(·), = ad
J(¯ u(·)) =
·kXen
inf
u(·)∈Uad [0,T ]
ad
J(u(·)).
3.
äkªàå¯K
157
0
¯ 2 > 0, ψ¯02 + |ψ(t)|
FT
½n 3.1. (C1)—(C3) ¤á§(¯y(·), u¯(·)) ´¯K (C)e ¯ : [0, T ] → IR ÷v `é§K3 ψ¯ ≤ 0 Ú ψ(·) n
∀ t ∈ [0, T ],
(3.1)
dψ¯ ¯ = −ψ¯0 fy0 (t, y¯(t), u¯(t)) − fy (t, y¯(t), u¯(t))ψ(t), dt
±9Xe^µ
¯ ψ¯0 f 0 (t, y¯(t), u ¯(t)) + hψ(t), f (t, y¯(t), u¯(t))i =
¯ max{ψ¯0 f 0 (t, y¯(t), u) + hψ(t), f (t, y¯(t), u)i} u∈U
3 [0, T ] þA??¤á©
(3.2)
DR A
£?nõ¼ê^4¯K, í4:7^ {´|^Û¼êò¯KzǑ ^4¯K©éu ¯K§ù{´(J©·Ǒ±35UI¥\I ¼ê ( 1.4) òkå¯KzǑ å¯K©, §du#5U I"y½ëY5§ù¯KÓ/J±?n ©^4 ,«?n{´|^v¼ê{ò^4¯KzǑCq ^ 4¯K©?nGkå`¯K{da q©ÙÌg´|^C©nòkå¯KzǑXCq å`¯K© Äk§·0±eM= (Ekeland) C©n §k'Ý þ (ål) m£§±ëw¼©Ûk'á© 4
5
y3§|^1w©Û 0,
0,
bx + c ≤ 0,
/ I: b + c ≥ 0, c ≥ 0. =
∀ x ∈ [0, 1].
(3.17)
DR A
bx + c ≥ 0,
a.e. x ∈ [0, 1].
dd (3.17),
u ¯(x) = bx + c,
då^
a.e. x ∈ [0, 1].
b + c = 1, 2 b + c = 1. 3 2 6 b+c≥0 .
) b = −4, c = 3, gñ / II: b + c ≤ 0, c ≤ 0. = dd (3.17)
bx + c ≤ 0,
u¯(x) = 0,
∀ x ∈ [0, 1].
a.e. x ∈ [0, 1].
3.
äkªàå¯K
169
u ¯(x) =
(
0,
x ∈ [0, −c/b],
bx + c,
då^
a.e. x ∈ [0, 1].
x ∈ (−c/b, 1],
b 1− 2 b 1+ 3
n
FT
w,å^gñ. / III: b + c > 0, c < 0. = bx + c 3 [0, 1] þkK , d7 k b > 0,
d (3.17),
c2 c +c 1+ = 1, 2 b3 b 2 c c c 1 + 1− 2 = . b3 2 b 6 c2 2 = , 2 b3 b c 1 = . 2b3 2b
DR A
1 + 2c + b 1 + 3c − 2b
5¿
3c c3 − 3 2b 2b 2c 1 c c3 1+ + − − 3 b 2 b b r 2c c c3 1+ + b b b3 2 1 > . b 2b 1+
=
≥
=
·Ògñ. / IV: b + c < 0, c > 0. = bx + c 3 [0, 1] þk K, d7 k b < 0,
d (3.17), u ¯(x) =
(
bx + c, x ∈ [0, −c/b],
0,
x ∈ (−c/b, 1],
a.e. x ∈ [0, 1].
1ÊÙ n
då^
2 2 c − c = 1, 2b 3 b 3 − c + c = 1. 3b2 2b2 6
) ±9 u¯(x) =
b = −4, (
FT
170
c = 8,
8 − 4x, x ∈ [0, 1/2], 0,
x ∈ (1/2, 1],
a.e. x ∈ [0, 1].
(3.18)
DR A
dþ¡?Ø, (3.18) ½Â u¯(·) ´÷vnÚ å^, Ï §7,´`. ·±TC©¯K¼Ǒ 83 . 5P
1.
nJÑÚïáÌ8õu L. S. Pontryagin£1908—1988). Pontryagin ´é ͶêÆ[§3ïÄnØ §®²3 ÿÀÆïÄ¡ ¤Ò©1952 §Pontryagin UC ïÄ, m©ïÄA^êƯK§AO´©§ÚnØ©1956
§Pontryagin ÚÓ¯JÑ n©1958 §Äkú Ù 'u5XÚm`ny²§1960 §¤
/ªny²©1961 §ÙÆ) V. G. Boltyanskii, R. V. Gamkrelidze Ú E. F. Mishchenko ÜÑ 5ZL§êÆn Ø6[43](k¥È)© 0, 0 ≤ t < tˆ ≤ T , 3 u(·) ≡ u U [t, T ], V (t, x) + ε(tˆ − t) ≥
Z
tˆ
f 0 (s, y(s), u(s)) ds + V (tˆ, y(tˆ)).
t
u´§5¿ V ∈ C ([0, T ] × IR )§·k 1
−ε
≤
=
=
ε,tˆ(·)
n
Z tˆ V (tˆ, y(tˆ)) − V (t, x) 1 − f 0 (s, y(s), u(s))ds tˆ − t tˆ − t t Z tˆ n 1 − Vt (s, y(s)) − hVx (s, y(s)), f (s, y(s), u(s))i tˆ − t t o −f 0 (s, y(s), u(s)) ds Z tˆ n 1 − Vt (s, y(s)) tˆ − t t
−
∈
Ä5y{Ú HJB §
≤
1 tˆ − t
185
o +H s, y(s), u(s), −Vx (s, y(s)) ds
Z tˆ n t
− Vt (s, y(s))
FT
2.
o + sup H s, y(s), u, −Vx (s, y(s)) ds u∈U tˆ ↓ t. (2.11) → −Vt (t, x) + sup H t, x, u, −Vx (t, x) ,
u∈U
þ¡4§·|^ ϕ = f, f ¤÷v 0
lim
sup
t↓s y∈IRn ,u∈U
±9éu y(·) = y(·; t, x, u
|ϕ(t, y, u) − ϕ(s, y, u)| = 0,
§
ε,tˆ(·))
(2.12)
lim sup |y(s) − x| = 0. tˆ↓t s∈[t,tˆ]
ù 'Xª±db (D2) ¥'u¼ê f !f ëY5Ú (1.8) ©(Ü (2.10) 9 (2.11) =(Ø© 2
DR A
0
éu~ 1.1§·k
sup H(t, x, u, p) = sup pu = |p|,
|u|≤1
l A HJB Ǒ
|u|≤1
∀(t, x, p) ∈ [0, T ) × IR × IR. −Vt + |Vx | = 0, V = x.
(2.13)
t=T
N´y V (t, x) = x − (T − t) ´§ (2.13) ²;)© e¡§·0XÛ|^ HJB §)5Ïé`©b ·®²l HJB § (2.8) )¼ê V ∈ C ([0, T ] × IR )©? 1
n
18Ù Ä5y{
186 n
FT
Ú§bé (t, x) ∈ [0, T ]×IR , ª (2.8) ¥þ(.3 u = u(t, x) , = H(t, x, u(t, x), −Vx (t, x)) = sup H(t, x, u, −Vx (t, x)), u∈U
∀ (t, x) ∈ [0, T ] × IRn .
(2.14)
·bé?Û (t, x) ∈ [0, T ) × IR , 3 y¯(·; t, x) ÷ve §µ n
(
y¯˙ (s) = f (s, y¯(s), u(s, y¯(s))), y¯(t) = x.
a.e. s ∈ [t, T ],
(2.15)
y3§½ (t, x)§P y¯(·) = y¯(·; t, x), ¿u ¯(s) = u(s, y¯(s; t, x)),
(2.16)
DR A
K
∀ s ∈ [t, T ],
d V (s, y¯(s)) ds = Vt (s, y¯(s)) + hVx (s, y¯(s)), f (s, y¯(s), u¯(s))i = −f 0 (s, y¯(s), u¯(s)),
é (2.17) l t T È©§·Ò V (t, x) = h(¯ y (T )) +
Z
T
s ∈ [t, T ].
(2.17)
f 0 (s, y¯(s), u¯(s))ds = J(¯ u(·); t, x).
t
ùL² (¯y(·), u¯(·)) ´¯K (D ) |`é©þ¡?Øw ·§ ·|^ HJB § ¼ê§o3/ªþ§ ·Ò±E¯K (D ) |`é©AO§|^ (2.14)§· ±E©¯K (D) |`©ù«{¡Ǒy {© |^y{)©¯K (D)§Nþ©Ǒ±eAÚ½µ tx
tx
2.
Ä5y{Ú HJB §
187
0
tx
FT
Ú½ 1. ) HJB § (2.8) ¼ê V (t, x)© Ú½ 2. ÏL (2.14) Ïé u(t, x)© Ú½ 3. (t, x) = (0, y )§, )§ (2.15) `é (¯ y (·), u¯(·))© o`5§Ä5y{Ìk±eA:µ(1) Ú\¯Ka (D ); (2) ½Â¼ê V (t, x); (3) Ñ`5n; (4) Ñ HJB §¿|^þ¡Ú½ 1—3 `é©|^ HJB §§· ±B/±e(ص ·K 2.4. (D1)—(D2) ¤á§
¼ê V ∈ C ([0, T ]×IR )© (¯y(·), u¯(·)) ´¯K(D) |`é§Kµ 1
max H(t, y¯(t), u, −Vx (t, x ¯(t))), u∈U
a.e. t ∈ [0, T ].
(2.18)
DR A
=
H(t, y¯(t), u ¯(t), −Vx (t, x ¯(t)))
n
y². ´é?Û t ∈ [0, T ], (¯y(·), u¯(·))(3 [t, T ] þ) Ǒ´ ¯K (D ) |`é, l t¯ y (t)
V (t, y¯(t)) =
Z
T
f 0 (s, y¯(s), u ¯(s)) ds + g(¯ y(T )).
t
þªmà'u t A??©u´éþªüà
Vt (t, y¯(t)) + hVx (t, y¯(t)), f (t, y¯(t), u ¯(t))i = −f 0 (t, y¯(t), u ¯(t)), a.e. [0, T ].
'þªÚ (2.8) = (2.18)©
2
§·25ww~K 1.1©d§V (t, x) = x − (T − t)© ùd (2.18)§u¯ ATA??÷v H(t, y¯(t), u¯(t), −Vx (t, y¯(t))) = max H(t, y¯(t), u, −Vx (t, y¯(t))), |u|≤1
18Ù Ä5y{
188
FT
Ù¥ y¯(·) ´Au u¯(·) G©d=
−Vx (t, y¯(t))¯ u(t) = max − Vx (t, y¯(t))u ,
l 5¿ V
x
|u|≤1
=1
§·k
u ¯(t) = −1,
ù·Ò `©
§3.
a.e. [0, T ].
a.e. [0, T ].
Ê5)
lþ!§|^Ä5yÏé`'
´(½ ¼ê V (· , ·)©XJ (i) V ∈ C ([0, T ] × IR ),
(ii)HJB § (2.8) k (²;) )§K V d HJB § (2.8) Ǒx©, ¢ ´ `5§(i) UÑؤá, HJB §Ǒؽo´k²;). XJü$), 'X` (2.8) A??¤á, K) 5 Øy. 3~ 1.2 ¥·®²w¼êؽ´ 1w© 0.
´¼ê V ´ Lipschitz ¼ê§ Ø´C ([0, T ] × IR) ¼ê©3 : (t, 0) ( ∀ t ∈ [0, T ))§V (t, x) ka©,¡§·k 1
x
sup H(t, x, u, p) = sup u∈U
|u|≤1
l A HJB §´
(t, x) ∈ [0, T ) × IR,
(3.1)
DR A
−vt + |xvx | = 0, v = x,
n o pux = |px|.
t=T
x ∈ IR.
·äóù§vk C ([0, T ]×IR) )©Ǒd§b v ∈ C ([0, T ]× IR) ´ (3.1) ), Kd (3.1) 1ª§·k 1
vx (T, x) = 1,
1
∀ x ∈ IR.
2d v ëY5, k ϕ : IR → [0, T ) x
vx (t, x) > 0,
△ ∀ (t, x) ∈ N = {(t, x) ϕ(x) ≤ t ≤ T }.
?Ú§ϕ ±¤Ǒó¼ê!
'u x ≥ 0 üNØ~©u´§d (3.1), v ÷v \ △ vt = xvx , (t, x) ∈ N + = N {(t, x) x ≥ 0, t ∈ [0, T ]}, v = x, x ∈ IR. t=T
(3.2)
18Ù Ä5y{
-
(
·k
τ = t,
FT
190
△
t
z = xe ,
Φ(τ, z) = v(τ, ze−τ ).
(3.3)
Φτ = vt + vx [−ze−τ ] = vt − xvx = 0.
dd§Φ Ø6u τ©u´·±r Φ(τ, z) ¤ Φ(z)©ù§d (3.3) v(t, x) = Φ(xet ),
|^ (3.2) ¥ªà^§·k
∀ (t, x) ∈ N
xe−T = v(T, xe−T ) = Φ(x),
Ïd§(Ü (3.4)—(3.5),
.
(3.4)
∀ x ≥ 0.
∀ (t, x) ∈ N
+
(3.5)
.
DR A
v(t, x) = xet−T ,
+
aq/§·k
v(t, x) = xeT −t ,
∀ (t, x) ∈ N \ N
+
.
l 3 N þ§v V ©du V Ø´ C (N ) ¼ê, Ï v Ǒ Ø´ C (N ) ¼ê©ùbgñ© lþ¡~f·±w·K 2.3 bLuǑ©Ǒ
Qkaqu·K 2.3 î35qvkǑb (ا·IÚ\±eVg© ½Â 3.1. ¼ê v ∈ C([0, T ] × IR ) ¡Ǒ (2.8) Ê5e )§XJ 1
1
n
v(T, x) ≤ h(x),
∀ x ∈ IRn ,
(3.6)
3.
Ê5)
191 1
n
n
FT
é?Û ϕ ∈ C ([0, T ] × IR ), v − ϕ 3,: (t, x) ∈ [0, T ) × IR 4§B¤á −ϕt (t, x) + sup H(t, x, u, −ϕx (t, x)) ≤ 0. u∈U
(3.7)
aq/§¼ê v ∈ C([0, T ] × IR ) ¡Ǒ´ (2.8) Ê5þ )§XJ3 (3.6)—(3.7) ¥ØÒ “≤/UǑ “≥”!“4” U Ǒ “4”©XJ v Q´Ê5e)q´Ê5þ)§K¡ v ǑÊ5 )© ·Ñ§3þ¡½Â¥§“4 (4)” ±^ “î 4 (î4)”!D “î (î)” O© ·òùd5y²3Öö©e¡(JL²+¼ê UØ´ HJB § (2.8) ²;)§§%´ (2.8) Ê5)© ½n 3.2. (D1)—(D2) ¤á§K¼ê V (· , ·) ´ (2.8) Ê5)© y². ϕ ∈ C ([0, T ] × IR )§V − ϕ 3,: (t, x) ∈ [0, T ) × IR 4©½ u ∈ U ©- y(·) = y(· ; t, x, u) ´Au u(·) ≡ u G;, Kd½n 2.2, ·k ( éu¿©C t
DR A
n
1
n
n
tˆ > t)
0
dd
V (t, x) − ϕ(t, x) − V (tˆ, y(tˆ)) + ϕ(tˆ, y(tˆ)) tˆ − t Z tˆ n o 1 ≤ f 0 (s, y(s), u)ds − ϕ(t, x) + ϕ(tˆ, y(tˆ)) tˆ − t t → f 0 (t, x, u) + ϕt (t, x) + hϕx (t, x), f (t, x, u)i, ≤
tˆ → t.
−ϕt (t, x) + H(t, x, u, −ϕx (t, x)) ≤ 0,
(3.8)
∀ u ∈ U.
18Ù Ä5y{
l
FT
192
−ϕt (t, x) + sup H(t, x, u, −ϕx (t, x)) ≤ 0. u∈U
(3.9)
,¡§XJ V − ϕ 3,: (t, x) ∈ [0, T ) × IR 4 , Kd V ½Â§é?Û ε > 0!tˆ > t (tˆ > t
¿©C t), ·±é u(·) = u (·) ∈ U [t, T ], ÷v n
ε,tˆ
0
≥ V (t, x) − ϕ(t, x) − V (tˆ, y(tˆ)) + ϕ(tˆ, y(tˆ)) Z tˆ ˆ ≥ −ε(t − t) + f 0 (s, y(s), u(s))ds t
+ϕ(tˆ, y(tˆ)) − ϕ(t, x).
DR A
ü>ر (tˆ − t), ¿- tˆ → t −ε
≤
=
≤
Z tˆ o 1 n − f 0 (s, y(s), u(s))ds − ϕ(tˆ, y(tˆ)) + ϕ(t, x) tˆ − t t Z tˆ n 1 − f 0 (s, y(s), u(s)) − ϕt (s, y(s)) (3.10) tˆ − t t o −hϕx (s, y(s)), f (s, y(s), u(s))i ds Z tˆ n 1 − ϕt (s, y(s)) tˆ − t t o + sup H s, y(s), u, −ϕx (s, y(s)) ds u∈U
→ −ϕt (t, x) + sup H(t, x, u, −ϕx (t, x)).
(3.11)
u∈U
aqu (2.11) y²§3þª4Ú½¥§·|^ f Ú f ëY5 ( Ù´|^ dd (2.12))©(Ü (3.9) Ú (3.10), · V ´ HJB § (2.8) Ê5)© 2 0
3.
Ê5)
193
FT
e¡·?ÚÄÊ5)5©Ä±e§ : −vt + H(t, x, −vx ) = 0, v = h, t=T
(3.12)
Ù¥ H : [0, T ] × IR × IR → IR Ú h : IR → IR ëY©´ (2.8) ´ (3.12) A~©§ (3.12) Ê5)½Âª½Â 3.1 Av koØÓ§Öög1Ñ©·k±e(J© ·K 3.3. v ∈ C([0, T ] × IR ), K v ´ (3.12) ²;)
= v ∈ C ([0, T ] × IR )
v ´ (3.12) Ê5)© y². 75 v ´ (3.12) ²;), K v ∈ C ([0, T ] × IR )©éu?Û ϕ ∈ C ([0, T ] × IR ), XJ v − ϕ 3 (t , x ) ∈ [0, T ) × IR 4§K·k n
n
n
n
1
n
1
n
1
n
0
0
ϕx (t0 , x0 ) = vx (t0 , x0 ).
DR A
ϕt (t0 , x0 ) = vt (t0 , x0 ),
n
l
−ϕt (t0 , x0 ) + H(t0 , x0 , −ϕx (t0 , x0 ))
=
−vt (t0 , x0 ) + H(t0 , x0 , −vx (t0 , x0 )) = 0.
ùL² v ´Ê5e)©Óny§v Ǒ´Ê5þ)© ¿©5 ϕ = v, K v − ϕ 3?Û: (t, x) ∈ [0, T ) × IR Q 4q4. l dÊ5)½Â§·k n
−vt (t, x) + H(t, x, −vx (t, x))
=
ùÒy² (Ø©
−ϕt (t, x) + H(t, x, −ϕx (t, x)) = 0.
2
uþã(ا·±òÊ5)ÀǑ (3.12) «2Â)©
18Ù Ä5y{
194
FT
Ê5)5
§4.
+ HJB § (2.8) Uvk²;)§´·®²w§ (¢kÊ5)©¯¢þ§¼ê V (· , ·) Ò´ (2.8) Ê5)© y3¯K´ (2.8) Ê5)´Ä´©XJ¹(¢Xd§ K HJB §Ê5)ÒǑx `¯K¼ê©!Ì 8Ò´éþã¯KÑ¡£© ½n 4.1. eãb¤á: (D2) ¼ê H : [0, T ] × IR × IR → IR Ú h : IR → IR ëY©
3ëY¼ê ω¯ : [0, +∞) × [0, +∞) → [0, +∞)§'uzCþü NØ~§
éu?¿ r ≥ 0, ω¯ (r, 0) = 0§ n
′
n
n
DR A
|H(t, x, p) − H(t, x, q)| ≤ L(1 + |x|)|p − q|,
(4.1)
|H(t, x, p) − H(t, y, p)| ≤ ω ¯ (|x| ∨ |y|, |x − y|(1 + |p|)), n
∀ t ∈ [0, T ], x, y, p, q ∈ IR ,
K§ (3.12) kÊ5)©
y². v(t, x) Ú vˆ(t, x) ´ (3.12) üÊ5)©Ǒy5§· Iy²
Ø L =
v(t, x) ≤ vˆ(t, x),
N , T
N
K^8B{, · (4.2)© y3§ T ∈ (0, ), P L , 1 − LT0 N = N (T0 ) ≡
∀ (t, x) ∈ (T −
1 , T ] × IRn , L
(4.3)
1 L
0
L0 =
(4.2)
´g,ê©ù§XJ·Uy²
v(t, x) ≤ vˆ(t, x),
(
∀ (t, x) ∈ [0, T ] × IRn .
(t, x) ∈ (T − T0 , T ) × IRn
|x| < L0 (t − T
+ T0 ) .
(4.4)
4.
Ê5)5
195
1 L
n
FT
Kéu (T − , T ] × IR ¥?Û:§ T ∈ (0, ) ¿©C , §Ò ½¹3 N (T ) ¥©u´Ǒy² (4.3), ·y² 1 L
0
0
h
sup
sup (t,x)∈N
i
v(t, x) − vˆ(t, x) ≤ 0.
(t,x)∈N
eþªØ¤á, K·k
1 L
h
i
v(t, x) − vˆ(t, x) ≥ γ ¯ > 0.
(4.5)
d (4.1) 9 (4.4), é?Û (t, x) ∈ N ±9 p, q ∈ IR , k n
|H(t, x, p) − H(t, x, q )|
y ε, δ > 0, ÷v
L(1 + |x|)|p − q |
≤
L(1 + L0 T0 )|p − q | = L0 |p − q |. ε + δ < L0 T0 .
C ∞ (IR)
(4.6)
(4.7)
DR A
2 K > 0 ±9 ζ ∈
≤
sup
K>
(t,x,s,y)∈N ×N
ζ (r ) =
0,
−K,
é?Û α, β, γ > 0, ·½Â
{v(t, x) − vˆ(s, y )},
(4.8)
ζ ′ (r ) ≤ 0,
(4.9)
r ≤ −δ,
r ≥ 0,
∀ r ∈ IR.
Φ(t, x, s, y )
=
v(t, x) − vˆ(s, y ) −
1 |x − y |2 α
1 − |t − s|2 + ζ hxi ε − L0 (t − T + T0 ) β
+ζ hy i ε − L0 (s − T + T0 ) + γ (t + s) − 2γT,
Ù¥ h·i ½ÂǑ
∀ (t, x, s, y ) ∈ N × N ,
ε
hz i ε = (|z |2 + ε2 )1/2 ,
z ∈ IRn .
du Φ ëY§ N × N ´;§· Φ 3 N × N þ3: (t , x , s , y ) ∈ N × N ©u´d 0
0
0
0
Φ(T, 0, T, 0) ≤ Φ(t0 , x0 , s0 , y0 ),
18Ù Ä5y{
(5¿ (4.7)—(4.9)) 0
= ≤
FT
196
v(T, 0) − vˆ(T, 0) + 2ζ (ε − L0 T0 ) 1 v(t0 , x0 ) − vˆ(s0 , y0 ) − |x0 − y0 |2 α 1 − |t0 − s0 |2 + ζ hx0 i ε − L0 (t0 − T + T0 ) β
(4.10)
+ζ hy0 i ε − L0 (s0 − T + T0 ) + γ (t0 + s0 ) − 2γT
0,
½ hy i − L (s ù§d (4.9) ±9 (4.10) = 0
0
− T + T0 ) > 0.
DR A
0 ε
0 < K − K + γ (t0 + s0 ) − 2γT ≤ 0.
ùÒ gñ©Ïd (4.12) ¤á©,¡§|^
Φ(t0 , x0 , t0 , x0 ) + Φ(s0 , y0 , s0 , y0 ) ≤ 2Φ(t0 , x0 , s0 , y0 ),
v(t0 , x0 ) − vˆ(t0 , x0 ) + 2ζ hx0 i ε − L0 (t0 − T + T0 )
+v(s0 , y0 ) − vˆ(s0 , y0 ) + 2ζ hy0 i ε − L0 (s0 − T + T0 ) +2γ (t0 + s0 ) − 4γT
≤
2v(t0 , x0 ) − 2ˆ v (s0 , y0 ) −
2 2 |x0 − y0 |2 − |t0 − s0 |2 α β
2ζ hx0 i ε − L0 (t0 − T + T0 )
l
+2ζ hy0 i ε − L0 (s0 − T + T0 ) + 2γ (t0 + s0 ) − 4γT. 2 2 |x0 − y0 |2 + |t0 − s0 |2 ≤ v(t0 , x0 ) − v(s0 , y0 ) α β +ˆ v (t0 , x0 ) − vˆ(s0 , y0 ) ≤ 2η(|t0 − s0 | + |x0 − y0 |),
(4.13)
Ê5)5
197
Ù¥ η(r ) =
´
1 2
sup |t−s|+|x−y|≤r (t,x,s,y)∈N ×N
n
FT
4.
o
|v(t, x) − v(s, y )| + |ˆ v(t, x) − vˆ(s, y )| .
©u´d N k.5, ·
lim η(r ) = 0
r→0+
△
η0 = sup η(r ) < +∞. r>0
l d (4.13) √
| x 0 − y0 | ≤
αη0 ,
?Ú(Ü (4.13) ±9 (4.14), ·k
|t0 − s0 | ≤
p
βη0 .
(4.14)
p 1 1 √ |x0 − y0 |2 + |t0 − s0 |2 ≤ η( αη0 + βη0 ). α β
△
△ε,δ = {(t, x) ∈ N
(4.15)
hxi ε ≤ L0 (t − T + T0 ) − δ}.
DR A
d (4.5) Ú (4.9), ε, δ, γ > 0 ¿©§·k sup
Φ(t, x, t, x)
(t,x)∈△ε,δ
=
sup
(t,x)∈△ε,δ
d §·k
[v(t, x) − vˆ(t, x) + 2γ (t − T )] ≥
sup
(t,x)∈△ε,δ
=
·äóµ3 r
0
Φ(t, x, t, x) ≤
γ ¯ > 0. 2
sup Φ(t, x, s, y )
N ×N
Φ(t0 , x0 , s0 , y0 ) ≤ v(t0 , x0 ) − vˆ(s0 , y0 ).
> 0,
(4.16)
(4.17)
é?Û 0 < α, β < r , ¤áX 0
(t0 , x0 , s0 , y0 ) ∈ N × N .
(4.18)
Öö5¿ùp: (t , x , s , y ) ´6u (α, β, ε, δ, γ) ÀJ©y3· 5y² (4.18)©XJ (4.18) ؤá§KU (4.12), ½k±e(صéu, (α , β ) → (0, 0), Φ 3 N × N þ: (t , x , s , y ) ÷v t = T, ½ s = T, ∀ m ≥ 1. d (4.14), ·k |x − y | → 0, t , s → T, m → ∞. 0
m
0
0
0
m
m
m
m
m
m
m
m
m
m
m
18Ù Ä5y{
Ïd§d (4.16)—(4.17), 0
§
(ϕ , ϕ ) 3 (t , x ) T´ (q, p)©|^ þeê·±ÑÊ5)d½Â© ½n 5.5. ¼ê v ∈ C([0, T ] × IR ) ´§ (2.8) Ê5)
= 0
1
0
0
t
x
0
0
n
∀ x ∈ IRn ,
v(T, x) = h(x),
±9é?Û (t, x) ∈ [0, T ) × IR , n
−q + sup H(t, x, u, −p) ≤ 0,
1,+ ∀ (q, p) ∈ Dt,x v(t, x),
u∈U
1,− ∀ (q, p) ∈ Dt,x v(t, x).
−q + sup H(t, x, u, −p) ≥ 0, u∈U
(5.9)
y². v ´ (2.8) Ê5), K ∀ (q, p) ∈ D v(t, x), dÚn 5.3, 3¼ê ϕ ∈ C ([0, T ] × IR ) v − ϕ 3 (t, x) ,
3T: (5.3) ¤á©l d½Â 3.1, (3.7) ¤á©ùÒ Ñ (5.9) ¥1'Xª©aq/y (5.9) ¥1'Xª© y (5.9) ¤á©XJ ϕ ∈ C ([0, T ] × IR ) 3 (t, x) ∈ [0, T ) × IR 4§K·± (ϕ (t, x), ϕ (t, x)) ∈ D v(t, x)©Ï d§|^ (5.9) ¥1'Xª§·± (3.7)©ùÒL² v ´ (2.8) Ê5e)©Óny v ´ (2.8) Ê5þ)©
DR A
1,+ t,x
1
n
1
n
t
x
n
1,+ t,x
2
½n 5.5 Ñ Ê5)d½Â©¯¢þ§(5.9) ¥1 ª´'uÊ5e)d^§ 1ªK´'uÊ5þ )d^©`5§mþ©´¹þ©8Ü ( (5.2) ¥1ª)§
üö±ØÓ©Ï e¡(JL²¼ ê V ±kr5©
5.
þ©Úe©
205
FT
½n 5.6. (D1)—(D2) ¤á, K¼ê V ´¼êa C([0, T ] ×IR ) ¥÷v±e^µé?Û (t, x) ∈ [0, T ) × IR , n
n
−q + sup H(t, x, u, −p) ≤ 0, u∈U −q + sup H(t, x, u, −p) ≥ 0, u∈U V (T, x) = h(x).
1,+ ∀ (q, p) ∈ Dt+,x V (t, x),
1,− ∀ (q, p) ∈ Dt+,x V (t, x),
(5.10)
y². é?Û (q, p) ∈ D V (t, x), ·ÜÚn 5.3(ii) ½Ún 5.4(ii) ¼ê ϕ(^ V O v), K·±|^½n 3.2 ¥ Ó?Ø (5.10)(5¿3p·3y² (3.8) 9 (3.10) ^'umm4)©,¡§d½n 4.1 ±9½n 5.5§¿5 ¿ D V (t, x) ⊆ D V (t, x) ±9 D V (t, x) ⊆ D V (t, x) ÷v (5.10) C([0, T ] × IR ) a¼ê5© 2 1,± t+,x
1,+ t,x
1,+ t+,x
1,− t,x
n
1,− t+,x
DR A
dþ¡½n§·w3^ (D1)—(D2) e, ^ (5.9) Ú (5.10) Ñ´½Â HJB § (2.9) Ê5)^ (TÊ5)´ §
TTÒ´¼ê V ), l §üöǑ½´d©¯¢ þ§3^ (D1)—(D2) e§(5.9) ¥üªf´©O (5.10) ¥ üªfd©½n 5.6 ¥(J ´~k©,
ù(J´'u HJB § (2.9) ù«'Aϧ§3 ^ (D1)—(D2) e©¯¢þ§ù J´Ø7§· ±é~§Ó(ةıe§µ F (t, x, v(t, x), vt (t, x), vx (t, x)) = 0,
(t, x) ∈ (a, b) × Ω,
(5.11)
½Â 5.7. −∞ < a < b < +∞, Ω ⊆ IR ´«©éu v ∈ C((a, b) × Ω)§·¡ v Ǒ (5.11) Ê5e)§XJ n
F (t, x, v(t, x), q, p) ≤ 0,
1,+ ∀ (t, x) ∈ (a, b) × Ω, (q, p) ∈ Dt,x v(t, x).
18Ù Ä5y{
206
F (t, x, v(t, x), q, p) ≥ 0,
FT
aq/§v ¡Ǒ (5.11) Ê5þ)§XJ
1,− ∀ (t, x) ∈ (a, b) × Ω, (q, p) ∈ Dt,x v(t, x).
XJ v Q´Ê5þ)q´Ê5e)§·Ò¡ǑÊ5)© ·ïá±eÚnµ Ún 5.8. v ∈ C((a, b) × Ω),
F ∈ C((a, b) × Ω × IR × IR × IRn ),
÷v
F (t, x, w, q, p) − F (t, x, w, qˆ, p) ≥ Λ(ˆ q − q),
(5.12)
DR A
∀ (t, x, w, p) ∈ (a, b) × Ω × IR × IRn , qˆ, q ∈ IR, qˆ ≥ q,
Ù¥ Λ > 0 ´~ê© (i) e v ´ (5.11) Ê5e), p ∈ D
1,+ x v(t, x),
F (t, x, v(t, x), q, p) ≥ 0,
q ∈ IR
÷v
K (q, p) ∈ D v(t, x)© ùp D v(t, x) L« t ½§¼ê v(t, ·) 3: x þ© (aq/§e¡ D v(t, x) L« t ½§¼ê v(t, ·) 3: x e©)© (ii) e v ´ (5.11) Ê5þ), p ∈ D v(t, x), q ∈ IR ÷v 1,+ t− ,x
1,+ x
1,− x
1,− x
K (q, p) ∈ D
F (t, x, v(t, x), q, p) ≤ 0,
©
1,− t− ,x v(t, x)
5.
þ©Úe©
207
1 1,+ x
0
FT
y². (i) v ´ (5.11) Ê5e)©Ø5§·b [−1, 1] ⊂ (a, b), B (0) ⊂ Ω ±9 Λ = 1, ùp B (0) ´ IR ¥ ü m¥© p ∈ D v(0, 0), q ∈ IR ÷v 1
0
F (0, 0, v(0, 0), q0 , p0 ) ≥ 0.
·¤y²Ò´ (q , p ) ∈ D 0
0
n
(5.13)
©P
1,+ t− ,x v(0, 0)
F˜ (t, x, q, p) = F (t, x, v(t, x), q, p),
∀ (t, x, q, p) ∈ (a, b) × Ω × IR × IRn ,
K F˜ ∈ C((a, b) × Ω × IR × IR )©P n
η(r) ≡
sup t2 +|x|2 +|p|2 ≤r2
|F˜ (t, x, q0 , p + p0 ) − F˜ (0, 0, q0 , p0 )|, r ≥ 0,
|t|≤1
DR A
K η(·) 3 [0, +∞) Ø~©d F˜ ëY5, η(r) → 0, r → 0 ½Â 1 r ω(r) ≡ 0,
K
(
Z
r
2r
{
1 ξ
Z
2ξ
+
.
XJ 0 < r < +∞, XJ r = 0,
√ η( θ)dθ}dξ + r,
ξ
ω(0) = 0, ω(·) ∈ C[0, +∞) ∩ C 1 (0, +∞), 0 < rω ′ (r) ≤ 2ω(2r),
∀ 0 < r < +∞,
(5.14)
|F˜ (t, x, q0 , p + p0 ) − F˜ (0, 0, q0 , p0 )| ≤ ω(t2 + |x|2 + |p|2 ),
? ε > 0©du p
∀ (t, x, p) ∈ [−1, 1] × IRn × IRn .
0
∈ Dx1,+ v(0, 0),
·k δ
ε
>0
v(0, x) − v(0, 0) − p0 · x ≤ ε|x|, ∀ |x| ≤ δε .
18Ù Ä5y{
208
ε
≥1
v(t, x) ≤ v(0, 0) + p0 · x + ε
À M
FT
u´d v ëY5§3~ê C
p t2 + |x|2 + q0 t + Cε |x|2 , (5.15)
∀ (t, x) ∈ {t = 0, |x| ≤ 1} ∪ {−1 ≤ t ≤ 0, |x| = 1}. 2
ε
= ω(18Cε2 )/ω( Cε 2 ) ε
±9 −ε ≤ t
·äó v(t, x)
≤ △
(5.16)
ε|t0 | . 4Mε ω(4)
(5.17)
p v(0, 0) + p0 · x + ε t2 + |x|2 + q0 t
+Cε |x|2 − Mε tω(t2 + |x|2 ) − ω(18ε2 )t ϕ(t, x), ∀ (t, x) ∈ Qε ,
(5.18)
DR A
=
< t1 < 0
ε , 2Mε
ω(4t20 ) ≤ |t1 | ≤
0
Ù¥ Q
©ÄK, ·k (tˆ, xˆ) ∈ Q ÷v
△
ε
= (t1 , 0) × B1 (0)
ε
v(tˆ, x ˆ) > ϕ(tˆ, x ˆ).
-
△
ψ(t, x) = (t − t1 )[v(t, x) − ϕ(t, x)],
5¿3 Q þ t ≤ 0§d (5.15), ·k
(5.19)
(t, x) ∈ Qε .
ε
ψ(t, x)|∂Qε ≤ 0,
¿d (5.19)
ψ(tˆ, x ˆ) > 0.
ù§3 (t˜, x˜) ∈ Q ε
ψ(t˜, x ˜) = max ψ(t, x) > 0. (t,x)∈Qε
(5.20)
þ©Úe©
= v(t, x) ≤
t˜ − t1 ˜ [v(t, x ˜) − ϕ(t˜, x ˜)] + ϕ(t, x), ∀ (t, x) ∈ Qε . t − t1
l dÚn 5.3(ii), (−
Ïd
209
FT
5.
1 1,+ ˜ [v(t˜, x ˜) − ϕ(t˜, x ˜)] + ϕt (t˜, x ˜), ϕx (t˜, x ˜)) ∈ Dt,x v(t, x ˜). ˜ t − t1
F˜ (t˜, x ˜, −
1 [v(t˜, x˜) − ϕ(t˜, x ˜)] + ϕt (t˜, x ˜), ϕx (t˜, x ˜)) ≤ 0. t˜ − t1
,¡§d (5.20), ·k
1 ψ(t˜, x ˜) > 0. [v(t˜, x ˜) − ϕ(t˜, x ˜)] = ˜ ˜ (t − t1 )2 t − t1
(5.21)
(5.22)
DR A
u´d (5.12), (5.13)—(5.15), (5.21)—(5.22), ¿5¿ t˜ ≤ 0 ≤
= ≤
≤
Mε ω(t˜2 + |˜ x|2 ) + ω(18ε2 ) ε(−t˜) p + Mε ω(t˜2 + |˜ x|2 ) + 2Mε t˜2 ω ′ (t˜2 + |˜ x|2 ) + ω(18ε2 ) t˜2 + |˜ x|2 q0 − ϕt (t˜, x ˜) 1 q0 − ϕt (t˜, x ˜) + [v(t˜, x ˜) − ϕ(t˜, x ˜)] ˜ t − t1 1 F˜ (t˜, x ˜, − [v(t˜, x˜) − ϕ(t˜, x ˜)] + ϕt (t˜, x ˜), ϕx (t˜, x ˜)) t˜ − t1 −F˜ (t˜, x ˜, q0 , ϕx (t˜, x ˜))
≤ −F˜ (t˜, x ˜, q0 , ϕx (t˜, x ˜))
≤ −F˜ (0, 0, q0 , p0 ) + ω(t˜2 + |˜ x|2 + |ϕx (t˜, x ˜) − p0 |2 )
≤ ω( t21 + |˜ x|2 + [ ε + 2Cε |˜ x| + 2Mε |t˜| |˜ x|ω ′ (t˜2 + |˜ x|2 ) ]2 ) 4Mε |t˜| |˜ x| 2 2 ˜2 + 2|˜ ≤ ω t21 + |˜ x|2 + [ ε + 2Cε |˜ x| + 2 ω(2 t x | ) ] . t˜ + |˜ x|2
(5.23)
18Ù Ä5y{
XJ |˜x| ≤ |t |, Kd (5.16), ·k 0
FT
210
4Mε |t˜| |˜ x| ω(2t˜2 + 2|˜ x|2 ) ≤ 2Mε ω(4t20 ) ≤ ε. t˜2 + |˜ x|2
XJ |˜x| > |t |, Kd (5.17), ·k 0
4Mε |t˜| |˜ x| 4Mε |t1 | ω(4) ≤ ε. ω(2t˜2 + 2|˜ x|2 ) ≤ |t0 | t˜2 + |˜ x|2
(Ü (5.23)—(5.25),
(5.24)
(5.25)
Mε ω(t˜2 + |˜ x|2 ) + ω(18ε2 )
2 ≤ ω(t21 + |˜ x|2 + 2ε2 + 2Cε |˜ x| ) ≤ ω(t21 + |˜ x|2 + 8ε2 + 8Cε2 |˜ x|2 )
≤ ω(9ε2 + 9Cε2 |˜ x|2 ).
XJ C |˜x|
2
≤ ε2 ,
K
DR A
2 ε
(5.26)
XJ C |˜x| 2 ε
2
ω(9ε2 + 9Cε2 |˜ x|2 ) ≤ ω(18ε2 ).
> ε2 ,
K
ω(9ε2 + 9Cε2 |˜ x|2 ) ≤ ω(18Cε2 ) ≤
ù§·ok
ω(18Cε2 ) ω(|˜ x|2 ) = Mε ω(|˜ x|2 ). ω(ε2 /Cε2 )
ω(9ε2 + 9Cε2 |˜ x|2 ) < ω(18ε2 ) + Mε ω(|˜ x|2 ).
Ïdd (5.26)—(5.27)
Mε ω(t˜2 + |˜ x|2 ) + ω(18ε2 ) < Mε ω(|˜ x|2 ) + ω(18ε2 ).
ù´gñ©Ïd (5.18) ¤á. l ·k lim
t→0− x→0
v(t, x) − v(0, 0) − q0 t − p0 · x ≤ ε + ω(18ε2 ). |t| + |x|
(5.27)
þ©Úe©
211
- ε → 0 +
lim
t→0− x→0
FT
5.
v(t, x) − v(0, 0) − q0 t − p0 · x ≤ 0. |t| + |x|
u´ (q , p ) ∈ D v(0, 0)© (ii) e v ´ (5.11) Ê5þ), K u ≡ −v ´eã§Ê5 e)µ 0
0
1,+ t− ,x
−F (t, x, −u, −ut, −ux ) = 0.
ù§d (i) (Ø©
2
e¡·Ñ½n 5.6 í2µ
½n 5.9. v ∈ C((a, b) × Ω), F ∈ C((a, b) × Ω × IR × IR × IR ) ÷v (5.12), K (i) v ´ (5.11) Ê5e)
=
DR A
n
F (t, x, v(t, x), q, p) ≤ 0,
∀ (t, x) ∈ (a, b) × Ω, (q, p) ∈ Dt1,+ + ,x v(t, x).
(ii) v
´ (5.11) Ê5þ)
=
F (t, x, v(t, x), q, p) ≥ 0,
∀ (t, x) ∈ (a, b) × Ω, (q, p) ∈ Dt1,− + ,x v(t, x).
y². (i) e v ´ (5.11) Ê5e)© (q, p) ∈ D K p ∈ D v(t, x)©·äó 1,+ x
F (t, x, v(t, x), q, p) ≤ 0.
1,+ t+ ,x v(t, x),
18Ù Ä5y{
ÄK
FT
212
F (t, x, v(t, x), q, p) > 0.
u´dÚn 5.8§(q, p) ∈ D v(t, x). l , (Üb^ (q, p) ∈ D v(t, x)§· (q, p) ∈ D v(t, x)©ùÊ5e)½Â gñ© (ii) (Øaqy© 2 1,+ t− ,x
1,+ t+ ,x
1,+ t,x
§6.
¼ê℄5
3!¥§·= 0¼ê,k5µ℄5© Äk§·ÑXe½Âµ ½Â 6.1. Ω ⊂ IR ´à8§ϕ : Ω → IR©·¡ ϕ ´f ℄, XJ3ÛÜëY ω, é?Û λ ∈ [0, 1] ±9 x, y ∈ Ω, ¤áX
DR A
n
λϕ(x) + (1 − λ)ϕ(y) − ϕ(λx + (1 − λ)y)
≤ λ(1 − λ)|x − y|ω(|x − y|, |x| ∨ |y|).
?Ú§XJk~ê C > 0 ω(s, r) ≤ Cs, ∀ s, r ≥ 0, K· ¡ ϕ ´ (r)℄© w,§ ϕ ´℄§½k, C > 0, ψ(x) ≡ ϕ(x) − C|x| ´Ï~¿Âe℄¼ê: 2
ψ(λx + (1 − λ)y) ≥ λψ(x) + (1 − λ)ψ(y), ∀ λ ∈ [0, 1], x, y ∈ Ω.
3¢C¼ê¥§·?Û (ÛÜk.) ℄ (à) ¼êÑ´ ÛÜ Lipschitz ëY©e¡ÚnL²ù(Jé2¼êa Ǒ¤á©
6.
¼ê℄5
213
FT
Ún 6.2. ϕ : X → IR ´ÛÜk.¼ê§÷vµ λϕ(x) + (1 − λ)ϕ(y) − ϕ(λx + (1 − λ)y) ≤ λ(1 − λ)|x − y|C(|x| ∨ |y|),
Ù¥ C(·) ´üNØ~¼ê, K
∀ x, y ∈ X, λ ∈ [0, 1], (6.1)
n o |ϕ(x) − ϕ(y)| ≤ 2 sup |ϕ(z)| + C(R + 1) |x − y|, |z|≤R+1
∀|x|, |y| ≤ R,
R > 0.
(6.2)
ùÒ´`§ϕ(·) ´ÛÜ Lipschitz ëY©AO§ ϕ ´f℄§ §½´ÛÜ Lipschitz ëY© y². éu?¿÷v |x|, |y| ≤ R x, y ∈ X §·5y² (6.2)©/ x = y ´²
© e¡§·b x 6= y© ξ = , ¿½Â
DR A
x−y |x−y|
w,
θ(t) = ϕ(y + (t − 1)ξ), θ(1) = ϕ(y),
t ∈ [0, |x − y| + 2].
θ(|x − y| + 1) = ϕ(x).
d (6.1)§é?Û s, t ∈ [0, |x − y| + 2] Ú λ ∈ [0, 1],
λθ(t) + (1 − λ)θ(s) − θ(λt + (1 − λ)s)
=
λϕ(y + (t − 1)ξ) + (1 − λ)ϕ(y + (s − 1)ξ) −ϕ(y + [λt + (1 − λ)s − 1]ξ)
≤
λ(1 − λ)|t − s|C(R + 1).
ùp§·|^ ±eOµ
|y + (t − 1)ξ| ≤ |y| + |t − 1| ≤ R + 1, t ∈ [0, 2], t−2 |y + (t − 1)ξ| = |y + (x − y) + ξ| ≤ R + 1, |x − y| t ∈ [2, |x − y| + 2].
(6.3)
18Ù Ä5y{
214
FT
Ó§éu |y + (s − 1)ξ| kÓ(J©éu?Û 0 ≤ t t ≤ |x − y| + 2, ·k 3
t2 =
ù§d (6.3), ( λ = θ(t2 ) ≥
ùÒ
1
< t2
t , A(·) ∈ L (t , T ; IR ), B(·) ∈ L 9 b(·) ∈ L (t , T ; IR )©5UIǑµ ∞
0
n×n
n
0
∞
(t0 , T ; IRn×k ),
±
DR A
0 2
(1.1)
J(u(·)) = J(u(·); t0 , y0 ) Z 1 T n = hQ(t)y(t), y(t)i + 2hS(t)y(t), u(t)i 2 t0 o 1 +hR(t)u(t), u(t)i dt + hGy(T ), y(T )i, 2
(1.2)
Ù¥ Q(·) ∈ L (t , T ; S ), S ∈ L (t , T ; IR ), R ∈ L (t , T ; S ) ±9 G ∈ S ©ùp S L« n × n é¡ N©3þ¡§¤kX êÑ´mCþ t k'©ØÚåØ)§·Ï~Ñ t© éu?Û M ∈ S , ·^ M ≥ 0 Ú M > 0 ©OL« M ´ ½Ú½, ^ M ≥ N (N ∈ S ) L« M − N ≥ 0©?Ú§é u M ∈ L (t , T ; S ), ·P ∞
n
0
n
∞
0
k×n
∞
0
n
n
n
∞
0
n
M ≥ 0 ⇐⇒ M (t) ≥ 0, a.e. t ∈ [t0 , T ], M > 0 ⇐⇒ M (t) > 0, a.e. t ∈ [t0 , T ], M ≫ 0 ⇐⇒ δ>0 M (t) ≥ δI,
3
a.e. t ∈ [t0 , T ].
k
2.
Ú?Ø
221
△
FT
3Ä5XÚg`¯K§·o´ÄXe #N8 U = L2 (t0 , T ; IRk ),
§´ Hilbert m©·'%`¯KQãXeµ ¯K (LQ). Ïé u¯(·) ∈ U , J(¯ u(·)) =
inf J(u(·)).
u(·)∈U
(1.3)
DR A
þã¯K¡Ǒ5g`¯K({¡ LQ ¯K)©· 6ØéÝ Q, G, R K5b©·Ú\eã½Âµ ½Â 1.1. (i) ¡¯K (LQ) ´k§XJ (1.3) mà´k © (ii) ¡¯K (LQ) ´ ()) XJ3 ( )¯ u(·) ∈ U ÷v (1.3)© §2.
Ú?Ø
3ïÄ LQ ¯K§G§55·±|^~ êC´úªòGCþ^Cþwª/L«Ñ5©?ÚN´ w§rGCþLª\5UI §LQ ¯KÒzǑ 3 Hilbert m U ¥ åg¼¯K© d~êC´úª§éu u(·) ∈ U , § (1.1) )Ǒ y(t) ≡ y(t; t0 , y0 , u(·)) = Φ(t, t0 )y0 +
Z
t
t0
Φ(t, s) B(s)u(s) + b(s) ds,
1ÔÙ 5XÚg`¯K
222
∂ Φ(t, s) = A(t)Φ(t, s), ∂t
FT
Ù¥ Φ(·, ·) ´§| (1.1) =£Ý , = Φ(s, s) = I,
ǑB姷P
Z t △ (Lu(·))(t) = Φ(t, s)B(s)u(s)ds, t0 △ b Lu(·) = (Lu(·))(T ), ∀ u(·) ∈ U , Z t △ Φ(t, s)b(s)ds, f (t) = Φ(t, t0 )y0 + t0
ù§·k
y(t) = f (t) + (Lu(·))(t),
∀t0 ≤ s ≤ t ≤ T .
∀ u(·) ∈ U ; t ∈ [t0 , T ],
∀ t ∈ [t0 , T ].
∀ t ∈ [t0 , T ].
(2.1)
é?Û h(·) ∈ Y = L (t , T ; IR ) N´ △
2
n
DR A
0
D
=
= =
Z
Z E h, (Lu(·)) ≡ Y
T
t0 Z T t0 Z T t0 Z T
hh(t), dt
Z
ds
Z h
Z
T
t0
hh(t), (Lu(·))(t)idt
t
Φ(t, s)B(s)u(s)dsidt
t0
t
hh(t), Φ(t, s)B(s)u(s)ids
t0 Z T s
hh(t), Φ(t, s)B(s)u(s)idt
T
B(s)⊤ Φ(t, s)⊤ h(t)dt, u(s)ids D E ≡ (L∗ h(·))(·), u(·) .
=
t0
s
U
ǑÒ´` L ∈ L (U , Y) f L ∗
(L h(·))(·) =
Z
·
∗
∈ L (Y, U )
Ǒ
T
B(·)⊤ Φ(s, ·)⊤ h(s)ds,
∀ h(·) ∈ Y.
(2.2)
2.
Ú?Ø
223 n
½=
b ∗ η)(·) = B(·)⊤ Φ(T, ·)⊤ η, (L
∗
∈ L (IRn , U )
∀ η ∈ IRn ,
D E b hη, Lu(·)i = B(·)⊤ Φ(T, ·)⊤ η, u(·) , U
u´§ÏLO
Ǒ
FT
aq/§ØJ Lb ∈ L (U , IR ) f Lb
(2.3)
∀ η ∈ IRn .
DR A
J(u(·)) E 1 nD ∗ b ∗ GL b + R)u(·), u(·) = (L QL + SL + L∗ S ⊤ + L 2 D U E ∗ ∗ b +2 (L Q + S)f (·) + L (Gf (T )), u(·) U D E o + Qf, f + hGf (T ), Gf (T )i Y E D E o 1 nD N u(·), u(·) + 2 H(·), u(·) +F , (2.4) ≡ 2 U U
Ù¥§ég,/§·r Q, R, S wUXeª½Ânfµ (Qy(·))(·) = Q(·)y(·), (Sy(·))(·) = S(·)y(·),
(Ru(·))(·) = R(·)u(·),
∀ y(·) ∈ Y,
∀ y(·) ∈ Y,
∀ u(·) ∈ U .
3 (2.4) ¥§H(·) Ú F ´ u(·) '©·wÏLrG CþLª\5UI§5`¯KÒz¤Ǒ Hilbert m U þg¼¯K©d§e¡ (JÒ´~g,¯© ½n 2.1. (i) XJ¯K (LQ) ´k§K N ≥ 0.
(2.5)
1ÔÙ 5XÚg`¯K
224
FT
¯K (LQ) ´ () )
= N ≥ 0§
3 ( ) u¯(·) ∈ U ÷v (ii)
Nu ¯(·) + H(·) = 0.
(2.6)
d§u¯(·) ´`© (iii) XJ N ≫ 0 (= N 3
N ∈ L (U ; U ))§K¯K (LQ) 3` u ¯(·),
deªÑµ −1
−1
u¯(·) = −N −1 H(·).
(2.7)
XJéu?Û y ∈ IR , ¯K (LQ) Ñ´), K Au b(·) = 0§¯K (LQ) Ǒ´)© y². (i) db§3~ê C > 0 (iv)
n
0
∀ u(·) ∈ U .
DR A
J(u(·)) ≥ −C,
AO
J(ℓu(·)) C ≥ − 2, ℓ2 ℓ
∀ u(·) ∈ U ; ℓ 6= 0.
u´d (2.4) ª§
D E N u(·), u(·)
U
J(ℓu(·)) ≥ 0. ℓ→∞ ℓ2
= lim
ùÒy² (2.5)© (ii) Äk§ u ¯(·) ∈ U ´¯K (LQ) `, K (5 ¿ N = N ) é?Û u(·) ∈ U , ·k ∗
0 ≤ lim
λ→0
E J(¯ u(·) + λu(·)) − J(¯ u(·)) D = Nu ¯(·) + H(·), u(·) . λ U
du U ´5m§l ½k D
E Nu ¯(·) + H(·), u(·)
U
= 0,
∀ u(·) ∈ U .
2.
Ú?Ø
225
J(u(·)) − J(¯ u(·))
= J(¯ u(·) + u(·) − u ¯(·)) − J(¯ u(·)) D E = Nu ¯(·) + H(·), u(·) − u¯(·) U E 1D ¯(·)), u(·) − u ¯(·) + N (u(·) − u 2 U E 1D = N (u(·) − u ¯(·)), u(·) − u ¯(·) ≥ 0, 2 U u ¯(·)
∀ u(·) ∈ U .
´`© (iii) (Ü (ii) ±9 N ≫ 0, XJ u ¯(·) ∈ U ´`§K (2.6) ¤á. l (2.7) á=©L5§XJ u ¯(·) ∈ U d (2.7) ½Â§KÓd (ii) §½´¯K (LQ) `© (iv) é, b(·)§¯K (LQ) é?Û y Ñ´) ©ǑQãB§ò (2.4) ¥ H(·) PǑ H(·; y , b(·)), Kdb, 3 u¯(·), u¯ (·) ∈ U ©O´¯K (LQ) Au y Ú 0 ` ©d (i) Ú (ii)§§´eã§)©
DR A
l
FT
Ïd§(2.6) ¤á©L5§XJ u¯(·) ∈ U ÷v (2.6)§Kd N ≥ 0 9 (2.4)
0
0
0
0
Nu ¯(·) + H(·; y0 , b(·)) = 0,
üª~=
(2.8)
Nu ¯0 (·) + H(·; 0, b(·)) = 0.
N (¯ u(·) − u ¯0 (·)) + H(·; y0 , 0) = 0.
(2.9)
ùÒL² b(·) = 0 §¯K (LQ) é?Û y ǑÑ´)© , §´d (2.8) )5 (2.9) )5, l y` 2 5© 0
XJ
R ≫ 0,
Q − S ⊤ R−1 S ≥ 0,
G ≥ 0,
(2.10)
K5UI±Ǒ J(u(·))
=
1 2
Z
T
t0
nD
FT
1ÔÙ 5XÚg`¯K
226
E (Q − S ⊤ R−1 S)y(t), y(t)
1h i 2 o + R 2 u(t) + R−1 Sy(t) dt
1 + hGy(T ), y(T )i. 2
(2.11)
l ±y² N ≫ 0 ¤á© (2.10) ¤á§¯K (LQ) ¡Ǒ´ IO LQ ¯K©Ïd§d½n 2.1(iii), IO LQ ¯K´ )©du N ´Ä/ª§Ø´éN´O§Ï |^ (2.7) O`3¢SA^¥´~ØB©Ǒd§·5 ?Ú©Û© ½n 2.2. ¯K(LQ) k|`é (¯y(·), u¯(·)), K
DR A
−1
R(t) ≥ 0,
3eã§) p¯(·): ÷v
(
a.e. t ∈ [t0 , T ],
p¯˙ (t) + A⊤ (t)¯ p(t) + Q(t)¯ y (t) + S(t)⊤ u¯(t) = 0, p¯(T ) = G¯ y (T ),
R(t)¯ u(t) + B(t)⊤ p¯(t) + S(t)¯ y (t) = 0,
a.e. t ∈ [t0 , T ].
(2.12)
(2.13)
(2.14)
y². (¯y(·), u¯(·)) ´¯K (LQ) |`é, Kd½n 2.1, (2.5)—(2.6) ¤á©|^ (2.1)—(2.4) m (2.6) µ 0 =
=
(N u ¯(·))(t) + H(t)
b ∗ GL b + R)¯ ((L∗ QL + SL + L∗ S ⊤ + L u(·))(t)
b ∗ (Gf (T )))(t) +((L∗ Q + S)f (·))(t) + (L
Ú?Ø
227
FT
2.
=
(L∗ Q¯ y(·))(t) + S(t)¯ y (t) + (L∗ S ⊤ u ¯(·))(t)
=
b ∗ G¯ +(L y (T ))(t) + R(t)¯ u(t) Z T B(t)⊤ Φ(s, t)⊤ Q(s)¯ y(s) + S(s)⊤ u ¯(s) ds t
+B(t)⊤ Φ(T, t)⊤ G¯ y (T ) +S(t)¯ y(t) + R(t)¯ u(t),
P p¯(t) =
K
Z
t
T
a.e. t ∈ [t0 , T ].
(2.15)
Φ(s, t)⊤ Q(s)¯ y(s) + S(s)⊤ u ¯(s) ds + Φ(T, t)⊤ G¯ y (T ),
B(t)⊤ p¯(t) + S(t)¯ y (t) + R(t)¯ u(t) = 0,
a.e. t ∈ [t0 , T ].
DR A
= (2.14) ¤á©Ó p¯(·) ÷v§ (2.13)© §d (2.5) (2.12)©ÖögCy²ù(Ø (ëSK 1)© 2
du (2.5) %º (2.12)§Ï (2.12) Ø=´¯K (LQ) ) 7^§Ǒ´¯K (LQ) k7^© ·K 2.3. ¯K(LQ) k§K (2.12) ¤á© uþã(ا3Ä LQ ¯K§8 ·o´b½ R ≥ 0© e¡~fL² (2.12) Ø´¯K (LQ) )¿©^§
LQ ¯Kk5îfu)5© ~ 2.1. ıeXÚµ (
y(t) ˙ = u(t),
y(0) = y0 ∈ IR,
t ∈ [0, T ],
±95UI J(u(·)) =
1 2
Z
T
FT
1ÔÙ 5XÚg`¯K
228
y(t)2 dt.
d§ (1.1)—(1.2) '§ R = 0© ε ∈ (0, T ) ±9 Kk
uε (t) = −
0 ≤ J(uε (·)) =
ù
1 2
Z
ε
0
0
y0 χ[0,ε] (t), ε
∀ t ∈ [0, T ],
ε → 0
t 2 y2ε y02 1 − dt = 0 → 0, ε 6
+
.
inf J(u(·)) = 0.
u(·)∈U
l A LQ ¯K´k©,¡§XJ y 6= 0, K±y ²A LQ ¯KØ´)©ù´ÏǑdé?Û u(·) ∈ U , þ k
DR A
0
y(·; y0 ; u(·)) 6= 0.
l ( 5¿ y(·; y ; u(·)) ëY ) 0
J(u(·)) =
1 2
Z
T
y(t)2 dt > 0,
0
∀ u(·) ∈ U .
Öö±N´/ÞÑ (2.12) ¤á´¯K (LQ) Ø´k ~f©8 §XJ R(t) ≥ 0 ´òz (=3 [t , T ] þ R ≫ 0 ؤ á)§K·¡¯K (LQ) ´ÛÉ(LQ ¯K)©Ïd§þ~¥ LQ ¯K´ÛÉ LQ ¯K©+k~fL²ÛÉ LQ ¯KEk U´)©´±e§3?Ú?Ø¥§·òb 0
R ≫ 0.
(2.16)
d§·±ò½n 2.2 §r¯K (LQ) 5 Hamilton XÚü:>¯KéXå5©
2.
Ú?Ø
229
FT
½n 2.4. (i) N ≥ 0,
(2.16) ¤á, K¯K (LQ) ´( ))
=eãü:>¯Kk()) (¯y(·), p¯(·)) ∈ Y ×
Y: −1 y (t) − BR−1 B ⊤ p¯(t) + b(t), y¯˙ (t) = (A − BR S)¯ p¯˙ (t) = −(A − BR−1 S)⊤ p¯(t) − (Q − S ⊤ R−1 S)¯ y (t), y¯(0) = y , p¯(T ) = G¯ y (T ). 0
d§
u ¯(t) = −R−1 [B ⊤ p¯(t) + S y¯(t)],
t ∈ [t0 , T ],
t ∈ [t0 , T ],
t ∈ [t0 , T ],
(2.17)
(2.18)
DR A
´`§ y¯(·) ǑA`;© (ii) N ≫ 0§
(2.16) ¤á, K (2.17) k)§
¯K (LQ) )§ dª (2.18) Ñ u¯(·) ´¯K (LQ) `© y². (i) Äk§¯K (LQ) ´ () )§(¯y(·), u¯(·)) Ǒ `é§Kd½n 2.2 (2.13)—(2.14)©du (2.16) ¤á, · l (2.14) )Ñ u¯(·) (2.18)©ò§\G§Ú (2.13) = (2.17)©L5§XJ (2.17) k) (¯y(·), p¯(·)), K^ (2.18) ½Â u ¯(·), \ (2.17) = (2.13)§
y¯(·) ǑAu u¯(·) G;© d (2.13) p¯(t) =
Z
t
T
Φ(s, t)⊤ Q(s)¯ y(s) + S(s)⊤ u ¯(s) ds + Φ(T, t)⊤ G¯ y (T ).
5¿ (2.18) =Ǒ (2.14)§r (2.15) íL5= Nu ¯(·) + H(·) = 0.
l d½n 2.1(ii) u¯(·) ´¯K (LQ) `© § 5Ü©y²´{ü©3dÑ©
1ÔÙ 5XÚg`¯K
230
ÖöØJ(Ü (i) Ú½n 2.1(iii) (Ø©
FT
(ii)
2
þã½nL²XJ N ≥ 0§R ≫ 0, K Hamilton XÚ (2.17) 9 (2.18) Ǒx LQ ¯K`)©AOéuIO LQ ¯ K§(2.18) Ñ §`©½n 2.4 ¥¼ê p¯(·) ´ ë`ÚA`;9ϼê©ù«éX´m ©eÚóÒ´Øù9ϼê§ïá` `G;méX©Ǒd§òÚÑͶ Riccati §© §3. Riccati
§Ú"`
DR A
Äk·5/ª/Ñ Riccati §©3 (2.17) ¥§¼ê p¯(·) Ú y¯(·) §´ÍÜ3姷Á^÷v p¯(t) = P (t)¯ y (t) + ϕ(t),
t ∈ [t0 , T ],
(3.1)
¼ê P (·) Ú ϕ(·) 5{z§|Lª©XJù¼ê± é, K`ÒäkXeG"/ªµ u ¯(t) = −R−1 [(B ⊤ P (t) + S)¯ y(t) + B ⊤ ϕ(t)],
t ∈ [t0 , T ].
(3.2)
ù/ª3A^¥´~k^©é (3.1) ü>§¿|^ (2.17)—(2.18) =
=
l
0 =
−(Q − S ⊤ R−1 S)¯ y − (A⊤ − S ⊤ R−1 B ⊤ )[P y¯ + ϕ] = p¯˙ P˙ y¯ + P [A¯ y + Bu ¯ + b + ϕ˙ n h i o P˙ y¯ + P A¯ y − BR−1 (B ⊤ P + S)¯ y + B ⊤ ϕ + b + ϕ. ˙ h P˙ + P (A − BR−1 S) + (A − BR−1 S)⊤ P
=
§Ú"`
231
i −P BR−1 B ⊤ P + Q − S ⊤ R−1 S y¯ h i +ϕ˙ + A⊤ − S ⊤ R−1 B ⊤ − P BR−1 B ⊤ ϕ + P b n o P˙ + P A + A⊤ P − (P B + S ⊤ )R−1 (B ⊤ P + S) + Q y¯
FT
3. Riccati
+ϕ˙ + [(A − BR−1 S)⊤ − P BR−1 B ⊤ ]ϕ + P b.
ù§XJe㧩Ok) P (·) Ú ϕ(·): ⊤ ˙ P (t) + P (t)A + A P (t) + Q
−[B ⊤ P (t) + S]⊤ R−1 [B ⊤ P (t) + S] = 0, P (T ) = G,
a.e. t ∈ [t0 , T ],
h i −1 ⊤ −1 ⊤ ϕ(t) ˙ + (A − BR S) − P (t)BR B ϕ(t) +P (t)b(t) = 0, a.e. t ∈ [t0 , T ], ϕ(T ) = 0,
(3.3) (3.4)
DR A
K (3.1) ¤á§
·±"` (3.2)©§ (3.3) ¡Ǒ¯K (LQ) Riccati §©ù´5~©§© (2.16) ¤á§3 (3.3) 1ªà§Ø P˙ ¼ê'uCþ P ´ÛÜ Lipschitz §Ï T§½´ÛÜ)©=½k , s < T , (3.3) 3 [s, T ] þk () )©
d5´ 3)3«S§P ½´é¡©´§`5§Øk õ^§T§3 [t , T ] )35´ØUy©ù p§·Öö5¿ 0
R ≫ 0,
R>0=
R(t) ≫ 0,
∀ t ∈ [t0 , T ],
(3.5)
m«O ( §1)© öØÓ, öØUy (3.3) ) 5© Äuþã©Û§·?ÚÑXe½nµ
1ÔÙ 5XÚg`¯K
232
FT
½n 3.1. R ≫ 0,
§ (3.3) 3 [t , T ] þk) P (·) ∈ C([t , T ]; S ), K¯K (LQ))§¿
äk`" (3.2), Ù¥ ϕ(·) d (3.4) (½©?Ú§·k 0
n
0
J(¯ u(·)) = =
inf J(u(·))
u(·)∈U
1 hP (t0 )y0 , y0 i + hϕ(t0 ), y0 i 2 Z 2 i 1 Th 1 + 2hϕ(t), b(t)i − R− 2 B ⊤ ϕ(t) dt. 2 t0
(3.6)
y². du R ≫ 0, (3.4) 3 [t , T ] þk) ϕ(·)© y¯(·) Ǒ ±eXÚ): 0
DR A
n h io ˙ (t) = A − BR−1 B ⊤ P (t) + S y¯(t) y ¯ −BR−1 B ⊤ ϕ(t) + b(t), t ∈ [t0 , T ], y¯(t ) = y . 0
0
¿ u¯(·) d (3.2) ѧK y¯(·) ǑAu u¯(·) ;©y3§? u(·) ∈ U , ¿P y(·) = y(· ; t , y , u(·)) ǑAG;©·k 0
= =
±9
0
hP (T )y(T ), y(T )i − hP (t0 )y0 , y0 i Z T d hP (t)y(t), y(t)i dt t0 dt Z T nD E [(P (t)B + S ⊤ )R−1 (B ⊤ P (t) + S) − Q]y(t), y(t) t0 o +2hB ⊤ P (t)y(t), u(t)i + 2hP (t)b(t), y(t)i dt,
=
−hϕ(t0 ), y0 i = hϕ(T ), y(T )i − hϕ(t0 ), y0 i Z T d hϕ(t), y(t)i dt dt t0
§Ú"`
=
Ïd
Z
E [P (t)B + S ⊤ )R−1 B ⊤ ϕ(t) − P (t)b(t), y(t) t0 o +hϕ(t), Bu(t) + b(t)i dt. T
nD
233
FT
3. Riccati
1 J(u(·)) − hP (t0 )y0 , y0 i − hϕ(t0 ), y0 i 2 Z Tn 1 = hRu, ui + h(P B + S ⊤ )R−1 (B ⊤ P + S)y, yi 2 t0
(3.7)
DR A
u´
+2h(B ⊤ P + S)y, ui + 2hR−1 (B ⊤ P + S)y, B ⊤ ϕi o +2hB ⊤ ϕ, ui + 2hϕ, bi dt Z i 2 1 T n − 12 h Ru + (B ⊤ P + S)y + B ⊤ ϕ = R 2 t0 2 o 1 − R− 2 B ⊤ ϕ + 2hϕ, bi dt.
J(¯ u(·))
=
≤
1 hP (t0 )y0 , y0 i + hϕ(t0 ), y0 i 2 Z 2 i 1 Th 1 + 2hϕ(t), b(t)i − R− 2 B ⊤ ϕ(t) dt 2 t0 J(u(·)).
ddá=± u¯(·) ´`§ÓǑ (3.6)© 2
þãy²'
´|^{ (3.7) ª©3½n 3.1 ¥§ ·wXJ Riccati § (3.3) Nk)§K¯K (LQ) ½k )©
``;m'Xd (3.2) Ñ© 5¿ P (·) Ú ϕ(·) Ñ´6u¯K (LQ) G§Ú5UI ¥Ý A, B, b, Q, S, P, G ±9ª:Ǒ T § (t , y ) ´ '©Ï d (3.2) Ñ`´== Gk' 0
0
1ÔÙ 5XÚg`¯K
234
FT
"©ùp§·2g|^ïÄm¯K 4¯K )©XJ½n¥ R ≫ 0 ^ (3.5) O§K/ªþ{E? 1©´dØUy (3.4) k) ϕ(·), =B (3.4) k) ϕ(·), ǑØ Uyd (3.2) Ѽê´á3 U ¥©±e(J?ÚÑ
Riccati §)N35Ú¯K (LQ) )5md 'X© ½n 3.2. R ≫ 0, K¯K(LQ)é?Û y Ñ)
= Riccati § (3.3) )3 [t , T ] þ3© 0
0
y². ¿©5µ3b R ≫ 0 e, XJ Riccati § (3.3) ) 3 [t , T ] þ3§Kd½n 3.1 ¯K (LQ) 3)© 75µXJ¯K (LQ) é?Û y Ñ3)§Kd ½n 2.1(iv), é?Û y ∈ IR , b(·) = 0 §¯K (LQ) E,´ ), l ·± 0
0
DR A
0
n
b(·) = 0.
(3.8)
d½n 2.1(i), N ≥ 0, u´d½n 2.4(i), eã§|3) (¯ y (·), p¯(·)) ∈ Y × Y:
−1 y − BR−1 B ⊤ p¯, t ∈ [t0 , T ], y¯˙ = (A − BR S)¯ −1 ⊤ ⊤ −1 p¯˙ = −(A − BR S) p¯ − (Q − S R S)¯ y, t ∈ [t0 , T ], y¯(t ) = y , p¯(T ) = G¯ y (T ). 0
0
(3.9)
d5 (¯y(·), p¯(·)) 'u y ´5. l k X(·) Ú Y (·) ÷v ( 0
y¯(t) ≡ y¯(t; y0 ) = X(t)y0 ,
p¯(t) ≡ p¯(t; y0 ) = Y (t)y0 ,
t ∈ [t0 , T ].
3. Riccati
§Ú"`
235
−1 ⊤ ˙ b t ∈ [t0 , T ], X(t) = AX(t) − BR B Y (t), ⊤ b b Y (t), Y˙ (t) = −QX(t) −A t ∈ [t0 , T ], X(t ) = I, Y (T ) = GX(T ), 0
Ù¥
FT
´ X(·) Ú Y (·) ´ýéëY§
÷v±e§|
(
·äó
b = A − BR−1 S, A b = Q − S ⊤ R−1 S. Q
det X(t) 6= 0,
ÄK§3 t
1
(3.10)
∈ [t0 , T ]
±9 y
t ∈ [t0 , T ].
0
∈ IRn , y0 6= 0
(3.11)
DR A
y¯(t1 ) ≡ X(t1 )y0 = 0.
d§·k5y² y¯(T ) = 0©Ø t < T ©5¿ y¯(·) ´` ;§A`Ǒ u¯(·) = −R [B p¯(·) + Sy¯(·)] (ë 2.18)§ Kd 1
−1
=
⊤
J(u(·); t0 , y0 ) Z 1 t1 n hQ(t)y(t), y(t)i + 2hS(t)y(t), u(t)i 2 t0 o 1 +hR(t)u(t), u(t)i dt + J(u|[t1 ,T ] (·); t1 , y(t1 )), 2
(¯y| (·), u¯| (·)) Ǒ´¯K (LQ) 3Ǒ (t , y¯(t )) `é (ëÄ5y`5n)©,¡§d (3.8), ± 9 y¯(t ) = 0 A f (·) Ǒ 0, l (2.4) ¥ H(·), F ǑǑ 0. u´§d (2.4) [t1 ,T ]
[t1 ,T ]
1
1
1
D E J(u(·); t1 , y¯(t1 )) = N1 u(·), u(·)
L2 (t1 ,T ;IRk )
,
1ÔÙ 5XÚg`¯K
236
1
1
0
FT
Ù¥ N Ǒ (2.4) ¥d t O t f N ©d½n 2.1 (i), N1 ≥ 0.
l d`5 u¯(t) = 0,
u´qd (3.8) 9G§
a.e. t ∈ [t1 , T ].
y¯(t) = 0,
t ∈ [t1 , T ].
AO p¯(T ) = G¯y(T ) = 0©u´d (3.9) 9)5§y¯(t) Ú p¯(t) 7,3 [t , T ] þðǑ 0, ùb y 6= 0 gñ, l (3.11) ¤á© y30
0
△
P (t) = Y (t)X(t)−1 ,
t ∈ [t0 , T ].
(3.12)
DR A
·òy² P (·) ´ Riccati § (3.3) )©Äk§P (t) ´é¡© ¯¢þ§d (3.10), O i dh ⊤ ⊤ X Y − Y X = 0, dt
Ïd§X
⊤
h i X ⊤ Y − Y ⊤ X
Y − Y ⊤ X ≡ 0.
l
t=T
t ∈ [t0 , T ],
= 0.
P = Y X −1 = (X ⊤ )−1 Y ⊤ = (Y X −1 )⊤ = P ⊤ ,
= P (t) ´é¡©,¡§O Ïd
d −1 b − X −1 BR−1 B ⊤ Y X −1 . X = −X −1 A dt
dY −1 d P˙ = X + Y X −1 dt dt h i b +A b⊤ Y ]X −1 − Y X −1 A b + X −1 BR−1 B ⊤ Y X −1 = −[QX b−A b⊤ P − P A b + P BR−1 B ⊤ P, = −Q
§Ú"`
3. Riccati
237
FT
(Ü P (T ) = G = P (·) ´ (3.3) )©
2
íØ 3.3. éuIOLQ¯K ( =ª (2.10) ¤á ), Riccati § (3.3) 3 [t , T ] þk) P (·)©d§¯K(LQ) ` u ¯(·) dª (3.2) Ñ©?Ú§k 0
P (t) ≥ 0,
∀ t ∈ [t0 , T ].
(3.13)
y². dudy N ≫ 0, d½n 2.1, ¯K (LQ) 3 `©Ïd, 2d½n 3.2, Riccati § (3.3) 3) P (·)© ¤±, ·Iy² (3.13)©Ø5§·Iy² t = t (ؤá©d½n 2.1(iv), ·§dAu b(·) = 0 ¯K (LQ) Ǒ´)©d©§)5§(3.4) ) ϕ(·) 7 ðǑ 0©u´§|^ (2.11) Ú (3.6) µ ∀ y ∈ IR , 0
n
DR A
0
=
hP (t0 )y0 , y0 i = 2J(¯ u(·)) Z T nD E (Q − S ⊤ R−1 S)¯ y (t), y¯(t) t0
1h i 2 o + R 2 u¯(t) + R−1 S y¯(t) dt + hG¯ y (T ), y¯(T )i ≥ 0.
l P (t ) ≥ 0© 0
2
þ¡(JL²éuIO LQ ¯K§Riccati §û½
LQ ¯K)©´§Öö5¿§) LQ ¯K¿Ø½´ IO©¯¢þ§|^½n 3.1, ·éN´E`) Ø´IO LQ ¯K©~X3 [0, T ] þ?| A, B, b, Q, S, R, G, R ≫ 0,
(2.10)(é?Û 0 < t < T ) ؤá ('X± G < 0)©du (3.3) ´ÛÜ)§Ï 3 t ∈ [0, T ) (3.3) 3 [t , T ] þk 0
0
0
1ÔÙ 5XÚg`¯K
238
FT
)©u´d½n 3.1, éu?Û y ∈ IR , A± (t , y ) Ǒ ¯K (LQ) ´)©´§d§¯K (LQ) Ø´I O© e¡·Ñ) LQ ¯K~f. 0
~ 3.1.
(
n
0
0
y˙ = y(t) + u(t), y(0) = 1,
1 J(u(·)) = 2
Z
0
1
1 u2 (t) dt + y 2 (1). 2
DR A
Á J(u(·)) . )µK¥§A = 1, B = 1, b = 0, Q = 0, S = 0, R = 1, G = 1. u´ù´IO LQ ¯K§k`. d§Riccati §Ǒ ( P˙ (t) + 2P (t) − P 2 (t) = 0, P (1) = 1.
) P (t) = 1 + e2 ÷v
.
2(t−1)
u¯(t) = −
t ∈ [0, 1],
du b = 0 á= ϕ = 0. l `
2 y¯(t), 1 + e2(t−1)
t ∈ [0, 1],
Ù¥ y¯ ´A`G. ?Ú§·k J(¯ u(·)) =
§4.
1 1 P (0)y 2 (0) = . 2 1 + e−2
« LQ ¯K
4.
« LQ ¯K
239
(
y(t) ˙ = Ay(t) + Bu(t), y(0) = y0 ,
5UIǑµ =
FT
!0 «5g`¯K©·ÄXê Ý Ǒ~/©Ä±eG§µ t ∈ [0, +∞),
(4.1)
J(u(·)) = J(u(·); y0 ) Z o 1 +∞ n hQy(t), y(t)i + 2hSy(t), u(t)i + hRu(t), u(t)i dt, 2 0
Ù¥ Q ∈ S , S ∈ IR , R ∈ S ©3Ä «g` ¯K§k«/ØÓ§éu n
k×n
k
△
u(·) ∈ U = L2 (0, +∞; IRk ),
DR A
5UI J(u(·)) ¿Ø´g,k½Â©ù´üöm «O©XJ A ¤kAÆÑäkK¢Ü§oN´y²éu ?Û u(·) ∈ U , J(u(·)) ´k¿Â©,¡§Ǒ±B/Þ Ñ~fL² A äk¢ÜKAƧ J(u(·); y ) k¿Â 8 U ⊆ U U6u y ©ù3?n¥´ØB©Ï 3!§·òb½ A ¤kAÆäkK¢Ü ©d§` ¯K´ ¯K (LQ) . Ïé u¯(·) ∈ U , 0
ad
0
1
∞
J(¯ u(·)) =
inf J(u(·)).
u(·)∈U
aquk«/§·½Âµ
1
(4.2)
XJ3Ý K A − BK AÆÑäkK¢Ü§K·±- u(·) = §r¯KzǑù/©
−Ky + v(·)
1ÔÙ 5XÚg`¯K
240
∞
∞
tA
y(t) ≡ y(t; y0 , u(·)) = e y0 +
P Y = L (0, +∞; IR ), 2
n
△ (Lu(·))(t) = △
Z
t
f (t) = etA y0 , Z ∗ (L h(·))(t) =
∀ t ∈ [0, +∞),
+∞
⊤
Z
t
e(t−s)A B(s)u(s)ds.
0
∀ u(·) ∈ U ; t ∈ [0, +∞),
B ⊤ e(s−t)A h(s)ds, t
∀ h(·) ∈ Y,
DR A
K
e(t−s)A Bu(s)ds,
0
FT
½Â 4.1. (i) ¡¯K (LQ) ´k§XJ (4.2) mà´ k© (ii) ¡¯K (LQ) ´ ()), XJ3 ( )¯ u(·) ∈ U ÷v (4.2)© d~êC´úª§éu u(·) ∈ U , § (4.1) )Ǒ
≡
E 1 nD ∗ J(u(·)) = (L QL + SL + L∗ S ⊤ + R)u, u 2 U D E D E o +2 (L∗ Q + S)f, u + Qf, f U Y E D E o 1 nD N u(·), u(·) + 2 H(·), u(·) +F , 2 U U
éAu½n 2.1, 2.2, ·K 2.3 Ú½n 2.4 ·k ½n 4.2. (i) XJ¯K (LQ) ´k§K ∞
N ≥ 0.
¯K (LQ) ´ ( ) )
= N ≥ 0§
3 ( ) u ¯(·) ∈ U ÷v (ii)
∞
Nu ¯(·) + H(·) = 0.
4.
« LQ ¯K
241
D E N u(·), u(·)
U
FT
d§u¯(·) ´`© (iii) XJ N ≫ 0, =3 δ > 0
D E ≥ δ u(·), u(·) , U
K¯K (LQ) ` u¯(·) deªÑµ ∞
∀ u(·) ∈ U ,
u¯(·) = −N −1 H(·).
d V (y0 ) = =
E 1 nD ∗ (L Q + S)e· A y0 , N −1 (L∗ Q + S)e· A y0 2 U 1 hP y0 , y0 i. 2
DR A
≡
inf J(u(·); y0 )
u(·)∈U
ùp P ∈ S © n
½n 4.3. (i) ¯K (LQ) k|`é (¯y(·), u¯(·)), K3 eã§) p¯(·): ∞
(
p¯˙ (t) + A⊤ p¯(t) + Q¯ y(t) + S ⊤ u ¯(t) = 0,
p¯(+∞) = 0,
÷v
R¯ u(t) + B ⊤ p¯(t) + S y¯(t) = 0,
(ii)
¯K (LQ) k§K ∞
R ≥ 0.
a.e. t ∈ [0, +∞).
1ÔÙ 5XÚg`¯K
242
FT
½n 4.4. (i) N ≥ 0,
R > 0, K¯K (LQ) ´ ( ) )
=eãü:>¯Kk ( ) ) (¯y(·), p¯(·)) ∈ Y × Y: y − BR B p¯ + b, u [0, +∞), y¯˙ = (A − BR S)¯ p¯˙ = −(A − BR S) p¯ − (Q − S R S)¯ y , u [0, +∞), ∞
−1
−1
−1
y¯(0) = y , 0
⊤
⊤
⊤
p¯(+∞) = 0.
−1
(4.3)
d§
u¯(t) = −R−1 [B ⊤ p¯(t) + S y¯(t)],
t ∈ [0, +∞),
(4.4)
´`§Ù¥ y¯(·) ǑA`;© (ii) N ≫ 0§
R > 0, K (4.3) k)§
¯K (LQ) )§ dª (4.4) Ñ u¯(·) ´¯K (LQ) "` © 5 4.1. 3½n 4.2 ¥§XJ¯K (LQ) é?Û y Ñ´ )§K|^T½n (i)—(ii), E,±N´/y²3 é¡ P ∞
DR A
∞
∞
V (y0 ) =
inf J(u(·); y0 ) =
u(·)∈U
1 hP y0 , y0 i. 2
0
(4.5)
Au½n 3.1 Ú 3.2 (J3ùpkØÓ§·dÑ ê Riccati §© ½n 4.5. (i) R > 0©±eê Riccati § P A + A⊤ P + Q − (B ⊤ P + S)⊤ R−1 (B ⊤ P + S) = 0,
(4.6)
ké¡) P ∈ S ©
A − BR (B P + S) AÆÑäkK¢ ܧK¯K (LQ) )§¿
äk`" n
−1
⊤
∞
u ¯(t) = −R−1 (B ⊤ P + S)¯ y(t),
t ∈ [0, +∞).
(4.7)
« LQ ¯K
243
?Ú§k J(¯ u(·)) =
FT
4.
1 hP y0 , y0 i. 2
(4.8)
R > 0©XJé?Û y , ¯K (LQ) k§Kê Riccati § (4.6) ké¡) P (4.5) ¤á© y². (i) ùÜ©y²±ì½n 3.1 y²§I 5¿´3y3ù«/e§ϕ(·) ØÑy§ A − BR (B P + S) ¤kAÆÑäkK¢Ü´Ǒ yd (4.7) (½G y¯(·) ÷ v (ii)
0
∞
−1
⊤
lim hP y¯(t), y¯(t)i → 0
t→+∞
d½n 4.2 Ú5 4.1, d (4.5) ¤á§u´|^e¡ `n — ·K 4.6 ±9 V (·) 1w5§·±y² (ii)
(4.9)
DR A
h∇V (y), Ayi + H(y, ∇V (y)) = 0, ∀ y ∈ IRn ,
Ù¥
H(y, p) = =
n o 1 infm hp, Bui + hQy, yi + 2hSy, ui + hRu, ui u∈IR 2 1 1 −1 ⊤ hQy, yi − hR (B p + Sy), B ⊤ p + Syi. (4.10) 2 2
du ∇V (y) = P y, ò§Ú (4.10) \ (4.9)§·
h[P A + A⊤ P + Q − (B ⊤ P + S)⊤ R−1 (B ⊤ P + S)]y, yi = 0, ∀ y ∈ IRn .
dd= (4.6)©
2
·K 4.6. A AÆþäkK¢Ü§N ≫ 0, R > 0, Ké ?Û t > 0, V (y0 )
=
nZ t 1 inf hQy(s; y0 , u(·)), y(s; y0 , u(·))i 2 u(·)∈U 0
FT
1ÔÙ 5XÚg`¯K
244
+2hSy(s; y0 , u(·)), u(s)i o +hRu(s), u(s)i ds + V (y(t; y0 , u(·))) .
~ 4.1. Á?ØXÚ 35UI
dy(t) = y(t) + u(t), y(0) = y0 dt
J(u(·)) =
1 2
Z
+∞
0
{3y 2 (t) + u2 (t)}dt
`. )µ·k A = 1, B = 1, Q = 3, R = 1. d§ê Riccati §Ǒ 3 + 2P − P 2 = 0,
DR A
) P = 3 ½ P = −1. 5¿þ¡ü¥§P = 3 ÷v A − BR (BP + S) < 0§ P = −1 Ø÷v. Ï P = −1. â ½n 4.5, k` −1
u¯(t) = −3¯ y(t),
t ∈ [0, +∞),
Ù¥ y¯ ´A`G. ?Ú§·k J(¯ u(·)) =
1 2 3 P y (0) = y02 . 2 2
5P
¯KÄk´ 1958 d Bellman-Glicksberg-Gross \±ïÄ ( [10] 1oÙ)§1960 §Kalman ïá LQ ¯KG"`§¿r Riccati §Ú\ nØ©
1. LQ
SK
245
§Riccati(1676—1754) 3*l&¥JÑ Xe©§
FT
2. 1720
y˙ = αy 2 + βtm
Ú
y˙ = αy 2 + βt + γt2 .
Ù éùa§?1 mïÄ©