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Daliang Zhao, Juan Mao, "New Controllability Results of Fractional Nonlocal Semilinear Evolution Systems with Finite Delay", Complexity, vol. 2020, Article ID 7652648, 13 pages, 2020. https://doi.org/10.1155/2020/7652648
New Controllability Results of Fractional Nonlocal Semilinear Evolution Systems with Finite Delay
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.
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 . In , 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 , 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, , Nirmala et al. , and Wang and Zhou , 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  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.
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 ). The fractional integral with order for a function can be defined asprovided that the right-side integral is pointwise defined on .
Definition 2 (see ). The Caputo fractional derivative with order for a function is written aswhere , provided the right-side integral is pointwise defined on .
Definition 3 (see ). 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 ). 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 .
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).
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 .
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 implies