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On the controllability of fractional functional integro-differential systems with an infinite delay in Banach spaces
Advances in Difference Equations volume 2013, Article number: 291 (2013)
In this manuscript, we study the sufficient conditions for controllability for fractional functional integro-differential systems involving the Caputo fractional derivative of order in Banach spaces. Our main approach is based on fractional calculus, the properties of characteristic solution operators, Mönch’s fixed point theorem via measures of noncompactness. Particularly, these results are under some weakly compactness conditions. An example is presented in the end to show the applications of the obtained abstract results.
MSC:26A33, 93B05, 47H08, 47H10.
The theory of fractional differential and integral equations have been proved to be valuable tools and effective in the modeling of many phenomena in various fields of engineering and scientific disciplines such as physics, chemistry, biology, control theory, signal and image processing, biophysics, blood flow phenomena, aerodynamics and so on. For more details on fractional calculus theory, one can see the monographs of Kilbas et al. , Lakshmikantham et al. , Miller and Ross  and Podlubny , Baleanu et al. , Tarasov , and the papers [7–15] as well as the references therein.
As it is known, the concept of controllability is an important property (quantitative and qualitative) of a control system, and the controllability property plays a crucial role in many control problems and both finite and infinite-dimensional spaces. Recently, the controllability for a fractional dynamical system has become a very active area to this field. Controllability of linear systems in finite dimensional spaces is well established in  and controllability of fractional evolution dynamical systems in a finite dimensional space is discussed, for example, in [7, 8, 14, 17].
Very recently, the authors Wang and Zhou  found some conditions guaranteeing the complete controllability of fractional evolution systems without assuming the compactness of characteristic solution operators by means of the Mönch fixed point technique and the measures of noncompactness. Wang et al.  established two sufficient conditions for nonlocal controllability for fractional evolution systems. These theorems guarantee the effectiveness of controllability results under some weakly noncompactness conditions. The authors Ji et al.  derived the nonlocal controllability of an impulsive differential evolution system and the evolution part is not compact by means of Mönch’s fixed point theorem via measures of noncompactness. Many authors investigated the controllability problem, there are different methods and techniques for various types of nonlinear systems and fixed point theorems; the details of these papers can be found in [11, 13–15, 21–26] and the references therein. From the above literature survey, to our knowledge, controllability of fractional functional integro-differential systems with infinite delay by using Mönch’s fixed point theorem via MNC properties and an abstract phase space have not been studied fully. Motivated by this fact and the articles [18–20], in this manuscript we are concerned with the controllability of fractional integro-differential systems with infinite delay of the form
where is the Caputo fractional derivative of order , A is the infinitesimal generator of a strongly continuous semigroup in a Banach space X, which means that there exists such that , , are given functions, where is a phase space to be defined later. The control function is given , a Banach space of admissible control functions, with U as a Banach space. Finally, B is a bounded linear operator from U into X. The histories , defined by , , belong to some abstract phase space .
The rest of this manuscript is organized as follows. In Section 2, we briefly present some basic notations and preliminaries. In Section 3, we establish some sufficient conditions for controllability of fractional integro-differential evolution systems with infinite delay. Finally, an example is given to illustrate the results reported in Section 4.
In this section, we mention notations, definitions, lemmas and preliminary facts needed to establish our main results.
Throughout this paper, we denote by X a Banach space with the norm . Let Y be another Banach space, let denote the space of bounded linear operators from X to Y. We also use norm of f whenever for some p with . Let denote the Banach space of functions which are Bochner integrable normed by . Let be the Banach space of continuous functions from J into X with the usual supremum norm for .
Now we define the abstract phase space , which was used in . Assume that is a continuous function with . For any , we define
and equip the space ℬ with the norm
Let us define
If is endowed with the norm
then it is clear that is a Banach space.
Now we consider the space
Set to be a seminorm in defined by
Assume , then for , . Moreover,
Let us recall the following known definitions. For more details, see [1, 5] and the references therein.
Definition 2.1 The fractional integral of order α with the lower limit zero for a function f is defined as
provided the right-hand side is point-wise defined on , where is the gamma function, which is defined by .
Definition 2.2 The Riemann-Liouville fractional derivative of order , , , is defined as
where the function has absolutely continuous derivative up to order .
Definition 2.3 The Caputo derivative of order α for a function can be written as
Remark 2.1 (i) If , then
The Caputo derivative of a constant is equal to zero.
If f is an abstract function with values in X, then integrals which appear in Definitions 2.1 and 2.2 are taken in Bochner’s sense.
A continuous function is said to be a mild solution of system (1.1)-(1.2) if on and the following integral equation
is satisfied, where and are called characteristic solution operators and given by
Here, is a probability density function defined on , that is,
The following results of and will be used throughout this paper.
Lemma 2.2 ()
The operators and have the following properties:
For any fixed , and are linear and bounded operators, that is, for any ,
and are strongly continuous.
For and any bounded subsets , and are equicontinuous if with respect to as for each fixed .
Definition 2.5 System (1.1) with (1.2) is said to be controllable on the interval J if for every continuous initial function , , there exists a control such that the mild solution of (1.1) with (1.2) satisfies .
Moreover, let us recall some definitions and properties of the measures of noncompactness.
Definition 2.6 ()
Let be the positive cone of an ordered Banach space . A function Φ defined on the set of all bounded subsets of the Banach space X with values in is called a measure of noncompactness (MNC) on X iff for all bounded subsets , where stands for the closed convex hull of Ω.
The MNC Φ is said to be:
monotone iff for all bounded subsets , of X, we have
nonsingular iff for every , ;
regular iff if and only if Ω is relatively compact in X.
One of the many examples of MNC is the noncompactness measure of Hausdorff β defined on each bounded subset Ω of X by
It is well known that MNC β enjoys the above properties and other properties (see [30, 31]): For all bounded subsets Ω, , of X,
, where ;
for any ;
If the map is Lipschitz continuous with constant k, then for any bounded subset , where Z is a Banach space.
Lemma 2.3 ()
If is bounded and equicontinuous, then is continuous for and
If is a sequence of Bochner integrable functions from J into X with the estimation for almost all and every , where , then the function belongs to and satisfies .
The following fixed point theorem, a nonlinear alternative of Mönch’s type, plays a key role in our proof of system (1.1)-(1.2).
Lemma 2.4 ([, Theorem 2.2])
Let D be a closed convex subset of a Banach space X and . Assume that is a continuous map which satisfies Monch’s condition, that is, ( is countable, is compact). Then F has a fixed point in D.
3 Controllability results
In this section, we establish the controllability results for problem (1.1)-(1.2). In order to prove the main theorem of this section, we list the following hypotheses.
A generates a strongly continuous semigroup in X.
For all bounded subsets and , as for each fixed .
(H2) The function satisfies:
is measurable for all and is continuous for a.e. .
There exist a constant and and a nondecreasing continuous function such that , for all , where Ω satisfies .
There exist a constant and such that, for any bounded subset , ,
where β is the Hausdorff MNC.
(H3) The function satisfies:
is measurable for and is continuous for a.e. .
There exists a constant such that for all .
There exists such that for any bounded subset ,
The linear operator is bounded, defined by has an inverse operator which takes values in and there exist two constants such that and . (For the construction of the operator W and its inverse, see .)
There exist a constant and such that for any bounded subset , .
For our convenience, let us take
Theorem 3.1 Assume that hypotheses (H1)-(H4) are satisfied. Then system (1.1)-(1.2) is controllable on J, provided that
Proof Using hypothesis (H4), for an arbitrary function , we define the control by
We show that, using this control, the operator defined by
has a fixed point. This fixed point is then a solution of a given system. Clearly, , which implies the fractional system (1.1)-(1.2) is controllable on J.
For , we define by
then . Let , . It is easy to see that x satisfies (2.1) if and only if y satisfies and
Let . For any ,
thus is a Banach space. Set for some , then is uniformly bounded, and for , from Lemma 2.1, we have
Define the operator by
Obviously, the operator Φ has a fixed point is equivalent to has one. So, our main goal is to show that has a fixed point and the proof is given in the following steps.
Step 1: There exists such that .
If this is not true, then for each positive number r, there exists a function . But does not belong to , i.e., for some .
Then by hypotheses (H2)(iii), (H3)(iii) and (H4)(ii) and Lemma 2.2(i), we have
Using (3.5) in (3.4),
Let and . Note that as . Now dividing both sides of (3.6) by r and taking the limit as , we get
Then, by (H2)(ii), we get . This is a contradiction. Hence, for some positive integer r, .
Step 2: is continuous on .
For this purpose, we assume that in . Define
Using (H2)(i), (ii), (H3)(i), (ii) and Lebesgue’s dominated convergence theorem, we obtain
Observing (3.7), (3.8) and (3.9), we have
which implies that is continuous on .
Step 3: is equicontinuous on J.
Indeed, let and . Then there is such that
Using Lemma 2.2, we can verify that the right-hand side of the above inequality tends to zero as . Therefore, is equicontinuous on J.
Step 4: The Mönch’s condition holds.
Suppose that is countable and . We show that , where β is the Hausdorff MNC. Without loss of generality, we may suppose that . Now we need to show that is relatively compact in X for each .
By Theorem 2.1, we have
Moreover, by Theorem 2.1 again
which implies, by Lemma 2.3,
where is defined in condition (3.10). Thus, from Mönch’s condition, we get
which implies that .
Hence, using Lemma 2.4, has a fixed point y in . Then is a mild solution of system (1.1)-(1.2) satisfying . Therefore system (1.1)-(1.2) is controllable on J. This completes the proof. □
As an application, we consider the following fractional control system:
where is a Caputo fractional partial derivative, , is continuous in t and ψ is continuous and satisfies certain smoothness conditions.
Let be endowed with the usual norm , and let be defined by ; , where . It is well known that A is an infinitesimal generator of a semigroup in X and is given by for , is not a compact semigroup on X with , where β is the Hausdorff MNC, and there exists such that . Moreover, is equicontinuous  for and .
Let be defined by
With this choice of A, B and f, system (4.1) can be rewritten as
For , the linear operator W is given by
If we assume that W satisfies (H4), then all the conditions of Theorem 3.1 are satisfied. Hence, system (4.1)-(4.3) is controllable on J.
In this paper, abstract results concerning the controllability of fractional functional integro-differential systems with infinite delay involving the Caputo derivative in a Banach space are obtained. Upon making some appropriate assumptions, by utilizing the ideas and techniques of solution operators, the theory of fractional calculus and Mönch’s fixed point theorem via measures of noncompactness, sufficient conditions for controllability results are obtained under some weakly compactness conditions. Finally, an example is presented in the end to show the applications of the obtained abstract results.
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The authors are grateful to the referees for careful reading of the manuscript and giving some useful remarks. The first author would like to thank Dr. K.V. Kupusamy, Chairman and Dr. Y. Robinson, Director at RVS Technical Campus, Coimbatore, Tamilnadu, India, for their constant encouragement and support during this research work.
The authors declare that they have no competing interests.
The two authors contributed equally to the manuscript and typed, read and approved the final version.
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Ravichandran, C., Baleanu, D. On the controllability of fractional functional integro-differential systems with an infinite delay in Banach spaces. Adv Differ Equ 2013, 291 (2013). https://doi.org/10.1186/1687-1847-2013-291
- Caputo fractional derivative
- measures of noncompactness
- fixed point theorem