NULL EXACT CONTROLLABILITY OF THE PARABOLIC
EQUATIONS WITH EQUIVALUED SURFACE
BOUNDARY CONDITION
ZHONGQI YIN
Received 25 January 2005; Revised 21 April 2005; Accepted 26 April 2005
This paper is devoted to showing the null exact controllability for a class of parabolic
equations with equivalued surface boundary condition. Our method is based on the duality argument and global Carleman-type estimate for a parabolic operator.
Copyright © 2006 Zhongqi Yin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let T > 0, Ω ⊂ Rn (n ∈ N) be a given bounded domain, ∂Ω = Γ0 ∪ Γ1 (Γ1 = ∅) (where
Γ0 is the interior boundary and Γ1 the outer boundary), Γ0 ∩ Γ1 = ∅. For simplicity, we
assume that Γ0 ,Γ1 ∈ C ∞ and ω = ∅ is a given subdomain of Ω. Denote the characteristic
function of ω by χω , and the unit outward normal vector of Ω by (n1 ,...,nn ). Put Q =
Ω × (0,T), Qω = ω × (0,T), and Σ = ∂Ω × (0,T). Let ai j (x) ∈ C 2 (Ω) satisfy ai j = a ji , and
for some Λ > 0, it holds that
ai j ξi ξ j ≥ Λ|ξ |2 ,
∀(x,ξ) ∈ Ω × Rn .
(1.1)
i, j
Here and henceforth, we denote ni, j =1 simply by i, j .
We consider the following controlled parabolic equation with equivalued surface
boundary condition:
∂y ∂
∂y
−
ai j (x)
= χω (x)b
∂t i, j ∂xi
∂x j
Γ0
y |Γ1 = 0,
∂y
ds = 0,
∂nA
in Q,
y |Γ0 = m(t),
y(x,0) = y0 (x) in Ω,
Hindawi Publishing Corporation
Journal of Applied Mathematics and Stochastic Analysis
Volume 2006, Article ID 62694, Pages 1–10
DOI 10.1155/JAMSA/2006/62694
(1.2)
2
Null controllability of a nonlocal parabolic equation
where y = y(x,t) is the state, b = b(x,t) is the control, m(t) ∈ L2 (0,T) is an unknown
function which depends only on the time variable t, y0 is the initial state, and
∂y
∂y
= ai j (x)
ni .
∂nA i, j
∂x j
(1.3)
In system (1.2), the state space is chosen as L2 (Ω), and the control space is L2 (ω). Let
Y y ∈ C [0,T];L2 (Ω)
L2 0,T;H 1 (Ω) | y |Γ1 = 0, y |Γ0 = m(·) ∈ L2 (0,T) .
(1.4)
It can be shown that for any y0 ∈ L2 (Ω) and b ∈ L2 (ω × (0,T)), system (1.2) admits one
and only one weak solution y ∈ Y (cf. [5, 6, 8]).
The null exact controllability problem of (1.2) is formulated as follows: for any given
y0 ∈ L2 (Ω), find a control b(x,t) ∈ L2 (ω × (0,T)) (if possible) such that the weak solution
y(·) ∈ Y satisfies y(T) = 0.
There are many concrete physical backgrounds for problem (1.2), for example, the
problem of resistivity well logging, the unstable temperature field around an underground
electric cable, and so on (cf. [5, 6]).
In recent years, great progress has been made in the exact controllability problem of the
linear and semilinear partial differential equations with Dirichlet or Neumann boundary condition, or other sorts of pointwise boundary value conditions ([1–4, 7, 9], and
the references cited therein). However, to the author’s best knowledge, there is no reference devoted to the same problem for the parabolic equations but with a spatial nonlocal
boundary condition. In this paper, we will show the null exact controllability for system
(1.2). By duality, the problem is reduced to the obtention of an observability inequality
for the corresponding adjoint equation, which in turn is derived by means of a global
Carleman-type estimate. Our method is stimulated by that in [4].
The rest of this paper is organized as follows. In Section 2, we state some preliminary
results and our main results. The final section, Section 3, is devoted to the proof of our
main theorem.
2. Main result
Throughout this paper, C denotes a positive constant depending only on λ, μ, Ω, T, and
ω, which may change from line to line.
To begin with, we fix ω0 to be a nonempty open subset of Ω such that ω0 ⊂ ω. Let
ψ ∈ C ∞ (Ω) satisfy ψ > 0 in Ω, ψ = 0 on ∂Ω, and |∇ψ(x)| > 0 for all x ∈ Ω0 = Ω\ω0 .
The existence of function ψ was proved in [7]. In this paper, we further assume that the
following technical condition holds:
∂ψ
∂n
2 ai j ni n j |Γ0 = Const.
(2.1)
i, j
This technical condition admits several interesting cases such as Ω = {x ∈ Rn | r < |x| < R}
Zhongqi Yin 3
j
for some 0 < r < R < ∞, ai j (x) = δi , and ∂ψ/∂n = Const on Γ0 . In this case, one may
choose ψ(x) = (|x|2 − r 2 )(R2 − |x|2 ).
The main result in this paper is stated as follows.
Theorem 2.1. Under the assumption (2.1), system (1.2) is null exact controllable.
By means of the usual duality argument (see, e.g., [3, 9]), the proof of Theorem 2.1
is easily reduced to the obtention of an observability inequality for the following adjoint
system of system (1.2):
−Lu ≡ −
u|Γ1 = 0,
Γ0
∂u
∂u ∂
−
ai j (x)
= 0 in Q,
∂t i, j ∂xi
∂x j
u|Γ0 = c(t),
∂u
ds = 0,
∂nA
(2.2)
u(x,T) = u(T) in Ω,
where u(T) ∈ L2 (Ω), and similarly to system (1.2), c(·) ∈ L2 (0,T) is an unknown function. More precisely, we need to show the following.
Theorem 2.2. Under the assumption (2.1), there is a constant C > 0 such that solutions
u ∈ Y of system (2.2) satisfy
u(0)
L2 (Ω)
≤ C u(x,t)L2 (ω×(0,T)) .
(2.3)
Remark 2.3. It would be quite interesting to drop the technical condition (2.1). But this
is by now an unsolved problem.
3. Proof of the main theorem
It suffices to prove Theorem 2.2. To this end, for any given parameters λ and μ, we set
−1 −1
α(x,t) = t(T − t)
ϕ(x,t) = t(T − t)
eμψ(x) − e2μψ C(Ω) ,
eμψ(x) ,
(3.1)
θ(x,t) = eλα(x,t) .
Clearly, Theorem 2.2 is an easy consequence of the following global Carleman-type
estimate for solutions of (2.2).
Theorem 3.1. Let (2.1) hold. Then there exist a constant μ1 and a function λ1 : R+ → (1, ∞)
such that for any μ > μ1 and λ > λ1 (μ), solutions u ∈ Y of system (2.2) satisfy
λ3
Q
θ 2 ϕ3 u2 dx dt + λ
≤ Cλ
3
Q
θ 2 ϕ|∇u|2 dx dt + λ−1
Q
θ 2 ϕ−1 u2t + (Δu)2 dx dt
(3.2)
2 3 2
Qω
θ ϕ u dx dt.
The rest of this section is devoted to prove Theorem 3.1. For this, we need the following
pointwise estimate for parabolic operator Lu, which is a special case of [4, Lemma 3.1].
4
Null controllability of a nonlocal parabolic equation
Lemma 3.2. Let u,α ∈ C 2 (Rn+1 ) and λ > 0 be given. Put θ = eλα and v = θu. Then it holds
that
2
2
2
θ |Lu| ≥ − λαt v −
+
ai j vxi vx j + λ ai j
i, j
2
j
x j αxi v
ai j vxi vt + λ2 ai j αxi αt v2
+ 2λ
2
2
+ λai j αxi x j v − λ ai j αxi αx j v
2
t
i
2
ai j am αxi vx vxm + ai j am αx xm
i
xi v
2
− 2ai j am αx vxi vxm
− 2ai j am αx xm vvxi + λai j am αxi αx xm v 2
+ λai j am
xm αx αxm v
2
2
− λ ai j am αx αxm v
2
xj
+ λαtt − λ
ai j
i, j
x j αxi t
+ ai j αxi x j t − λ αi j αxi αx j
t
− 2λ ai j αxi αt x j + 4λai j αxi x j αt
+ 2λ
2λ ai j x j am αxi αx xm − ai j am αx xm x j xi − λ am xm ai j αxi αx x j
i, j,,m
− λ ai j am αxi x j αx xm − 2λai j am αxi x j αx xm
2
+ λ ai j am αxi αx j αx
2ai j am αx
+ 2λ
i, j,,m
x j vxi vxm
xm
2
− 2λ ai j am αxi αx j αx xm
v2
− ai j am xm αx vxi vx j + ai j am αx xm vxi vx j .
(3.3)
Proof of Theorem 3.1. The main idea of our proof is to use the pointwise estimate in
Lemma 3.2. The proof is divided into several steps.
Step 1. Recall that θ = eλα , v = θu. We claim that
Q
θ 2 (Lu)2 dx dt −
≥ 2Λ2 λ3 μ4
Q
Q
div V dx dt + Cλ3 μ3
Q
ϕ3 v2 dx dt + Cλμ2
ϕ3 |∇ψ |4 v2 dx dt + 2Λ2 λμ2
Q
Q
ϕ|∇v|2 dx dt
(3.4)
ϕ|∇v|2 |∇ψ |2 dx dt,
where V = (V1 ,V2 ,...,Vn ), and
Vj = − 2
ai j vxi vt + λ2 ai j αxi αt v2
i
+ 2λ
ai j am αxi vx vxm − 2ai j am αx vxi vxm + λai j am
i,,m
2
2
− λ ai j am αxi αx αxm v + λai j am αxi αx xm v
− 2ai j am αx xm vvxi + ai j am αx xm xi v 2 .
2
xm αxi αxm v
2
(3.5)
Zhongqi Yin 5
By the definition of α and ϕ in (3.1), it is easy to see that
− λαt v 2 +
Q
ai j vxi vx j + λ ai j
2
2
2
2
x j αxi v − λ ai j αxi αx j + λai j αxi x j v
i, j
dt = 0.
(3.6)
t
Let us estimate the last three “energy” terms of first order in the right-hand side of
(3.3). First,
4λ
a
a
α
v
v
dx
dt
i
j
m
x
x
x
xj
i
m
Q
i, j,,m
2
ai j am x j μψx ϕvxi vxm + ai j am μψx x j ϕ + μ ψx ψx j ϕ vxi vxm dx dt = 4λ
Q
i, j,,m
≤ Cλμ
2
Q
ϕ|∇v| dx dt + Cλμ
2
Q
ϕ|∇v|2 dx dt,
(3.7)
where C is a positive constant for λ large enough.
Next, it is easy to see that
− 2λ
≤
a
a
α
v
v
dx
dt
Cλμ
ϕ|∇v|2 dx dt,
i j m xm x xi x j
Q
Q
i, j,,m
Q
2λ
ai j am αx xm vxi vx j dx dt
i, j,,m
=
Q
2λ
μai j am ψx xm ϕvxi vx j + μ ai j am ψx ψxm ϕvxi vx j dx dt
i, j,,m
≥ −Cλμ
Q
ϕ|∇v|2 dx dt + 2λμ2 Λ2
Q
ϕ|∇ψ |2 |∇v|2 dx dt.
It remains to deal with the “energy” term of zero order, that is,
right-hand side of (3.3). Similarly to [4], we have
−2
ᏽ
λ3
2ai j am αxi αx j αx xm v2 − ai j am αxi αx j αx
i, j,,m
=−
ᏽ
2λ3 μ3
(3.8)
2
xm v
2
Q {· · · }v
2 dx dt,
in the
dx dt
2ai j am ψxi ψx j ψx xm ϕ3 v2 + 2μai j am ψxi ψx j ψx ψxm ϕ3 v2
i, j,,m
− ai j am ψxi ψx j ψx xm ϕ3 v 2 − 3μai j am ψxi ψx j ψx ψxm ϕ3 v 2 dx dt
≥ −Cλ3 μ3 ϕ3 v 2 dx dt + 2λ3 μ4 Λ2 ϕ3 |∇ψ |4 v 2 dx dt.
ᏽ
ᏽ
(3.9)
6
Null controllability of a nonlocal parabolic equation
Therefore, for large λ, the following estimate holds:
λαtt − λ
Q
ai j
i, j
+ 2λ
x j αxi t
− λ ai j αxi αx j t + ai j αxi x j t − 2λ ai j αxi αt x j + 4λai j αxi x j αt
2
λ ai j am αxi αx j αx
i, j,,m
+ 2λ ai j
xm
x j am αxi αx xm
− λ ai j am αxi x j αx xm − λ am xm ai j αxi αx j x j
− 2λ2 ai j am αxi αx j αx xm
− 2λai j am αxi x j αx xm − ai j am αx xm x j
xi
≥ −Cλ3 μ3 ϕ3 v 2 dx dt + 2λ3 μ4 Λ2 ϕ3 |∇ψ |4 v 2 dx dt.
ᏽ
v2 dx dt
Q
(3.10)
Step 2. From (3.4), one finds
C
θ 2 (Lu)2 dx dt −
Q
Q
≥ λ 3 μ4
div V dx dt + λ3 μ3
Q
Q
ϕ3 |∇ψ |4 v2 dx dt + λμ2
Q
ϕ3 v2 dx dt + λμ2
Q
ϕ|∇v|2 dx dt
(3.11)
ϕ|∇ψ |2 |∇v|2 dx dt.
Set
Qω0 = ω0 × (0,T).
(3.12)
Noting that |∇ψ(x)| > 0 for all x ∈ Ω0 = Ω \ ω0 , by (3.11), it is easy to see that
2
C
Q
2
θ (Lu) dx dt −
Q
≥ λ 3 μ4
Q
3 3
div V dx dt + λ μ
Q ω0
ϕ3 v2 dx dt + λμ2
Q
3 2
ϕ v dx dt + λμ
2
2
Q ω0
ϕ|∇v| dx dt
ϕ|∇v|2 dx dt.
(3.13)
Returning v to eλα u in (3.13), we arrive at
C
Q
θ 2 (Lu)2 dx dt −
≥ λ3
Q
Q
div V dx dt + λ3
θ 2 ϕ3 u2 dx dt + λ
Q
Q ω0
θ 2 ϕ3 u2 dx dt + λ
Q ω0
θ 2 ϕ|∇u|2 dx dt
θ 2 ϕ|∇u|2 dx dt.
(3.14)
Step 3. The purpose of this step is to get rid of the fourth term in the left-hand side of
(3.14). To this end, we multiply Lu by χθ 2 ϕu and then integrate it over Q, where χ ∈
C0∞ (ω), χ = 1 in ω0 , and χ = 0 in Ω\ω. Then, we obtain
∂u 2
χθ ϕudx dt +
Q ∂t
∂
Q i, j
∂xi
ai j (x)
∂u
χθ 2 ϕudx dt =
∂x j
Q
(Lu)χθ 2 ϕudx dt.
(3.15)
Zhongqi Yin 7
Using integration by parts, it is easy to deduce from (3.15) that
2
λ
Q ω0
2
2
θ ϕ|∇u| dx dt ≤ C
Q
2
θ (Lu) dx dt + λ
3
2 3 2
Qω
θ ϕ u dx dt .
(3.16)
Combining (3.13) and (3.16), we end up with
C
Q
θ 2 (Lu)2 dx dt + λ3
Qω
≥ λ3
Q
θ 2 ϕ3 u2 dx dt −
Q
θ 2 ϕ3 u2 dx dt + λ
Q
div V dx dt
(3.17)
θ 2 ϕ|∇u|2 dx dt.
Step 4. This step is to estimate the “divergence” term div V . Denote the terms on the
right-hand side of (3.5) by Ii , i = 1,2,...,9. First,
I1 + I2 =
Q j
2
=λ μ
ai j vxi vt + λ2 ai j αxi αt v2
2
i
T
0
2
2
θ ϕt c (t)dt
Γ0 i, j
xj
T
2
ai j ψxi n j ds − 2λ μ
0
2
2
θ ϕαt c (t)dt
Γ0 i, j
ai j ψxi n j ds.
(3.18)
Next,
I3 + I5 + I9 =
Q j
2λ
2
ai j am αxi vx vxm − 2ai j am αx vxi vxm − λ ai j am αxi αxm v
i,,m
= −4λ3 μ3
T
− 4λ2 μ2
−2
T
0
0
T
0
ϕ2 θ 2 c(t)dt
λμϕθ 2 dt
Γ i, j,,m
dx dt
xj
ϕ3 c2 (t)θ 2 dt
2
Γ0 i, j,,m
Γ0 i, j,,m
ai j am ψxi ψx ψxm n j ds
ai j am ψx ψxi uxm n j ds
ai j am ψx uxi uxm n j ds.
(3.19)
Further,
I4 =
Q j
= 2λμ2
2λ
T
0
ai j am αx xm
i
T
0
0
xi v
Γ0 i, j,,m
θ 2 ϕc2 (t)dt
T
2
dx dt
xj
θ 2 ϕc2 (t)dt
+ 2λμ
+ 2λμ3
Γ0 i, j,,m
θ 2 ϕc2 (t)dt
ai j am
ai j am
Γ0 i, j,,m
xi ψx ψxm n j ds
xi ψx xm n j ds
ai j am ψxi ψx ψxm n j ds
8
Null controllability of a nonlocal parabolic equation
+ 2λμ2
T
0
+ 2λμ
2
T
2
T
0
T
Γ0 i, j,,m
θ 2 ϕc2 (t)dt
0
2
θ ϕc (t)dt
+ 2λμ
Γ0 i, j,,m
Γ0 i, j,,m
2
2
θ ϕc (t)dt
0
≥ Cλμ3
θ 2 ϕc2 (t)dt
Γ0 i, j,,m
ai j am ψxi x ψxm n j ds
ai j am ψx ψxi xm n j ds
ai j am ψxi x x n j ds
ai j am ψxi ψx ψxm n j ds.
(3.20)
Further,
I6 = −
Q j
= −4λ2 μ3
− 4λμ2
− 4λμ2
− 4λμ
4λ
T
0
T
0
T
0
T
ai j am αx xm vvxi
i,,m
θ 2 ϕ2 c2 (t)dt
θ 2 ϕ2 c2 (t)dt
0
Γ0 i, j,,m
Γ0 i, j,,m
θ 2 ϕc(t)dt
Γ0 i, j,,m
θ 2 ϕc(t)dt
dx dt
xj
Γ0 i, j,,m
ai j am ψxi ψx ψxm n j ds
ai j am ψxi ψx xm n j ds
(3.21)
ai j am ψx ψxm uxi n j ds
ai j am ψx xm uxi n j ds.
Finally,
I7 + I8 =
Q j
= 2λ2 μ3
2λ
T
0
+ 2λ2 μ2
+ 2λ2 μ2
2
ai j am αxi αx xm v + ai j am
i,,m
θ 2 ϕ2 c2 (t)dt
T
0
T
0
2
xm αxi αxm v
2
dx dt
)
xj
Γ0 i, j,,m
θ 2 ϕ2 c2 (t)dt
θ 2 ϕ2 c2 (t)dt
ai j am ψxi ψx ψxm n j ds
Γ0 i, j,,m
Γ0 i, j,,m
(3.22)
ai j am ψxi ψx xm n j ds
ai j am
xm ψxi ψxm n j ds.
Now, combining (3.18)–(3.22) and using the technical condition (2.1), we conclude
that, for large λ and μ, it holds that
Q
div V dx dt ≥ Cλ3 μ3
T
0
θ 2 ϕ3 c2 (t)dt − 4λμΛ
T
0
θ 2 ϕdt
Γ
|∇u|2
i, j
ai j ψxi n j ds.
(3.23)
Zhongqi Yin 9
Step 5. It remains to estimate
that
vt +
ai j vxi
2
Q (λϕ)
−1 2
θ (u2t +
≤ C λ2 α2t v 2 +
xj
i, j
i, j (ai j uxi )x j )
2
λ ai j
2 dx dt.
For this, we observe
2
2 2
2 2 2
2
x j αxi v + λ ai j αxi x j v
i, j
+ λ4 a2i j α2xi α2x j v2 + λ2 a2i j α2i j α2xi vx2j + ai j
2
2
x j vxi
(3.24)
.
This implies
(λϕ)
−1
vt +
ai j vxi
i, j
2
xj
≤ C(λϕ)−1 λ2 α2t v 2 +
2
λ ai j
i, j
2
2 2
2 2 2
2
x j αxi v + λ ai j αxi x j v
+ λ4 a2i j α2xi α2x j v2 + λ2 a2i j α2i j α2xi vx2j + ai j
2
2
x j vxi
(3.25)
≤ C λ3 μ4 ϕ3 v 2 + λμϕ|∇v |2 .
Noting that
2
(λϕ)−1 vt ai j vxi
Q i, j
=2
x j dx dt
ai j (λϕ) vxi vt
Q
i, j
−1
+
x j dx dt − 2
Q
ai j t (λϕ)−1 vxi vx j dx dt +
Q
−1
ai j (λϕ)x j vxi vt dx dt+
Q
ai j (λϕ)−1 vt
Q
ai j (λϕ)−t 1 vxi vx j dx dt
xi vx j dx dt ,
(3.26)
we get
2
Q
(λϕ)
−1
i, j
1
ai j vxi x j vt dx dt ≤
2
−1 2
Q
(λϕ) vt dx dt + Cλ
T
+ λμ
0
2
θαt c (t)dt
Q
Γ0 i, j
ϕ|∇v|2 dx dt
(3.27)
ai j ψxi n j ds.
By (3.24)–(3.27), we obtain that
Q
(λϕ)−1 vt2 + vxi x j
2 dx dt
≤C
Q
(Lu)2 θ 2 dx dt + λ3
T
+ λμ
0
θαt c2 (t)dt
Γ0 i, j
Q ω0
μ3 ϕ3 v2 dx dt + λ
ai j ψxi n j ds.
Q ω0
μϕ|∇v|2 dx dt
(3.28)
10
Null controllability of a nonlocal parabolic equation
This gives
Q
(λϕ)−1 vt2 + (Δv)2 dx dt
≤C
Q
(Lu)2 θ 2 dx dt + λ3
T
+ λμ
0
θαt c2 (t)dt
Γ0 i, j
Q ω0
μ3 ϕ3 v2 dx dt + λ
Q ω0
μϕ|∇v|2 dx dt
(3.29)
ai j ψxi n j ds.
It is easy to see that the last term of the above inequality can be absorbed by (3.23) for
large λ and μ. Finally, replacing v by eλα u and combining (2.2), (3.16), (3.17), (3.23), and
(3.29), we get the desired estimate (3.2). This completes the proof of Theorem 3.1.
Acknowledgments
This work is partially supported by FANEDD of China (Project No 200119), NCET of
China under Grant NCET-04-0882, and the NSF of China under Grants 10371084 and
10525105. The author gratefully acknowledges Professor Xu Zhang for his guidance, encouragement, and suggestions.
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Zhongqi Yin: Department of Engineering and Technology of Caotang School,
Sichuan Normal University, Chengdu 610072, China
E-mail address: zhongqiyin@sohu.com