# Approximate Controllability of the Degenerate System with the First-Order Term

Authors: Runmei Du
Journal: Journal of Function Spaces
 (2015)

---

## Abstract

We consider the approximate controllability of the degenerate system with the first-order term. The first-order term in the equation  cannot be controlled by the diffusion term. The system is shown to be approximately controllable by constructing a control by means of its conjugate problem.


---

## Body

## 1. Introduction

In this paper, we investigate the approximate controllability of the equation(1)
∂
u
∂
t
-
div
⁡
a
x
∇
u
+
b
→
x
·
∇
u
+
c
x
u
=
h
x
,
t
χ
ω
x
,
x
,
t
∈
Q
T
=
Ω
×
0
,
T
,where Ω is a bounded domain of R
n, ω is an open and nonempty subset of Ω, a
∈
C
(
Ω
¯
)
∩
C
1
(
Ω
) and is positive in Ω, b
→
=
(
b
1
,
b
2
,
…
,
b
n
)
∈
W
1
,
∞
(
Ω
;
R
n
), c
∈
L
∞
(
Ω
), h
∈
L
2
(
Q
T
) is the control function, and χ
ω is the characteristic function of ω. We note that a
(
x
) may be allowed to vanish at some points on the later boundary ∂
Ω, and thus (1) may be degenerate on the set {
(
x
,
t
)
∈
∂
Ω
×
(
0
,
T
)
:
a
(
x
)
=
0
}, a portion of the lateral boundary.In recent years, various controllability problems for linear and nonlinear differential equations have been considered. There are a great number of results on constrained controllability (see [1–3] and the references therein) and unconstrained controllability (see [4–6] and the references therein). Among these, some authors have investigated the null controllability of one-dimensional linear and semilinear equations with boundary degeneracy. In particular, the null controllability of the following degenerate equation is considered:(2)
∂
u
∂
t
-
∂
∂
x
x
α
∂
u
∂
x
+
x
α
b
x
,
t
∂
u
∂
x
+
c
x
,
t
u
=
h
x
,
t
χ
x
0
,
x
1
x
,
x
,
t
∈
0,1
×
0
,
T
,where b
,
c
∈
L
∞
(
(
0,1
)
×
(
0
,
T
)
), 0
<
x
0
<
x
1
<
1. Equation (2) may be used to describe some physical models (see [7, 8] and the references therein). In [7–14], the degeneracy of (2) is divided into weak one and strong one according to the value of α, and different boundary conditions are proposed for the two cases. More precise, the boundary value condition is(3)
u
0
,
t
=
u
1
,
t
=
0
,
t
∈
0
,
T
,in the weakly degenerate case with 0
<
α
<
1, while it is(4)
x
α
∂
u
∂
x
0
,
t
=
u
1
,
t
=
0
,
t
∈
0
,
T
,in the strongly degenerate case with α
≥
1. On the other hand, the following initial value condition is proposed for both cases:(5)
u
x
,
0
=
u
0
x
,
x
∈
0,1
.Then, system (2), (3) or (4), (5) is null controllable if 0
<
α
<
2 [7–9, 12, 14], while it is not if α
≥
2 [13]. However, the first-order term in (2) is controlled by the diffusion term. Further, the equation(6)
∂
u
∂
t
-
∂
∂
x
x
α
∂
u
∂
x
+
∂
∂
x
b
x
,
t
u
+
c
x
,
t
u
=
h
x
,
t
χ
x
0
,
x
1
x
,
x
,
t
∈
0,1
×
0
,
T
,has been investigated. Different from (2), the convection term in (6) cannot be controlled by the diffusion term. In [15], the authors studied the null controllability for system (6), (3), and (5) only in the case 0
<
α
<
1
/
2.Since the above one-dimensional degenerate systems may be not null controllable, a natural question is whether the systems are approximately controllable. More generally, the multidimensional degenerate systems have been investigated. In [16], the authors considered the equation (7)
∂
u
∂
t
-
div
⁡
a
x
∇
u
+
c
x
u
=
h
x
,
t
χ
ω
x
,
x
,
t
∈
Q
T
.Similar to [16], the lateral boundary ∂
Ω is decomposed into three parts: the nondegenerate boundary Σ
1, the weakly degenerate boundary Σ
2, and the strongly degenerate boundary Σ
3. The boundary value condition is prescribed on Σ
=
Σ
1
∪
Σ
2; namely,(8)
u
x
,
t
=
0
,
x
,
t
∈
Σ
×
0
,
T
.Then the authors proved the approximate controllability of system (7), (8) and the initial value condition(9)
u
x
,
0
=
u
0
x
,
x
∈
Ω
,where u
0
∈
L
2
(
Ω
).In the present paper, we assume(10)
∫
Ω
a
-
1
x
d
x
<
+
∞
,which means that the strongly degenerate boundary Σ
3 is empty. For example, if n
=
1, Ω
=
(
0,1
), a
(
x
)
=
x
α, then (10) implies α
=
0 (the nondegenerate case) or 0
<
α
<
1 (the weakly degenerate case). In this case, we consider (1) subject to (11)
u
x
,
t
=
0
,
x
,
t
∈
∂
Ω
×
0
,
T
,
(12)
u
x
,
0
=
u
0
x
,
x
∈
Ω
,where u
0
∈
L
2
(
Ω
). In the present paper, our method is similar to [5, 16–18]. The control is constructed via the conjugate problem.The paper is organized as follows. In Section2, we establish the well-posedness of system (1), (11), and (12) under condition (10). The approximate controllability of the system is proved in Section 3 subsequently.


## 2. The Well-Posedness of the Problem

In this section, we establish the well-posedness of problem (1), (11), and (12) in case (10). More generally, let us consider the problem(13)
∂
u
∂
t
-
div
⁡
a
x
∇
u
+
b
→
x
·
∇
u
+
c
x
u
=
f
x
,
t
,
x
,
t
∈
Q
T
,
u
x
,
t
=
0
,
x
,
t
∈
∂
Ω
×
0
,
T
,
u
x
,
0
=
u
0
x
,
x
∈
Ω
,where f
∈
L
2
(
Q
T
).The weak solution of problem (13) is defined as follows.Definition 1.
A functionu
(
x
,
t
) is called a weak solution of problem (13), if u
∈
C
(
[
0
,
T
]
;
L
2
(
Ω
)
)
∩
B and for any function φ
∈
L
∞
(
Q
T
)
∩
B with ∂
φ
/
∂
t
∈
L
2
(
Q
T
) and φ
(
·
,
T
)
|
Ω
=
0, the following integral equality holds: (14)
∬
Q
T
-
u
∂
φ
∂
t
+
a
∇
u
·
∇
φ
+
b
→
x
·
∇
u
φ
+
c
u
φ
d
x
d
t
=
∬
Q
T
f
φ
d
x
d
t
+
∫
Ω
u
0
x
φ
x
,
0
d
x
.Here, we use B to denote the closure of the set C
0
∞
(
Q
T
) with respect to the norm(15)
u
B
=
∬
Q
T
a
x
u
x
,
t
2
+
∇
u
x
,
t
2
d
x
d
t
,
u
∈
B
.As to the setB, we give the following remark whose proof can be found in [19] Corollary 2.1 and Remark 2.1.Lemma 2.
Ifu
∈
B, then (16)
u
∂
Ω
×
0
,
T
=
0in the trace sense.Next, we establish the well-posedness for problem (13).Theorem 3.
For anyf
∈
L
2
(
Q
T
) and u
0
∈
L
2
(
Ω
), problem (13) admits a unique weak solution u satisfying(17)
u
L
∞
0
,
T
;
L
2
Ω
+
a
∇
u
2
L
1
Q
T
≤
C
f
L
2
Q
T
+
u
0
L
2
Ω
,where C
>
0 is a constant depending only on T, b
→
W
1
,
∞
(
Ω
;
R
n
), and c
L
∞
(
Ω
). Moreover, if f
∈
C
0
∞
(
Q
T
) and u
0
∈
C
0
∞
(
Ω
), then ∂
u
/
∂
t
∈
L
2
(
Q
T
) and u
∈
L
∞
(
Q
T
).Proof.
First, we prove the existence. For any0
<
ɛ
<
1, we take u
ɛ
,
0
∈
C
0
∞
(
Ω
) such that (18)
u
ɛ
,
0
⟶
u
0
in
L
2
Ω
.Consider the problem(19)
∂
u
ɛ
∂
t
-
div
⁡
a
x
+
ɛ
∇
u
ɛ
+
b
→
x
·
∇
u
ɛ
+
c
x
u
ɛ
=
f
x
,
t
,
x
,
t
∈
Q
T
,
(20)
u
ɛ
x
,
t
=
0
,
x
,
t
∈
∂
Ω
×
0
,
T
,
(21)
u
ɛ
x
,
0
=
u
ɛ
,
0
x
,
x
∈
Ω
.According to the classical theory on parabolic equations, problem (19)–(21) admits a unique weak solution u
ɛ
∈
H
1
(
0
,
T
;
L
2
(
Ω
)
)
∩
L
2
(
0
,
T
;
H
1
(
Ω
)
).
Multiply (19) by u
ɛ and then integrate over Ω
×
0
,
t
(
0
<
t
<
T
) to get (22)
1
2
∫
Ω
u
ɛ
2
x
,
t
d
x
+
∫
0
t
∫
Ω
a
x
+
ɛ
∇
u
ɛ
x
,
s
2
d
x
d
s
=
∫
0
t
∫
Ω
1
2
∑
i
=
1
n
∂
b
i
∂
x
i
-
c
u
ɛ
2
x
,
s
d
x
d
s
+
∫
0
t
∫
Ω
f
u
ɛ
d
x
d
s
+
1
2
∫
Ω
u
ɛ
,
0
2
x
d
x
.Using the Hölder inequality and the Grönwall inequality, we can get(23)
u
ɛ
L
∞
0
,
T
;
L
2
Ω
+
a
x
+
ɛ
∇
u
ɛ
2
L
1
Q
T
≤
C
f
L
2
Q
T
+
u
0
L
2
Ω
,where C
>
0 is a constant depending only on T, b
→
W
1
,
∞
(
Ω
;
R
n
), and c
L
∞
(
Ω
). From estimate (23), there exist a subsequence of {
u
ɛ
}, denoted by itself, and a function u
∈
C
(
0
,
T
;
L
2
(
Ω
)
)
∩
B, such that(24)
u
ɛ
⇀
u
in
L
2
Q
T
,
a
x
+
ɛ
1
/
2
∇
u
ɛ
⇀
a
x
1
/
2
∇
u
in
L
2
Q
T
;
R
n
.
Sinceu
ɛ is the weak solution of problem (19)–(21), the following integral equality holds for any function φ
∈
L
∞
(
Q
T
)
∩
B with ∂
φ
/
∂
t
∈
L
2
(
Q
T
) and φ
(
·
,
T
)
|
Ω
=
0:(25)
∬
Q
T
-
u
ɛ
∂
φ
∂
t
+
a
x
+
ɛ
∇
u
ɛ
·
∇
φ
+
b
→
x
·
∇
u
ɛ
φ
+
c
u
ɛ
φ
d
x
d
t
=
∬
Q
T
f
φ
d
x
d
t
+
∫
Ω
u
ɛ
,
0
x
·
φ
x
,
0
d
x
.Note that b
i
(
x
)
a
-
1
/
2
(
x
)
φ
∈
L
2
(
Q
T
) due to (10). Let ɛ
→
0
+ in (25) to get (26)
∬
Q
T
-
u
∂
φ
∂
t
+
a
∇
u
·
∇
φ
+
b
→
x
·
∇
u
φ
+
c
u
φ
d
x
d
t
=
∬
Q
T
f
φ
d
x
d
t
+
∫
Ω
u
0
x
φ
x
,
0
d
x
,which means that u is the weak solution of problem (13).
Moreover, iff
∈
C
0
∞
(
Q
T
) and u
0
∈
C
0
∞
(
Ω
), the maximum principle yields(27)
u
ɛ
L
∞
Q
T
≤
C
,where C is independent of ɛ. Denote v
ɛ
=
∂
u
ɛ
/
∂
t. Then v
ɛ satisfies(28)
∂
v
ɛ
∂
t
-
div
⁡
a
x
+
ɛ
∇
v
ɛ
+
b
→
x
·
∇
v
ɛ
+
c
x
v
ɛ
=
∂
f
∂
t
x
,
t
,
x
,
t
∈
Q
T
,
v
ɛ
x
,
t
=
0
,
x
,
t
∈
∂
Ω
×
0
,
T
,
v
ɛ
x
,
0
=
div
⁡
a
x
+
ɛ
∇
u
0
-
b
→
x
·
∇
u
0
-
c
x
u
0
+
f
x
,
0
,
x
∈
Ω
.From (23), we have(29)
v
ɛ
L
2
Q
T
≤
C
,where C is independent of ɛ. It follows from (27) and (29) that there exists a subsequence of {
u
ɛ
}, denoted by itself, such that (30)
u
ɛ
⇀
u
weakly  star  in
L
∞
Q
T
,
∂
u
ɛ
∂
t
⇀
∂
u
∂
t
in
L
2
Q
T
.Thus, u
∈
L
∞
(
Q
T
) and ∂
u
/
∂
t
∈
L
2
(
Q
T
).
Finally, we prove the uniqueness by the Holmgren method. Letu
¯ and u
~ be two weak solutions of problem (13) and denote (31)
w
x
,
t
=
u
¯
x
,
t
-
u
~
x
,
t
,
x
,
t
∈
Q
T
.Then w
∈
C
(
[
0
,
T
]
;
L
2
(
Ω
)
)
∩
B and for any function φ
∈
L
∞
(
Q
T
)
∩
B with ∂
φ
/
∂
t
∈
L
2
(
Q
T
) and φ
(
·
,
T
)
|
Ω
=
0, the following integral equality holds:(32)
∬
Q
T
-
w
∂
φ
∂
t
+
a
∇
w
·
∇
φ
+
∑
i
=
1
n
b
i
D
i
w
φ
+
c
w
φ
d
x
d
t
=
0
.For any g
∈
C
0
∞
(
Q
T
), the above existence result shows that the problem (33)
-
∂
ψ
∂
t
-
div
⁡
a
x
∇
ψ
-
b
→
x
·
∇
ψ
+
c
x
-
d
i
v
b
→
x
ψ
=
g
,
x
,
t
∈
Q
T
,
ψ
x
,
t
=
0
,
x
,
t
∈
∂
Ω
×
0
,
T
,
ψ
x
,
T
=
0
,
x
∈
Ω
,admits a weak solution ψ
∈
L
∞
(
Q
T
)
∩
B with ∂
ψ
/
∂
t
∈
L
2
(
Q
T
). Taking φ
=
ψ in (32), we get (34)
∬
Q
T
w
g
d
x
d
t
=
0
.This leads to (35)
w
x
,
t
=
0
,
a.e.
x
,
t
∈
Q
Towing to the arbitrariness of g
∈
C
0
∞
(
Q
T
). Therefore, (36)
u
¯
x
,
t
=
u
~
x
,
t
,
a.e.
x
,
t
∈
Q
T
;namely, the weak solution of problem (13) is unique. The proof is complete.


## 3. Approximate Controllability of the Control System

In this section, we investigate the approximate controllability of control system (1), (11), and (12).First, we consider control system (1), (11) with null initial data; namely,(37)
u
x
,
0
=
0
,
x
∈
Ω
.The study on the approximate controllability of the control system is related to its conjugate problem(38)
-
∂
v
∂
t
-
div
⁡
a
x
∇
v
-
b
→
x
·
∇
v
+
c
x
-
d
i
v
b
→
x
v
=
0
,
x
,
t
∈
Q
T
,
v
x
,
t
=
0
,
x
,
t
∈
∂
Ω
×
0
,
T
,
v
x
,
T
=
v
T
x
,
x
∈
Ω
.Define a mapping (39)
L
:
L
2
Ω
⟶
L
1
ω
×
0
,
T
,
v
T
⟼
v
χ
ω
x
,where v is the weak solution of conjugate problem (38). Then the mapping L satisfies the following:(a)
L is a continuous linear operator from L
2
(
Q
T
) to L
1
(
ω
×
(
0
,
T
)
);
(b)
ifL
v
T
=
0
a
.
e
.
(
x
,
t
)
∈
ω
×
(
0
,
T
), then it holds v
=
0
a
.
e
.
(
x
,
t
)
∈
Q
T
.Property (a) follows from Theorem3, and property (b) can be deduced from the unique continuation of the nondegenerate parabolic equation [20, 21].Fixu
1
∈
L
2
(
Ω
) and ɛ
>
0. For v
T
∈
L
2
(
Ω
), we introduce a functional (40)
J
v
T
=
1
2
∫
0
T
∫
ω
L
v
T
x
,
t
d
x
d
t
2
+
ɛ
v
T
L
2
Ω
-
∫
Ω
u
1
x
v
T
x
d
x
.For this functional, we have the following proposition.Proposition 4.
J
(
·
) is a strictly convex and continuous functional defined on L
2
(
Ω
) and satisfies(41)
lim
inf
v
T
L
2
Ω
→
+
∞
⁡
J
v
T
v
T
L
2
Ω
≥
ɛ
.Furthermore, the functional J
(
·
) achieves its minimum at a unique point v
^
T in L
2
(
Ω
) and(42)
v
^
T
=
0
⟺
u
1
L
2
Ω
≤
ɛ
.Proof.
One can easily prove thatJ
(
·
) is strictly convex by the linearity of L and the convexity of L
2
(
Ω
) norm. Moreover, the continuity of J
(
·
) can be derived from Theorem 3 and the continuity of L.
Now we prove (41) by contradiction. Otherwise, there exists a sequence {
v
T
(
k
)
}
k
=
1
∞
⊂
L
2
(
Ω
) satisfying(43)
l
i
m
k
→
∞
v
T
k
L
2
Ω
=
+
∞
,
l
i
m
k
→
∞
J
v
T
k
v
T
k
L
2
Ω
<
ɛ
.For k
=
1,2
,
…, we denote v
~
T
(
k
)
=
v
T
(
k
)
/
v
T
(
k
)
L
2
(
Ω
)
. There exist a subsequence of {
v
~
T
(
k
)
}
k
=
1
∞, denoted by {
v
~
T
(
k
i
)
}
i
=
1
∞, and v
~
T
∈
L
2
(
Ω
), such that v
~
T
L
2
(
Ω
)
≤
1 and (44)
v
~
T
k
i
⇀
v
~
T
in
L
2
Ω
,
as
i
⟶
+
∞
.Then, it follows from Theorem 3 that (45)
v
~
k
i
⇀
v
~
in
L
2
Q
T
,
as
i
⟶
+
∞
,where v
~ and v
~
(
k
i
) are the weak solution of conjugate problem (38) with v
T
=
v
~
T and v
T
=
v
~
T
(
k
i
), respectively. Additionally, (43) yields (46)
lim
k
→
∞
⁡
∫
0
T
∫
ω
L
v
~
T
k
i
x
,
t
d
x
d
t
=
0
.Hence (47)
∫
0
T
∫
ω
L
v
~
T
x
,
t
d
x
d
t
=
0
.This and (b) lead to v
~
=
0 in Q
T and thus v
~
T
=
0 in Ω. Therefore, (48)
lim
i
→
∞
⁡
J
v
T
k
i
v
T
k
i
L
2
Ω
≥
lim
i
→
∞
⁡
ɛ
-
∫
Ω
u
1
x
v
~
T
k
i
x
d
x
v
T
k
i
L
2
Ω
≥
ɛ
,which contradicts (43) and completes the proof of (41).
From (41), we get that (49)
lim
inf
v
T
L
2
Ω
→
+
∞
⁡
J
v
T
v
T
L
2
Ω
≥
ɛ
>
0
=
J
0
.This, together with the strict convexity and the continuity of J
(
·
), implies that the functional J
(
·
) achieves its minimum at a unique point in L
2
(
Ω
).
Finally, we prove (42). On the one hand, if u
1
L
2
(
Ω
)
≤
ɛ, it follows from the Hölder inequality that (50)
J
v
T
≥
0
,
∀
v
T
∈
L
2
Ω
,and thus v
^
T
=
0. On the other hand, if v
^
T
=
0, then (51)
1
τ
J
τ
u
1
≥
0
,
∀
τ
>
0
;that is, (52)
τ
2
∫
0
T
∫
ω
L
u
1
x
,
t
d
x
d
t
2
+
ɛ
u
1
L
2
Ω
-
∫
Ω
u
1
2
d
x
≥
0
,
∀
τ
>
0
.Letting τ
→
0
+ yields u
1
L
2
(
Ω
)
≤
ɛ. The proof is complete.For the functionalJ
(
·
), we have the following lemma (Proposition 3.2 [16]).Lemma 5.
For anyv
T
∈
L
2
(
Ω
) with v
T
≠
0, (53)
∂
J
v
T
=
ξ
∈
L
2
Ω
:
t
h
e
r
e
e
x
i
s
t
s
z
∈
sgn
⁡
L
v
T
χ
ω
x
s
a
t
i
s
f
y
i
n
g
∫
Ω
ξ
x
θ
T
x
d
x
=
∫
0
T
∫
ω
L
v
T
x
,
t
d
x
d
t
·
∫
0
T
∫
ω
z
x
,
t
L
θ
T
x
,
t
d
x
d
t
+
ɛ
∫
Ω
v
T
x
v
T
L
2
Ω
θ
T
x
d
x
-
∫
Ω
u
1
x
θ
T
x
d
x
,
∀
θ
T
∈
L
2
Ω
.Here we say z
∈
sgn
⁡
(
s
), if z
(
x
,
t
)
=
s
(
x
,
t
)
/
|
s
(
x
,
t
)
| when s
(
x
,
t
)
≠
0, while |
z
(
x
,
t
)
|
≤
1 when s
(
x
,
t
)
=
0.Now we can get the approximate controllability of control system (1), (11), and (12) with null initial data by Proposition 4 and Lemma 5.Theorem 6.
Problem (1), (11), and (37) is approximate controllability. That is to say, for any given number ɛ
>
0 and function u
1
∈
L
2
(
Ω
), there exists a control h
∈
L
2
(
Q
T
), such that(54)
u
x
,
T
-
u
1
x
L
2
Ω
≤
ɛ
.Proof.
Since we can takeh
=
0 to get (54) when u
1
L
2
(
Ω
)
≤
ɛ, we only consider the case u
1
L
2
(
Ω
)
>
ɛ.
From Proposition4, there exists v
^
T
∈
L
2
(
Ω
) with v
^
T
≠
0 realizing the minimum of J
(
·
). Note that J
(
·
) is subdifferentiable at v
^
T. Therefore, 0
∈
∂
J
(
v
^
T
). By Lemma 5, for any θ
T
∈
L
2
(
Ω
), there exists z
∈
sgn
⁡
(
v
^
)
χ
ω
(
x
) such that(55)
v
^
L
1
D
T
∫
0
T
∫
ω
z
x
,
t
θ
n
x
,
t
d
x
d
t
+
∫
Ω
ɛ
v
^
T
x
v
^
T
L
2
Ω
-
u
1
θ
T
n
x
d
x
=
0
,where v
^ and θ
n are the weak solutions of conjugate problem (38) with v
T
=
v
^
T and v
T
=
θ
T
n
(
x
), respectively. Here, θ
T
n
∈
C
0
∞
(
Ω
) satisfying (56)
θ
T
n
⟶
θ
T
in
L
2
Ω
,
as
n
⟶
∞
.
From the definition of the weak solutionu to problem (1), (11), and (37) with h
=
v
^
L
1
(
ω
×
(
0
,
T
)
)
z, we have(57)
∬
Q
T
-
u
∂
θ
n
∂
t
d
x
d
t
+
∬
Q
T
a
∇
u
·
∇
θ
n
d
x
d
t
+
∬
Q
T
b
→
x
·
∇
u
θ
n
+
∬
Q
T
c
u
θ
n
d
x
d
t
=
∬
Q
T
v
^
L
1
ω
×
0
,
T
z
χ
ω
θ
n
d
x
d
t
-
∫
Ω
u
x
,
T
θ
T
n
d
x
.On the other hand, since θ
n is the weak solution to problem (38) with v
T
=
θ
T
n
(
x
), we obtain(58)
∬
Q
T
-
∂
θ
n
∂
t
u
d
x
d
t
+
∬
Q
T
a
∇
θ
n
·
∇
u
d
x
d
t
+
∬
Q
T
b
→
x
θ
n
·
∇
u
+
∬
Q
T
c
θ
n
u
d
x
d
t
=
0
.Combining (57), (58), and (55) yields (59)
∫
Ω
u
x
,
T
θ
T
n
x
d
x
=
∫
Ω
u
1
-
ɛ
v
^
T
x
v
^
T
L
2
Ω
θ
T
n
x
d
x
.Let n
→
∞ to get (60)
∫
Ω
u
x
,
T
θ
T
x
d
x
=
∫
Ω
u
1
-
ɛ
v
^
T
x
v
^
T
L
2
Ω
θ
T
x
d
x
,which implies (54) owing to the arbitrariness of θ
T
∈
L
2
(
Ω
). The proof is complete.Finally, we prove the approximate controllability of control system (1), (11), and (12).Theorem 7.
Control system (1), (11), and (12) is approximately controllable. That is to say, for any given initial datum u
0
∈
L
2
(
Ω
), the desired datum u
1
∈
L
2
(
Ω
), and the admissible error value ɛ
>
0, there exists a control h
∈
L
2
(
Q
T
) such that the weak solution u of problem (1), (11), and (12) satisfies(61)
u
x
,
T
-
u
1
x
L
2
Ω
≤
ɛ
.Proof.
It follows from Theorem6 that there exists a control h
∈
L
2
(
Q
T
) such that the weak solution w
~ to the problem (62)
∂
w
~
∂
t
-
div
⁡
a
x
,
t
∇
w
~
+
b
→
x
·
∇
w
~
+
c
x
,
t
w
~
=
h
x
,
t
χ
ω
x
,
x
,
t
∈
Q
T
,
w
~
x
,
t
=
0
,
x
,
t
∈
∂
Ω
×
0
,
T
,
w
~
x
,
0
=
0
,
x
∈
Ω
,satisfies(63)
w
~
x
,
T
-
u
1
x
-
w
x
,
T
L
2
Ω
≤
ɛ
,where w is the weak solution to the problem(64)
∂
w
∂
t
-
div
⁡
a
x
,
t
∇
w
+
b
→
·
∇
w
+
c
x
,
t
w
=
0
,
x
,
t
∈
Q
T
,
w
x
,
t
=
0
,
x
,
t
∈
∂
Ω
×
0
,
T
,
w
x
,
0
=
u
0
x
,
x
∈
Ω
.
Then we can get (61) from (63) and the uniqueness result in Theorem 3. The proof is complete.From the proof of Theorems6 and 7, it is worthy to note the following.Remark 8.
The controls obtained in Theorems6 and 7 are both quasi bang-bang controls.



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*Source: 151484-2015-08-05.xml*