Dưới gradient của hàm giá trị trong bài toán điều khiển tối ưu chứa tham số

F : X ⇒ Y X Y
R
R := R ∪ {±∞}
R
n
n
X
∗
X
x
∗
, x x
∗
∈ X
∗
x ∈ X
T
∗
T
L
p
([0, 1], R
n
)
x : [0, 1] → R
n

1

0
|x(t)|
p
dt < +∞
W
1,p
([0, 1], R
n
)
x : [0, 1] → R
n
˙x ∈ L
p
([0, 1], R
n
)
C
0
([0, 1], R
m
)
v : [0, 1] → R
m
C
1
([0, 1], R
n
)
x : [0, 1] → R
n

˙x ∈ C
1
([0, 1], R
n
)
T
−1
T
|x| x ∈ R
n
x
0
x ∈ C
0
([0, 1], R
n
)
x
1
x ∈ C
1
([0, 1], R
n
)
B
X
(x, ρ) x ρ X
∂U U
x ∈ X x X
∅

A ⊂ B (B ⊃ A) A B
A ⊆ B A B
A ∩ B A B
A ∪ B A B
A × B A B
A + B A B
|A| A
∃x x
∀x x
intA A
M(n, m) n × m
M
n,m
(R) R M(n, m)

N(z
0
; Ω) Ω z
0
N(z
0
; Ω) Ω z
0
∇
z
f(¯z, ¯w)) z f (¯z, ¯w)

∂ϕ(¯z) ϕ ¯z

∂

+
ϕ(¯z) ϕ ¯z
domF F
gphF F
S(w) w
h.k.
A
T
A
A := B A B
✷
F : E
1
→ 2
E
2
F
domF = {z ∈ E
1
|F (z) = ∅}
gphF = {(z, v) ∈ E
1
× E
2
|v ∈ F (z)}.

Z
Z
∗
Z ϕ : Z →
¯
R
¯z ∈ Z ϕ(¯z)  ≥ 0
ˆ
∂

ϕ(¯z) := {z
∗
∈ Z
∗
| lim inf
z→¯z
ϕ(z) − ϕ(¯z) − z
∗
, z − ¯z
z − ¯z
≥ −}
 ϕ ¯z z
∗
∈

∂

ϕ(¯z)
 ϕ ¯z  = 0


∂ϕ(¯z) :=

∂
0
ϕ(¯z) ϕ ¯z
z
∗
∈

∂ϕ(¯z) ϕ ¯z
ˆ
∂
+
ϕ(¯z) := −
ˆ
∂(−ϕ)(¯z) ϕ ¯z
Ω Z ¯z ∈ Ω  ≥ 0
 Ω ¯z

N

(¯z; Ω) :=

z
∗
∈ Z
∗
| lim sup
z
Ω

−→¯z
z
∗
, z − ¯z
z − ¯z
≤ 

.
 = 0

N
0
(¯z; Ω) Ω ¯z

N(¯z; Ω) z
∗
∈ Z
∗
Ω ¯z 
k
→ 0
+
z
k
→ ¯z z
∗
k
→ z
∗
z

∗
k
∈

N

k
(z
k
; Ω) k
Ω ¯z N(¯z; Ω)

N(¯z; Ω) ⊂ N(¯z; Ω) Ω

N(¯z; Ω) = N(¯z; Ω) = {z
∗
∈ Z
∗
|z
∗
, z − ¯z ≤ 0, ∀z ∈ Ω}.
α : [a, b] −→ R
[a, b] M > 0
n

i=1
|α(x
i
) − α(x
i−1

)| ≤ M
P = {x
0
, x
1
, , x
n
} [a, b].
X, Y
D ⊂ X
1. h : D → Y
¯x ∈ D η > 0  ≥ 0
h(x) − h(¯x) ≤ x − ¯x, ∀x ∈ B(¯x, η) ∩ D.
2. F : D → 2
Y
(¯x, ¯y) ∈ gphF := {(x, y) ∈ X × Y |y ∈ F (x)}
h : D → Y h ¯x
h(¯x) = ¯y h(x) ∈ F (x) x ¯x D.
X W Z
X
∗
W
∗
Z
∗
M : Z → X T : W → X
M
∗
: X
∗

→ Z
∗
T
∗
: X
∗
→ W
∗
M T f : Z × W →
¯
R
Ω Z w ∈ W
H(w) = {z ∈ Z| Mz = T w}.
h(w) := inf
z∈H(w)∩Ω
f(z, w).
ˆ
S(w) = {z ∈ H(w) ∩ Ω| h(w) = f(z, w)}
w ∈ W ¯z
¯w ¯z ∈

S( ¯w)
h
¯w
h
¯w ∈ dom
ˆ
S
(i) f (¯z, ¯w)
(ii) M

(iii) Ω Z intΩ = ∅

∂h( ¯w) ⊆

z
∗
∈

N(¯z;Ω)
[∇
w
f(¯z, ¯w) + T
∗
((M
∗
)
−1
(∇
z
f(¯z, ¯w) + z
∗
))].
ˆ
S
( ¯w, ¯z)

∂h( ¯w) =

z
∗

∈

N(¯z;Ω)
[∇
w
f(¯z, ¯w) + T
∗
((M
∗
)
−1
(∇
z
f(¯z, ¯w) + z
∗
))].
u ∈ C
0
([0, 1], R
m
)
x ∈ C
1
([0, 1], R
n
)
g(x(1)) +

1
0

L(t, x(t), u(t), θ(t))dt
˙x(t) = A(t)x(t) + B(t)u(t) + T (t)θ(t) h.k. t ∈ [0, 1],
x(0) = α,
u ∈ U.
C
0
([0, 1], R
m
)
v : [0, 1] → R
m
C
1
([0, 1], R
n
)
x : [0, 1] → R
n
˙x ∈ C
0
([0, 1], R
n
)
x ∈ C
1
([0, 1], R
n
)
x
1

= x
0
+  ˙x
0
x
0
= max
t∈[0,1]
|x(t)|.
x, u
(α, θ) ∈ R
n
× C
0
([0, 1], R
k
)
g : R
n
→
¯
R L : [0, 1] × R
n
× R
m
× R
k
→
¯
R

A(t) = (a
ij
(t))
n×n
B(t) = (b
ij
(t))
n×m
T (t) = (c
ij
(t))
n×k
U C
0
([0, 1], R
m
)
X = C
1
([0, 1], R
n
), U = C
0
([0, 1], R
m
),
Θ = C
0
([0, 1], R
k

), W = R
n
× Θ
w = (α, θ) ∈ W V (w) V : W →
¯
R
V
w = (α, θ) ∈ W
J(x, u, w) = g(x(1)) +

1
0
L(t, x(t), u(t), θ(t))dt,
G(w) = {z = (x, u) ∈ X × U| },
K = X × U.
V (w) := inf
z∈G(w)∩K
J(z, w).
w ∈ W S(w)
X = W
1,p
([0, 1], R
n
), U = L
p
([0, 1], R
m
), Θ = L
p
([0, 1], R

k
).
W
1,p
([0, 1], R
n
)
x : [0, 1] → R
n
˙x ∈ L
p
([0, 1], R
n
)
V ¯w = (¯α,
¯
θ)
(H1) L : [0, 1] × R
n
× R
m
× R
k
→
¯
R g : R
n
→
¯
R

L(·, x, u, v) (x, u, v) ∈ R
n
× R
m
× R
k
L(t, ·, ·, ·) g(·) t ∈ [0, 1]
(H2) A : [0, 1] → M
n,n
(R) B : [0, 1] → M
n,m
(R)
T : [0, 1] → M
n,k
(R)
(H3) (x, u) ∈ S(w).
A : X → X B : U → X
M : X × U → X T : W → X
Ax = x −

(·)
0
A(τ)x(τ )dτ,
Bu = −

(·)
0
B(τ)u(τ)dτ,
M(x, u) = Ax + Bu,
T (α, θ) = α +


(·)
0
T (τ )θ(τ)dτ.
(H2) (H3)
G(w)
= {(x, u) ∈ X × U| x = α +

(·)
0
Axdτ +

(·)
0
Budτ +

(·)
0
T θdτ }
= {(x, u) ∈ X × U| x −

(·)
0
Axdτ −

(·)
0
Budτ = α +

(·)

0
T θdτ }
= {(x, u) ∈ X × U| M(x, u) = T (w)}.
U
∗
X
∗
C
0
([0, 1], R
m
) C
1
([0, 1], R
n
)
u
∗
∈ U
∗
u
∗
, u = a, u(0) +
m

i=1

1
0
u

i
(t)dµ
i
(t),
a ∈ R
m
µ
i
(t) µ
m
(t)
0 µ
i
(0) = 0, ∀i = 1, m
x
∗
∈ X
∗
x
∗
, x = b, x(0) + c, ˙x(0) +
n

i=1

1
0
˙x
i
(t)dλ

i
(t),
b, c ∈ R
n
λ
i
(t) λ
n
(t)
0 λ
i
(0) = 0, ∀i = 1, n
u
∗
(a, µ) x
∗
(b, c, λ)
µ = (µ
1
, , µ
m
) λ = (λ
1
, , λ
n
) .
T
∗
: X
∗

→ W
∗
, A
∗
: X
∗
→ X
∗
, B
∗
: X
∗
→ U
∗
M
∗
: X
∗
→ X
∗
× U
∗
T A B
M
(a) M T
(b) M
(c) T
∗
(b, c, λ) =


b, T
T
(0)c, T
T
(t)λ(t) −

t
0
˙
T
T
(τ)λ(τ )dτ

;
(d) M
∗
(b, c, λ) =

A
∗
(b, c, λ), B
∗
(b, c, λ)

B
∗
(b, c, λ) =

− B
T

(0)c, −B
T
(t)λ(t) +

t
0
˙
B
T
(s)λ(s)ds

,
A
∗
(b, c, λ) = (b
∗
, c
∗
, λ
∗
),
b
∗
= b − A
T
(0)c −

1
0
A

T
(t)dλ(t), c
∗
= c,
λ
∗
(t) = λ(t) +

t
0
A
T
(s)λ(s)ds − t

1
0
A
T
(s)dλ(s) −

t
0
ϕ
1
(s)ds,
ϕ
1
(s) =

s

0
˙
A
T
(τ)λ(τ )dτ.
(a) x(0) + || ˙x||
0
||x||
0
+ || ˙x||
0
x ∈ X A,
k
0
, k
1
Ax
1
≤ x
1
+ max
t∈[0,1]
|A(t)x(t)|
≤ x
1
+ k
0
x
0
≤ x

1
+ k
0
x
1
≤ k
1
x
1
.
B Bu
1
≤ k
2
u
0
k
2
k
3
M(x, u)
1
≤ k
3
(x
1
+ u
0
).
M T

(b) y X, Ax = y

˙x = Ax + ˙y
x(0) = α.
x ∈ X
A B
M
(c) T
∗
(b, c, λ) = (d, a, µ)
(b, c, λ), T (α, θ) = (d, a, µ), (α, θ) .
b, α + c, T (0)θ(0) +

1
0
T (t)θ(t)dλ(t)
= d, α + a, θ(0) +

1
0
θ(t)dµ(t).
b, α +

T
T
(0)c, θ(0)

+

1

0
T (t)θ(t)dλ(t)
= d, α + a, θ(0) +

1
0
θ(t)dµ(t).

1
0
T (t)θ(t)dλ(t) = T (t)θ(t)λ(t)|
1
0
−

1
0
λ(t)d(T (t)θ(t))
= T (t)θ(t)λ(t)|
1
0
−

1
0
λ(t)T (t)dθ(t) −

1
0
λ(t)θ(t)dT (t)

= T (t)θ(t)λ(t)|
1
0
− λ(t)T (t)θ(t)|
1
0
+

1
0
θ(t)d(λ(t)T (t))
−

1
0
λ(t)θ(t)
˙
T (t)dt
=

1
0
θ(t)d(T
T
(t)λ(t)) −

1
0
˙
T

T
(t)λ(t)θ(t)dt
=

1
0
θ(t)d

T
T
(t)λ(t) −

t
0
˙
T
T
(τ)λ(τ )dτ

.
b, α +

T
T
(0)c, θ(0)

+

1
0

θ(t)d

T
T
(t)λ(t) −

t
0
˙
T
T
(τ)λ(τ )dτ

= d, α + a, θ(0) +

1
0
θ(t)dµ(t).



d = b
a = T
T
(0)c
µ = T
T
(t)λ(t) −

t

0
˙
T
T
(τ)λ(τ )dτ.
(d) B
∗
(b, c, λ) (b, c, λ) ∈ X
∗
.
B
∗
(b, c, λ) = (a, µ).
(b, c, λ), Bu = (a, µ), u , ∀u ∈ U.
b, 0 + c, −B(0)u(0) −

1
0
B(t)u(t)dλ(t) = a, u(0) +

1
0
u(t)dµ(t).

−B
T
(0)c, u(0)

−


1
0
B(t)u(t)dλ(t) = a, u(0) +

1
0
u(t)dµ(t).

1
0
B(t)u(t)dλ(t) = B(t)u(t)λ(t)|
1
0
−

1
0
λ(t)d(B(t)u(t))
= B(t)u(t)λ(t)|
1
0
−

1
0
λ(t)B(t)du(t) −

1
0
λ(t)u(t)dB(t)

= B(t)u(t)λ(t)|
1
0
− λ(t)B(t)u(t)|
1
0
+

1
0
u(t)d(B
T
(t)λ(t))
−

1
0
λ(t)u(t)
˙
B(t)dt
=

1
0
u(t)d(B
T
(t)λ(t)) −

1
0

˙
B
T
(t)λ(t)u(t)dt
=

1
0
u(t)d(B
T
(t)λ(t)) −

1
0
u(t)d


t
0
˙
B
T
(s)λ(s)ds

=

1
0
u(t)d


B
T
(t)λ(t) −

t
0
˙
B
T
(s)λ(s)ds

.

−B
T
(0)c, u(0)

+

1
0
u(t)d

−B
T
(t)λ(t) +

t
0
˙

B
T
(s)λ(s)ds

= a, u(0) +

1
0
u(t)dµ(t),
u ∈ U.

a = −B
T
(0)c
µ = −B
T
(t)λ(t) +

t
0
˙
B
T
(s)λ(s)d(s).
A
∗
(b, c, λ) (b, c, λ) ∈ X
∗
.
A

∗
(b, c, λ) = (b
∗
, c
∗
, λ
∗
).
x ∈ X,
(b, c, λ), Ax = (b
∗
, c
∗
, λ
∗
), x .
b, x(0) + c, ˙x(0) − A(0)x(0) +

1
0
( ˙x(t) − A(t)x(t))dλ(t)
= b
∗
, x(0) + c
∗
, ˙x(0) +

1
0
˙x(t)dλ

∗
(t).

b − A
T
(0)c, x(0)

+ c, ˙x(0) +

1
0
˙x(t)dλ(t) −

1
0
A(t)x(t))dλ(t)
= b
∗
, x(0) + c
∗
, ˙x(0) +

1
0
˙x(t)dλ
∗
(t).

1
0

A(t)x(t)dλ(t) = A(t)x(t)λ(t)|
1
0
−

1
0
λ(t)d(A(t)x(t))
= A(t)x(t)λ(t)|
1
0
−

1
0
λ(t)A(t)dx(t) −

1
0
λ(t)x(t)dA(t)
= A(1)x(1)λ(1) −

1
0
λ(t)A(t) ˙x(t)dt −

1
0
˙
A

T
(t)λ(t)x(t)dt
= A(1)x(1)λ(1) −

1
0
˙x(t)d


t
0
A
T
(s)λ(s)ds

−

1
0
x(t)d


t
0
˙
A
T
(s)λ(s)ds

= A(1)x(1)λ(1) −


1
0
˙x(t)d


t
0
A
T
(s)λ(s)ds

−

x(t)

t
0
˙
A
T
(s)λ(s)ds




1
0
+


1
0


t
0
˙
A
T
(s)λ(s)ds

dx(t)
= A(1)x(1)λ(1) − x(1)

1
0
˙
A
T
(t)λ(t)dt +

1
0
˙x(t)d


t
0
ϕ
1

(s)ds

−

1
0
˙x(t)d


t
0
A
T
(s)λ(s)ds

,
ϕ
1
(s) =

s
0
˙
A
T
(t)λ(t)dt.
A(1)x(1)λ(1) − x(1)

1
0

˙
A
T
(t)λ(t)dt
= A(1)x(1)λ(1) − x(1)

λ(t)A
T
(t)|
1
0
−

1
0
A
T
(t)dλ(t)

= A(1)x(1)λ(1) − A(1)x(1)λ(1) + x(1)

1
0
A
T
(t)dλ(t)
=

x(1),


1
0
A
T
(t)dλ(t)

=

x(0) +

1
0
˙x(t)dt,

1
0
A
T
(t)dλ(t)

=

x(0),

1
0
A
T
(t)dλ(t)


+


1
0
˙x(t)dt,

1
0
A
T
(t)dλ(t)

=


1
0
A
T
(t)dλ(t), x(0)

+

1
0
˙x(t)d

t


1
0
A
T
(s)dλ(s)

.

b − A
T
(0)c −

1
0
A
T
(t)dλ(t), x(0)

+ c, ˙x(0)
+

1
0
˙x(t)d

λ(t) − t

1
0
A

T
(s)dλ(s) +

t
0
A
T
(s)λ(s)ds −

t
0
ϕ
1
(s)ds

= b
∗
, x(0) + c
∗
, ˙x(0) +

1
0
˙x(t)dλ
∗
(t).
b
∗
= b − A
T

(0)c −

1
0
A
T
(t)dλ(t),
c
∗
= c,
λ
∗
(t) = λ(t) − t

1
0
A
T
(s)dλ(s) +

t
0
A
T
(s)λ(s)ds −

t
0
ϕ
1

(s)ds.

K = X × U
G(w) = {(x, u) ∈ X × U| M(x, u) = T (w)}.