Controllability of Neutral Fractional Functional Equations with Impulses and Infinite Delay

and Applied Analysis 3 The Mittag-Leffler type function in two arguments is defined by the series expansion Eq,p (z) = ∞ ∑ k=0 z k Γ (qk + p) = 1 2πi ∫ C μ q−p e μ


Introduction
Control theory is an area of application-oriented mathematics which deals with the analysis and design of control systems.In particular, the concept of controllability plays an important role in various areas of science and engineering.More precisely, the problem of controllability deals with the existence of a control function, which steers the solution of the system from its initial state to a final state, where the initial and final states may vary over the entire space.Control problems for various types of deterministic and stochastic dynamical systems in infinite dimensional systems have been studied in [1][2][3][4][5][6].
On the other hand, the impulsive differential systems can be used to model processes which are subject to abrupt changes, and which cannot be described by the classical differential systems [7].Moreover, impulsive control which is based on the theory of impulsive equations, has gained renewed interests due its promising applications towards controlling systems exhibiting chaotic behavior.Therefore, the controllability problem for impulsive differential and integrodifferential systems in Banach spaces has been studied extensively (see [8] and the references therein).Moreover, fractional calculus has received great attention, because fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various processes [9].Also, the study of fractional differential equations has emerged as a new branch of applied mathematics, which has been used for construction and analysis of mathematical models in various fields of science and engineering [10].Therefore, the problem of the existence of solutions for various kinds of fractional differential systems has been investigated in [11][12][13].Very recently, Dabas and Chauhan [14] studied the existence, uniqueness, and continuous dependence of mild solution for an impulsive neutral fractional order differential equation with infinite delay by using the fixed point technique and solution operator on a complex Banach space.
Recently, many authors pay their attention to study the controllability of fractional evolution systems [15,16].Wang and Zhou [17] investigated the complete controllability of fractional evolution systems without involving the compactness of characteristic solution operators.Kumar and Sukavanam [18] derived a new set of sufficient conditions for the approximate controllability of a class of semilinear delay control systems of fractional order by using contraction principle and the Schauder fixed point theorem.Sakthivel et al. [19] studied the controllability results for a class of fractional neutral control systems with the help of semigroup theory and fixed point argument.The minimum energy control problem for infinite-dimensional fractional-discrete time linear systems is discussed in [20].Debbouche and Baleanu [21] derived a set of sufficient conditions for the controllability of a class of fractional evolution nonlocal impulsive quasilinear delay integrodifferential systems by using the theory of fractional calculus and fixed point technique.
However, controllability of impulsive fractional integrodifferential equations with infinite delay has not been studied via the theory of solution operator.Motivated by this consideration, in this paper, we investigate the exact controllability of a class of fractional order neutral integrodifferential equations with impulses and infinite delay in the following form: where    is the Caputo fractional derivative of order , 0 <  < 1;  : () ⊂  →  is an infinitesimal generator of the solution operator, {  ()} ≥0 is defined on a Banach space  with the norm ‖ ⋅ ‖  ; the control function (⋅) is given in  2 (, ),  is a Banach space;  is a bounded linear operator from  into ; the histories   : (−∞, 0] →  are defined by   () = ( + ) belongs to an abstract phase space B ℎ defined axiomatically, and   :  → ,  = 1, 2, . . .,  are bounded functions.Also, the fixed times , and ( +  ) = lim ℎ → 0 (  + ℎ) and ( −  ) = lim ℎ → 0 (  − ℎ) denote the right and left limits of () at  =   , respectively.Further,  :  × B ℎ → ,  :  × B ℎ ×  →  are given functions; the term () is given by () = ∫  0 (, )(), where  ∈ (,  + ) is the set of all positive continuous functions on  = {(, ) ∈  2 : 0 ≤  ≤  ≤ }.The main aim of this paper is to obtain some suitable sufficient conditions for the controllability results corresponding to admissible control sets without assuming the semigroup is compact.Further, we address the approximate controllability issue for the considered fractional systems.In order to prove the controllability results, we follow a technique similar to that of [14,19] with some necessary modifications.
Definition 1 (see [10]).The Caputo derivative of order  for a function  : [0, ∞) →  can be written as for  − 1 <  < ,  ∈ .If 0 <  ≤ 1, then The Laplace transform of the Caputo derivative of order  > 0 is given as Abstract and Applied Analysis 3 The Mittag-Leffler type function in two arguments is defined by the series expansion where  is a contour which starts and ends at −∞ and encircles the disc ‖‖ ≤ || 1/2 counter clockwise.The Laplace transform of the Mittag-Leffler function is given as follows: and for more details (see [14]).
Definition 2 (see [9]).A closed and linear operator  is said to be sectorial if there are constants  ∈ ,  ∈ [/2, ], and  > 0, such that the following two conditions are satisfied: Definition 3 (see [11]).Let  be a linear closed operator with domain () defined on .One can call  the generator of a solution operator if there exist  ≥ 0 and strongly continuous functions   : R + → () such that {  : Re  > } ⊂ () and In this case,   is called the solution operator generated by .
Consider the space and there exist  ( −  ) and  ( +  ) where |   is the restriction of  to   = (  ,  +1 ],  = 0, 1, 2 . . ., .Let ‖ ⋅ ‖ B  be a seminorm in B  defined by Lemma 4 (see [14]).If the functions  :  × B ℎ → ,  :  × B ℎ ×  →  satisfy the uniform Hölder condition with the exponent  ∈ (0, 1] and  is a sectorial operator, then if  is a solution of the following fractional integral equation: . . . where   () is the solution operator generated by  given by where B denotes the Bromwich path.
Let   (; ) be the state value of system (1) at terminal time  corresponding to the control  and the initial value  ∈ B ℎ .Introduce the set R(, ) = {  (; )(0) : (⋅) ∈  2 (, )}, which is called the reachable set of system (1) at terminal time .Definition 5.The fractional control system (1) is said to be exactly controllable on the interval  if R(, ) = .
Assume that the linear fractional differential control system is exactly controllable.It is convenient at this point to introduce the controllability operator associated with (13) as where  * denotes the adjoint of , and  *  () is the adjoint of   ().It is straightforward that the operator Γ  0 is a linear bounded operator [19].Lemma 6.If the linear fractional system (13) is exactly controllable if and only then for some  > 0 such that ⟨Γ  0 , ⟩ ≥ ‖‖ 2 , for all  ∈  and consequently ‖( In order to define the concept of mild solution for the control problem (1), by comparison with the impulsive neutral fractional equations given in [14], we associate problem (1) to the integral equation . . .

Controllability Results
In this section, we formulate and prove a set of sufficient conditions for the exact controllability of impulsive neutral fractional control differential system (1) by using the solution operator theory, fractional calculations, and fixed point argument.To prove the controllability result, we need the following hypotheses: (H1) There exists a constant  > 0 such that (H5) The linear fractional system ( 13) is exactly controllable.
Theorem 8. Assume that the hypotheses (H1)-(H5) are satisfied, then the fractional impulsive system (1) is exactly controllable on  provided that where Proof.For an arbitrary function (⋅), choose the feedback control function as follows: . . .
However, the concept of exact controllability is very limited for many dynamic control systems, and the approximate controllability is more appropriate for these control systems instead of exact controllability.Taking this into account, in this paper, we will also discuss the approximate controllability result of the nonlinear impulsive fractional control system (1).The control system is said to be approximately controllable if, for every initial data  and every finite time horizon  > 0, an admissible control process can be found such that the corresponding solution is arbitrarily close to a given square integrable final condition.Further, approximate controllable systems are more prevalent, and often, approximate controllability is completely adequate in applications.In recent years, for deterministic and stochastic control systems including delay term, there are several papers devoted to the study of approximate controllability [23][24][25].Sukavanam and Kumar [26] obtained a set of conditions which ensure the approximate controllability of a class of semilinear fractional delay control systems.Recently, Sakthivel et al. [23] formulated and proved a new set of sufficient conditions for approximate controllability of fractional differential equations by using the fractional calculus theory and solutions operators.It should be mentioned that the approximate controllability of (32) is equivalent to the convergence of function (, Γ  0 ) to zero, as  → 0 + in the strong operator topology (see [23,27] and references therein).
Theorem 11.Assume that conditions (H1)-(H4) hold and that the family {  () :  > 0} is compact.In addition, assume that the function  is uniformly bounded and the linear system associated with the system (1) is approximately controllable, then the nonlinear fractional control system with infinite delay where . . .
One can easily show that for all  > 0, the operator Ψ has a fixed point by employing the technique used in Theorem 8 with some changes.
Let x (⋅) be a fixed point of Ψ. Further, any fixed point of Ψ is a mild solution of (1) under the control and satisfies . . .