Journal of Mathematics

Volume 2019, Article ID 8603878, 7 pages

https://doi.org/10.1155/2019/8603878

## Approximate Controllability of Fractional Nonlinear Hybrid Differential Systems via Resolvent Operators

Mathematics Department, Al-Azhar University-Gaza, State of Palestine

Correspondence should be addressed to Mohammed M. Matar; moc.liamtoh@rattam_demmahom

Received 26 November 2018; Revised 17 February 2019; Accepted 20 February 2019; Published 25 March 2019

Guest Editor: Thabet Abdeljawad

Copyright © 2019 Mohammed M. Matar. 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 obtain sufficient conditions for the approximate controllability of a fractional nonlinear hybrid differential system. The results are obtained by using resolvent and sectorial operators technique via Dhage fixed point theorem.

#### 1. Introduction

Fractional differential systems have described many practical dynamical phenomena more efficiently than the corresponding integer-order systems; hence they have attracted the attention of many researchers in such fields (see [1–10] and references cited therein).

One of these systems is the fractional control system with all its branches such as stability, controllability, and observability. In the recent years, many investigations on the controllability problems of fractional behaviour have extensively appeared with various applications on linear and nonlinear systems. Particularly, the researches have focused on exact (complete) and approximate controllability (see the articles [11–17] and the references therein).

The fractional control systems involving a linear closed (unbounded) operator which generates resolvent operators were considered recently by many authors [15, 18–20]. The lack of the semigroup property of the generated resolvent operator was the most popular difficulty that has been faced by the interested researchers. However, some authors used the idea of analytic sectorial operators to overcome this problem. For more details, we refer the reader to the papers [20–23] and references therein. To the best of our knowledge, there is not any investigation in the controllability problem via resolvent operators applied on hybrid systems such as the system that has been discussed in the article [24].

In this article, we study the approximate controllability for a fractional hybrid differential system of the formwhere is the Caputo fractional derivative of order such that , and are two real Hilbert spaces, is the infinitesimal generator of a resolvent operator , is a bounded linear operator, , denotes the -order fractional integral, and are given functions such that does not vanish on

#### 2. Preliminaries

Let be the space of all -valued continuous functions defined on with the norm , and let be the space of -valued Bochner integrable functions defined on with the norm , where

Now, let us recall some basic preliminaries on fractional calculus [25] and operator theory [26].

*Definition 1. *The fractional-order integral of a function (or ) of order is defined by

*Definition 2. *The Caputo fractional derivative of order of a function (or ) is defined by

*Definition 3. *The Laplace transform of a function is given by

The Laplace transform of the Caputo fractional derivative is given by The Laplace transform of the fractional integral is given by

The inverse Laplace transform of a function is given byfor some suitable path to ensure the existence of the integral.

The resolvent operator of an operator is defined as

The resolvent set is the set of all regular values of such that is injective, bounded linear operator.

The following fixed point theorem, which is due to Dhage [27], is essential tool for the proof of the main result.

Theorem 4. *Let be a nonempty bounded closed convex subset of a Banach algebra Let and be continuous operators satisfying the following:*(a)* is completely continuous,*(b)* is Lipschitzian with a Lipschitz constant ,*(c)* implies for all , and*(d)*, where .**Then the operator equation has a solution in .*

*3. Fractional Control Systems via Resolvent Operators*

*Let be a linear operator defined on the subspace , the domain of to the space An operator is said to be closed if and only if its domain is a complete space with respect to the norm An operator is said to be densely defined if its domain is dense in . The denseness of the domain is necessary and sufficient for the existence of the adjoint. The adjoint operator of unbounded operators can be defined as bounded operators. For more details on these topics, the reader may refer to [26, 28].*

*Next, we introduce some information about solution operators [29].*

*Consider system*

*which has an integral solution given by*

*Definition 5. *Let be a closed and densely defined operator on . A family of bounded linear operators in is called a solution operator (or -resolvent) generated by if the following conditions are satisfied:(S1) is strong continuous on and , where is the identity operator.(S2) and for all and (S3) is a solution of the integral equation (10).

*Moreover, a solution operator is called compactif for every , is a compact operator. If is a solution operator of system (9), then by (S3), we deduce that*

*where consists of all for which the limit exists. We call as the infinitesimal generator of or simply we say that generates the solution operator .*

*Definition 6. *Let be a closed linear operator. is said to be a sectorial operator of type if there exist , and such that the solution operator of exists throughout the sector and

*Hereafter, we assume that is a sectorial operator of type that generates the solution operator In this case, we can write the solution operator of system (9) as*

*with being a suitable path in a sector *

*Lemma 7. The linear fractional systemhas an integral solution given by*

*Proof. *Letting , for any , then system (14) is equivalent to systemApplying the Laplace transform to system (16), we have and that impliesTherefore, Now, taking the inverse Laplace transform, we get the solution (15). This finishes the proof.

*We define a mild solution for system (1).*

*Definition 8. *A function is called a mild solution of system (1) if it satisfies for any .

*We introduce some preliminaries about controllability (see [3, 11–13, 18–20, 27]). We assume that is a mild solution (we call it now as state function) of the fractional differential system (1) corresponding to a control .*

*Definition 9. *System (1) is said to be approximately controllable on if for every desired final state and , there exists a control such that satisfies

*The set is called the reachability set of system (1). Therefore, the fractional system (1) is said to be approximately controllable on if , where denotes the closure of . If the used control function is fixed, the symbol is used instead of *

*We define the controllability operator as*

*Then is a bounded linear operator defined on . The adjoint operator of is given by*

*The controllability Gramnian is defined by*

*Following the idea, as in [20], the suggested control function for system (1) can be written in the form. where*

*4. Approximate Controllability*

*We prove the approximate controllability of the fractional control system (1) by using the mild solution (20) and the control defined by (26). More precisely, we prove the existence of at least one state satisfying (20) and (26) following the same arguments presented in [20], but using Dhage fixed point theorem. For this lets and where is given by (26).*

*If is compact -semigroup, then the Cauchy operator defined as is also compact. Unfortunately, the resolvent operator does not have the property of semigroups which leads to the impossibility of obtaining the compactness of the Cauchy operator . However, we can prove the continuity of the solution operator in the case of analytic operators by which we can prove the compactness of the Cauchy operator .*

*Let be a fixed positive real number such that Clearly, is a bounded closed and convex set. We need the following assumptions:(H1) is compact analytic operator such that (H2) is continuous and there exists a positive constant such that , for all (H3) is continuous and there exists a function such that , for all (H4) is a linear bounded operator and there exists such that .(H5)Let , such that (H6), *

*Theorem 10. Assume that conditions (H1)-(H6) are satisfied. Then system (1) has a mild solution on .*

*Proof. *We show the operators and satisfying the hypotheses of Dhage fixed point theorem. For the sake of clarity, we split the proof into two main steps.*Step 1*. Firstly, we prove the continuity of and . Let be a sequence in with in . By the hypotheses (H2) and (H3), we obtain the convergence of and to and , respectively, for any . HenceThus, by Lebsegue Dominated Convergence Theorem and the fact that , for some and for any , we have ThenAs and using again dominated convergence theorem we have Thus and are continuous on Next, we show that is bounded on . In fact, for all , we haveThen, the inequality holds for all .

The last thing in this step, we show that is a compact operator. It is sufficient to prove that is compact. But this has been proved in many articles see, for example, ([20]: Theorem 3.3) by using the Ascoli-Arzela theorem. Hence we conclude that is compact. Therefore, is completely continuous. By following the same arguments presented in [13], we can prove.....*Step 2*. The hypothesis (H3) shows that the operator is Lipschitz with Lipschitzian constant Next, we show that whenever for all For this, letting and , we haveIn consequence, this implies that

*If is chosen large enough such that , then we ensure that Therefore, all hypotheses of Dhage Theorem are satisfied; then there exists a fixed point satisfying the operator equation , which is a solution of system (1).*

*Next result, we investigate the approximate controllability of the fractional control system (1). We introduce the following extra conditions:(H7) as in the strong operator topology.(H8)The sequence is bounded in *

*Theorem 11. Assume that conditions (H1)-(H8) are satisfied. Then, the fractional system (1) is approximately controllable on .*

*Proof. *In virtue of Theorem 10, there exists a mild solution such thatwhereTherefore,Hence,Now, by condition (H2), we havewhich implies that the sequence is bounded in the Hilbert space Together with (H8), there exist subsequences of and in converging weakly to some points , respectively. LetThus, Using the compactness of , we can deduce that the mappingfrom to is compact. So, we obtain thatas This implies that as In view of (40) and condition (H5), we obtain thatHence as , which implies that the fractional system (1) is approximately controllable on . This finishes the proof.

*Example 12. *Consider the fractional control system Let and , where and are absolutely continuous, , Then is an infinitesimal generator of an analytic semigroup , , which can be written in the formwhere is the orthonormal basis for It can be shown that is also a generator of a compact analytic operator , , given bywhere is the Wright function (see [19, 21]). Setting , , , and , then system (48) is equivalent to system (1) for any The operator satisfies the hypothesis (H1) such that . Simple calculations lead to , , and Therefore, if we choose such that and that (H6) and (H7) are both satisfied, then, using Theorems 10, and 11, we ensure that system (48) is approximately controllable on

*Data Availability*

*The data used to support the findings of this study are included within the article.*

*Conflicts of Interest*

*The author declares that they have no conflicts of interest.*

*References*

- T. Abdeljawad, F. Jarad, and J. Alzabut, “Fractional proportional differences with memory,”
*The European Physical Journal Special Topics*, vol. 226, no. 16-18, pp. 3333–3354, 2017. View at Publisher · View at Google Scholar · View at Scopus - J. Alzabut, S. Tyagi, and S. Abbas, “Discrete fractional-order BAM neural networks with leakage delay: existence and stability results,”
*Asian Journal of Control*, 2018. View at Publisher · View at Google Scholar - D. Boyadzhiev, H. Kiskinov, and A. Zahariev, “Integral representation of solutions of fractional system with distributed delays,”
*Integral Transforms and Special Functions*, vol. 29, no. 9, pp. 725–744, 2018. View at Publisher · View at Google Scholar · View at MathSciNet - H. Kiskinov and A. Zahariev, “Asymptotic stability of delayed fractional system with nonlinear perturbation,”
*AIP Conference Proceedings*, vol. 2048, no. 1, Article ID 050014, 2018. View at Publisher · View at Google Scholar - R. L. Magin,
*Fractional Calculus in Bioengineering*, Begell, House Publisher, Conn, USA, 2006. - M. M. Matar and E. S. Abu Skhail, “On stability of nonautonomous perturbed semilinear fractional differential systems of order
*α∈*(1,2),”*Journal of Mathematics*, vol. 2018, Article ID 1723481, 10 pages, 2018. View at Publisher · View at Google Scholar · View at MathSciNet - I. M. Stamova, G. T. Stamov, and J. O. Alzabut, “Global exponential stability for a class of impulsive BAM neural networks with distributed delays,”
*Applied Mathematics & Information Sciences*, vol. 7, no. 4, pp. 1539–1546, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - V. E. Tarasov, “Fractional hydrodynamic equations for fractal media,”
*Annals of Physics*, vol. 318, no. 2, pp. 286–307, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - G. M. Zaslavsky,
*Hamiltonian Chaos and Fractional Dynamics*, Oxford University Press, Oxford, UK, 2005. View at MathSciNet - H. Zhou, J. Alzabut, and L. Yang, “On fractional Langevin differential equations with anti-periodic boundary conditions,”
*The European Physical Journal Special Topics*, vol. 226, no. 16-18, pp. 3577–3590, 2017. View at Publisher · View at Google Scholar · View at Scopus - 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 Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - K. Balachandran, M. Matar, and J. J. Trujillo, “Note on controllability of linear fractional dynamical systems,”
*Journal of Control and Decision*, vol. 3, no. 4, pp. 267–279, 2016. View at Publisher · View at Google Scholar · View at MathSciNet - K. Balachandran, J. Y. Park, and J. J. Trujillo, “Controllability of nonlinear fractional dynamical systems,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 75, no. 4, pp. 1919–1926, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Matar, “Controllability of fractional semilinear mixed Volterra-Fredholm integrodifferential equations with nonlocal conditions,”
*International Journal of Mathematical Analysis*, vol. 4, no. 21-24, pp. 1105–1116, 2010. View at Google Scholar · View at MathSciNet · View at Scopus - M. M. Matar and H. N. Abu Ghalwa, “Approximate controllability of nonlocal fractional integrodifferential control systems of order 1<
*α*<2,”*Acta Mathematica Universitatis Comenianae*, vol. 88, no. 1, pp. 131–144, 2019. View at Google Scholar - M. M. Matar, “On controllability of linear and nonlinear fractional integrodifferential systems,”
*Fractional Differential Calculus*, vol. 9, no. 1, pp. 19–32, 2019. View at Google Scholar - H. Yang and E. Ibrahim, “Approximate controllability of fractional nonlocal evolution equations with multiple delays,”
*Advances in Difference Equations*, vol. 2017, article 272, 15 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet - Y.-K. Chang, A. Pereira, and R. Ponce, “Approximate controllability for fractional differential equations of Sobolev type via properties on resolvent operators,”
*Fractional Calculus and Applied Analysis*, vol. 20, no. 4, pp. 963–987, 2017. View at Publisher · View at Google Scholar · View at MathSciNet - L. Chen, Z. Fan, and G. Li, “On a nonlocal problem for fractional differential equations via resolvent operators,”
*Advances in Difference Equations*, vol. 2014, article 251, pp. 1–12, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Fan, “Approximate controllability of fractional differential equations via resolvent operators,”
*Advances in Difference Equations*, vol. 2014, article 54, 2014. View at Publisher · View at Google Scholar - Z. Fan and G. Mophou, “Nonlocal problems for fractional differential equations via resolvent operators,”
*International Journal of Differential Equations*, vol. 2013, Article ID 490673, 9 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - Z. Fan and G. Mophou, “Existence and optimal controls for fractional evolution equations,”
*Nonlinear Studies. The International Journal*, vol. 20, no. 2, pp. 163–172, 2013. View at Google Scholar · View at MathSciNet - T. Lian, Z. Fan, and G. Li, “Approximate controllability of semilinear fractional differential system of order 1<q<2 via resolvent operators,”
*Filomat*, vol. 31, no. 18, pp. 5769–5781, 2017. View at Publisher · View at Google Scholar · View at MathSciNet - N. Mahmudov and M. M. Matar, “Existence of mild solution for hybrid differential equations with arbitrary fractional order,”
*TWMS Journal of Pure and Applied Mathematics*, vol. 8, no. 2, pp. 160–169, 2017. View at Google Scholar · View at MathSciNet - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, New York, NY, USA, Elsevier, 2006. View at MathSciNet - D. R. Smart,
*Fixed Point Theorems*, Cambridge University Press, London, UK, 1980. View at MathSciNet - B. C. Dhage, “On a fixed point theorem in Banach algebras with applications,”
*Applied Mathematics Letters*, vol. 18, no. 3, pp. 273–280, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - E. Kreyszig,
*Introductory Functional Analysis with Applications*, John Wiley & Sons, New York, NY, USA, 1978. View at MathSciNet - J. Prüss,
*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

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