Observability Estimate for the Fractional Order Parabolic Equations on Measurable Sets

Abstract

We establish an observability estimate for the fractional order parabolic equations evolved in a bounded domain of . The observation region is , where and are measurable subsets of and (0,), 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).

1. Introduction

Letbe a bounded domain in,, with real analytic boundary. Letbe a Lebesgue measurable subset with positive measure, and denote the characteristic function ofby. Let. Letbe a Lebesgue measurable subset with positive measure, and denote the characteristic function ofby. Now, we define an unbounded operatorinas follows:

Let,, be the eigenvalues of, and let  be the corresponding eigenfunctions satisfied that,, which constitutes an orthonormal basis of. It is well known that we can define a class of operatorinas follows: Moreover, the operator is a self-adjoint operator andis an infinitesimal generator of a strong continuous semigroup. Now, we consider the following linear controlled fractional order parabolic equation: where,is a linear bounded operator indefined by,  , andis a control function taken from the space. We denoteto be the unique solution of (3) corresponding to the controland the initial value. We denoteandto be the usual norm and the inner product in, 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, with, generates the rotationally invariantstable Lévy process. For, this process is the normal Brownian motionon(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 timeif for any, there exists a control function, such that the solutions of (3) matches 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 regionis 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, whendo not contain a ball. Recently, the null controllability for the parabolic equations withthat 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.

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: Thus, the exact null controllability property is equivalent to the existence of a constantsuch that the following inequality holds for every solution of (5): Inequality (6) is called observability inequality, and the best constantin inequality (6) will be referred as the observability constant. In this work, we discuss the internal observability estimate for the adjoint system (5) whenandare measurable subsets ofand, 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  ,, is a bounded domain with a real analytic boundary andis a Lebesgue measurable set with positive measure. Let, and letbe a Lebesgue measurable set with positive measure. Let. Then, there exists a constantsuch that, for any data, the solution of (5) satisfied

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, whereandare measurable subsets ofand, 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 operatoris the generator of a semigroup of contraction in, which we denote by,. Indeed, the semigroup can be written as follows: From this, it follows that for any (see [12]).

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

Lemma 2. Letbe a measurable set with a positive measure, and letbe a density point for. Then for each, there exists asuch that the sequence, given by satisfies

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

Lemma 3. Letbe a bounded domain in,. Suppose thathave real analytic boundary. Then, for each subsetwith positive measure, there exist two positive constantsand, which only depend on, such that for each finiteand each choice of the coefficientswith.

This conclusion can be found in the literature [8].

Next, for each, we defineand. Indeed, for each,.

Lemma 4. For any, it always holds that

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

3. The Proof of Main Result

Proof. Let. Then,. Letbe a density point for. By Lemma 2, for, there exists a and a sequence satisfied (10) and (11).
We now define a sequence subset ofas follows: In fact,is a subset of, and By (11), it follow that Letbe the first natural number satisfying; namely,. Letbe a positive number satisfying Taking, by (8), it follows that, for any, Combining with (16) and (12), this shows that For each, we can write, whereand. Takingin (19), it follows that By the definition of, it is easily to see that This, together with (20), deduces The last step is based on the energy decay property of. Along with Lemma 4, we derive that Therefore, Thus, This leads to Summing (26) fromto, it follows that where Actually, by (17), we can derive that This, together with (27), shows that Now, we are in the position to prove (7). The solution of (5) can be written as follows: Along with (30), we can deduce that This completes the proof of the main result.

Conflict of Interests

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).