Scholarly article on topic 'Observability Estimate for the Fractional Order Parabolic Equations on Measurable Sets'

Observability Estimate for the Fractional Order Parabolic Equations on Measurable Sets Academic research paper on "Mathematics"

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Academic research paper on topic "Observability Estimate for the Fractional Order Parabolic Equations on Measurable Sets"

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 361904, 5 pages http://dx.doi.org/10.1155/2014/361904

Research Article

Observability Estimate for the Fractional Order Parabolic Equations on Measurable Sets

Guojie Zheng1,2 and M. Montaz Ali2,3

1 College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China

2 School of Computational and Applied Mathematics, University of the Witwatersrand (Wits), Johannesburg 2050, South Africa

3 TCSE, Faculty ofEngineeringandBuilt Environment, University of the Witwatersrand (Wits), Johannesburg2050, SouthAfrica

Correspondence should be addressed to Guojie Zheng; guojiezheng@yeah.net Received 10 January 2014; Accepted 3 March 2014; Published 27 March 2014 Academic Editor: Sheng-Jie Li

Copyright © 2014 G. Zheng and M. M. Ali. 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.

We establish an observability estimate for the fractional order parabolic equations evolved in a bounded domain H of R". The observation region is Fxm, where to and F are measurable subsets of H 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).

Moreover, the operator Aa is a self-adjoint operator and -Aa is an infinitesimal generator of a strong continuous semigroup |Sa(i)}ta0. Now, we consider the following linear controlled fractional order parabolic equation:

dty (x, t) + Aay (x, t) = Bu (x, t), (x,t) eHx (0, T], y (x, 0) = y0 (x), x£ H,

where a > 1/2, B is a linear bounded operator in L2(H) defined by Bu = Xf(^) ' Xw(x)u, e L2(H), and u(-,t) is a control function taken from the space L2(0,T;L2(H)). We denote y(-;y0,u) to be the unique solution of (3) corresponding to the control u and the initial value y0. We denote || • || and {■, •) to be the usual norm and the inner product in L2(H), 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 -Aa, with

1. Introduction

Let H be a bounded domain in R", n > 1, with real analytic boundary. Let to c H be a Lebesgue measurable subset with positive measure, and denote the characteristic function of to by xa. Let T > 0. Let F c (0, T) be a Lebesgue measurable subset with positive measure, and denote the characteristic function of F by \F. Now, we define an unbounded operator A in L2(H) as follows:

D(A) = H2 (H)nH0Q (H), Av = -Av, for any v e D (A).

Let |Ai}°°1, 0 < < A2 < •••, be the eigenvalues of A = -A, and let {ei}'^1 be the corresponding eigenfunc-tions satisfied that lle^x)!^^ = 1, i = 1,2,3,..., which constitutes an orthonormal basis of L2(H). It is well known that we can define a class of operator Aa (a > 0) in L2(H) as follows:

D(Aa) =

v eL2 (H) | v =

2« I 12

V;e:, y A; |V;I < >X>

11'/ , t >= 1 >=1

Aav = y^*viei, where v = yv i=1 i=1

a e (0,1], generates the rotationally invariant 2a stable Levy process. For a = 1, this process is the normal Brownian motion Bt on R" (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 time T if for any y0 e L2(Q), there exists a control function u e L2(0, T; L2(Q)), such that the solutions of (3) matches

y(T;y0,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 region w is 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, when w do not contain a ball. Recently, the null controllability for the parabolic equations with w that 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 for a e (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:

dtf(x,t)=Aaf(x,t) (x,t) eQx[0,T), f (x, T) = <p0 (x), x e Q.

Thus, the exact null controllability property is equivalent to the existence of a constant C = C(T) > 0 such that the following inequality holds for every solution of (5):

\\v(x>0)\\2L2(n) <C(T) \f\2dxdt.

Inequality (6) is called observability inequality, and the best constant C(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) when w and F are measurable subsets of Q and (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 that Q c R", n > 1, is a bounded domain with a real analytic boundary and w c Q is a Lebesgue measurable set with positive measure. Let T > 0, and let F c (0,T) bea Lebesgue measurable set with positive measure.

Let a > 1/2. Then, there exists a constant C = C(Q, T, w, F, a) such that, for any data f0 e L2(Q), thesolution of (5) satisfied

(x'0)\\i2(n) < C (Q,T,w,F,a) \<p(x,t)\2dxdt.

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 region F x w, where w and F are measurable subsets of Q and (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 Section 2, 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 -Aa is the generator of a semigroup of contraction in L2(Q), which we denote by Sa(t), a > 1/2. Indeed, the semigroup can be written as follows:

(t)y = ^e-x°t{f,e,)ei, for yeL2 (Q). (8) ;=1

From this, it follows that

IIs«mlm IML^),

for any a > 1/2 (see [12]).

Throughout the rest of the paper, the following notation will be used. For each measurable set A c R, |A| stands for its Lebesgue measure in R. The following lemma is quoted from [13].

Lemma 2. Let E c [0, T] bea measurable set with a positive measure, and let I be a density point for E. Thenfor each z > 1, there exists a l1 e (I, T) such that the sequence {IjlTOv given by

satisfies

=U-(k-I),

\En(li+1,l,)\>2(l, -l,+1).

Next, we recall the following results, which play a key role in this paper.

Lemma 3. Let Q be a bounded domain in Rn, n> 1. Suppose that Q have real analytic boundary. Then, for each subset w c Q with positive measure, there exist two positive constants C1 >1 and C2 > 0, which only depend on Q, w, such that

Xh\2 <ciec^ f

X, <r Jw

YUiei(x)

for each finite r > 0 and each choice of the coefficients |fl'}^.<r with e R.

This conclusion can be found in the literature [8]. Next, for each r > 0, we define Xr = span|ej(x)}A sr and Xf = span|e;(x)}A >r. Indeed, for each r > 0, L2(Q) = Xr ® Xf.

Lemma 4. For any e Xf, it always holds that

0|ll2(n)

>0||l2(n)'

This lemma can be easily obtained by (8) and (9).

3. The Proof of Main Result

Proof. Let E = ji | t = r-s, where s e Fj. Then, |E| = |F| > 0. Let I be a density point for E. By Lemma 2,for z > 1, there exists a Z1 e (/, r) and a sequence j/¡j^ satisfied (10) and (11). We now define a sequence subset of (0, r) as follows:

E, = {i-^ | ieEn (/i+1 + /,)}. (14)

In fact, E, is a subset of (/i+1, /i+1 + (3/4)(Z, - Zi+1)), and

|E,| =

En| Z.+ + , h

= E n {(/i+1, Z,)\( /1+1, /1+1 + )}

By (11), it follow that

h - ¿,+1

Let (0 be the first natural number satisfying <'0 > 1/(2a - 1); namely, f0 = [1/(2a - 1)] + 1. Let b > 1 be a positive number satisfying

( Ç ) (i - !)(/1 - /) > 4C2fc(,+1)/2 +4 ln (8C1z),

for i = i0, i0 + 1, i0 + 2,____

Taking r = fc', by (8), it follows that, for any ÇeX,.,

r;i+1+(3/4)ft-;i+1)

Xe, to

s. ( zi+i + 4(h - WK

fii+1+(3/4)(ii-ii+1)

* I Xe, (i)||Sa(i)?||L2(n)di.

•'i+i

Combining with (16) and (12), this shows that 3

Z, - ¿,+1

< |E,| •

s. ( /,+1 + 4(z, - z«+1 ))* s. ( /,+1 + 3(z, - U)^

< I Xe, (i) ||Sa(i)?||l2(n)di

< C^2^

■ii+i+(3/4)(i,-ii+i)

Xe, (i)||sa(i)?||l2(„)di.

For each 0 e L2(Q), we can write 0 = + 02, where e Xr. and 02 e X^. Taking $ = S„((Z, - Z^)/^ in (19), it follows that

(13) h - h

llS.(^)01|lL2(n)

< C1e'

.CzV^i

s. ( i„1 + 4( i,-1,+,)).S„ ( ^ ) (

i,+i+(3/4)(i,-i,+i)

Xe, to

^e^ | xe,

h - ^+1

X IM^Hl^-

By the definition of £', it is easily to see that

h - 1 = Xe to,

for any i e | /i+1 + ^ ^'+1, Z;

This, together with (20), deduces

JawrilL2(n)

< C1eC2V?'

•%i+№H ,+i)/4)

*(llS« Will**) + IIS« (i)02||L2(n))di

<C1eC2V^ xe (0 ||Sa(i)0||L2(„)di

Ji,+i+((i,-i,+i )/4) ( )

+ C1eC2V^ (/, - Z,+1)||sa (/,+1 + ^ )02

di. (22)

The last step is based on the energy decay property of Sa(i). Along with Lemma 4, we derive that

< C1eC2^^ [

+ C1eC2^ (/, - /¡+1) e-r,°1((',-',+i)/4)||Sa(I,+1)0

'I Xe (i) HS«(i)0||l2(„)di

I (i) ||Sa(i)0||i2(w)di

+ Ciec^ (/. -/,+i)e

-^(№-¡+1 )/4)

||S«(^i+1)02||L2(n)-

Therefore,

h _ki.

||SOT(/i)0||i2(O) < C^ I (i) ||Sa(i)0||i2(„)di

+ Ci eC^ (/,-/,+i ^ ((l, -H.+l)/4)||Sa(/1+i )*

«V'>+^2NL2(n)

< CieC2V?'

4 M aVW2||L2(n) H

v_ (^) (

I Xe (i) ||Sa(i)0||i2(„)di

•^j+l

+ Ci eC^ (/,-Z,+i ^((H -k'+l)/4)||Sa(/1+i )*

av i+i /r2 ||l2(0)

'aVi+i^Hi2 (n)'

i'r||L2(n) l,

<Cie^ I' ^ (i) ||Sa(i)0||i2(„)di •^j+l

+ (/, - /,+i) e^«*"^ (Ci^ + 1) ||Sa(/,+i)0||L2(n).

This leads to

||i2(n)

CieC2^

| Xe (i)||Sa(i)0||L2(„)di.

i+i )t ||l2(o)

Summing (26) from <'0 to to, it follows that

/ _ / to

i—i0

j0T Xe (i) ||Sa(i)0||i2(„)di,

, _ ¿i+i _ ^i+2 _ C^2^ + 1 (? _ ) p-rf((ij-ij+i)/4) 4Ci eC2V^ CieC2V7 (ii Ji+i)e '

i _ (0, (0 + 1, <'0 + 2

Actually, by (17), we can derive that

> 0, for any i = (0, (q + 1, (0 + 2. This, together with (27), shows that

IK(r)0lL2(n) £ )0||i2(n)

4CX ec2V^ (- N

h0 _ Je "

Now, we are in the position to prove (7). The solution of (5) can be written as follows:

y(i) = Sa (r-i)^Q. Along with (30), we can deduce that

>o ¡o + l ^

This completes the proof of the main result.

Conflict of Interests

(32) □

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their sincere thanks to the referees for their providing several important references and for their valuable suggestions. This work was partially supported by the National Research Foundation of South Africa under Grant CPR2010030300009918, the National Natural Science Foundation of China (U1204105, 61203293), and the Key Foundation of Henan Educational Committee (13A120524,12B120006).

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