Complete Controllability of Fractional Neutral Differential Systems in Abstract Space
By using fractional power of operators and Sadovskii fixed point theorem, we study the complete controllability of fractional neutral differential systems in abstract space without involving the compactness of characteristic solution operators introduced by us.
Recently, fractional differential systems have been proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, and so forth (see [1–5]). There has been a great deal of interest in the solutions of fractional differential systems in analytic and numerical sense. One can see the monographs of Kilbas et al. , Miller and Ross , Podlubny , Lakshmikantham et al. , Tarasov , Wang et al. [11–13] and the survey of Agarwal et al.  and the reference therein. In order to study the fractional systems in the infinite dimensional space, the first important step is how to introduce a new concept of mild solutions. A pioneering work has been reported by EI-Borai  and Zhou and Jiao .
In recent years, controllability problems for various types of nonlinear fractional dynamical systems in infinite dimensional spaces have been considered in many publications. An extensive list of these publications focused on the complete and approximate controllability of the fractional dynamical systems can be found (see [17–34]). Although the controllability of fractional differential systems in abstract space has been discussed, Hernández et al.  point out that some papers on controllability of abstract control systems contain a similar technical error when the compactness of semigroup and other hypotheses is satisfied, more precisely, in this case the application of controllability results are restricted to the finite dimensional space. Ji et al.  find some conditions guaranteeing the controllability of impulsive differential system when the Banach space is nonseparable and evolution systems are not compact, by means of Möch fixed point theorem and the measure of noncompactness. Meanwhile, Wang et al. [19, 20] have researched the complete controllability of fractional evolution systems without involving the compactness of characteristic solution operators. Neutral differential equations arise in many areas of applied mathematics and for this reason these equations have received much attention in the last decades. Sakthivel and Ren  have established a new set of sufficient conditions for the complete controllability for a class of fractional order neutral systems with bounded delay under the natural assumption that the associated linear control is completely controllable. To the author’s knowledge, there are few papers on the complete controllability of the abstract neutral fractional differential systems with unbounded delay.
In the present paper, we introduce a suitable concept of the mild solutions including characteristic solution operators and which are associated with operators semigroup and some probability density functions . Then also without involving the compactness of characteristic solution operators, we obtain the controllability of the following abstract neutral fractional differential systems with unbounded delay: where the state variable takes values in Banach space , belongs to some abstract phase space , and is the phase space to be specified later. The control function is given in , with as a Banach spaces. is a bounded linear operator from to . The operator is a generator of a uniformly bounded analytic semigroup in which , are appropriate functions.
Throughout this paper will be a Banach space with norm and is another Banach space, denote the space of bounded linear operators from to . We also use to denote the of norm of whenever for some with . Let denote the Banach space of functions which are Bochner integrable normed by . Let be the infinitesimal generator of a uniformly bounded analytic semigroup . Let , then it is possible to define the fractional power , for , as a closed linear operator on its domain . Furthermore, the subspace is dense in and the expression defines a norm on . Hereafter we denote by the Banach space normed with . Then for each , the Banach space, and for and the imbedding is compact whenever the resolvent operator of is compact. For a uniformly bounded analytic semigroup the following properties will be used:(a)there is a such that for all .(b)for any , there exists a positive constant such that For more details about the above preliminaries, we can refer to .
Although the semigroup is only the uniformly bounded analytic semigroup but not compact, we can also give the definition of mild solution for our problem by using the similar method introduced in .
Definition 1. We say that a function is a mild solution of the system (1) if , the restriction of to the interval is continuous and for each , the function is integrable and satisfies the following integral equation: where and are called characteristic solution operators and are given by and for , , Here, is a probability density function defined on , that is, , , and .
Definition 2 (complete controllability). The fractional system (1) is said to be completely controllable on the interval if, for every initial function and there exists a control such that the mild solution of (1) satisfies .
The following results of and will be used throughout this paper.
Lemma 3. The operators and have the following properties:(i) for any fixed , and are linear and bounded operators, that is, for any , (ii) and are strongly continuous and there exists such that , for any ;(iii) for and any bounded subsets , and are equicontinuous if with respect to as for each fixed .
To end this section, we recall Kuratowski’s measure of noncompactness, which will be used in the next section to study the complete controllability via the fixed points of condensing operator.
Definition 4. Let be a Banach space and the bounded set of . The Kuratowski’s measure of noncompactness is the map defined by here .
Lemma 5. Let and be two bounded sets of a Banach space . Then(i) if and only if is relatively compact;(ii) if ;(iii).
Lemma 6 (sadovskii’s fixed point theorem). Let be a condensing operator on a Banach space , that is, is continuous and takes bounded sets into bounded sets, and for every bounded set of with . If for a convex closed and bounded set of , then has a fixed point in .
3. Complete Controllability Result
To study the system (1), we assume the function represents the history of the state from up to the present time and belongs to some abstract phase space , which is defined axiomatically. In this article, we will employ an axiomatic definition of the phase space introduced by Hale and Kato  and follow the terminology used in . Thus, will be a linear space of functions mapping into endowed with a seminorm . We will assume that satisfies the following axioms:(A) If , , is continuous on and , then for every the following conditions hold:(i) is in ;(ii); (iii).
Here is a constant, , is continuous and is locally bounded, and are independent of .(B) For the function in (A), is a -valued continuous function on .(C) The space is complete.
Now we give the basic assumptions on the system (1).
(i) generates a uniformly bounded analytic semigroup in ; (ii) for all bounded subsets and , as for each fixed . is continuous function, and there exists a constant and such that the function is -valued and satisfies the Lipschitz condition: for , , , and the inequality holds for .The function satisfies the following conditions:(i)for each , the function is continuous and for each the function is strongly measureable;(ii) for each positive number , there is a positive function such thatThe linear operator is bounded, from into is defined by and there exists a bounded invertible operator defined on and there exist two positive constants such that , . For all bounded subsets , the set where is relatively compact in for arbitrary and .
Proof. Using the assumption , for arbitrary function define the control
It will be shown that when using this control the operator defined by
has a fixed point . Then is a mild solution of system (1), and it is easy to verify that , which implies that the system is controllable.
Next we will prove that has a fixed point using the fixed point theorem of Sadovskii .
Let be the function defined by then and the map is continuous. We can assume . For each . We can denote by the function defined by If satisfies (18), we can decompose it as , , which implies for every and the function satisfies Moreover . Let be the operator on defined by Obviously the operator has a fixed point is equivalent to has a fixed point, so it turns out to prove that has a fixed point. For each positive number , let then for each , is clearly a bounded closed convex set in . Since by (3) and (10) the following relation holds: then from Bocher’s theorem  it follows that is integrable on , so is well defined on .
In order to make the following process clear we divide it into several steps.
Step 1. We claim that there exists a positive number such that .
If it is not true, then for each positive number , there is a function , but , that is, for some . However, on the other hand, we have where is the corresponding control of , . Since there holds where Dividing on both sides by and taking the low limit, we get This contradicts (16). Hence for some positive number , .
Now, we define operator and on as for all , respectively.
We prove that is contraction, while is completely continuous.
Step 2. is contraction.
Let . Then, for each , and by axiom and (15), we have Thus and is contraction.
Step 3. is completely continuous.
Let with in , then for each , and by and , we have as .
Since , then by the dominated convergence theorem we have as , that is, is continuous.
Next we prove that the family is an equicontinuous family of functions. To do this, let , then Noting that We see that tends to zero independently of as since for and any bounded subsets , is equicontinuous.
Hence, maps into an equicontinuous family functions.
It remains to prove that is relatively compact in . let be fixed, , for , we define and , for .
Clearly, is compact, and hence, it is only to consider . For each , arbitrary , define where Then the sets are relatively compact in since the condition . It comes from the following inequalities:
Therefore, is relatively compact in for all .
Thus, the continuity of and relatively compact of imply that is a completely continuous operator.
These arguments enable us to conclude that is a condense mapping on , and by the fixed point theorem of Sadovskii there exists a fixed point for on . In fact, by Step 1–Step 3 and Lemma 3, we can conclude that is continuous and takes bounded sets into bounded sets. Meanwhile, it is easy to see since is relatively compact. Since and , we can obtain for every bounded set of with , that is, is a condense mapping on . If we define , it is easy to see that is a mild solution of (1) satisfying . Then the proof is completed.
Remark 8. In order to describe various real-world problems in physical and engineering sciences subject to abrupt changes at certain instants during the evolution process, impulsive fractional differential equations always have been used in the system model. So we can also consider the complete controllability for (1) with impulses.
Remark 9. Since the complete controllability steers the systems to arbitrary final state while approximate controllability steers the system to arbitrary small neighborhood of final state. In view of the definition of approximate controllability in , we can deduce that the considered systems (1) is also approximate controllable on the interval .
4. An Example
As an application of Theorem 7, we consider the following system:
To write system (40) to the form of (1), let and defined by with domain , the generates a uniformly bounded analytic semigroup which satisfies the condition . Furthermore, has a discrete spectrum, the eigenvalues are , , with the corresponding normalized eigenvectors . Then the following properties hold.
(i) If , then
(ii) For each , In particular, .
(iii) The operator is given by on the space .
Here we take the phase space , which contains all classes of functions such that is Lebesgue measurable and is Lebesgue integrable on where is a positive integrable function. The seminorm in is defined by From , under some conditions is a phase space verifying (A), (B), (C), and in this case (see  for the details).
We assume the following conditions hold.(a) The function is measurable and .(b) The function is measurable, and let .(c) The function , is measurable, with , the function for each is measurable in .(d)The function defined by belongs to .(e)The linear operator is defined by
and has a bounded invertible operator defined .
We define by and , where From (a) and (c) it is clear that and are bounded linear operators on . Furthermore, , and . In fact, from the definition of and (b) it follows that , where . From (b) we know that is a bounded linear operator with . Hence , which implies the assertion. Therefore, from Theorem 7, the system (40) is completely controllable on under the above assumptions.
In this paper, by using the uniformly boundedness, analyticity, and equicontinuity of characteristic solution operators and the Sadovskii fixed point theorem, we obtained the complete controllability of the abstract neutral fractional differential systems with unbounded delay in a Banach space. It shows that the compactness of the characteristic solution operators can be weakened to equicontinuity. Our theorem guarantees the effectiveness of complete controllability results under some weakly compactness conditions.
The authors are highly grateful for the referee’s careful reading and comments on this paper. The present Project is supported by NNSF of China Grant no. 11271087, no. 61263006, and Guangxi Scientific Experimental (China-ASEAN Research) Centre no. 20120116.
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