




u
t
− Lu = f(x, t); (x, t) ∈ Ω × (0, T ),
u|
∂Ω×(0,T ]
= 0,
u|
t=0
= ϕ.
1.





u
t
− Lu = f(x, t); (x, t) ∈ Ω × (0, T ),
u|
∂Ω×(0,T ]
= 0,
u|
t=0
= ϕ.
(1.1)
u(x, T ; f, ϕ) = ψ

(x)
f ∈ L
2
(Q
T
)
J(f) =
1
2
u(., T ; f, ϕ) − ψ
T
(x)
2
L
2
(Ω)
(1.3)
1
J(f) + αf
2
L
2
(Q
T
)
(1.4)
α > 0 ψ
T
f(x, t) = f(x) ∈ L
2

(Q
T
) =
L
2
((0, T ); H
1
(Ω)) u(x, t) L
2
(Q
T
)
∂u/∂x
i
, i = 1, , n Q
T
(u, v)
H
1,0
(Q
T
)
=

Q
T
(uv + u
x
v
x

1,0
(Q
T
)
∂Q
T
Q
T
L
∞
(Q
T
)
Q
T
a
ij
(x, t), i, j = 1, 2, n; a
i
(x, t), b
i
(x, t), i = 1, 2, , n; a(x, t)
a
ij
, a
i
, b
i
∈ L
∞

a
2
i
,




n

i=1
b
2
i
, |a| ≤ µ.
u(x, t) H
1
,
0
0
(Q
T
)
u ∈ H
1,0
0
(Q
T
)


i
u
x
i
η + auη)dxdt =

Ω
ϕη(x, 0)dx +

Q
T
fηdxdt,
∀η ∈ H
1,0
(Q
T
).
ϕ ∈ L
2
(Ω), f ∈ L
2
(Q
T
)
u(x, t) C([0, T ]; L
2
(Ω)) ∩ H
1,0
(Q
T

N
(x, t) =
N

k=1
u
N
k
(t)ψ
k
(x), (2.1)
(u
N
t
, ψ
l
) +
n

i=1
(
n

j=1
a
ij
u
N
x
j

l
); l = 1, 2, , N, (2.3)
L
2
(Ω) f
f
N
=
N

k=1
(f, ψ
k
)ψ
k
, (2.4)
ψ
k
(f
N
, ψ
l
) = (f, ψ
l
) (2.5)
u
N
u
N


0
(Q
T
)
a
i
= b
i
= 0; i =
1, 2, n,
J(f
N
) =
1
2

N

k=1
u
N
k
(T )ψ
k
− ψ
δ
T

2
, (2.6)

N
∈H
N
J(f
N
)
H
N
(ψ
1
, ψ
2
, , ψ
N
).
3.
f ∈ L
2
(Ω) f
N
=
N

k=1
f
k
ψ
k
→ f L
2

T

2
L
2
(Ω)
− u(., T ; f
N
, ϕ) − ψ
δ
T

2
L
2
(Ω)
= u(., T ; f, ϕ) − u(., T; f
N
, ϕ)
2
L
2
(Ω)
+ 2(u(., T ; f, ϕ) − u(., T ; f
N
, ϕ), u(., T ; f
N
, ϕ) − ψ
δ
T

, ϕ) −ψ
δ
T
 ≤ u(., T ; f
N
, ϕ)  +  ψ
δ
T

|J(f) − J(f
N
)| → 0 N → ∞.

f
N
∗
(u
∗
, f
∗
)
f
N
∗
f
∗
L
2
(Ω) u(., t; f
N

∗
N ≥ N
∗
|J(f
ε
) − J(f
εN
)| <
ε
2
.
−
ε
2
< J(f
ε
) − J(f
εN
) <
ε
2
J(f
εN
) < J(f
ε
) +
ε
2
.
J

N
≥ J
∗
.
lim
N→∞
J
∗
N
lim
N→∞
J
∗
N
= J
∗
. (3.1)
0 ≤ J(f
N
∗
) − J
∗
≤ |J(f
N
∗
) − J
∗
N
| + |J
∗

∗
f
∗
, H
1
0
(Ω) L
2
(Ω), f
N
∗
f
∗
L
2
(Ω). u(x, t; f
N
∗
, ϕ) u(x, t; f
N
∗
, ϕ)
L
2
(Q
T
).


