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

*is 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 [1, 2] 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. *en, we consider the approximate controllability of FNDED by means of the approximate sequence method, and simultaneously, some applicable sufficient conditions are obtained.


Introduction
e 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 integerorder 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][6][7][8][9][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][12][13][14][15][16][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][8][9][10][11][12][14][15][16][17][18][19][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. e 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.

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 [1,2] 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. e FNDED and order in [1,2] is debated as follows: where p: 0 ≤ p ≤ 1, C D p t denotes the Caputo fractional derivative, q(q ≥ 0), and the linear densely unbounded closed operator A: D(A)⊆E ⟶ E is the (p, q)− regularized family defined on the Banach space E. Here, the state x(·) is evaluated in Banach space E. Let U be a Banach space of admissible control functions. e variable u(·) takes a value in L 2 ([0, b]; U), B: U ⟶ E, which is linear and bounded. In addition, f: e 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). en, 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 ‖ · ‖ E and L(E) be the space of all linear bounded operators on E. Let In the following, we recall some definitions to be adopted in the entire work.
Definition 1. (see [1,2]). e Riemann-Liouville fractional integral of a function z and order α and from lower limit 0 can be denoted by Definition 2. (see [1,2]). e Caputo derivative of order α for a function z ∈ C([0, ∞), R) can be defined as Remark 1 (see [1,2] , then the Caputo derivative Definition 3. (see [15,16]). Let q ≥ 0 and 0 ≤ p ≤ 1. A is a linear closed operator, and its domain D(A) is in Banach space E. We say that A is the generator of a (p, q)-regularized operator family S p (t) t ≥ 0 ⊂ L(E) if the three formulas are established in the following: (a) S p (t) is strongly continuous on R + , and S p (0) � I.
For every x ∈ D(A), the following equation holds: where S p (t) is the (p, q)-regularized operator family generated by A. On the basis of the operator S p , the two operators P p , Q p : Mathematical Problems in Engineering e following basic statement is deduced from [22].

Lemma 2.
(see [15,16] In view of Lemma 2, from (9) and (11), we have Combining (9) and (10), we obtain At first, we put forward a new form of the mild solution for the system in the following. Lemma 3. Let 0 ≤ p ≤ 1 and g ∈ L 1 (J, E); if x satisfies the equation then x is denoted by the integral equation Proof. Taking the Riemann-Liouville integral to both sides of the first equality of (14), we have From Lemma 1, we can get that at is, Let η > 0. Taking the Laplace transform for (18), we yield Hence, we obtain en, applying Laplace inverse transform on (20) and combining with the result of Lemma 2 and the Laplace transform of the convolution, we achieve is completes the proof. Because of Lemma 3, we naturally deduce a new representation of the mild solution for (14).
is reputedly a mild solution for the fractional neutral linear system with damping (14) if there exists g ∈ L 1 (J, E) satisfying To gain the global existence of the mild solution for FNDED (1), we give hypothesis S(0). S(0): A is the infinitesimal generator of an exponentially bounded (p, q)− regularized operator family S p (t) on E, Mathematical Problems in Engineering namely, there exist two real numbers M ≥ 1 and ω ≥ 0 satisfying Remark 2. From hypothesis S(0) and formulas (9) and (10), we have

Theorem 1.
Under conditions S(0)-S(2), the fractional neutral differential equations with damping (1) has one and only one mild solution belonging to the space . Firstly, we prove that Ω maps B r into itself. Obviously, Ωx 0 ∈ B r . For t ∈ (0, b], we achieve is is because which means ‖(Ωx)(t)‖ ≤ r, Ωx⊆B r . Next, we demonstrate that Ω is a contraction mapping on B r . In fact, for x(·), y(·) ∈ B r , (30)

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 L and I from If we let the combination (x, u) be the mild solution for FNDED (1) with u ∈ L 2 ([0, b], U), then we represent it as x(t) � x(t; x 0 , x 1 , u), and the terminal item x(b) can be written as where So, the reachable set K b (f) which is composed of all possible final states at time b is Mathematical Problems in Engineering (34) us, the approximate controllability for FNDED (1) means the set K b (f) is dense on space E. at is to say, the definition of approximate controllability is acquired.
Definition 5. Let x 0 , x 1 ∈ E. We called the fractional neutral differential equations with damping (1) is approximately controllable on [0, b] if, for any ε > 0 and x b ∈ E, there is a control function u N ε ∈ U satisfying And we add some of the postulated conditions. S(3): for each given ε > 0 and v(·), w(·) ∈ L 2 ([0, b]; E), there is u(·) ∈ U satisfying where ‖Bu(·)‖ L 2 (J,U) ≤ M u ‖v(·)‖ L 2 (J,U) , and M u is a positive real number irrelevant of v(·) and holds Because condition (37) implies (26), the existence is still met when inequality (26) is changed into (37) in eorem 1. Next, to demonstrate the conclusion of the approximate controllability remains true for (1), we give the following two lemmas.