Bài toán điều khiển được hệ phương trình rời rạc tuyến tính

R
n
n
Z
+
A A
I
A

A
|A| A
F
−
(M) F M
A A
˙x(t) = t (t, x(t), u(t)) t ≥ 0,
x(k + 1) = f (k, x(k), u(k)) , k ∈ Z
+
,
x(.) u(.)
x
0
, x

1
u(t)
x
0
x
1
x
0
t
0
x
1
t
1
x(k + 1) = f(k, x(k), u(k)), k ∈ Z
+
.
x(0) = x
0
u
k
= (u(0), u(1), . . . , u(k − 1), . . .) ,
x(1) = f (0, x
0
, u(0))
x(2) = f (1; f (0, x
0
, u(0)) , u(1))
x(3) = f (2, f (1; f (0, x
0

, u(0)) , u(1)) , u(2))
. . . .
f (k, x(k), u(k))
f (k, x(k), u(k))
f (k, x(k), u(k)) = A(k)x(k) + B(k)u(k), k ∈ Z
+
.
x(k + 1) = A(k)x(k) + B(k)u(k), k ∈ Z
+
.
x(0) = x
0
u
k
= (u(0), u(1), . . . , u(k − 1)) ,
x(k) k > 0
x(k) = F (k, 0)x
0
+
k−1

s=0
F (k, s + 1)B(s)u(s),
F (k, s)
x(k + 1) = A(k)x(k), k ∈ Z
+
.
F (k, s)
F (k, s) = A(k − 1) × . . . × A(s), k ≥ s ≥ 0,
F (k, k) = I.

A(.), B(.)
x(k + 1) = Ax(k) + Bu(k).
F (k, s) = A
k−s
, k ≥ s ≥ 0
x(k) = A
k
x
0
+
k−1

s=0
A
k−s−1
Bu(s).
x(k + 1) = A(k)x(k) + B(k)u(k), k ∈ Z
+
,
x(k) ∈ R
n
u(k) ∈ R
m
n ≥ m
A(k), B(k), k ∈ Z
+
(n × n) (n × m)
u(k); k = 0, 1, 2, R
m
x(0) = x

0
u(k)
x (k, x
0
, u) k > 0
x (k, x
0
, u) = F (k, 0)x
0
+
k−1

i=0
F (k, i + 1)B(i)u(i),
F (k, i)



F (k, i) = A(k − 1)A(k − 2) . . . A(i), k > i,
F (k, k) = I.
x
0
, x
1
∈ R
n
(x
0
, x
1

)
k
1
> 0
u(k) x (k, x
0
, u)
x (0, x
0
, u) = x
0
, x (k
1
, x
0
, u) = x
1
.
x
0
, x
1
k
1
> 0 (x
0
, x
1
) k
1

V (0) ⊂ R
n
V (0)
x
1
∈ R
n
k
1
> 0 (0, x
1
) k
1
0
x
0
∈ R
n
k
1
> 0 (x
0
, 0)
k
1
u
k
= (u(0), u(1), . . . , u(k − 1)) ∈ R
km
.

k

k
=

x =
k−1

i=0
F (k, i + 1)B(i)u(i) : u
k
∈ R
km

.
0 k
C
k
= {x : −F (k, 0)x ∈ 
k
} .
0
 =
∞
∪
k>0

k
, C =
∞

∪
k>0
C
k
,

0
= {0} .
 = R
n
0 C = R
n
0 ∈ int 
0 0 ∈ int C
X
ρ(x, y) : X × X → R
+
ρ(x, y) = 0 x = y.
ρ(x, y) = ρ(y, x).
ρ(x, y) ≤ ρ(x, z) + ρ(z, y).
{x
n
} α > 0
N > 0 n, m > N ρ (x
n
, x
m
) < α
(X, ρ)
X

X
X =
∞

i=1
M
i
,
int
¯
M
i
0
= ∅ i
0
≥ 1
X
(X, ρ) ρ(.)
ρ(x, y) = ρ(x − y, 0),
ρ(αx, 0) = |α| ρ(x, 0), ∀α ∈ R C x ∈ X,
ρ(x, 0) x
x = ρ(x, 0).
X, Y A :
X → Y
A(αx + βy) = αAx + βAy,
α, β ∈ R, (x, y) ∈ X × Y
A A
A = {x : Ax ∈ Y } .
A Im A
Im A = {y = Ax, x ∈ A} .

L(X, Y )
A = sup
x≤1
Ax .
Y = R A ∈ L(X, Y )
X X
∗
X x
∗
, x
x
∗
(x) x
∗
∈ X
∗
x ∈ X
X M ⊂ X M
x, y ∈ M λ ∈ [0, 1]
λx + (1 − λ)y ∈ M.
M M M
X = R
n
M =

y =
n

i=1
a

i
x
i
, ∀x
i
∈ M,
n

i=1
a
i
= 1, a
i
≥ 0

.
M M
M
M =

y =
n

i=1
a
i
x
i
, ∀n ≥ 1, ∀a
i

∈ R, x
i
∈ M

.
X M ⊂ X
λM ⊂ M, ∀λ > 0.
M 0 ∈ M M
M = {δx : ∀δ > 0, ∀x ∈ M}.
M M
+
M
+
= {x
∗
∈ X : x
∗
, x ≥ ∀x ∈ M} .
M =

(x
1
, x
2
) ∈ R
2
: x
1
≥ 0, x
2

∈ R

M
+
=

(x
1
, x
2
) ∈ R
2
: x
1
≥ 0, x
2
= 0

.
M =

(x
1
, x
2
) ∈ R
2
: x
1
≤ 0, x

2
= 0

M
+
=

(x
1
, x
2
) ∈ R
2
: x
1
≤ 0, x
2
∈ R

.
M ⊂ X

¯
M

+
= M
+
M
+

¯
M = X M
+
= {0}
x ∈
¯
M x
∗
, x ≥ 0, ∀x
∗
∈ M
+
.
M
1
⊆ M
2
M
+
2
⊆ M
+
1
M
int M = ∅
{A
i
}
i∈N
∈ L(X, X) A

i
A
j
= A
j
A
i
N
A
i
(int M) ⊂ int M, ∀i ∈ N.
M = X.
f
∗
∈ M
+
⊂ X
∗
{A
∗
i
}
i∈N
A
∗
i
f
∗
= λ
i

f
∗
, λ
i
> 0, i ∈ N.
A ∈ L(X, X) X M
X M = X, int M = ∅
AM ⊂ M f
∗
∈ M
+
⊂ X
∗
f
∗
= 0
A
∗
f
∗
= λf
∗
, λ ≥ 0.



x(k + 1) = A(k)x(k) + B(k)u(k); k ∈ Z
+
,
x(k) ∈ R

n
, u(k) ∈ R
m
,
A(k), B(k) (n × n) (n × m)
C(k) = [F (k, k)B(k − 1), F (k, k − 1)B(k − 2), . . . , F (k, 1)B(0)] ; k ∈ Z
+
,
F (k, s)
x(k + 1) = A(k)x(k), k ∈ Z
+
.
k
0
≥ 1
C(k
0
) = n.
L
k
u
k
=
k−1

i=0
F (k, i + 1)B(i)u(i).
L
k
: R

km
→ R
n

k
= L
k

R
km

= Im L
k
.
∞
∪
k=1
L
k

R
km

= R
n
.
k
0
≥ l
L

k
0

R
k
0
m

= R
n
= 
k
0
,
(n × n)
D(k
0
) = C(k
0
)C

(k
0
)
D
−1
(k) x
1
∈ R
n

u(k) = B

(k)F

(k
0
, k + 1) D
−1
(k
0
) x
1
.
x (k
0
) =
k
0
−1

i=0
F (k
0
, i + 1) B(i)B

(i)F

(k
0
, i + 1) D

−1
(k
0
) x
1
= D (k
0
) D
−1
(k
0
) x
1
= x
1
.
0 x
1
∈ R
n

A, B
k
0
≥ 1

B, AB, , A
k
0
−1

B

= n.
A, B
C (k
0
) =

B, AB, , A
k
0
−1
B

.




x
1
(k + 1) = x
1
(k) + x
2
(k) + u(k),
x
2
(k + 1) = x
1

(k) − x
2
(k) + u(k).
A =


1 1
1 −1


, B =


1
1


⇒ AB =


1 1
1 −1




1
1



=


2
0


,
C(2) = [B, AB] =


1 2
1 0


.
C(2) [B, AB] = 2.
0



x
1
(k + 1) = x
1
(k) − x
2
(k) + u(k),
x
2

(k + 1) = x
1
(k) − x
2
(k) + u(k).
A =


1 −1
1 −1


, B =


1
1


⇒ AB =


1 −1
1 −1




1
1



=


0
0


,
C(2) = [B, AB] =


1 0
1 0


.
C(2) [B, AB] = 1 < 2.
k
0
≥ 1
F (k
0
, 0) ⊆ Im C (k
0
) .
C = R
n
C

k
= [−F (k, 0)]
−
(
k
) ,
F
−
(M) M
F
−
(M) = {x ∈ R
n
: F x ∈ M} .
k
0
> 0
C
k
0
= R
n
[−F (k
0
, 0)]
−
(
k
0
) = R

n
.
k
0

A, B
k
0
≥ 1
Im A
k
0
⊆ Im

B, AB, , A
k
0
−1
B

.