Approximate Controllability for a Kind of Fractional Neutral Differential Equations with Damping

Abstract

This paper gains several meaningful results on the mild solutions and approximate controllability for a kind of fractional neutral differential equations with damping (FNDED) and order belonging to in Banach spaces. At first, a new expression for the mild solutions of FNDED via the (p, q)-regularized operator family and the technique of Laplace transform is acquired. Then, we consider the approximate controllability of FNDED by means of the approximate sequence method, and simultaneously, some applicable sufficient conditions are obtained.

1. Introduction

The primary description of the fractional-order derivative was proposed by Riemann and Liouville toward the end of the nineteenth century, but the notion of the arbitrary derivative and integral which generalized the classical integer-order derivative and integral was presented by Leibniz and Liouville in 1695. However, until the late 1960s when many phenomena on physics, engineering technology, and economics were more accurately described by fractional differential equations (FDEs), scientists began to show great interest on fractional order calculus. For example, they are widely adopted for nonlinear oscillations of earthquakes and in the fluid-dynamic traffic model. In practice, FDEs are deemed to optimize the traditional differential equation model. About the elementary theory of fractional differential and evolution systems, one can refer to Podlubny [1], Kilbas et al. [2], Zhou [3, 4], and [5–10] and the references cited therein.

In the particles’ realistic movement, the resistance to motion is unavoidable, so it is suitable to add the damping character in mathematical models and control systems. Recently, a great deal of meaningful conclusions for the mathematical models with damping influence have been presented by the researchers [11–17].

As we all know, the controllability exerts a momentous effect on control theory and engineering technology. It lies in the fact that it is bound up with quadratic optimal control, observer design, and pole assignment. For this reason, the controllability has been actively investigated by many investigators, and an impressive progress has been made in recent years [7–12, 14–20]. Controllability of the deterministic systems in infinite dimensional spaces has been broadly investigated. In some results of the controllability for systems described by fractional differential models, the fixed-point and the approximate sequence method are felicitously used. Nonetheless, as demonstrated by Triggiani [21], for many parabolic partial differential systems, the conditions of complete controllability are very finite. The research to approximate controllability is more proper for the practical systems than to complete controllability. For the past few years, as regards differential dynamical systems in Banach spaces, several results are achieved about the approximate controllability [8, 19]. However, as far as we know, the approximate controllability of the fractional neutral differential equations with damping and order belonging to [1, 2] is still relatively infrequent, so it is more interesting and necessary to study it.

2. Preliminaries and Notations

Inspired by the aforementioned analysis, the approximate controllability for a kind of fractional neutral differential equations with damping of order belonging to in the Banach space is studied in our work. We acquire several sufficient conditions to pledge the approximate controllability of the FNDED via the contraction mapping theory and approximate sequence method. The FNDED and order in [1, 2] is debated as follows:where , denotes the Caputo fractional derivative, , and the linear densely unbounded closed operator is the regularized family defined on the Banach space . Here, the state is evaluated in Banach space . Let U be a Banach space of admissible control functions. The variable takes a value in , , which is linear and bounded. In addition, is nonlinear, which will be explained in detail later; and is denoted by ; . The relevant linear neutral system of FNDED (1) is

The structure of this paper is given as follows. In Section 2, we review several fundamental concepts and provide a new form of the mild solution for FNDED (1). Then, in Sections 3 and 4, we acquire several meaningful results for the existence and uniqueness of the mild solution and, furthermore, the approximate controllability for FNDED (1).

Let E be a Banach space with norm and be the space of all linear bounded operators on E. Let , be the space of E-valued Bochner integrable function , and

In the following, we recall some definitions to be adopted in the entire work.

Definition 1. (see [1, 2]). The Riemann–Liouville fractional integral of a function and order and from lower limit 0 can be denoted by

Definition 2. (see [1, 2]). The Caputo derivative of order for a function can be defined as

Remark 1 (see [1, 2]). If , then

Lemma 1. (see [2]). Let . If or , then the Caputo derivative

Definition 3. (see [15, 16]). Let and . A is a linear closed operator, and its domain is in Banach space E. We say that A is the generator of a -regularized operator family if the three formulas are established in the following:(a) is strongly continuous on , and .(b) and for all .(c)For every , the following equation holds:where is the -regularized operator family generated by A. On the basis of the operator , the two operators , : are reminded:The following basic statement is deduced from [22].

Lemma 2. (see [15, 16]). Let A be a densely linear closed operator in Banach space E. Then, A is the generator of a -regularized operator family if and only if there is a strong continuous operator and constant satisfying andIn view of Lemma 2, from (9) and (11), we haveCombining (9) and (10), we obtainAt first, we put forward a new form of the mild solution for the system in the following.

Lemma 3. Let and ; if satisfies the equationthen is denoted by the integral equation

Proof. Taking the Riemann–Liouville integral to both sides of the first equality of (14), we haveFrom Lemma 1, we can get thatThat is,Let . Taking the Laplace transform for (18), we yieldHence, we obtainThen, applying Laplace inverse transform on (20) and combining with the result of Lemma 2 and the Laplace transform of the convolution, we achieveThis completes the proof.
Because of Lemma 3, we naturally deduce a new representation of the mild solution for (14).

Definition 4. A function is reputedly a mild solution for the fractional neutral linear system with damping (14) if there exists satisfyingTo gain the global existence of the mild solution for FNDED (1), we give hypothesis S(0).
: A is the infinitesimal generator of an exponentially bounded regularized operator family on E, namely, there exist two real numbers and satisfying

Remark 2. From hypothesis S(0) and formulas (9) and (10), we have(i).(ii).(iii).

3. Existence and Uniqueness of Mild Solutions

In this segment, firstly, we consider the existence and uniqueness of mild solutions for FNDED (1). To this end, we impose the following conditions.

: is continuous and satisfiesfor every , and , , and .

: for almost every , the nonlinear item is continuous, and there is satisfyingfor all and .

Theorem 1. Under conditions –, the fractional neutral differential equations with damping (1) has one and only one mild solution belonging to the space with , and for every control item , provided that

Proof. Define the mapping byLet , , , , and , which is a closed and bounded subset of . For each , we prove has a fixed point in . Firstly, we prove that maps into itself. Obviously, . For , we achieveThis is becausewhich means .
Next, we demonstrate that is a contraction mapping on . In fact, for ,Sincedue to , is a contraction mapping. Consequently, has the unique fixed point belonging to the space . (27) is the unique mild solution of FNDED (1).

4. Approximate Controllability

In this segment, we acquire several appropriate sufficient conditions of the approximate controllability for FNDED (1) by virtue of the approximate technique and the iterative approach.

Define two continuous linear operators and from to by

If we let the combination be the mild solution for FNDED (1) with , then we represent it as , and the terminal item can be written aswhere .

So, the reachable set which is composed of all possible final states at time is

Thus, the approximate controllability for FNDED (1) means the set is dense on space . That is to say, the definition of approximate controllability is acquired.

Definition 5. Let . We called the fractional neutral differential equations with damping (1) is approximately controllable on if, for any and , there is a control function satisfyingAnd we add some of the postulated conditions.
: for each given and , there is satisfyingwhere , and is a positive real number irrelevant of and holdsBecause condition (37) implies (26), the existence is still met when inequality (26) is changed into (37) in Theorem 1. Next, to demonstrate the conclusion of the approximate controllability remains true for (1), we give the following two lemmas.

Lemma 4. Let be two pairs relevant to FNDED (1). Then, in terms of conditions S(0)–S(3), the following result holds:

Proof. The mild solution of FNDED (1) in satisfiesWe define asLet , . Apparently, meets (1) if and only if , and for each , we acquireNow, let us take and ; then,Recombining the formula, we getThis completes the proof.

Theorem 2. If hypotheses )– hold, the fractional neutral differential equations with damping (1) is approximately controllable on J.

Proof. is the infinitesimal generator of a -regularized operator family in a Banach space, and its domain is dense in Banach space . So, in accordance with the definition of approximate controllability, it is only needed to demonstrate that . Next, we should prove that, for every and , there is withwhere satisfiesBy the theory of the fractional resolvent, we deduce that for , which implies that for . It is evident that there is one such thatCondition (S3) suggests for each and , there is such thatFurthermore, for , we determine again by condition S(3) and (36) with the following two properties:where .
Thus, we may define in and derive the following inequality:By the same way, we obtain the sequence satisfyingwhere . Because hypothesis S(3) holds, we can acquire that the sequence is a Cauchy sequence in , and in this way, we can gain a function item satisfyingOn account of the mapping is linear-continuous, for any given , we can find a real integer number satisfyingConsequently, by inequalities (50) and (52), we derivewhere .
This means that . Thus, ; therefore, the fractional neutral differential equations with damping (1) is approximately controllable on [0, b].

Theorem 3. Postulate the range of operator B is denoted by and it is dense in . If conditions S(0)–S(3) are true, the fractional neutral differential equations with damping (1) is approximately controllable on [0, b].

Proof. Because is dense in , for every function and , there is , where , satisfyingNow, we haveThus, from (50), we haveIt indicates that condition S(3) holds if we select . Therefore, FNDED (1) is approximately controllable on [0, b] by using Theorem 2.

Data Availability

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was funded by Program for Innovative Research Team in Huainan Normal University (No. XJTD202008); Natural Science Foundation of Anhui Province (Nos. 2008085MF200 and 2008085QA19); Natural Science Fund of Colleges and Universities in Anhui Province (Nos. KJ2018A0470 and KJ2019A0696); Humanities and Social Sciences Fund of Colleges and Universities in Anhui Province (No. SK2018A0522).