Chỉnh hóa phương trình Parabolic ngược thời gian với hệ số phụ thuộc thời gian bằng phương pháp Tikhonov

X Y
A : X → Y
A(λu + µv) = λAu + µAv, ∀u, v ∈ X, λ, µ ∈ R.
A : X → Y
A := sup{Au
Y
|u
X
 1} < ∞.
H
., .
A : H → H
A
∗
: H → H Au, v = u, A
∗
v, ∀u, v ∈ H
A
A A
∗
= A.
u, v ∈ H
u, v = 0.
{w
k
}
k1

⊂ H

w
k
, w
l
 = 0, (k, l = 1, 2, , k = l),
w
k
 = 1, (k = 1, 2, ).
u ∈ H {w
k
}
k1
⊂ H
u =
+∞

k=1
u, w
k
w
k
u
2
=
+∞

k=1
|u, w

k
|
2
.
A : H → H
A
f : Ω → F Ω
E F f x
0
Ω S ∈ L(E, F )
f(x
0
+ h) − f(x
0
) − S(h) = o(h).
∀ > 0, ∃δ > 0, ∀h : h < δ
f(x
0
+ h) − f(x
0
) − S(h)  (h).
lim
h→0
f(x
0
+ h) − f(x
0
) − S(h)
h
= 0.

f Ω Ω
f x
0
E
S f

(x
0
) Df(x
0
) f x
0
S ∈ L(E, F )
T ∈ L(E, F ) (1.1)
S T f x
0
∈ Ω
f(x
0
+ h) − f(x
0
) − S(h) = o(h)
f(x
0
+ h) − f(x
0
) − T (h) = o(h).
S(h) − T (h)
= (f(x
0

+ h) − f(x
0
) − S(h)) − (f(x
0
+ h) − f(x
0
) − T (h))
= o(h) + o(h) = o(h).
lim
h→0
S(h) − T (h)
h
= 0 h ∈ E h = 0
S(th) − T (th)
th
=
tS(h) − tT (h)
|t|h
=
S(h) − T (h)
h
∀t ∈ R, t = 0
S(h) − T (h)
h
= lim
t→0
S(th) − T (th)
th
= 0.
S(h) − T(h) = 0. S(h) = T(h), ∀h = 0

S(h) = T (h), ∀h ∈ E S ≡ T
f Ω f

: Ω → L(E, F )
Ω  x → f

(x) ∈ L(E, F ).
f

f
C
1
f ∈ C
1
Ω.
f

(x
0
) ∈ L(E, F ) f x
0
E = R
Ψ(T ) = T(1), T ∈ L(R, F )
L(R, F ) F
T ∈ L(R, F ) T (1)
f : (a, b) → F x
0
∈ (a, b)
lim
h→0

[
f(x
0
+ h) − f(x
0
)
h
−f

(x
0
)(1)] = lim
h→0
f(x
0
+ h) − f(x
0
) − f

(x
0
)(h)
h
= 0.
f

(x
0
) f


(x
0
)(1)
f

(x
0
) = lim
h→0
f(x
0
+ h) − f(x
0
)
h
f : Ω → F, f = const, f

= 0 Ω
f ∈ L(E, F ) f x
0
∈ E f

(x
0
) = f
f x
0
∈ E
lim
h→0

f(x
0
+ h) − f(x
0
) − S(h)
h
= 0
⇔ lim
h→0
f(x
0
) + f(h) − f(x
0
) − S(h)
h
= 0 f ∈ L(E, F ) .
f = S ∈ L(E, F )
lim
h→0
f(x
0
) + f(h) − f(x
0
) − S(h)
h
= lim
h→0
0
h
= 0.

f

(x
0
) = f
E
f = S \Ω S : E
1
×E
2
→ F
(x
0
1
, x
0
2
) ∈ Ω
f((x
0
1
, x
0
2
) + (h
1
, h
2
)) − f(x
0

1
, x
0
2
) − S(x
0
1
, h
2
) − S(h
1
, x
0
2
)
= S(x
0
1
+ h
1
, x
0
2
+ h
2
) − S(x
0
1
, x
0

2
) − S(x
0
1
, h
2
) − S(h
1
, x
0
2
)
= S(x
0
1
, x
0
2
) + S(x
0
1
, h
2
) + S(h
1
, x
0
2
) + S(h
1

, h
2
)
− S(x
0
1
, x
0
2
) − S(x
0
1
, h
2
) − S(h
1
, x
0
2
)
= S(h
1
, h
2
)  S h
1
 h
2
 = o((h
1

, h
2
)).
f (x
0
1
, x
0
2
)
f

(x
0
1
, x
0
2
)(h
1
, h
2
) = S(x
0
1
, h
2
) + S(h
1
, x

0
2
), ∀(h
1
, h
2
) ∈ E
1
× E
2
.
f = S \ Ω Ω ⊂ E
1
× × E
n
S ∈ L(E
1
, , E
n
; F ) f (x
1
, , x
n
) ∈ Ω
f

(x
1
, , x
n

)(h
1
, h
n
) = S(h
1
, x
1
, , x
n
) + + S(x
1
, , x
n−1
, h
n
),
∀(h
1
, , h
n
) ∈ E
1
× × E
n
E F
Ω E x
0
∈ Ω
f, g : Ω → F x

0
αf + βg x
0
α β ∈ R
(αf + βg)

(x
0
) = αf

(x
0
) + βg

(x
0
).
f : Ω → R g : Ω → R x
0
gf : Ω → R
x
0
(gf)

(x
0
) = g

(x
0

)f(x
0
) + g(x
0
)f

(x
0
).
g(x
0
) = 0 f/g x
0

f
g


(x
0
) =
g(x
0
)f

(x
0
) − f(x
0
)g


(x
0
)
g
2
(x
0
)
.
0  lim
h→0
(αf + βg)(x
0
+ h) − (αf + βg)(x
0
) − αf

(x
0
)(h) − βg

(x
0
)(h)
h
= lim
h→0
αf(x
0

+ h) − αf(x
0
) − αf

(x
0
)(h)
h
+ lim
h→0
βg(x
0
+ h) − βg(x
0
) − βg

(x
0
)(h)
h
 lim
h→0
|α|f(x
0
+ h) − f(x
0
) − f

(x
0

)(h)
h
+ lim
h→0
|β|g(x
0
+ h) − g(x
0
) − g

(x
0
)(h)
h
 0 f, g x
0
.
(αf + βg)

(x
0
) = αf

(x
0
) + βg

(x
0
)

gf x
0
A = lim
h→0
g(x
0
+ h)f(x
0
+ h) − g(x
0
)f(x
0
) − g

(x
0
)f(x
0
) − g(x
0
)f

(x
0
)
h
= 0.
0  A  lim
h→0
f(x

0
+ h)[g(x
0
+ h) − g(x
0
) − g

(x
0
)(h)]
h
+ lim
h→0
g(x
0
)[f(x
0
+ h) − f(x
0
) − f

(x
0
)(h)]
h
+ lim
h→0
f(x
0
+ h)g


(x
0
)(h) − f(x
0
)g

(x
0
)(h)
h
.
g x
0
0  A
1
= lim
h→0
f(x
0
+ h)[g(x
0
+ h) − g(x
0
) − g

(x
0
)(h)]
h

= |f(x
0
)| lim
h→0
g(x
0
+ h) − g(x
0
) − g

(x
0
)(h)
h
= |f(x
0
)|.0 = 0.
A
1
= 0
f x
0
0  A
2
= lim
h→0
g(x
0
)[f(x
0

+ h) − f(x
0
) − f

(x
0
)(h)]
h
= |g(x
0
)| lim
h→0
f(x
0
+ h) − f(x
0
) − f

(x
0
)(h)
h
= |g(x
0
)|.0 = 0.
A
2
= 0
0  A
3

= lim
h→0
f(x
0
+ h)g

(x
0
)(h) − f(x
0
)g

(x
0
)(h)
h
= lim
h→0
[f(x
0
+ h) − f(x
0
)]g

(x
0
)(h)
h
= lim
h→0

|f(x
0
+ h) − f(x
0
)|
g

(x
0
)(h)
h
 lim
h→0
|f(x
0
+ h) − f(x
0
)|
g

(x
0
)h
h
= lim
h→0
|f(x
0
+ h) − f(x
0

)|g

(x
0
)
= |f(x
0
) − f(x
0
)|g

(x
0
) = 0.
A
3
= 0
0  A  0 A = 0
g(x
0
) = 0
1
g
x
0
lim
h→0
1
h






1
g(x
0
+ h)
−
1
g(x
0
)
+
g

(x
0
)(h)
g
2
(x
0
)





= lim

h→0
1
h





g(x
0
+ h) − g(x
0
)
g(x
0
+ h)g(x
0
)
−
g

(x
0
)(h)
g
2
(x
0
)






= lim
h→0
g(x
0
+ h) − g(x
0
) − g

(x
0
)(h)
g(x
0
+ h)g(x
0
)
= 0.
f
g
= f.
1
g
x
0

f

g


(x
0
) =
g(x
0
)f

(x
0
) − f(x
0
)g

(x
0
)
g
2
(x
0
)
.
E F G
U ⊂ E V ⊂ F x
0
∈ U f : U →
V, g : V → G x

0
y
0
= f(x
0
) g◦f : U → G
x
0
(g ◦f)

(x
0
) = g

(f(x
0
))f

(x
0
).
f x
0
lim
h→0
f(x
0
+ h) − f(x
0
) − f


(x
0
)(h)
h
= 0.
x = x
0
+ h
lim
x−x
0
→0
f(x) − f(x
0
) − f

(x
0
)(x − x
0
)
(x − x
0
)
= 0.
ϕ(x − x
0
) = f(x) − f(x
0

) − f

(x
0
)(x − x
0
).
lim
x→x
0
ϕ(x − x
0
)
x − x
0

= 0
f(x) − f(x
0
) = f

(x
0
)(x − x
0
) + ϕ(x − x
0
).
g y
0

g(y) − g(y
0
) = g

(y
0
)(y −y
0
) + ψ(y −y
0
).
lim
y→y
0
ψ(y −y
0
)
y −y
0

= 0
g ◦f(x) − g ◦f(x
0
) = g(f(x)) − g(f(x
0
))
= g

(f(x
0

))(f(x) − f(x
0
)) + ψ(f(x) − f(x
0
))
= g

(f(x
0
))[f

(x
0
)(x − x
0
) + ϕ(x − x
0
)] + ψ(f(x) − f(x
0
))
= g

(f(x
0
))f

(x
0
)(x − x
0

) + g

(f(x
0
))ϕ(x − x
0
) + ψ(f(x) − f(x
0
)).
g ◦f x
0
B = lim
x→x
0
g

(f(x
0
))ϕ(x − x
0
) + ψ(f(x) − f(x
0
))
(x − x
0
)
= 0.
0  B  lim
x→x
0

g

(f(x
0
))ϕ(x − x
0
)
(x − x
0
)
+ lim
x→x
0
ψ(f(x) − f(x
0
))
(x − x
0
)
 lim
x→x
0
g

(f(x
0
))
ϕ(x − x
0
)

x − x
0

+ lim
x→x
0
ψ(f(x) − f(x
0
))
f(x) − f(x
0
)
.
f(x) − f(x
0
)
x − x
0

 0.
B = 0
g ◦f x
0
(g ◦f)

(x
0
) = g

(f(x

0
))f

(x
0
)
X
K X
∗
= L(X, K)
X X
∗
X
{x
n
} X x
0
x
n
→ x
0
n → ∞ x
n
− x
0
 → 0 n → ∞
{x
n
} ⊂ X x
0

∈ X ∀f ∈ X
∗
f(x
n
) → f(x
0
) n → ∞ {x
n
} x
0
x
n
 x
0
X
X x
n
 x x
n
→ x
M X {x
n
} ⊂ M x
n
 x
x
n
→ x
x
n

 x x  lim inf x
n

{x
n
} X x ∈ X
f ∈ X
∗
f f(x
n
) → f(x) n → ∞ x
n
 x
{x
n
} ⊂ X, x, y ∈ X x
n
 x, x
n
 y
x = y x
n
 x ∀f ∈ X
∗
f(x
n
) → f(x) n → ∞
|f(x
n
) −f(x)| → 0 n → ∞ x

n
 y ∀f ∈ X
∗
|f(x
n
) − f(y)| → 0 n → ∞
0  |f(x) − f(y)| = |f(x) − f(x
n
) + f(x
n
) − f(y)|
 |f(x) − f(x
n
)| + |f(x
n
) − f(y)| → 0, n → ∞.
|f(x) − f(y)| = 0, ∀f ∈ X
∗
f(x) = f(y), ∀f ∈ X
∗
f(x − y) = 0, ∀f ∈ X
∗
x − y = 0
g ∈ X
∗
g = 1, g(x −y) = x −y = 0
x − y = 0 x = y
X k K
{e
1

, e
2
, , e
k
} x ∈ X x =
k

i=1
x
i
e
i
{x
n
} X
x
n
=
k

i=1
x
n
i
e
i
f
i
(y) = y
i

, ∀y ∈ X, ∀i = 1, , k f
i
∈ X
∗
, ∀i = 1, , k
x
n
 x f
i
(x
n
) → f(x), ∀i = 1, , k x
n
i
→ x
i
, ∀i = 1, , k
x
n
−x =

k

i=1
|x
n
i
− x
i
|

2

1/2
→ 0, n → ∞ x
n
→ x
x
n
x ∃ > 0 : x
n
k
−x  , ∀k
M {x
n
k
i
}
y y = x ε  x
n
k
i
−x → 0 i → ∞
x
n
→ x
{x
n
} X ∀f ∈ X
∗
f(x

n
) =
x
n
(f) K (x
n
)
n∈N
{x
n
}
sup
n∈N
x
n
 < +∞.
X
x = 0
x = 0
f ∈ X
∗
f = 1 x = x, f = lim
n→∞
x
n
, f  lim inf
n→∞
x
n
.

X
·, · (ϕ
n
) X ϕ ∈ X ϕ
n
 ϕ
n → ∞
ϕ
n
, ψ → ϕ, ψ, n → ∞
ψ ∈ X
φ ϕ
n
ϕ −φ, ψ = 0
ψ ∈ X ψ = ϕ − φ ϕ = φ
H {e
n
:
n ∈ N} H {e
n
}
∀ m, n ∈ N
∗
, m = n
e
m
− e
n

2

= e
m
− e
n
, e
m
− e
n
 = e
m
, e
m
 − 2e
m
, e
n
 + e
n
, e
n

= e
m

2
+ e
n

2
= 2.

e
m
− e
n
 =
√
2 m = n {e
n
}
{e
n
} H
f ∈ H
∗
a ∈ H f(x) = a, x, ∀ x ∈ H
f(e
n
) = a, e
n
, ∀ n ∈ N
∗
a ∈ H {e
n
: n ∈ N}
H a =
+∞

n=1
a, e
n

e
n
a
2
=
+∞

n=1
|a, e
n
|
2
+∞

n=1
|a, e
n
|
2
|a, e
n
| → 0 n → ∞ a, e
n
 →
0 n → ∞ f(e
n
) → 0 = f(0) n → ∞, ∀ f ∈ H
∗
e
n

0
X
(ϕ
n
) ϕ ∈ X (ψ
n
)
ψ ∈ X (ϕ
n
, ψ
n
) ϕ, ψ.
(ϕ
n
) ϕ ∈ X (ϕ
n
) ϕ
(ϕ
n
) ϕ ∈ X.
(ϕ
n
) ϕ ∈ X (ϕ
n
)
∃M > 0 : ϕ
n
  M, ∀n ∈ N.
|ϕ
n

, ψ
n
 − ϕ, ψ|  |ϕ
n
, ψ
n
| − |ϕ
n
, ψ| + |ϕ
n
, ψ| − |ϕ, ψ|
 ϕ
n
.ψ
n
− ψ + |ϕ
n
, ψ − ϕ, ψ|
 M.ψ
n
− ψ + |ϕ
n
, ψ − ϕ, ψ|.
n → ∞
lim
n→∞
ϕ
n
, ψ
n

 = ϕ, ψ.
(ϕ
n
, ψ
n
) → ϕ, ψ.
ϕ
n
−ϕ
2
= ϕ
n
− ϕ, ϕ
n
− ϕ = ϕ
n

2
−ϕ
n
, ϕ−ϕ, ϕ
n
+ϕ
2
.
lim
n→∞
ϕ
n
, ϕ = ϕ, ϕ = lim

n→∞
ϕ, ϕ
n
 lim
n→∞
ϕ
n
 = ϕ
lim
n→∞
ϕ
n
− ϕ = 0
ϕ
n
→ ϕ.
ϕ(x) X
x
0
{x
n
} x
n
 x
0
ϕ(x
0
)  lim inf ϕ(x
n
)

ϕ(x)
T ∈ L(X, Y ) T
ϕ
n
 ϕ T (ϕ
n
)  T (ϕ) n → ∞
ϕ
n
 ϕ lim sup
n→∞
ϕ
n
  ϕ
ϕ
n
 ϕ ψ ∈ Y
T ϕ
n
, ψ = ϕ
n
, T
∗
ψ → ϕ, T
∗
ψ = T ϕ, ψ.
T (ϕ
n
)  T (ϕ) n → ∞
ϕ

n
 ϕ ϕ
n
, ϕ → ϕ, ϕ = ϕ
2
n → ∞
|ϕ
n
, ϕ|  ϕ
n
.ϕ ϕ
2

lim sup
n→∞
ϕ
n
.ϕ ϕ  lim sup
n→∞
ϕ
n

{ϕ
n
}
n∈N
X ϕ
n
  l
{e

j
: j ∈ N} X := span{ϕ
n
: n ∈ N}.
ϕ
n
, e
1
 ϕ
n
1
(k)
, e
1
.
ϕ
n
1
(k)
, e
2
 n
2
(k) n
1
(k) ϕ
n
2
(k)
, e

2

n
l
(k) l ∈ N
ϕ
n
l
(k)
, e
l
 n
l+1
(k) n
l
(k)
ϕ
n
l
(l)
, e
k
 ξ
k
C l → ∞ k ∈ N
ϕ :=

k∈N
ξ
k

e
k
X ϕ  l
K ∈ N
K

k=1
|ξ|
2
= lim
l→∞
K

k=1
|ϕ
n
l
(l)
, e
k
|
2
 lim sup
l→∞
ϕ
n
l
(l)

2

 l.
ϕ
n
l
(l)
, ψ → ϕ, ψ ψ ∈ X ψ ∈ X  > 0
K ∈ N
∞

k=K+1
|ψ, e
k
|
2
 (

4
)
2
. L > 0





ϕ − ϕ
n
l
(l)
,

K

k=1
ψ, e
k
e
k








2
, l  L.
|ϕ − ϕ
n
l
(l)
, ψ|   l  L
K X
K
F : D(F ) ⊂ X → Y grF :=
{(ϕ, F (ϕ)) : ϕ ∈ D(F )} X ×Y ϕ
n
 ϕ
F (ϕ
n

)  g ϕ ∈ D(F ) F(ϕ) = g
F D(F ) F
D(F )
K ⊂ X K

u
t
+ A(t)u = 0, 0 < t < T,
u(T ) − f  , f ∈ H,  > 0
H ·, ·  · 
A(t) (0  t  T ) : D(A(t)) ⊂ H → H
H f ∈ H 
u : [0, T ] → H

u
t
+ A(t)u = 0, 0 < t < T,
u(T ) − f  .
u(t)
u
t
+ A(t)u = 0 0 < t < T ν(t)
[0, T ]
u(t)  cu(T )
ν(t)
u(0)
1−ν(t)
, ∀t ∈ [0, T ]
c
A(t) t

u(t) A(t)
Lu =
du
dt
+ A(t)u = 0, 0 < t ≤ T,
k, c
−
d
dt
A(t)u(t), u(t)  2A(t)u
2
− c (A(t) + k)u(t), u(t).
a
1
(t) [0, T ] a
1
(t) 
c, ∀t ∈ [0, T ]
−
d
dt
A(t)u(t), u(t)  2A(t)u
2
− a
1
(t) (A(t) + k)u(t), u(t).
t ∈ [0, T ]
a
2
(t) = exp



t
0
a
1
(τ)dτ

, a
3
(t) =

t
0
a
2
(ξ)dξ,
ν(t) =
a
3
(t)
a
3
(T )
.
t ∈ [0, T ]
u(t)  e
kt−kT ν(t)
u(T )
ν(t)

u(0)
1−ν(t)
.
A(t)
(H
1
) 0  t  T A(t)
σ(A(t)) ⊂ Σ
ω
= {λ ∈ C; |argλ| < ω}, 0  t  T,
ω 0 < ω <
π
2
(λ − A(t))
−1
 
M
|λ|
, λ ∈ Σ
ω
, 0  t  T,
M  1
(H
2
) D(A(t)) t A(t)
(H
3
) t ∈ [0, T ] A(t)
u(t) Lu =
du

dt
+ A(t)u = 0, 0 < t  T k
[0, T ] a
1
(t)
−
d
dt
A(t)u(t), u(t)  2A(t)u
2
− a
1
(t) (A(t) + k)u(t), u(t).
(H
1
) − (H
2
)
N > 0
A(t)(A(t)
−1
− A(s)
−1
)  N|t − s|, 0  s, t  T.
A(t)
H
A(t)(A(t)
−1
−A(s)
−1

) = −(A(t)−A(s))A(s)
−1
 = (A(t)−A(s))A(s)
−1
.
A(t)
A(t)A(s)
−1
(t, s) ∈ [0, T ] ×[0, T ]
N > 0
(A(t) − A(s))A(s)
−1
  N|t − s|, 0  s, t  T.
u(0)
E
u(0)  E.
v
t
+ A(t)v = 0, 0 < t < T, v(0) = v
0
v
0
v(T ) − ϕ
2
+

ε
E

2

v(0)
2
.
A(t) (0  t  T)
U(t, 0) (0  t 
T ) H U(0, 0) = I v(t)
v
t
+ A(t)v = 0, 0 < t  T v(t) = U(t, 0)v(0)
ω(t, v)

u
t
+ A(t)u = 0, 0 < t < T
u(0) = v
z(t, v)

u
t
− A(t)u = 0, 0 < t < T
u(T ) = ω(T, v) −ϕ.
J(v) := ω(T, v) − ϕ
2
+

ε
E

2
v

2
.
J(v)
J

(v)h = 2 z(0, v), h + 2

ε
E

2
v, h, ∀h ∈ H.
J(v + h) − J(v)
= ω(T, v + h) − ϕ
2
− ω(T, v) − ϕ
2
+

ε
E

2

v + h
2
− v
2

= ω(T, h) + ω(T, v) − ϕ

2
− ω(T, v) − ϕ
2
+

ε
E

2

2 v, h + h
2

= 2 ω(T, v) − ϕ, ω(T, h)+ ω(T, h)
2
+

ε
E

2

2 v, h + h
2

= 2 z(T, v), ω(T, h) + ω(T, h)
2
+

ε

E

2

2 v, h + h
2

.

T
0
ω
t
(t, h), z(t, v)dt =

T
0
−A(t)ω(t, h), z(t, v)dt
= −

T
0
ω(t, h), A(t)z(t, v)dt
= −

T
0
ω(t, h), z
t
(t, v)dt.


T
0
ω
t
(t, h), z(t, v)dt =

T
0

d
dt
ω(t, h), z(t, v)

dt
−

T
0
ω(t, h), z
t
(t, v)dt.

T
0

d
dt
ω(t, h), z(t, v)


dt = 0.
ω(T, h), z(T, v) = ω(0, h), z(0, v)
z(T, v), ω(T, h) = z(0, v), h.
J(v + h) − J(v) = 2 z(0, v), h + ω(T, h)
2
+

ε
E

2

2 v, h + h
2

.
f(t) := ω(t, h)
2
, t ∈ [0, T ]
f

(t) = −2 A(t)ω(t, h), ω(t, h)  0, ∀t ∈ [0, T ].
ω(T, h)
2
= f(T )  f(0) = ω(0, h)
2
= h
2
ω(T, h)
2

= o(h).
z
1
(t, v), z
2
(t, −ϕ)

u
t
− A(t)u = 0, 0 < t < T,
u(T ) = ω(T, v)

u
t
− A(t)u = 0, 0 < t < T
u(T ) = −ϕ.
J

(v + h) − J

(v)
= 2 z(0, v + h), · − 2 z(0, v), · + 2

ε
E

2
(v + h, · − v, ·)
= 2 z(0, v + h) − z(0, v), · + 2


ε
E

2
h, ·
= 2 z
1
(0, v + h) + z
2
(t, −ϕ) − z
1
(0, v) − z
2
(t, −ϕ), · + 2

ε
E

2
h, ·
= 2 z
1
(0, v + h) − z
1
(0, v), · + 2

ε
E

2

h, ·
= 2 z
1
(0, h), · + 2

ε
E

2
h, ·.

J

(v)h, h

= 2 z
1
(0, h), h + 2

ε
E

2
h
2
= 2 z
1
(0, h), ω(0, h) + 2

ε

E

2
h
2
.

T
0
ω
t
(t, h), z
1
(t, h)dt =

T
0
−A(t)ω(t, h), z
1
(t, h)dt
= −

T
0
ω(t, h), A(t)z
1
(t, h)dt
= −

T

0
ω(t, h), z
1t
(t, h)dt.

T
0
ω
t
(t, h), z
1
(t, h)dt =

T
0

d
dt
ω(t, h), z
1
(t, h)

dt
−

T
0
ω(t, h), z
1t
(t, h)dt.


T
0

d
dt
ω(t, h), z
1
(t, h)

dt = 0.

T
0

d
dt
ω(t, h), z
1
(t, h)

dt = 0
⇔ ω(0, h), z
1
(0, h) = ω(T, h), z
1
(T, h)
⇔ z
1
(0, h), ω(0, h) = z

1
(T, h), ω(T, h)
⇔ z
1
(0, h), ω(0, h) = ω(T, h), ω(T, h)
⇔ z
1
(0, h), ω(0, h) = ω(T, h)
2
 0.

J

(v)h, h

= 2ω(T, h)
2
+ 2

ε
E

2
h
2
 0.

J

(v)h, h


= 0 h = 0 J
U(T, 0) U(T, 0) : H → H
U(T, 0)(v(0)) = v(T ) U(T, 0)
H (H, U(T, 0)(H))
U(T, 0)
I := inf
v(0)∈X
U(T, 0)v(0) − ϕ
2
+

ε
E

2
v(0)
2
(ψ
n
) X
U(T, 0)(ψ
n
) − ϕ
2
+

ε
E


2
ψ
n

2
 I +
1
n
.
ε
E
> 0 ψ
n
ψ
n(k)
ψ ∈ X U(T, 0)(ψ
n(k)
)
U(T, 0)(ψ
n(k(l))
)
U(T, 0) U(T, 0)(ψ
n(k(l))
)  U(T, 0)(ψ)
l → ∞
U(T, 0)(ψ) − ϕ
2
+

ε

E

2
ψ
2
 lim sup
n→∞
{U(T, 0)(ψ
n
) − ϕ
2
+

ε
E

2
ψ
n

2
}  I.
ψ
lim
v→+∞
J(v) = +∞
(ϕ
n
) (v
n

) ϕ
n
→ ϕ v
n
ϕ ϕ
n
(v
n
)
v
n
v
n
(T ) − ϕ
n

2
+

ε
E

2
v
n
(0)
2
 v(T ) − ϕ
2
+


ε
E

2
v(0)
2
v(0) ∈ X (v
n
(0)) (v
n
(T ))
(v
m
) (v
n
) ¯v
v
m
(0)  ¯v(0) v
m
(T )  ¯v(T ).
¯v(0)  lim inf v
m
(0),
¯v(T ) − ϕ  lim inf v
m
(T ) − ϕ
m
.

¯v(T ) − ϕ
2
+

ε
E

2
¯v(0)
2
 lim inf

v
m
(T ) − ϕ
m

2
+

ε
E

2
v
m
(0)
2

 lim sup


v
m
(T ) − ϕ
m

2
+

ε
E

2
v
m
(0)
2

 lim
m→∞

˜v(T ) − ϕ
m

2
+

ε
E


2
˜v(0)
2

= ˜v(T ) − ϕ
2
+

ε
E

2
˜v(0)
2
˜v(0) ∈ X ¯v
lim
m→∞

v
m
(T ) − ϕ
m

2
+

ε
E

2

v
m
(0)
2

= ¯v(T ) − ϕ
2
+

ε
E

2
¯v(0)
2
.