New Controllability Results of Fractional Nonlocal Semilinear Evolution Systems with Finite Delay

Zhao, Daliang; Mao, Juan

doi:https://doi.org/10.1155/2020/7652648

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Volume 2020 | Article ID 7652648 | https://doi.org/10.1155/2020/7652648

New Controllability Results of Fractional Nonlocal Semilinear Evolution Systems with Finite Delay

Daliang Zhao¹and Juan Mao²

Academic Editor: Quanxin Zhu

Received09 May 2020

Accepted19 Jun 2020

Published13 Jul 2020

Abstract

In the present paper, sufficient conditions ensuring the complete controllability for a class of semilinear fractional nonlocal evolution systems with finite delay in Banach spaces are derived. The new results are obtained under a weaker definition of complete controllability we introduced, and then the Lipschitz continuity and other growth conditions for the nonlinearity and nonlocal item are not required in comparison with the existing literatures. In addition, an appropriate complete space and a corresponding time delay item are introduced to conquer the difficulties caused by time delay. Our main tools are properties of resolvent operators, theory of measure of noncompactness, and Mönch fixed point theorem.

1. Introduction

In recent decades, various research studies on fractional differential systems are growing vigorously, which are chiefly due to the enormous scope and extensive applications of fractional calculus theory in mathematics, biology, physics, economics, and engineering science [1–9].

As we all know, time delay effects exist widely in various fields such as communication security, weather predicting, and population dynamics. The difficulties in the study of these fields lie in the time lag effect for the systems caused by the delay. One can see [10–20], for further details. It should be noted that, in the case of research on fractional delay evolution systems, it is still in the initial stage. On the contrary, fractional calculus has also been applied to controllability issues in recent years. As one of the most important notions in control theory of mathematics, controllability has significant influence on the fields of control and engineering. We point out that complete controllability of infinite dimensional systems with noncompact semigroup is an important research direction, and there have been many outstanding achievements in this regard such as [21–26]. For more details of other research results on control theory, please refer to [27–40] and the reference therein.

As we have seen, there is scarcely any results on complete controllability of fractional nonlocal evolution equations with delay in Banach spaces, except [21, 23, 24]. However, the Lipschitz and certain growth conditions on nonlocal item and nonlinearity are still necessary for [21]. In [23], the authors supposed the nonlinear function and nonlocal item to be Lipschitz continuous and the semigroup generated by the considered system to be compact. In [24], nonlinear function was imposed some growth conditions, and the resolvent operator used to define the mild solution was compact. However, it is a little pity that these suppositions are usually difficult to be made to work in numerous specific applications. In fact, many excellent achievements concerning complete controllability of various nonlinear differential systems, such as those in Debbouche and Baleanu, [11], Nirmala et al. [22], and Wang and Zhou [25], have been derived during recent years. However, the limitation also lies in that the nonlinear functions here are all provided with the Lipschitz continuity or other growth assumptions. In addition, the compactness or other assumptions of semigroup actually prevents us from studying complete controllability in the infinite dimensional space.

The current paper, inspired by the aforementioned analyses, is to address the complete controllability of the following fractional nonlocal semilinear evolution equations with finite delay in Banach spaces:where denotes the Caputo derivative of order . , , and is a bounded linear operator, where and are Banach spaces. is a closed linear unbounded operator on with dense domain . . and are given functions satisfying certain appropriate conditions that will be given later.

The proposed fractional evolution system (1) here, which generalizes the case of integral (first) order differential equations studied in [41] about the complete controllability, has more extensive and valid applications in contrast to the abovementioned literatures [11, 21–25] as follows. First of all, under the new concept of complete controllability we drew into, that is, a weaker definition of complete controllability than the existing notion, the nonlinear function and nonlocal item here are only allowed to be continuous instead of other restrictions such as Lipschitz continuity and certain growth conditions. Secondly, compactness of the semigroup and the resolvent operator used to define the mild solution for system (1) is no longer needed. Last but not the least, obstacles in the estimation of Kuratowski measures of noncompactness caused by time delay have been conquered in a new complete space with corresponding defined delay function.

An outline of the current paper is as follows. Some essential preparations are made in Section 2. In Section 3, we derive a sufficient condition on complete controllability for the addressed systems. In Section 4, we give an example to illustrate the obtained new results.

2. Preliminaries

Let be the space of continuous functions from into provided with the supreme norm . Similarly, denotes the Banach space of continuous functions from to with the usual supreme norm. If , we denote the norm of this space simply by . The domain is endowed with the graph norm , where denotes the norm of Banach space . For , stands for the space of all the continuous functions from into equipped with the norm , where . Throughout this paper, denotes the space of bounded linear operators from into Banach space provided with the operator norm .

In order to conquer the inconveniences caused by time delay in the study of complete controllability, for each , , and the function in (1), we draw into the function defined byfor each . Simple check indicates , where stands for a complete integrable space consists of integrable functions from into .

Remark 1. Under the function defined in (2) in complete space , we can get over the difficulties caused by time delay in the process of estimating Kuratowski measures of noncompactness (for details, please see Lemmas 4 and 5).
Next, we list the well-known definitions as follows.

Definition 1 (see [5]). The fractional integral with order for a function can be defined asprovided that the right-side integral is pointwise defined on .

Definition 2 (see [5]). The Caputo fractional derivative with order for a function is written aswhere , provided the right-side integral is pointwise defined on .

Definition 3 (see [42]). The bounded linear operator on is defined as a resolvent operator of the following integral equation:where scalar kernel and , provided that it satisfies(i) is strongly continuous, (ii) commutes with , that is, , and for each and each (iii), for all ,

Definition 4 (see [42]). Suppose for each , and there exists a function , which satisfyThen, we call a differentiable resolvent operator of (5).
Next, we focus on the following equation:where , and list the definition of mild solution of (7) as below on the basis of literature [42].

Definition 5. We call a mild solution for (7) if , which satisfiesfor each .
Taking advantage of properties of the differentiable resolvent operator, we present the equivalent definition of mild solution for (7).

Lemma 1 (see [42]). Assume that is a differentiable resolvent operator for (7) and . Then,is a mild solution of (7).

Before going further, we now recall below a few relevant properties of Kuratowski measures of noncompactness, which will play a critical role in our next proof of complete controllability. For further details, please see [43].

Lemma 2. Let be a Banach space and be the Kuratowski measures of noncompactness which is given by for a bounded subset in .(I)Let be bounded sets of and . Then,(i) is relatively compact(ii)(iii)(II)Assume that is a countable set of strongly measurable functions from into Banach space , and there exists a function such that , then is integrable on , which satisfies

For the convenience of expression, the Kuratowski measures of noncompactness of a bounded subset in spaces , and are all written as , on the premise of no confusion.

Lemma 3 (Mönch). If there is a closed and convex set , where is a Banach space, , and suppose that is continuous and satisfies countable, is relatively compact. Then, has a fixed point in .

To end this section, we give the following useful lemmas.

Lemma 4. Assume that converges to in as . Then, converges to in for every as .

Proof. Based on the definition in (2), we knowwhich implies thatThis completes the proof.

From the definition of Kuratowski measures of noncompactness in Lemma 2, we can infer the following lemma.

Lemma 5. Let be any bounded countable sequence in . Then, for each , one haswhere .

3. Main Results

In the sequel, we assume that admits a differentiable resolvent operator on . Based on Definition 5 and the Riemann–Liouville standard fractional integral, the mild solution of system (1) can be defined as follows.

Definition 6. For each , a function is said to be a mild solution of the considered system (1) on , if for any andwhere .

Definition 7 (complete controllability). We shall call system (1) is completely controllable on , if for each initial function and , there exist a corresponding control and a real number such that the mild solution of (1) on satisfies .

Remark 2. In comparison with the existing notion in [11, 21, 22, 25, 41], in which is equal to , our concept in which is weaker can be regarded as an extension of the present notion of complete controllability.
Before presenting and proving the main results, we firstly list the hypotheses as below:(H1), and satisfies(i) maps bounded sets in into bounded sets in .(ii)There exist a constant and a function such that, for any bounded subsets (H2)(i) The linear operator is bounded, and there exists a constant satisfying .(ii)Linear operators , denoted by from to are defined aswhere has invertible operators taking values in , which satisfy, for some constant , , and there is a constant and a function satisfyingfor any bounded subset .(H3)(i) For every , is a strict -set contraction and for every , is continuous with respect to each .(ii) and there exists a positive constant such that for every .(H4)The following estimation is valid:whereand is the function mentioned in Definition 4.In the following, let be a fixed constant satisfying . From (H1), takeFor simplicity, letand setBy means of conditions (H2) and (H4), for any and any , define a feedback control functionwhereDenotethen is clearly a closed convex set in . From Lemma 1, we thus define an operator which is given byFor the purpose of simplifying our next proof processes, we give the following conclusions.

Lemma 6. Assume and . Then, , and

Proof. For and such that , we havewhich implies that and . The conclusion follows.

Lemma 7. Assume that conditions (H1) (i) and (H2)–(H4) hold. Then, the operator is equicontinuous on .

Proof. Firstly, we prove that . In view of assumptions (H1) (i), (H2), and (H3) (ii), we can deriveObserving (H2), for any and , we deduceThen, this together with (24) indicatesOn the contrary,which infers thatTherefore, we can obtainIt is obvious that for any . Then, we conclude that .
Next, we shall demonstrate that is equicontinuous on . For arbitrary and with , we present the following discussions.(i)If , thenClearly,whereNext, we shall show that is independent of as . In fact, with regard to , we deriveNotice thatAccording to Definition 4, we deduceFrom the proof process of Lemma 6 and (32), it follows that(ii)If , then from the continuity of and (H3) (i), one has(iii)If , thenThus, it implies that , as , for all . Consequently, is equicontinuous on .

Lemma 8. Let conditions (H1) (i), (H2), (H3) (i), and (H4) be satisfied. Then, the operator is continuous.

Proof. It follows from Lemma 7 that . Next, we show that is continuous. Let be a sequence such that in as .
In view of condition (H1) (i) and Lebesgue dominated convergence theorem, we haveTherefore, this together with the continuity of impliesConsequently, for each , we haveFrom the analogous proof of equicontinuous for operator in Lemma 7 and the well-known Ascoli–Arzelà theorem, it is not difficult to obtain , as , i.e., is continuous on . This completes the proof.

We present now our main results of this paper.

Theorem 1. If assumptions (H1)–(H4) hold, then the fractional evolution (1) is completely controllable on .

Proof. Consider the operator defined as (28). In view of Lemma 1, it is enough to prove that when using the control , the operator has a fixed point which is exactly a mild solution of (1) on . From the verified fact , it follows that the control steers system (1) from the initial function to in finite time . The complete controllability on of system (1) is thus proved. To this end, we shall take advantage of Mönch fixed point theorem. The continuity of operator is given by Lemma 8. In the next step, we demonstrate that Mönch’s condition holds for operator .
Let . It is not difficult to check that . Assume that bounded set is countable and , and we shall prove that . From Lemma 7, it is easy to derive that is equicontinuous on . Notice that , so is also equicontinuous on .
For any , letwhereNo loss of generality, suppose . From hypothesis (H1) (ii) and Lemma 5, for any , it follows thatThen, this indicates from Lemma 2 and Hölder inequality thatwhich together with Lemma 2, (H2) (ii), and (H3) (i) impliesThus, by using Hölder inequality again, one hasConsequently, from the abovementioned, we can deriveFor , also in view of Lemma 2, one obtainsThen, by (55) and (56), we can inferBesides, for , we have from (H3) (i) thatOn the contrary, from the equicontinuity of on , it follows thatConsequently, by (24) and (57)–(59), one can derivewhich deduces . Due to Lemma 2 (I) (i), we know that is relatively compact. Then, from Lemma 3, has at least one fixed point . This shows that the complete controllability on for system (1) is valid. The proof is now completed.

Remark 3. (i) Different from some papers [21, 25, 44] utilizing the definition of mild solution by probability density functions which are initially presented by El-Borai [3], the way of making use of differentiable resolvent operators to define mild solution in this paper can avoid the complexity of definitions and properties related to the probability density functions and its associated characteristic solution operators; sometimes, the limitation of fractional order lying in (0, 1) due to the probability density functions can be extricated. (ii) In a comparative way, nonlocal item in this paper has better application effect in physics. In practical applications, it may be given by , where is a given constant and . At time , we have , which is exactly the case in [44].

4. An Example

As an application of our abstract results, we consider the following fractional partial differential nonlocal evolution equations with delay of the form:where , , denotes the characteristic function of some subinterval , and satisfies for . is a fixed constant for satisfying . , which satisfies for , , and for . Thus, is an infinitesimal generator of a noncompact semigroup which is given by for . Let

Under these assumptions, partial system (61) can be regarded as a control problem of the abstract form:

For any and for any bounded set , we have that , which indicates that is a strict -set contraction with . In addition, it is not difficult to verify that satisfy other assumptions of Theorem 1. Thus, system (61) is completely controllable on .

5. Conclusions

In this work, complete controllability of a class of semilinear fractional nonlocal evolution systems with finite delay in Banach spaces is investigated by using properties of resolvent operators, theory of measure of noncompactness, and Mönch fixed point theorem. Under a weaker concept of complete controllability and a proper complete space we introduced, the controllability results of the addressed system are obtained without Lipschitz continuity and other growth conditions imposed on the nonlinearity and nonlocal item. In fact, the nonlinear function is only supposed to be continuous. Then, we improve and generalize some analogous results of fractional evolution systems such as [11, 21–25].

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

This research was supported by the National Natural Science Foundation of China under Grant 61873150 and a project of Shandong Province Higher Educational Science and Technology Program of China under Grant J18KA233 hosted by the author Daliang Zhao.

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Copyright © 2020 Daliang Zhao and Juan Mao. 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.

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