- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 812501, 8 pages

http://dx.doi.org/10.1155/2013/812501

## Controllability of Impulsive Fractional Functional Integro-Differential Equations in Banach Spaces

^{1}Department of Mathematics, RVS Faculty of Engineering, RVS Technical Campus, Coimbatore 641 402, Tamil Nadu, India^{2}Departamento de Análisis Matemático, University of La Laguna, 38271 La Laguna, Tenerife, Spain

Received 13 May 2013; Revised 6 August 2013; Accepted 14 August 2013

Academic Editor: Mitsuru Sugimoto

Copyright © 2013 C. Ravichandran and J. J. Trujillo. 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

This paper is concerned with the controllability problem for a class of mixed type impulsive fractional integro-differential equations in Banach spaces. Sufficient conditions for the controllability result are established by using suitable fixed point theorem combined with the fractional calculus theory and solution operator under some weak conditions. The example is given in illustrate the theory. The results of this article are generalization and improved of the recent results on this issue.

#### 1. Introduction

In the past few decades, the fractional calculus, that is, calculus of integrals and derivatives of any arbitrary real or complex order has gained considerable popularity and importance, based on the wide applications in engineering and sciences such as fluid flow, rheology, dynamical processes in self-similar and porous structures, diffusive transport akin to diffusion, and electrical networks. For more details about fractional calculus theory and fractional differential equations with applications see the monographs of Baleanu et al. [1, 2], Kilbas et al. [3], Lakshmikantham et al. [4], Miller and Ross [5], Podlubny [6], and the papers of [7–14].

Differential equations with impulsive conditions constitute an important field of research due to their numerous applications in ecology, medicine biology, electrical engineering, and other areas of science and technology. There has been a significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments, see for instance the monographs by Lakshmikantham et al. [15], Bainov and Simeonov [16], Samoilenko and Perestyuk [17] and the papers of [18–24].

On the other hand, the controllability for fractional dynamical system has become an interesting research area to this field and one of the fundamental concepts in modern mathematical control theory. Very recently, the authors shu et al. [25] studied the existence of solutions for impulsive fractional differential equations, assuming the operator to be a sectorial, and the results are obtained by using Banach contraction theorem and Leray-schauder’s alternative fixed point theorem. Shu et al. [26] established the existence and uniqueness of solutions for class of fractional partial semilinear functional differential equations with finite delay, here assuming is the infinitesimal generator of an analytic semigroup and by using Banach fixed point theorem.

Tomar and Dabas [27] extended the results of [25] into a controllability of impulsive fractional semilinear evolution equations with nonlocal conditions with as the -resolvent family and the results are obtained by using Banach contraction principle. Many researchers [28–39] investigated the existence and controllability problem combined with fractional derivative with (or without) impulsive conditions. From above the collection of the literature survey, up till now, there is no work reported on this topic, and inspired by the above mentioned works [25–27, 40] we will establish the controllability of impulsive fractional mixed type functional integro-differential equations with finite delay of the form where is Caputo fractional derivative of order , is the bounded linear operator of an -resolvent family defined on a Banach space , is a bounded linear operator, , , and are given functions, where such that is continuous everywhere except for a finite number of points s at which and exists and , , , , and represent the right and left limits of at , respectively.

For any continuous function, is defined on the interval and any . We denote by be the element of defined by Here, represents the history of the time , upto the present time . For , then .

We consider the problems (1)–(3) to study the controllability results using the solution operator and fixed-point theorems. The rest of this paper is organized as follows. In Section 2, we present some necessary definitions and preliminary results that will be used to prove our main results. The proof of our main results is given in Section 3. Finally, an example is included in Section 4.

#### 2. Preliminaries

In this section, we mention some definitions and properties required for establishing our results. Let be a complex Banach space with its norm denoted as , and represents the Banach space of all bounded linear operators from into , and the corresponding norm is denoted by . Let denote the space of all continuous functions from into with supremum norm denoted by . In addition, represents the closed ball in with the center at and the radius .

A two-parameter function of the Mittag-Leffler type is defined by the series expansion where Ha is a Hankel path, that is, a contour which starts and ends at and encircles the disc contour clockwise. For short, . It is an entire function which provides a simple generalization of the exponent function: and the cosine function: and plays an important role in the theory of fractional differential equations. The most interesting properties of the Mittag-Leffler functions are associated with their Laplace integral see [3, 6, 41] for more details.

*Definition 1 (see [40]). *Caputo derivative of order for a function is defined as
for , . If , then
The Laplace transform of the Caputo derivative of order is given as

*Definition 2 (see [42]). *Let be a closed and linear operator with domain defined on a Banach space and . Let be the resolvent set of . We call the generator of an -resolvent family if there exists and a strongly continuous function such that and
In this case, is called the -resolvent family generated by .

*Definition 3 (see [43]). *Let be a closed and linear operator with domain defined on a Banach space and . Let be the resolvent set of . We call the generator of an -resolvent family if there exists and a strongly continuous function such that and
In this case, is called the solution operator generated by .

The concept of the solution operator is closely related to the concept of a resolvent family [44, Chapter 1]. For more details on -resolvent family and solution operators, we refer to [44, 45] and the references therein.

#### 3. Controllability Results

In this section, we present and prove the controllability for the system (1)–(3). In order to prove the controllability results, we need the following results which are taken from [25, 41]. If and , then for any and , we have Let , , where is the Banach space of bounded linear operators from into equipped with its natural topology. So, we have Let us consider the set functions , , and there exist and , with . Endowed with the norm the space is a Banach space.

Lemma 4 (see [25, 27, 40]). *If satisfies the uniform Hölder condition with the exponent and is a sectorial operator, then the unique solution of the Cauchy problem
**
is given by
**
where
** denotes the Bronwich path, is called the -resolvent family, and is the solution operator generated by . *

Now, we define the mild solution of a system (1)–(3).

*Definition 5. *A function is called a mild solution of the system (1)–(3) if on ; and satisfies the following integral equation:
From Lemma 4 we can verify that definition.

Note that, mild solution depends on control functions . The solution of (1)–(3) under a control denoted by is called the trajectory (state) function of (1) under . The set of all possible terminal states, denoted by is called the reachable set of system (1) at terminal time .

*Definition 6. *The system (1)–(3) is said to be controllable on if .

Now we list the following hypothesis: is continuous and there exist functions such that is continuous and there exists a constant such that for all is continuous and there exists a constant such that for all The linear operators , defined by has an invertible operator taking values in and there exists a positive constant such that and (For the construction of the operator and its inverse, see [46]). The function is continuous and there exists such that

Theorem 7. *If the hypotheses are satisfied, then the impulsive fractional integro-differential system (1)–(3) is controllable on provided
*

*Proof. *Let be any arbitrary function, now to transfer the system (1) from initial state to consider the control
We define the operator by
Note that is well defined on .

For our convenience, let us take
and for .
From our assumptions, we have
Let us take and . From (13) we have

For , and by using (13)–(29) we have
Similarly, for
and for
Thus, for all , we have
Since + , then is a contraction, and so by Banach fixed point theorem there exists a unique fixed point such that . This fixed point is then a solution of the system (1)–(3), and clearly, , which implies that the system is controllable on . This completes the proof.

#### 4. Example

Consider the following fractional partial functional integro-differential equations of the form
where . The the above example resembles the control system (1)–(3), if we take (i) as the state space and as the state. (ii) Input trajectory as the control, where is any Banach space.(iii) is defined by are absolutely continuous and ,and . Then , , where , is the orthogonal set of eigen vectors of . It is well known that is the infinitesimal generator of an analytic semigroup in and that is given by
for all , and every . From these expression it follows that, is a uniformly bounded compact semigroup, so that, is a compact operator for , that is, .(iv) by for almost every .(v) is any function satisfying assumption (H_{3}).

Therefore, the the system (36) can be written to the abstract form (1)–(3), and all the conditions of Theorem 7 are satisfied. We can conclude that the system (36) is controllable on .

#### 5. Conclusions

In this article, abstract results concerning the controllability of impulsive fractional functional integro-differential equations involving Caputo fractional derivative in Banach spaces are obtained. By using fractional calculus theory and some standard fixed point theorem, we derived the controllability results. An example is provided to show the effectiveness of the proposed results.

#### Acknowledgment

The authors wish to thank the referees for their several detailed remarks that led to a substantial improvement in the content as well as in the presentation of the paper. The work is partially supported by the Government of Spain and the FEDER project MTM2010-16499. C. Ravichandran would like to thank Dr. K. V. Kupusamy, Chairman, and Dr. Y. Robinson, Director at RVS Technical Campus, Coimbatore, for their constant encouragement and support for this research work.

#### References

- D. Baleanu, Z. B. Gunvenc, and J. A. T. Machdo,
*New Trends in Nanotechnology and Fractional Calculus Applications*, Springer, New York, NY, USA, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus Models and Numerical Methods*, vol. 3 of*Series on Complexity, Nonlinearity and Chaos*, World Scientific, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - A. Kilbas, H. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, Elsevier, Amsterdam, The Netherlands, 2006. View at MathSciNet - V. Lakshmikantham, S. Leela, and J. Vasundhara Devi,
*Theory of Fractional Dynamic Systems*, Cambridge Scientific, 2009. - K. S. Miller and B. Ross,
*An Introduction to the Fractional Calculus and Fractional Differential Equations*, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet - I. Podlubny,
*Fractional Differential Equations*, Academic Press, New York, NY, USA, 1999. View at MathSciNet - M. Belmekki and M. Benchohra, “Existence results for fractional order semilinear functional differential equations with nondense domain,”
*Nonlinear Analysis*, vol. 72, no. 2, pp. 925–932, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. F. Luchko, M. Rivero, J. J. Trujillo, and M. P. Velasco, “Fractional models, non-locality, and complex systems,”
*Computers & Mathematics with Applications*, vol. 59, no. 3, pp. 1048–1056, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. S. Jesus and J. A. Tenreiro Machado, “Application of integer and fractional models in electrochemical systems,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 248175, 17 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Hernández, D. O'Regan, and K. Balachandran, “On recent developments in the theory of abstract differential equations with fractional derivatives,”
*Nonlinear Analysis*, vol. 73, no. 10, pp. 3462–3471, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Hernández, D. O'Regan, and K. Balachandran, “Existence results for abstract fractional differential equations with nonlocal conditions via resolvent operators,”
*Indagationes Mathematicae*, vol. 24, no. 1, pp. 68–82, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Ravichandran and D. Baleanu, “Existence results for fractional neutral functional integro-differential evolution equations with infinite delay in Banach spaces,”
*Advances in Difference Equations*, vol. 215, pp. 1–12, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - J. P. C. dos Santos, V. Vijayakumar, and R. Murugesu, “Existence of mild solutions for nonlocal Cauchy problem for fractional neutral integro-differential equation with unbounded delay,”
*Communications in Mathematical Analysis*, vol. 14, no. 1, pp. 59–71, 2013. View at MathSciNet - J. R. Wang, W. Wei, and Y. Zhou, “Fractional finite time delay evolution systems and optimal controls in infinite-dimensional spaces,”
*Journal of Dynamical and Control Systems*, vol. 17, no. 4, pp. 515–535, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov,
*Theory of Impulsive Differential Equations*, World Scientific, Singapore, 1989. View at MathSciNet - D. D. Bainov and P. S. Simeonov,
*Systems with Impulse Effect*, Ellis Horwood, Chichester, UK, 1989. View at MathSciNet - A. M. Samoilenko and N. A. Perestyuk,
*Impulsive Differential Equations*, vol. 14, World Scientific, Singapore, 1995. View at Publisher · View at Google Scholar · View at MathSciNet - A. Anguraj, M. Mallika Arjunan, and E. Hernández, “Existence results for an impulsive neutral functional differential equation with state-dependent delay,”
*Applicable Analysis*, vol. 86, no. 7, pp. 861–872, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - V. Vijayakumar, S. Sivasankaran, and M. Mallika Arjunan, “Existence of global solutions for second order impulsive abstract functional integrodifferential equations,”
*Dynamics of Continuous, Discrete & Impulsive Systems*, vol. 18, no. 6, pp. 747–766, 2011. View at Zentralblatt MATH · View at MathSciNet - S. Sivasankaran, M. Mallika Arjunan, and V. Vijayakumar, “Existence of global solutions for second order impulsive abstract partial differential equations,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 74, no. 17, pp. 6747–6757, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Vijayakumar, S. Sivasankaran, and M. M. Arjunan, “Existence of solutions for second-order impulsive neutral functional integro-differential equations with infinite delay,”
*Nonlinear Studies*, vol. 19, no. 2, pp. 327–343, 2012. View at MathSciNet - V. Vijayakumar, K. Alagiri Prakash, and R. Murugesu, “Existence of global solutions for second order impulsive differential equations with nonlocal conditions,”
*Nonlinear Studies*, vol. 20, no. 3, pp. 1–13, 2013. - C. Cuevas, E. Hernández, and M. Rabelo, “The existence of solutions for impulsive neutral functional differential equations,”
*Computers & Mathematics with Applications*, vol. 58, no. 4, pp. 744–757, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Hernández and H. R. Henríquez, “Impulsive partial neutral differential equations,”
*Applied Mathematics Letters*, vol. 19, no. 3, pp. 215–222, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - X. B. Shu, Y. Z. Lai, and Y. Chen, “The existence of mild solutions for impulsive fractional partial differential equations,”
*Nonlinear Analysis*, vol. 74, no. 5, pp. 2003–2011, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. B. Shu, Y. Z. Lai, and F. Xu, “Existence and uniqueness of mild solution for abstract fractional functional differential equations,”
*Dynamics of Continuous, Discrete & Impulsive Systems*, vol. 18, no. 3, pp. 371–382, 2012. View at Zentralblatt MATH · View at MathSciNet - N. K. Tomar and J. Dabas, “Controllability of impulsive fractional order semilinear evolution equations with nonlocal conditions,”
*Journal of Nonlinear Evolution Equations and Applications*, vol. 5, pp. 57–67, 2012. - A. Debbouche and D. Baleanu, “Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems,”
*Computers & Mathematics with Applications*, vol. 62, no. 3, pp. 1442–1450, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Balachandran and J. Y. Park, “Controllability of fractional integrodifferential systems in Banach spaces,”
*Nonlinear Analysis. Hybrid Systems*, vol. 3, no. 4, pp. 363–367, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Chen, A. Chen, and X. Wang, “On the solutions for impulsive fractional functional differential equations,”
*Differential Equations and Dynamical Systems*, vol. 17, no. 4, pp. 379–391, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Q. Chen, H. S. Ahn, and D. Xue, “Robust controllability of interval fractional order linear time invariant systems,”
*Signal Processing*, vol. 86, no. 10, pp. 2794–2802, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - J. Dabas, A. Chauhan, and M. Kumar, “Existence of the mild solutions for impulsive fractional equations with infinite delay,”
*International Journal of Differential Equations*, vol. 2011, Article ID 793023, 20 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. L. Guo, “Controllability and observability of impulsive fractional linear time-invariant system,”
*Computers & Mathematics with Applications*, vol. 64, no. 10, pp. 3171–3182, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Balachandran and J. Kokila, “On the controllability of fractional dynamical systems,”
*International Journal of Applied Mathematics and Computer Science*, vol. 22, no. 3, pp. 523–531, 2012. View at MathSciNet - K. Balachandran and J. Kokila, “Controllability of non-linear implicit fractional dynamical systems,”
*IMA Journal of Applied Mathematics*, pp. 1–9, 2013. View at Publisher · View at Google Scholar - K. Balachandran, V. Govindaraj, L. Rodriguez-Germa, and J. J. Trujillo, “Controllability results for nonlinear fractional order dynamical systems,”
*Journal of Optimization Theory and Applications*, vol. 156, no. 1, pp. 33–44, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Balachandran, V. Govindaraj, L. Rodriguez-Germa, and J. J. Trujillo, “Controllability of nonlinear higher order fractional dynamical systems,”
*Nonlinear Dynamics*, vol. 71, no. 4, pp. 605–612, 2013. View at Publisher · View at Google Scholar - J. A. Machado, C. Ravichandran, M. Rivero, and J. J. Trujillo, “Controllability results for impulsive mixed-type functional integro-differential evolution equations with nonlocal conditions,”
*Fixed Point Theory and Applications*, vol. 66, pp. 1–16, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - H. Wang, “Existence results for fractional functional differential equations with impulses,”
*Journal of Applied Mathematics and Computing*, vol. 38, no. 1-2, pp. 85–101, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - A. Chauhan and J. Dabas, “Existence of mild solutions for impulsive fractional-order semilinear evolution equations with nonlocal conditions,”
*Electronic Journal of Differential Equations*, vol. 107, pp. 1–11, 2011. View at Zentralblatt MATH · View at MathSciNet - E. Bazhiekova,
*Fractional evolution equations in Banach spaces [Ph.D. thesis]*, Eindhoven University of Technology, 2001. - D. Araya and C. Lizama, “Almost automorphic mild solutions to fractional differential equations,”
*Nonlinear Analysis*, vol. 69, no. 11, pp. 3692–3705, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. P. Agarwal, B. De Andrade, and G. Siracusa, “On fractional integro-differential equations with state-dependent delay,”
*Computers & Mathematics with Applications*, vol. 62, no. 3, pp. 1143–1149, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - J. Pruss,
*Evolutionary Integral Equations and Applications*, vol. 87 of*Monographs in Mathematics*, Birkhäuser, Basel, Switzerland, 1993. View at Publisher · View at Google Scholar · View at MathSciNet - C. Lizama, “Regularized solutions for abstract Volterra equations,”
*Journal of Mathematical Analysis and Applications*, vol. 243, no. 2, pp. 278–292, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. D. Quinn and N. Carmichael, “An approach to nonlinear control problems using fixed-point methods, degree theory and pseudo-inverses,”
*Numerical Functional Analysis and Optimization*, vol. 7, no. 2-3, pp. 197–219, 1985. View at Publisher · View at Google Scholar · View at MathSciNet