Journal of Applied Mathematics

Volume 2012 (2012), Article ID 931975, 17 pages

doi:10.1155/2012/931975

Research Article

Exact Null Controllability for Fractional Nonlocal Integrodifferential Equations via Implicit Evolution System

Amar Debbouche¹ and Dumitru Baleanu2,3

¹Department of Mathematics, Faculty of Science, Guelma University, Guelma, Algeria

²Department of Mathematics and Computer Science, Cankaya University, Ankara, Turkey

³Institute of Space Sciences, P.O. Box, MG-23, 76900 Magurele-Bucharest, Romania

Received 1 May 2012; Revised 15 July 2012; Accepted 16 July 2012

Academic Editor: F. Marcellán

Copyright © 2012 Amar Debbouche and Dumitru Baleanu. 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.

Abstract.

We introduce a new concept called implicit evolution system to establish the existence results of mild and strong solutions of a class of fractional nonlocal nonlinear integrodifferential system, then we prove the exact null controllability result of a class of fractional evolution nonlocal integrodifferential control system in Banach space. As an application that illustrates the abstract results, two examples are provided.

1. Introduction

In this paper, we study the fractional nonlocal integrodifferential system of the form

where , . Let be the infinitesimal generator of a -semigroup in a Banach space , and is a family of linear closed operators defined on dense sets , , respectively, in into . It is assumed that , , and are given abstract functions. Here, and .

Basic researches in differential equations have showed that many phenomena in nature are modeled more accurately using fractional derivatives and integrals; for more detail, we can refer to [1–13] and the references therein. There are many applications where the fractional calculus can be used, for example, viscoelasticity, electrochemistry, diffusion processes, control theory, heat conduction, electricity, mechanics, chaos, and fractals [14].

Controllability is a fundamental concept in mathematical control theory and plays an important role in both finite and infinite dimensional spaces, that is, systems represented by ordinary differential equations and partial differential equations, respectively. So it is natural to extend this concept to dynamical systems represented by fractional differential equations. Several fractional partial differential equations and integrodifferential equations can be expressed abstractly in some Banach spaces, in many cases, the accurate analysis, design and assessment of systems subjected to realistic environments must take into account the potential of random loads and randomness in the system properties. Randomness is intrinsic to the mathematical formulation of many phenomena such as fluctuations in the stock market or noise in communication networks. Fu studied the controllability results of some kinds of neutral functional differential systems, see [15, 16]. In our previous work [17], we established the controllability of fractional evolution nonlocal impulsive quasilinear delay integrodifferential systems.

The existence results to evolution equations with nonlocal conditions in Banach space were studied first by Byszewski [18, 19]; subsequently, many authors have been studied the same question, see for instance [20–23].

Deng [24] indicated that, using the nonlocal condition to describe for instance, the diffusion phenomenon of a small amount of gas in a transparent tube can give better result than using the usual local Cauchy problem . Let us observe also that, since Deng’s papers, the function is considered

where are given constants and .

In this paper, we introduce a new concept in the theory of Semigroup named “implicit evolution system” to show the reader “what is the main difference between the solutions of fractional () and classical (first order) homogeneous evolution equation?” which is based on the work [17] and Pazy [25]. A new form of nonlocal condition is also presented.

Our paper is organized as follows. Section 2 is devoted to a review of some essential results which will be used in this work to obtain our main results. In Section 3, we use the theory of semigroups [25] in order to introduce our new concept that is called implicit evolution system. In Section 4, we establish the existence, uniqueness, and regularity of mild solutions of a class of fractional evolution nonlinear integrodifferential systems with nonlocal conditions in Banach space. In Section 5, we prove the exact null controllability of a class of fractional evolution nonlocal integrodifferential control systems; the last section deals to give examples that provide the abstract results.

2. Preliminary Results

Definition 2.1. The fractional integral of order with the lower limit zero for a function is defined as

provided the right side is pointwise defined on , where is the gamma function.

Riemann-Liouville derivative of order with the lower limit zero for a function can be written as

The Caputo derivative of order for a function can be written as

Remark 2.2. (1) If , then

(2) The Caputo derivative of a constant is equal to zero.

(3) If is an abstract function with values in , then integrals which appear in Definition 2.1 are taken in Bochner’s sense.

Definition 2.3. By a strong solution of the nonlocal Cauchy problem (1.1), (1.2), we mean a function with values in such that

(i)is a continuous function in and ,(ii) exists and continuous on , , and satisfies (1.1) on and (1.2).

It is suitable to rewrite (1.1), (1.2) in the form

see also [26, 27].

Let and be two Banach spaces such that is densely and continuously embedded in . We denote by every Banach space endowed with the usual norm, which is given by , for . The space of all bounded linear operators from to is denoted by . We recall some definitions and known facts from Pazy [25].

Definition 2.4. Let be a linear operator in , and let be a subspace of . The operator defined by and for is called the part of in .

Definition 2.5. Let be a subset of and for every and , and let be the infinitesimal generator of a -semigroup , , on . The family of operators is stable if there are constants and such that

for every finite sequences , .

The stability of implies that

and any finite sequences , , .

Definition 2.6. Let , be the -semigroup generated by . A subspace of is called -admissible if is invariant subspace of , and the restriction of to is a -semigroup in .

Let be a subset of such that for every , is the infinitesimal generator of a -semigroup , on . We make the following assumptions.

(H₁) The family is stable.

(H₂) is -admissible for , and the family of parts of in is stable in .

(H₃) For , , is a bounded linear operator from to and is continuous in the norm .

(H₄) There is a constant such that

holds for every , and .

In the next section, we will introduce a new concept in the theory of semigroups.

3. Implicit Evolution System

Let be a subset of and a family of operators satisfying the conditions (H₁)–(). If has values in , then there is a unique evolution system , , , in satisfying

(i) for , where and are stability constants,(ii) for and ,(iii) for and .

Remark 3.1. (1) If is the identity and , then is the explicit evolution system given in Pazy [25] and in Zaidman [28].

(2) Since, in our case, is dependent of and , so we call it an implicit evolution system generated by .

(3) For nonautonomous differential equations in a Banach space, the implicit evolution system is similar to our concept -resolvent family.

(4) We can deduce that (1.1)-(1.2) is well posed if and only if is the generator of the implicit evolution system .

Further, we assume the following.

(H₅) For every satisfying for , we have

and is strongly continuous in for .

(H₆) is reflexive.

(H₇) For every , .

(H₈) The operator exists in for any with and

where is a positive constant independent of both and .

(H₉) is Lipschitz continuous in and bounded in , that is, there exist constants and such that

For the conditions (H₉) and (H₁₀), let be taken as both and .

(H₁₀) is continuous, and there exist constants and such that

(H₁₁) is continuous, and there exist constants and such that

Let us take , , .

(H₁₂) There exist positive constants and such that

By a mild solution of (1.1), (1.2), we mean a function with values in and satisfying the integral equation

(H₁₃) Further, there exists a constant such that for every with values in and every we have

4. Existence Results

Theorem 4.1. Let and , . If is the generator of an implicit evolution system and the assumptions (H₅)() are satisfied, then (1.1), (1.2) has a unique mild solution on .

Proof. Let be a nonempty closed subset of defined by

Consider a mapping on defined by

For , we have

Thus, maps into itself. Now, we will show that is a strict contraction on which will ensure the existence of a unique continuous function satisfying (3.7) on .

If , then

Thus,

which means that is a strict contraction map from into , and therefore by the Banach contraction principle there exists a unique fixed point such that . Hence, is a unique mild solution of (1.1), (1.2) on .

Theorem 4.2. Assume the following.

(i)Conditions (H₁)(H₁₃) hold.(ii)The functions and are uniformly Hölder continuous in for every element in .(iii)There are numbers , and such that

for all and all .

Then, the problem (1.1), (1.2) has a unique strong solution on .

Proof. Applying Theorem 4.1, the problem (1.1), (1.2) has a mild solution . Now, we will show that is a unique strong solution of the considered problem on .

According to (ii), is uniformly Hölder continuous in for every element in , also (iii) implies that and are uniformly Hölder continuous on ([20, 26]).

Set

Clearly is uniformly Hölder continuous in .

Consider the following nonlocal Cauchy problem:

From Pazy, (4.8) has a unique solution on given by

Noting that [21], each term on the right hand side of (4.9) belongs to , thus , using the uniqueness of , we have that . Hence, is the unique strong solution of (1.1), (1.2) on .

In next section, some results are obtained from Sakthivel et al. [29, 30].

5. Exactly Null Controllability Results

Consider fractional nonlocal evolution integrodifferential control system of the form

where the unknown takes values in the Banach space , the control function belongs to the spaces , a Banach space of admissible control functions with , a Banach space. Further, is a bounded linear operator from into , the function is given, and the others terms are defined as above.

For all and admissible control , the problem (5.1) admits a mild solution given by

Definition 5.1. We will say that system (5.1) is exactly null controllable on the interval if for a11 , there exists a control , such that the mild solution of (5.1) corresponding to verifies and .

In order to prove the controllability result, in addition, we consider the following conditions.

(H₁₄)   is continuous, and there exist constants and such that for all , , we have

(H₁₅) Let

where and , and let

where , , and .

(H₁₆) The bounded linear operator defined by

has an induced inverse operator which takes values in /ker  and there exist positive constants , , such that and .

Theorem 5.2. If hypotheses (H₁)(H₁₆) are satisfied, then the control nonlocal fractional integrodifferential system (5.1) is exactly null controllable on .

Proof. Let , , .

We define an operator by

Using the hypothesis (H₁₄), for an arbitrary function , we define the control

Using this controller, we will show that the operator has a fixed point. This fixed point is then a solution of (5.2).

Clearly, , which means that the control steers system (5.1) from the initial state to origin in time , provided we can obtain a fixed point of the nonlinear operator .

Now, we show that maps into itself.

We have

Thus, maps into itself. Now, for , we have

Therefore, is a contraction mapping, and hence there exists a unique fixed point , such that . Any fixed point of is a mild solution of (5.1) on which satisfies . Thus, system (5.1) is exactly null controllable on .

6. Examples

To illustrate the abstract results, we give the following examples.

Example 6.1. Consider the nonlinear integropartial differential equation of fractional order

with nonlocal condition

where , , , , , is an -dimensional multi-index, , and , , is given by

Let be the set of all square integrable functions on . We denote by the set of all continuous real-valued functions defined on which have continuous partial derivatives of order less than or equal to . By , we denote the set of all functions with compact supports. Let be the completion of with respect to the norm

It is supposed that the following hold.

(i) The operator is uniformly elliptic on . In other words, all the coefficients , , are continuous and bounded on , and there is a positive number such that

for all and all , , , and .

(ii) All the coefficients , , satisfy a uniform Hölder condition on . Under these conditions, the operator with domain of definition generates an evolution operator defined on , and it is well known that is dense in and the initial function is an element in Hilbert space , see [26, page 438]. Applying Theorem 4.1, this achieves the proof of the existence of mild solutions of the problem (6.1), (6.2). In addition,

(iii) If the coefficients , , satisfy a uniform Hölder condition on and the operators and satisfy.

There are numbers , and , such that

for all , , and all . Applying Theorem 4.2, we deduce that (6.1), (6.2) has a unique strong solution.

The second example is concerned with the controllability result.

Example 6.2. Consider the fractional nonlocal evolution integropartial differential control system of the form

where , , and the functions , are continuous.

Let us take

Put , where and is continuous.

We define by with domain are absolutely continuous, , . Assume that generates an evolution system such that for every positive numbers and , and .

Also, define by , for all and .

Assume that the linear operator that is given by

has a bounded invertible operator in /ker.

Let us assume that the nonlinear functions , , and satisfy the following Lipschitz conditions

where , , , , and .

All the conditions stated in Theorem 5.2 are satisfied. Hence, system (6.7) is exactly null controllable on .

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