# Observability Estimate for the Fractional Order Parabolic Equations on Measurable Sets

Authors: Guojie Zheng; M. Montaz Ali
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We establish an observability estimate for the fractional order parabolic equations evolved in a bounded domainΩ of ℝ
n. The observation region is F
×
ω, where ω and F are measurable subsets of Ω and (0,T), respectively, with positive measure. This inequality is equivalent to the null controllable property for a linear controlled fractional order parabolic equation. The building of this estimate is based on the Lebeau-Robbiano strategy and a delicate result in measure theory provided in Phung and Wang (2013).


---

## Body

## 1. Introduction

LetΩbe a bounded domain inℝ
n,n
≥
1, with real analytic boundary. Letω
⊂
Ωbe a Lebesgue measurable subset with positive measure, and denote the characteristic function ofωbyχ
ω. LetT
>
0. LetF
⊂
(
0
,
T
)be a Lebesgue measurable subset with positive measure, and denote the characteristic function ofFbyχ
F. Now, we define an unbounded operatorAinL
2
(
Ω
)as follows:
(1)
D
(
A
)
=
H
2
(
Ω
)
∩
H
0
1
(
Ω
)
,
A
v
=
-
Δ
v
,
for
any
v
∈
D
(
A
)
.Let{
λ
i
}
i
=
1
∞,0
<
λ
1
<
λ
2
≤
⋯, be the eigenvalues ofA
=
-
Δ, and let  {
e
i
}
i
=
1
∞be the corresponding eigenfunctions satisfied that∥
e
i
(
x
)
∥
L
2
(
Ω
)
=
1,i
=
1,2
,
3
,
…, which constitutes an orthonormal basis ofL
2
(
Ω
). It is well known that we can define a class of operatorA
α
(
α
≥
0
)inL
2
(
Ω
)as follows:(2)
D
(
A
α
)
=
{
v
∈
L
2
(
Ω
)
∣
v
=
∑
i
=
1
∞
‍
v
i
e
i
,
∑
i
=
1
∞
‍
λ
i
2
α
|
v
i
|
2
<
∞
}
,
A
α
v
=
∑
i
=
1
∞
‍
λ
i
α
v
i
e
i
,
where
v
=
∑
i
=
1
∞
‍
v
i
e
i
.
Moreover, the operator A
α is a self-adjoint operator and-
A
αis an infinitesimal generator of a strong continuous semigroup{
S
α
(
t
)
}
t
≥
0. Now, we consider the following linear controlled fractional order parabolic equation:
(3)
∂
t
y
(
x
,
t
)
+
A
α
y
(
x
,
t
)
=
B
u
(
x
,
t
)
,
(
x
,
t
)
∈
Ω
×
(
0
,
T
]
,
y
(
x
,
0
)
=
y
0
(
x
)
,
x
∈
Ω
,
whereα
>
1
/
2,Bis a linear bounded operator inL
2
(
Ω
)defined byB
u
=
χ
F
(
t
)
·
χ
ω
(
x
)
u,  y
0
∈
L
2
(
Ω
), andu
(
·
,
t
)is a control function taken from the spaceL
2
(
0
,
T
;
L
2
(
Ω
)
). We denotey
(
·
;
y
0
,
u
)to be the unique solution of (3) corresponding to the controluand the initial valuey
0. We denote∥
·
∥and〈
·
,
·
〉to be the usual norm and the inner product inL
2
(
Ω
), respectively.In recent years, extensive research has been devoted to the study of differential equations with fractional orders due to their importance for applications in various branches of applied sciences and engineering. Many important phenomena in signal processing, electromagnetics, crowded systems, and fluid mechanics are well described by fractional differential equation (see [1]). In this paper, we always discuss the fractional Laplacian. The fractional Laplacian-
A
α, withα
∈
(
0,1
], generates the rotationally invariant2
αstable Lévy process. Forα
=
1, this process is the normal Brownian motionB
tonℝ
n(see [2]).Now, we will focus on the issue of what the controllable property is for the controlled system (3). System (3) is said to be null controllable in timeTif for anyy
0
∈
L
2
(
Ω
), there exists a control functionu
∈
L
2
(
0
,
T
;
L
2
(
Ω
)
), such that the solutions of (3) matches
(4)
y
(
T
;
y
0
,
u
)
=
0
.
The problem of null controllability of parabolic equations has also been the object of numerous studies. Extensive related references can be found in [3–7] and the rich works cited therein. Especially, we refer to [5] for a null controllability result for the parabolic equations which plays a crucial role in establishing the main result in our paper. In the above works, the control regionωis always assumed to contain an open ball. The reason is that the main technique used in the argument, Carleman inequality, is required to construct weight functions. The construction of such functions seems to be not possible, whenωdo not contain a ball. Recently, the null controllability for the parabolic equations withωthat is a measurable subset of positive measure has been established in [8], where an inequality involving measurable sets for a class of real analytic functions was set up in a skillful way. On the other hand, the classical null controllability for some fractional order parabolic equation was studied in [9, 10]. In particular, in [9] the authors proved that one-dimension problem is not controllable from the boundary forα
∈
(
0
,
1
/
2
].By the classical duality argument [11], the controllable properties can be transformed into observability problems on the adjoint system. The adjoint system for (3) may be described as follows:
(5)
∂
t
φ
(
x
,
t
)
=
A
α
φ
(
x
,
t
)
(
x
,
t
)
∈
Ω
×
[
0
,
T
)
,
φ
(
x
,
T
)
=
φ
0
(
x
)
,
x
∈
Ω
.
Thus, the exact null controllability property is equivalent to the existence of a constantC
=
C
(
T
)
>
0such that the following inequality holds for every solution of (5):
(6)
∥
φ
(
x
,
0
)
∥
L
2
(
Ω
)
2
≤
C
(
T
)
∫
F
‍
∫
ω
‍
|
φ
|
2
d
x
d
t
.
Inequality (6) is called observability inequality, and the best constantC
(
T
)in inequality (6) will be referred as the observability constant. In this work, we discuss the internal observability estimate for the adjoint system (5) whenωandFare measurable subsets ofΩand(
0
,
T
), respectively, with positive measure. To the best of our knowledge, this observability estimate has not been studied in the past publications.The main result of the paper is presented as follows.Theorem 1.
Suppose thatΩ
⊂
ℝ
n,n
≥
1, is a bounded domain with a real analytic boundary andω
⊂
Ωis a Lebesgue measurable set with positive measure. LetT
>
0, and letF
⊂
(
0
,
T
)be a Lebesgue measurable set with positive measure. Letα
>
1
/
2. Then, there exists a constantC
=
C
(
Ω
,
T
,
ω
,
F
,
α
)such that, for any dataφ
0
∈
L
2
(
Ω
), the solution of (5) satisfied
(7)
∥
φ
(
x
,
0
)
∥
L
2
(
Ω
)
2
≤
C
(
Ω
,
T
,
ω
,
F
,
α
)
∫
F
‍
∫
ω
‍
|
φ
(
x
,
t
)
|
2
d
x
d
t
.Observability inequality (7) in Theorem 1 allows for estimating the total energy of the solutions of (5) at time 0 in terms of the partial energy localized in the observation regionF
×
ω, whereωandFare measurable subsets ofΩand(
0
,
T
), respectively, with positive measure. This inequality is equivalent to the null controllability property for the controlled system (3).We proceed as follows. In Section2, we give some preliminary results. Section 3 is devoted to the proof of Theorem 1.


## 2. Preliminary Results

In this section, we will introduce some notions and preliminary results. Based on classical semigroup theory, we see that the operator-
A
αis the generator of a semigroup of contraction inL
2
(
Ω
), which we denote byS
α
(
t
),α
>
1
/
2. Indeed, the semigroup can be written as follows:
(8)
S
α
(
t
)
ψ
=
∑
i
=
1
∞
‍
e
-
λ
i
α
t
〈
ψ
,
e
i
〉
e
i
,
for
ψ
∈
L
2
(
Ω
)
.
From this, it follows that
(9)
∥
S
α
(
t
)
ψ
∥
L
2
(
Ω
)
≤
e
-
λ
1
α
t
∥
ψ
∥
L
2
(
Ω
)
,
for anyα
>
1
/
2 (see [12]).Throughout the rest of the paper, the following notation will be used. For each measurable setA
⊆
ℝ,|
A
|stands for its Lebesgue measure in  ℝ. The following lemma is quoted from [13].Lemma 2.
LetE
⊆
[
0
,
T
]be a measurable set with a positive measure, and letlbe a density point forE. Then for eachz
>
1, there exists al
1
∈
(
l
,
T
)such that the sequence{
l
i
}
i
=
1
∞, given by
(10)
l
i
+
1
=
l
+
1
z
i
(
l
1
-
l
)
,
satisfies
(11)
|
E
∩
(
l
i
+
1
,
l
i
)
|
≥
1
2
(
l
i
-
l
i
+
1
)
.Next, we recall the following results, which play a key role in this paper.Lemma 3.
LetΩbe a bounded domain inR
n,n
≥
1. Suppose thatΩhave real analytic boundary. Then, for each subsetω
⊂
Ωwith positive measure, there exist two positive constantsC
1
>
1andC
2
>
0, which only depend onΩ
,
ω, such that
(12)
∑
λ
i
≤
r
‍
|
a
i
|
2
≤
C
1
e
C
2
r
∫
ω
‍
|
∑
λ
i
≤
r
‍
a
i
e
i
(
x
)
|
2
d
x
for each finiter
>
0and each choice of the coefficients{
a
i
}
λ
i
≤
rwitha
i
∈
ℝ.This conclusion can be found in the literature [8].Next, for eachr
>
0, we defineX
r
=
span
⁡
{
e
i
(
x
)
}
λ
i
≤
randX
r
⊥
=
span
⁡
{
e
i
(
x
)
}
λ
i
>
r. Indeed, for eachr
>
0,L
2
(
Ω
)
=
X
r
⊕
X
r
⊥.Lemma 4.
For anyξ
0
∈
X
r
⊥, it always holds that
(13)
∥
S
α
(
t
)
ξ
0
∥
L
2
(
Ω
)
<
e
-
r
α
t
∥
ξ
0
∥
L
2
(
Ω
)
.This lemma can be easily obtained by (8) and (9).


## 3. The Proof of Main Result

Proof.
LetE
=
{
t
|
t
=
T
-
s
,
where
s
∈
F
}. Then,|
E
|
=
|
F
|
>
0. Letlbe a density point forE. By Lemma 2, forz
>
1, there exists a l
1
∈
(
l
,
T
)and a sequence {
l
i
}
i
=
1
∞ satisfied (10) and (11).
We now define a sequence subset of(
0
,
T
)as follows:
(14)
E
i
=
{
t
-
l
i
-
l
i
+
1
4
∣
t
∈
E
∩
(
l
i
+
1
+
l
i
-
l
i
+
1
4
,
l
i
)
}
.
In fact,E
iis a subset of(
l
i
+
1
,
l
i
+
1
+
(
3
/
4
)
(
l
i
-
l
i
+
1
)
), and
(15)
|
E
i
|
=
|
E
∩
(
l
i
+
1
+
l
i
-
l
i
+
1
4
,
l
i
)
|
=
|
E
∩
{
(
l
i
+
1
,
l
i
)
∖
(
l
i
+
1
,
l
i
+
1
+
l
i
-
l
i
+
1
4
)
}
|
.
By (11), it follow that
(16)
|
E
i
|
≥
l
i
-
l
i
+
1
4
.
Leti
0be the first natural number satisfyingi
0
>
1
/
(
2
α
-
1
); namely,i
0
=
[
1
/
(
2
α
-
1
)
]
+
1. Letb
>
1be a positive number satisfying
(17)
(
b
α
z
)
i
(
1
-
1
z
)
(
l
1
-
l
)
>
4
C
2
b
(
i
+
1
)
/
2
+
4
ln
⁡
(
8
C
1
z
)
,
(
b
α
z
)
i
(
1
-
1
z
)
(
l
1
-
i
)
ii
for
i
=
i
0
,
i
0
+
1
,
i
0
+
2
,
…
.
Takingr
i
=
b
i, by (8), it follows that, for anyξ
∈
X
r
i,
(18)
∫
l
i
+
1
l
i
+
1
+
(
3
/
4
)
(
l
i
-
l
i
+
1
)
‍
χ
E
i
(
t
)
∥
S
α
(
l
i
+
1
+
3
4
(
l
i
-
l
i
+
1
)
)
ξ
∥
L
2
(
Ω
)
d
t
≤
∫
l
i
+
1
l
i
+
1
+
(
3
/
4
)
(
l
i
-
l
i
+
1
)
‍
χ
E
i
(
t
)
∥
S
α
(
t
)
ξ
∥
L
2
(
Ω
)
d
t
.
Combining with (16) and (12), this shows that
(19)
l
i
-
l
i
+
1
4
∥
S
α
(
l
i
+
1
+
3
4
(
l
i
-
l
i
+
1
)
)
ξ
∥
L
2
(
Ω
)
≤
|
E
i
|
·
∥
S
α
(
l
i
+
1
+
3
4
(
l
i
-
l
i
+
1
)
)
ξ
∥
L
2
(
Ω
)
≤
∫
l
i
+
1
l
i
+
1
+
(
3
/
4
)
(
l
i
-
l
i
+
1
)
‍
χ
E
i
(
t
)
∥
S
α
(
t
)
ξ
∥
L
2
(
Ω
)
d
t
≤
C
1
e
C
2
r
i
∫
l
i
+
1
l
i
+
1
+
(
3
/
4
)
(
l
i
-
l
i
+
1
)
‍
χ
E
i
(
t
)
∥
S
α
(
t
)
ξ
∥
L
2
(
ω
)
d
t
.
For eachϕ
∈
L
2
(
Ω
), we can writeϕ
=
ϕ
1
+
ϕ
2, whereϕ
1
∈
X
r
iandϕ
2
∈
X
r
i
⊥. Takingξ
=
S
α
(
(
l
i
-
l
i
+
1
)
/
4
)
ϕ
1in (19), it follows that
(20)
l
i
-
l
i
+
1
4
∥
S
α
(
l
i
)
ϕ
1
∥
L
2
(
Ω
)
≤
l
i
-
l
i
+
1
4
∥
S
α
(
l
i
+
1
+
3
4
(
l
i
-
l
i
+
1
)
)
∘
S
α
(
l
i
-
l
i
+
1
4
)
ϕ
1
∥
L
2
(
Ω
)
≤
C
1
e
C
2
r
i
∫
l
i
+
1
l
i
+
1
+
(
3
/
4
)
(
l
i
-
l
i
+
1
)
‍
χ
E
i
(
t
)
≤
C
1
e
C
2
r
i
∫
l
i
+
1
‍
×
∥
S
α
(
t
+
l
i
-
l
i
+
1
4
)
ϕ
1
∥
L
2
(
ω
)
d
t
≤
C
1
e
C
2
r
i
∫
l
i
+
1
+
(
(
l
i
-
l
i
+
1
)
/
4
)
l
i
‍
χ
E
i
(
t
-
l
i
-
l
i
+
1
4
)
≤
C
1
e
C
2
r
i
∫
l
i
+
1
‍
×
∥
S
α
(
t
)
ϕ
1
∥
L
2
(
ω
)
d
t
.
By the definition ofE
i, it is easily to see that
(21)
iiiiiii
χ
E
i
(
t
-
l
i
-
l
i
+
1
4
)
=
χ
E
(
t
)
,
for
any
t
∈
(
l
i
+
1
+
l
i
-
l
i
+
1
4
,
l
i
)
.
This, together with (20), deduces
(22)
l
i
-
l
i
+
1
4
∥
S
α
(
l
i
)
ϕ
1
∥
L
2
(
Ω
)
≤
C
1
e
C
2
r
i
∫
l
i
+
1
+
(
(
l
i
-
l
i
+
1
)
/
4
)
l
i
‍
χ
E
(
t
)
≤
C
1
e
C
2
r
i
∫
l
i
+
1
l
i
‍
×
(
∥
S
α
(
t
)
ϕ
∥
L
2
(
ω
)
+
∥
S
α
(
t
)
ϕ
2
∥
L
2
(
Ω
)
)
d
t
≤
C
1
e
C
2
r
i
∫
l
i
+
1
+
(
(
l
i
-
l
i
+
1
)
/
4
)
l
i
‍
χ
E
(
t
)
∥
S
α
(
t
)
ϕ
∥
L
2
(
ω
)
d
t
+
C
1
e
C
2
r
i
(
l
i
-
l
i
+
1
)
∥
S
α
(
l
i
+
1
+
l
i
-
l
i
+
1
4
)
ϕ
2
∥
L
2
(
Ω
)
d
t
.
The last step is based on the energy decay property ofS
α
(
t
). Along with Lemma 4, we derive that
(23)
l
i
-
l
i
+
1
4
∥
S
α
(
l
i
)
ϕ
1
∥
L
2
(
Ω
)
≤
C
1
e
C
2
r
i
∫
l
i
+
1
+
(
(
l
i
-
l
i
+
1
)
/
4
)
l
i
‍
χ
E
(
t
)
∥
S
α
(
t
)
ϕ
∥
L
2
(
ω
)
d
t
+
C
1
e
C
2
r
i
(
l
i
-
l
i
+
1
)
e
-
r
i
α
(
(
l
i
-
l
i
+
1
)
/
4
)
∥
S
α
(
l
i
+
1
)
ϕ
2
∥
L
2
(
Ω
)
≤
C
1
e
C
2
r
i
∫
l
i
+
1
l
i
‍
χ
E
(
t
)
∥
S
α
(
t
)
ϕ
∥
L
2
(
ω
)
d
t
+
C
1
e
C
2
r
i
(
l
i
-
l
i
+
1
)
e
-
r
i
α
(
(
l
i
-
l
i
+
1
)
/
4
)
∥
S
α
(
l
i
+
1
)
ϕ
2
∥
L
2
(
Ω
)
.
Therefore,
(24)
l
i
-
l
i
+
1
4
∥
S
α
(
l
i
)
ϕ
∥
L
2
(
Ω
)
≤
C
1
e
C
2
r
i
∫
l
i
+
1
l
i
‍
χ
E
(
t
)
∥
S
α
(
t
)
ϕ
∥
L
2
(
ω
)
d
t
+
C
1
e
C
2
r
i
(
l
i
-
l
i
+
1
)
e
-
r
i
α
(
(
l
i
-
l
i
+
1
)
/
4
)
∥
S
α
(
l
i
+
1
)
ϕ
2
∥
L
2
(
Ω
)
+
l
i
-
l
i
+
1
4
∥
S
α
(
l
i
)
ϕ
2
∥
L
2
(
Ω
)
≤
C
1
e
C
2
r
i
∫
l
i
+
1
l
i
‍
χ
E
(
t
)
∥
S
α
(
t
)
ϕ
∥
L
2
(
ω
)
d
t
+
C
1
e
C
2
r
i
(
l
i
-
l
i
+
1
)
e
-
r
i
α
(
(
l
i
-
l
i
+
1
)
/
4
)
∥
S
α
(
l
i
+
1
)
ϕ
2
∥
L
2
(
Ω
)
+
l
i
-
l
i
+
1
4
e
-
r
i
α
(
l
i
-
l
i
+
1
)
∥
S
α
(
l
i
+
1
)
ϕ
2
∥
L
2
(
Ω
)
.
Thus,
(25)
l
i
-
l
i
+
1
4
∥
S
α
(
l
i
)
ϕ
∥
L
2
(
Ω
)
≤
C
1
e
C
2
r
i
∫
l
i
+
1
l
i
‍
χ
E
(
t
)
∥
S
α
(
t
)
ϕ
∥
L
2
(
ω
)
d
t
+
(
l
i
-
l
i
+
1
)
e
-
r
i
α
(
(
l
i
-
l
i
+
1
)
/
4
)
(
C
1
e
C
2
r
i
+
1
)
∥
S
α
(
l
i
+
1
)
ϕ
∥
L
2
(
Ω
)
.
This leads to
(26)
l
i
-
l
i
+
1
4
C
1
e
C
2
r
i
∥
S
α
(
l
i
)
ϕ
∥
L
2
(
Ω
)
-
C
1
e
C
2
r
i
+
1
C
1
e
C
2
r
i
(
l
i
-
l
i
+
1
)
e
-
r
i
α
(
(
l
i
-
l
i
+
1
)
/
4
)
∥
S
α
(
l
i
+
1
)
ϕ
∥
L
2
(
Ω
)
≤
∫
l
i
+
1
l
i
‍
χ
E
(
t
)
∥
S
α
(
t
)
ϕ
∥
L
2
(
ω
)
d
t
.
Summing (26) fromi
0to∞, it follows that
(27)
l
i
0
-
l
i
0
+
1
4
C
1
e
C
2
r
i
0
∥
S
α
(
l
i
0
)
ϕ
∥
L
2
(
Ω
)
+
∑
i
=
i
0
∞
‍
k
i
∥
S
α
(
l
i
+
1
)
ϕ
∥
L
2
(
Ω
)
≤
∫
0
T
‍
χ
E
(
t
)
∥
S
α
(
t
)
ϕ
∥
L
2
(
ω
)
d
t
,
where
(28)
k
i
=
l
i
+
1
-
l
i
+
2
4
C
1
e
C
2
r
i
+
1
-
C
1
e
C
2
r
i
+
1
C
1
e
C
2
r
i
(
l
i
-
l
i
+
1
)
e
-
r
i
α
(
(
l
i
-
l
i
+
1
)
/
4
)
,
k
i
=
l
i
+
1
-
l
i
+
2
4
C
1
e
C
2
r
i
+
1
-
C
1
e
C
2
r
i
+
1
C
1
e
C
2
r
i
ii
i
=
i
0
,
i
0
+
1
,
i
0
+
2
…
.
Actually, by (17), we can derive that
(29)
k
i
>
0
,
for
any
i
=
i
0
,
i
0
+
1
,
i
0
+
2
…
.
This, together with (27), shows that
(30)
∥
S
α
(
T
)
ϕ
∥
L
2
(
Ω
)
≤
∥
S
α
(
l
i
0
)
ϕ
∥
L
2
(
Ω
)
≤
4
C
1
e
C
2
r
i
0
l
i
0
-
l
i
0
+
1
∫
E
‍
∥
S
α
(
t
)
ϕ
∥
L
2
(
ω
)
d
t
.
Now, we are in the position to prove (7). The solution of (5) can be written as follows:
(31)
φ
(
t
)
=
S
α
(
T
-
t
)
φ
0
.
Along with (30), we can deduce that
(32)
∥
φ
(
x
,
0
)
∥
L
2
(
Ω
)
≤
4
C
1
e
C
2
r
i
0
l
i
0
-
l
i
0
+
1
∫
F
∥
φ
(
t
)
∥
L
2
(
x
,
ω
)
d
t
.
This completes the proof of the main result.



---

*Source: 361904-2014-03-27.xml*