Controllability for Retarded Semilinear Neutral Control Systems of Fractional Order in Hilbert Spaces

: In this paper, we discuss the approximate controllability for a class of retarded semilinear neutral control systems of fractional order by investigating the relations between the reachable set of the semilinear retarded neutral system of fractional order and that of its corresponding linear system. The research direction used here is to ﬁnd the conditions for nonlinear terms so that controllability is maintained even in perturbations. Finally, we will show a simple example to which the main result can be


Introduction
Let H and V be two complex Hilbert spaces so that V is a dense subspace of H. In this paper, we study the approximate controllability of the following semilinear retarded neutral functional differential control system of fractional order: d α dt α [x(t) + g(t, x t )] = Ax(t) + 0 −h a(s)A 1 x(t + s)ds + F(t, x(t)) + Bu(t), t > 0, where h > 0, 1/2 < α < 1, (φ 0 , φ 1 ) ∈ H × L 2 (−h, 0; V), a(·) is Hölder continuous, and g, F are provided with functions that satisfy some assumptions described later. Moreover, A is a densely defined closed linear operator that generates an analytic semigroup and A 1 is generally unbounded and closed linear operator that satisfies domain D(A 1 ) ⊃ D(A). For a given s ∈ [0, T], x s : [−h, 0] → H is defined as x s (r) = x(s + r) for r ∈ [−h, 0]. The controller B is a bounded linear operator from U to H, where U is a Banach space of control variables. This kind of systems occurs in many real practical mathematical models that arise in dynamic systems, science and engineering applications. For example, fractional differential equations are treated as another model of nonlinear differential equations [1][2][3][4], and the nonlinear vibrations of earthquake can be modeled as fractional derivatives [5]. There has been a significant development in applications and theory of fractional differential equations in recent years, see [6][7][8][9] and the references therein. Most of the work on neutral initial value problems dominated by delayed semilinear parabolic equation have been devoted to the existence of solutions and the control problems. Recently, the existence theory and applications for fractional neutral evolution equations has been devoted in [7,10], and the existence and approximation of solutions to fractional evolution equation in Muslim [11]. In addition, Sukavanam et al. [12] have been studied approximate controllability of delayed fractional order semilinear equatins.
In this paper, we introduce a different approach from previous works. by assuming either the boundedness of an induced inverse of the controllability operator on quotient space as in [13] or the constraint of the Lipschitz constant of nonlinear term as in [12]. Our research direction is to take advantage of the regularity and variations of constant formula for solutions of the given systems using the fundamental solution that appears in a linear system. The result assert the equivalence condition between the controllability for the retarded neutral control system of fractional order and one for the associated the linear system excluded the nonlinear term.
The paper is organized as follows-in Section 2, we deal with the regularity and structure for solutions of semilinear fractional order retarded neutral functional differential equations and introduce basic properties. In Section 3, we will obtain the equivalent relations between the reachable set of the semilinear fractional order retarded neutral functional differential equation and that of its corresponding linear system. Finally, we will show a simple example to which the main result can be applied.

Retarded Linear Equations
Let V be a Hilbert space densely and continuously embedded in H. The norms of V and H are denoted by || · || and | · |, respectively. The norm of dual space V * is denoted by || · || * . For simplicity, we can consider that ||u|| * ≤ |u| ≤ ||u||, u ∈ V.
Let b(·, ·) be a bounded sesquilinear form defined in V × V satisfying Gårding's inequality: Let A be the operator derived from the sesquilinear form −b(·, ·): where (·, ·) denotes also the duality pairing between V and V * . From (2) it follows that for each u ∈ V Re (Au, u) ≥ c 0 ||u|| 2 .
Thanks to the Lax-Milgram theorem, we know that A is a bounded linear operator from V to V * . The realization of A in H, which is the restriction of A to domain is also denoted by A. It is proved in Theorem 3.6.1 of [14] that A generates an analytic semigroup S(t) = e tA in both H and V * , and there exists a constant M 0 such that for every u ∈ D(A). By identifying the dual of H with H, we may consider the following sequence where each space is continuous injection.
First, we consider the following linear time delayed functional differential equation with forcing term k: where A 1 is generally unbounded and closed linear operator that satisfies domain D(A 1 ) ⊃ D(A). In order to construct the fundamental solution of (5), we need to impose the following two condition: The function a(·) is assumed to be real valued and Hölder continuous of order ρ in the interval [−h, 0]: According to Nakaglri [16,17], the fundamental solution W(·) to (5) is by definition a bounded the operator valued function satisfying where S(·) is the semigroup generated by A. For each t > 0, we introduce the operator valued function U t (·) defined by Then (5) is represented If X and Y are two Banach space, L(X, Y) is the collection of all bounded linear operators from X into Y, and L(X, X) is simply written as L(X). From Proposition 4.1 of [18] or Theorem 1 of [19], it follows the following results.

Semilinear Fractional Order Differential Equations
Consider the following retarded neutral differential system with fractional order: where 0 < α < 1, k is a forcing term, A and A 1 are the linear operators defined as in We will set Π = L 2 (−h, 0; V).

Definition 1.
Let Γ be the Gamma function. The fractional integral of order α > 0 with the lower limit 0 from a function f is defined by The fractional derivative of order α > 0 in the Caputo sense with the lower limit 0 from a We refer [20] for the fundamental consequences about fractional integrals and fractional derivative.
The mild solution of System (8) is represented as (see [10,21]): By using the fundamental solution W(·) described by (6) in the sense of Nakaglri [10], (8) is also represented by Throughout this section, we assume c 1 = 0 in (2) to keep things simple without losing generality. So we have the closed half plane {λ : Re λ ≥ 0} ⊂ ρ(A). Hence, it is possible to define the fractional power A α for α > 0 so that the subspace D(A α ) with a norm is dense in H. It is also well known fact that A α is a closed operator with its domain dense and D(A α ) ⊃ D(A β ) for 0 < α < β. Due to the well known fact that A −α is a bounded operator, there is a constant C −α > 0 such that Lemma 3. For t > 0, there is a positive constant C α such that the following inequalities hold for all t > 0: Proof. From ( [14] [Lemma 3.6.2]), it follows that there exists a constant C such that the following inequalities hold: where S(t) is an analytic semigroup generated by A. The relation (10) is immediately from the inequalities (11) and (12) by results of fractional power of A and the definition of W(t).
We need the following assumptions on System (8) to establish our results.
nonlinear mapping satisfying the following: There exist positive constants L 0 , L 1 , L 2 such that Here, the operator F is the nonlinear part of quasilinear equations as seen in Yong and Pan [22].

Assumption 3.
Let g : [0, T] × Π → H be a nonlinear mapping so that there is a L g > 0 satisfying the following conditions: for all s, t ∈ [0, T], and x,x ∈ Π.
Proof. From Assumption 2, it is easily seen that The proof of (13) is the same argument.
Proof. The first inequality (14) is easy to verify. Moreover, since the proof of (15) is completed.

Remark 1.
Here, we note that by using interpolation theory, we have Thus, there exists a constant C 1 > 0 such that By virtue of Theorem 2.1 of [23], we have the following result on the corresponding linear equation of (8).
Using Lemma 1 we can follow the argument of Proposition 1 term by term to deduce the following result.

Approximate Reachable Sets
Let U is a Banach space of control variables and let the controller B be a bounded linear operator from U to H. In this section, we concern with the approximate controllability of the following semilinear neutral control system with delays of fractional order: The solution x(t) = x(t; g, F, u) of (3,1) is the following form: Noting that for T > 0, φ ∈ H × Π and u ∈ L 2 (0, T; U), we define reachable sets as follows. (19) is said to be H-approximately controllable for the initial value φ (resp. in time T) if R(φ) = H ( resp. R T (φ) = H).
(2) The linear system corresponding (19) is said to be H-approximately controllable for the initial value φ (resp. in time T) if L(φ) = H ( resp. L T (φ) = H).
Then we have where L T (0) V is the closure of L T (0) in V. Therefore, if the linear system (19) with g ≡ 0 and F ≡ 0 is V-approximately controllable, then the nonlinear system (19) is also H-approximately controllable.
Noting that L T (0) V = L T (0) ∩ V and V is dense in H, from Theorems 1 and 2, we obtain the following control results of (19).

Corollary 1. Under Assumptions 1-3, we have
Therefore, the H-approximate controllability of linear system (19) with g ≡ 0 and F ≡ 0 is equivalent to the condition for the H-approximate controllability of the nonlinear system (19).

Example
Let and let U be a Banach space of control variables. Consider the following retarded neutral differential system of fractional order in Hilbert space H: where h > 0, a 1 (·) is Hölder continuous, B ∈ L(U, H), and A 1 ∈ L(H). Let a(u, v) = π 0 du(y) dy dv(y) dy dy. Then The eigenvalue and the eigenfunction of A are λ n = −n 2 and z n (y) = (2/π) 1/2 sin ny, respectively. Moreover, (a1) {z n : n ∈ N} is an orthogonal basis of H and The nonlinear mapping F is the nonlinear part of quasilinear equations considered by Yong and Pan [22]. Define g : [0, T] × Π → H as g(t, x t ) = ∞ ∑ n=1 t 0 e n 2 t ( 0 −h a 2 (s)x(t + s)ds, z n )z n , , t > 0.
Then it can be checked that Assumption 3 is satisfied. Indeed, for x ∈ Π, we know It is immediately seen that Assumption 3 has been satisfied. Thus, all the conditions stated in Theorems 1 and 2 have been satisfied for the Equation (44). Therefore, by Theorems 1 and 2, we get that the approximate controllability of the general retarded linear differential system corresponding to (44) with g ≡ 0 and F ≡ 0 is equivalent to the condition for the H-approximate controllability of the semilinear system (44) for any α > 1 2 .

Conclusions
This paper deals with the approximate controllability for a class of retarded semilinear neutral control systems of fractional order by investigating the relations between the reachable set of the semilinear retarded neutral system of fractional order and that of its corresponding linear system. The research direction used here is to find more general hypotheses of nonlinear terms so that controllability is maintained even in perturbations. The technique used here is to take advantage of the regularity and basic properties for solutions of the given systems using the fundamental solution that appears in a linear system. The result assert the equivalence condition between the controllability for the retarded neutral control system of fractional order and one for the associated the linear system excluded the nonlinear term, which can be also applied to the functional analysis concerning nonlinear control problems.