Controllability of Neutral Fractional Functional Equations with Impulses and Infinite Delay

Abstract

We examine the controllability problem for a class of neutral fractional integrodifferential equations with impulses and infinite delay. More precisely, a set of sufficient conditions are derived for the exact controllability of nonlinear neutral impulsive fractional functional equation with infinite delay. Further, as a corollary, approximate controllability result is discussed by assuming compactness conditions on solution operator. The results are established by using solution operator, fractional calculations, and fixed point techniques. In particular, the controllability of nonlinear fractional control systems is established under the assumption that the corresponding linear control system is controllable. Finally, an example is given to illustrate the obtained theory.

1. 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 [1–6].

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 [11–13]. 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: 𝐷^𝑞_𝑡𝑥(𝑡)+𝑔𝑡,𝑥_𝑡=𝐴𝑥(𝑡)+𝑔𝑡,𝑥_𝑡+𝐽_𝑡^1−𝑞𝐵𝑢(𝑡)+𝑓𝑡,𝑥_𝑡(𝑡),[],𝐻𝑥𝑡∈𝐽=0,𝑏,𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘𝑥,𝑘=1,2,…,𝑚,₀=𝜙∈ℬ_ℎ, (1) where 𝐷^𝑞_𝑡 is the Caputo fractional derivative of order 𝑞, 0<𝑞<1; 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is an infinitesimal generator of the solution operator, {𝑆_𝑞(𝑡)}_𝑡≥0 is defined on a Banach space 𝑋 with the norm ‖⋅‖_𝑋; the control function 𝑢(⋅) is given in 𝐿²(𝐽,𝑈), 𝑈 is a Banach space; 𝐵 is a bounded linear operator from 𝑈 into 𝑋; the histories 𝑥_𝑡∶(−∞,0]→𝑋 are defined by 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃) belongs to an abstract phase space ℬ_ℎ defined axiomatically, and 𝐼_𝑘∶𝑋→𝑋, 𝑘=1,2,…,𝑚 are bounded functions. Also, the fixed times 𝑡_𝑘 satisfy 0≤𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑚<𝑡_𝑚+1≤𝑏, Δ𝑥(𝑡_𝑘)=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘), and 𝑥(𝑡⁺_𝑘)=lim_ℎ→0𝑥(𝑡_𝑘+ℎ) and 𝑥(𝑡⁻_𝑘)=lim_ℎ→0𝑥(𝑡_𝑘−ℎ) denote the right and left limits of 𝑥(𝑡) at 𝑡=𝑡_𝑘, respectively. Further, 𝑔∶𝐽×ℬ_ℎ→𝑋, 𝑓∶𝐽×ℬ_ℎ×𝑋→𝑋 are given functions; the term 𝐻𝑥(𝑡) is given by 𝐻𝑥(𝑡)=∫^𝑡₀𝐺(𝑡,𝑠)𝑥(𝑠)𝑑𝑠, where 𝐺∈𝐶(𝐷,𝑅⁺) is the set of all positive continuous functions on 𝐷={(𝑡,𝑠)∈𝑅²∶0≤𝑠≤𝑡≤𝑏}. 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.

2. Preliminaries

In this section, we will recall some basic definitions and lemmas which will be used in this paper. Let 𝐿(𝑋) denote the Banach space of bounded linear operators from 𝑋 into 𝑋 with the norm ‖⋅‖_𝐿(𝑋). Let 𝐶(𝐽,𝑋) denote the space of all continuous functions from 𝐽 into 𝑋 with the norm ‖𝑥‖=sup_𝑡∈𝐽‖𝑥(𝑡)‖.

Now, we present the abstract space ℬ_ℎ [7]. Let ℎ∶(−∞,0]→(0,+∞) be a continuous function with 𝑙=∫⁰_−∞ℎ(𝑡)𝑑𝑡<+∞. For any 𝑎>0, define ℬ={𝜑∶[−𝑎,0]→𝑋 such that 𝜑(𝑡) is bounded and measurable} and equip the space ℬ with the norm ‖𝜑‖_[−𝑎,0]=sup_{𝑠∈[−𝑎,0]}‖𝜑(𝑠)‖, 𝜑∈ℬ. Further, define the space ℬ_ℎ={𝜑∶(−∞,0]→𝑋, for any 𝑐>0, 𝜑|_[−𝑐,0]∈ℬ with 𝜑(0)=0 and ∫⁰_−∞ℎ(𝑠)‖𝜑‖_[𝑠,0]𝑑𝑠<+∞}. If ℬ_ℎ is endowed with the norm ‖𝜑‖_ℬℎ=∫⁰_−∞ℎ(𝑠)‖𝜑‖_[𝑠,0]𝑑𝑠, 𝜑∈ℬ_ℎ, then (ℬ_ℎ,‖⋅‖_ℬℎ) is a Banach space.

We assume that the phase space (ℬ_ℎ,‖⋅‖_ℬℎ) is a seminormed linear space of functions mapping (−∞,0] into 𝑋 and satisfying the following fundamental axioms [22]. (A1) If 𝑥∶(−∞,𝑏]→𝑋,𝑏>0, is continuous on 𝐽 and 𝑥₀∈ℬ_ℎ, then for every 𝑡∈𝐽, the following conditions hold: (i)𝑥_𝑡∈ℬ_ℎ, (ii)‖𝑥(𝑡)‖≤𝐿‖𝑥_𝑡‖_ℬℎ, (iii)‖𝑥_𝑡‖_ℬℎ≤𝐶₁(𝑡)sup_{0≤𝑠≤𝑡}‖𝑥(𝑠)‖+𝐶₂(𝑡)‖𝑥₀‖_ℬℎ, where 𝐿>0 is a constant; 𝐶₁∶[0,𝑏]→[0,∞) is continuous, 𝐶₂∶[0,∞)→[0,∞) is locally bounded, and 𝐶₁, 𝐶₂ are independent of 𝑥(⋅). (A2) For the function 𝑥(⋅) in (A1), 𝑥_𝑡 is a ℬ_ℎ-valued function on [0,𝑏]. (A3) The space ℬ_ℎ is complete.

Definition 1 (see [10]). The Caputo derivative of order 𝑞 for a function 𝑓∶[0,∞)→𝑅 can be written as 𝐷^𝑞_𝑡𝑓(𝑡)=1Γ𝑛−𝑞^𝑡₀(𝑡−𝑠)^{𝑛−𝑞−1}𝑓^(𝑛)(𝑠)𝑑𝑠=𝐼^𝑛−𝑞𝑓^𝑛(𝑡), (2) for 𝑛−1<𝑞<𝑛, 𝑛∈𝑁. If 0<𝑞≤1, then 𝐷^𝑞_𝑡𝑓(𝑡)=1Γ1−𝑞^𝑡₀(𝑡−𝑠)^−𝑞𝑓⁽¹⁾(𝑠)𝑑𝑠. (3) The Laplace transform of the Caputo derivative of order 𝑞>0 is given as 𝐿𝐷^𝑞_𝑡𝑓(𝑡)∶𝜆=𝜆^𝑞𝑓(𝜆)−^𝑛−1_𝑘=0𝜆^{𝑞−𝑘−1}𝑓^(𝑘)(0);𝑛−1<𝑞<𝑛. (4)

The Mittag-Leffler type function in two arguments is defined by the series expansion 𝐸_𝑞,𝑝(𝑧)=^∞_𝑘=0𝑧^𝑘Γ=1𝑞𝑘+𝑝2𝜋𝑖_𝐶𝜇^𝑞−𝑝𝑒^𝜇𝜇^𝑞−𝑧𝑑𝜇,𝑞,𝑝>0,𝑧∈ℂ, (5) where 𝐶 is a contour which starts and ends at −∞ and encircles the disc ‖𝜇‖≤|𝑧|^1/2 counter clockwise. The Laplace transform of the Mittag-Leffler function is given as follows: ^∞₀𝑒^−𝜆𝑡𝑡^𝑝−1𝐸_𝑞,𝑝𝜔𝑡^𝑞𝜆𝑑𝑡=^𝑞−𝑝𝜆^𝑞,−𝜔Re𝜆>𝜔^1/𝑞,𝜔>0, (6) and for more details (see [14]).

Definition 2 (see [9]). A closed and linear operator 𝐴 is said to be sectorial if there are constants 𝜔∈𝑅, 𝜃∈[𝜋/2,𝜋], and 𝑀>0, such that the following two conditions are satisfied: (i)𝜌(𝐴)⊂∑_(𝜃,𝜔)={𝜆∈𝐶∶𝜆≠𝜔,|arg(𝜆−𝜔)|<𝜃}, (ii)‖𝑅(𝜆,𝐴)‖_𝐿(𝑋)≤𝑀/|𝜆−𝜔|, 𝜆∈∑_(𝜃,𝜔).

Definition 3 (see [11]). Let 𝐴 be a linear closed operator with domain 𝐷(𝐴) defined on 𝑋. One can call 𝐴 the generator of a solution operator if there exist 𝜔≥0 and strongly continuous functions 𝑆_𝑞∶ℝ⁺→𝐿(𝑥) such that {𝜆^𝑞∶Re𝜆>𝜔}⊂𝜌(𝐴) and 𝜆^𝑞−1𝜆^𝑞𝐼−𝐴⁻¹𝑥=^∞₀𝑒^−𝜆𝑡𝑆_𝑞(𝑡)𝑥𝑑𝑡,Re𝜆>𝜔,𝑥∈𝑋. (7) In this case, 𝑆_𝑞 is called the solution operator generated by 𝐴.

Consider the space ℬ_𝑏=𝑥∶(−∞,𝑏]→𝑋suchthat𝑥|_𝐽𝑘∈𝐶𝐽_𝑘𝑡,𝑋andthereexist𝑥⁻_𝑘and𝑥𝑡⁺_𝑘with𝑥𝑡_𝑘=𝑥𝑡⁻_𝑘,𝑥₀=𝜙∈ℬ_ℎ,𝑘=0,1,2,…,𝑚, (8) where 𝑥|_𝐽𝑘 is the restriction of 𝑥 to 𝐽_𝑘=(𝑡_𝑘,𝑡_𝑘+1], 𝑘=0,1,2…,𝑚. Let ‖⋅‖_ℬ𝑏 be a seminorm in ℬ_𝑏 defined by ‖𝑥‖_ℬ𝑏=sup_𝑠∈𝐽‖𝑥(𝑠)‖+𝜙_ℬℎ,𝑥∈ℬ_𝑏. (9)

Lemma 4 (see [14]). If the functions 𝑔∶𝐽×ℬ_ℎ→𝑋, 𝑓∶𝐽×ℬ_ℎ×𝑋→𝑋 satisfy the uniform Hölder condition with the exponent 𝛽∈(0,1] and 𝐴 is a sectorial operator, then a piecewise continuously differentiable function 𝑥∈ℬ⁰_𝑏 is a mild solution of 𝐷^𝑞_𝑡𝑥(𝑡)+𝑔𝑡,𝑥_𝑡=𝐴𝑥(𝑡)+𝑔𝑡,𝑥_𝑡+𝐽_𝑡^1−𝑞𝑓𝑡,𝑥_𝑡(𝑡),[],𝐻𝑥𝑡∈𝐽=0,𝑏,𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘𝑥,𝑘=1,2,…,𝑚,₀=𝜙∈ℬ_ℎ, (10) if 𝑥 is a solution of the following fractional integral equation: 𝑥(𝑡)=𝜙(𝑡)(],𝑆,𝑡∈−∞,0_𝑞(𝑡)𝜙(0)+𝑔0,𝜙−𝑔𝑡,𝑥_𝑡+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝐻𝑥(𝑠)𝑑𝑠,𝑡∈0,𝑡₁,𝑆_𝑞(𝑡)𝜙(0)+𝑔0,𝜙+𝑆_𝑞𝑡−𝑡₁𝐼₁𝑥𝑡⁻₁+𝑆_𝑞𝑡−𝑡₁𝑔𝑡₁,𝑥_𝑡1+𝐼₁𝑥_𝑡⁻₁−𝑔𝑡₁,𝑥_𝑡1−𝑔𝑡,𝑥_𝑡+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝐻𝑥(𝑠)𝑑𝑠,𝑡∈𝑡₁,𝑡₂,⋮𝑆_𝑞(𝑡𝜙(0)+)+𝑔0,𝜙_𝑚𝑖∑₌₁𝑆_𝑞𝑡−𝑡_𝑖𝐼_𝑖𝑥𝑡⁻_𝑖+∑^𝑚_𝑖=1𝑆_𝑞𝑡−𝑡_𝑖𝑔𝑡_𝑖,𝑥_𝑡𝑖+𝐼_𝑖𝑥_𝑡⁻_𝑖−𝑔𝑡_𝑖,𝑥_𝑡𝑖−𝑔𝑡,𝑥_𝑡+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝐻𝑥(𝑠)𝑑𝑠,𝑡∈𝑡_𝑚,𝑏, (11) where 𝑆_𝑞(𝑡) is the solution operator generated by 𝐴 given by 𝑆_𝑞(𝑡)=𝐸_𝑞,1𝐴𝑡^𝑞=12𝜋𝑖𝐵_𝑟𝑒^𝜆𝑡𝜆^𝑞−1𝜆^𝑞−𝐴𝑑𝜆, (12) where 𝐵_𝑟 denotes the Bromwich path.

Let 𝑥_𝑏(𝜙;𝑢) be the state value of system (1) at terminal time 𝑏 corresponding to the control 𝑢 and the initial value 𝜙∈ℬ_ℎ. Introduce the set ℛ(𝑏,𝜙)={𝑥_𝑏(𝜙;𝑢)(0)∶𝑢(⋅)∈𝐿²(𝐽,𝑈)}, which is called the reachable set of system (1) at terminal time 𝑏.

Definition 5. The fractional control system (1) is said to be exactly controllable on the interval 𝐽 if ℛ(𝑏,𝜙)=𝑋.

Assume that the linear fractional differential control system 𝐷^𝑞_𝑡𝑥(𝑡)(𝑡)+(𝑡)[],𝑥(0)(0)=𝐴𝑥𝐵𝑢)(,𝑡∈0,𝑏=𝜙 (13) is exactly controllable. It is convenient at this point to introduce the controllability operator associated with (13) as Γ^𝑏₀=^𝑏₀𝑆_𝑞(𝑏−𝑠)𝐵𝐵^∗𝑆^∗_𝑞(𝑏−𝑠)𝑑𝑠, (14) where 𝐵^∗ denotes the adjoint of 𝐵, and 𝑆^∗_𝑞(𝑡) is the adjoint of 𝑆_𝑞(𝑡). It is straightforward that the operator Γ^𝑏₀ is a linear bounded operator [19].

Lemma 6. If the linear fractional system (13) is exactly controllable if and only then for some 𝛾>0 such that ⟨Γ^𝑏₀𝑥,𝑥⟩≥𝛾‖𝑥‖², for all 𝑥∈𝑋 and consequently ‖(Γ^𝑏₀)⁻¹‖≤1/𝛾.

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 𝑥(𝑡)=𝜙(𝑡)(],𝑆,𝑡∈−∞,0_𝑞(𝑡)𝜙(0)+𝑔0,𝜙−𝑔𝑡,𝑥_𝑡+∫^𝑡₀𝑆_𝑞()(𝑠)+∫𝑡−𝑠𝐵𝑢𝑑𝑠_𝑡0𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝐻𝑥(𝑠)𝑑𝑠,𝑡∈0,𝑡₁,𝑆_𝑞(𝑡)𝜙(0)+𝑔0,𝜙+𝑆_𝑞𝑡−𝑡₁𝐼₁𝑥𝑡⁻₁+𝑆_𝑞𝑡−𝑡₁𝑔𝑡₁,𝑥_𝑡1+𝐼₁𝑥_𝑡⁻₁−𝑔𝑡₁,𝑥_𝑡1−𝑔𝑡,𝑥_𝑡+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝐵𝑢(𝑠)𝑑𝑠+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝐻𝑥(𝑠)𝑑𝑠,𝑡∈𝑡₁,𝑡₂,⋮𝑆_𝑞(𝑡𝜙(0)+)+𝑔0,𝜙_𝑚𝑖∑₌₁𝑆_𝑞𝑡−𝑡_𝑖𝐼_𝑖𝑥𝑡⁻_𝑖+∑^𝑚_𝑖=1𝑆_𝑞𝑡−𝑡_𝑖𝑔𝑡_𝑖,𝑥_𝑡𝑖+𝐼_𝑖𝑥_𝑡⁻_𝑖−𝑔𝑡_𝑖,𝑥_𝑡𝑖−𝑔𝑡,𝑥_𝑡+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝐵𝑢(𝑠)𝑑𝑠+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝐻𝑥(𝑠)𝑑𝑠,𝑡∈𝑡_𝑚,𝑏. (15)

Definition 7. A function 𝑥∶(−∞,𝑏]→𝑋 is said to be a mild solution for the system (1) if for each 𝑢∈𝐿²(𝐽,𝑈), 𝑥₀=𝜙∈ℬ_ℎ on (−∞,0]; Δ𝑥|_{𝑡=𝑡𝑘}=𝐼_𝑘(𝑥(𝑡⁻_𝑘)), 𝑘=1,…,𝑚, the restriction of 𝑥(⋅) to the interval [0,𝑏)⧵{𝑡₁,…,𝑡_𝑚} is continuous, and the integral equation (15) is satisfied.

3. 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 𝑀>0 such that 𝑆_𝑞(𝑡)_𝐿(𝑋)≤𝑀,∀𝑡∈[0,𝑏]. (16)(H2) The function 𝑔∶𝐽×ℬ_ℎ→𝑋 is continuous, and there exists a constant 𝐿_𝑔>0 such that 𝑔𝑡₁,𝜓₁−𝑔𝑡₂,𝜓_2𝑋≤𝐿_𝑔𝑡₁−𝑡₂+𝜓₁−𝜓_2ℬℎ,𝑡_𝑖∈𝐽,𝜓_𝑖∈ℬ_ℎ,𝑖=1,2. (17)(H3) There exist constants 𝜇₁>0 and 𝜇₂>0 such that 𝑓𝑡,𝜑,𝑥−𝑓𝑡,𝜓,𝑦_𝑋≤𝜇₁𝜑−𝜓_ℬℎ+𝜇₂𝑥−𝑦_𝑋,𝑡∈𝐽,𝜑,𝜓∈ℬ_ℎ,𝑥,𝑦∈𝑋. (18)(H4)𝐼_𝑘∈𝐶(𝑋,𝑋), and there exist constants 𝜌>0 such that 𝐼_𝑘(𝑥)−𝐼_𝑘𝑦_𝑋≤𝜌𝑥−𝑦_𝑋,𝑥,𝑦∈𝑋,foreach𝑘=1,…,𝑚. (19)(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 𝐽 provided that 1𝐿=1+𝛾𝑀²_𝐵𝑀²𝑚𝑀𝜌+𝑚𝑀𝐿_𝑔𝐶^∗₁2+𝜌+𝐿_𝑔𝐶^∗₁+𝑀𝑏𝜇₁𝐶^∗₁+𝜇₂𝐻^∗<1, (20) where 𝐶^∗₁=sup_0<𝜏<𝑏𝐶₁(𝜏) and 𝐻^∗=sup_{𝑡∈[0,𝑏]}∫^𝑡₀𝐺(𝑡,𝑠)𝑑𝑠<∞.

Proof. For an arbitrary function 𝑥(⋅), choose the feedback control function as follows:𝑢_𝑥(𝑡)=𝐵^∗𝑆^∗_𝑞𝑡₁−𝑡Γ^𝑡1₀⁻¹×𝑥_𝑡1−𝑆_𝑞𝑡₁𝜙(0)+𝑔0,𝜙+𝑔𝑡₁,𝑥_𝑡1−∫^𝑡1₀𝑆_𝑞𝑡₁−𝑠×𝑓𝑠,𝑥_𝑠,𝐻𝑥(𝑠)𝑑𝑠,𝑡∈0,𝑡₁,𝐵^∗𝑆^∗_𝑞𝑡₂−𝑡Γ^𝑡2₀⁻¹×𝑥_𝑡1−𝑆_𝑞𝑡₂𝜙(0)+𝑔0,𝜙−𝑆_𝑞𝑡₂−𝑡₁𝐼₁𝑥𝑡⁻₁−𝑆_𝑞𝑡₂−𝑡₁𝑔𝑡₁,𝑥_𝑡1+𝐼₁𝑥_𝑡⁻₁−𝑔𝑡₁,𝑥_𝑡1+𝑔𝑡₂,𝑥_𝑡2−∫^𝑡2₀𝑆_𝑞𝑡₂−𝑠×𝑓𝑠,𝑥_𝑠,𝐻𝑥(𝑠)𝑑𝑠,𝑡∈𝑡₁,𝑡₂,⋮𝐵^∗𝑆^∗_𝑞(𝑏−𝑡)Γ^𝑏₀⁻¹×𝑥_𝑏−𝑆_𝑞(𝑏)𝜙(0)+𝑔0,𝜙−∑^𝑚_𝑖=1𝑆_𝑞𝑏−𝑡_𝑖𝐼_𝑖𝑥𝑡⁻_𝑖−∑^𝑚_𝑖=1𝑆_𝑞𝑏−𝑡_𝑖𝑔𝑡_𝑖,𝑥_𝑡𝑖+𝐼_𝑖𝑥_𝑡⁻_𝑖−𝑔𝑡_𝑖,𝑥_𝑡𝑖+𝑔𝑏,𝑥_𝑏−∫^𝑏₀𝑆_𝑞(𝑏−𝑠)𝑓𝑠,𝑥_𝑠,𝐻𝑥(𝑠)𝑑𝑠,𝑡∈𝑡_𝑚,𝑏 (21)and define the operator Φ∶ℬ_𝑏→ℬ_𝑏 by(𝑡)=𝜙(𝑡)(],𝑆Φ𝑥)(,𝑡∈−∞,0_𝑞(𝑡)𝜙(0)+𝑔0,𝜙−𝑔𝑡,𝑥_𝑡+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝐵𝑢_𝑥(𝑠)𝑑𝑠+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝐻𝑥(𝑠)𝑑𝑠,𝑡∈0,𝑡₁,𝑆_𝑞(𝑡)𝜙(0)+𝑔0,𝜙+𝑆_𝑞𝑡−𝑡₁𝐼₁𝑥𝑡⁻₁+𝑆_𝑞𝑡−𝑡₁𝑔𝑡₁,𝑥_𝑡1+𝐼₁𝑥_𝑡⁻₁−𝑔𝑡₁,𝑥_𝑡1−𝑔𝑡,𝑥_𝑡+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝐵𝑢_𝑥(𝑠)𝑑𝑠+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝐻𝑥(𝑠)𝑑𝑠,𝑡∈𝑡₁,𝑡₂,⋮𝑆_𝑞(𝑡)𝜙(0)+𝑔0,𝜙+∑^𝑚_𝑖=1𝑆_𝑞𝑡−𝑡_𝑖𝐼_𝑖𝑥𝑡⁻_𝑖+∑^𝑚_𝑖=1𝑆_𝑞𝑡−𝑡_𝑖𝑔𝑡_𝑖,𝑥_𝑡𝑖+𝐼_𝑖𝑥_𝑡⁻_𝑖−𝑔𝑡_𝑖,𝑥_𝑡𝑖−𝑔𝑡,𝑥_𝑡+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝐵𝑢_𝑥(𝑠)𝑑𝑠+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝐻𝑥(𝑠)𝑑𝑠,𝑡∈𝑡_𝑚,𝑏. (22)It should be noted that the control (21) transfers the system (1) from the initial state 𝜙 to the final state 𝑥_𝑏 provided that the operator Φ has a fixed point. In order to prove the exact controllability result, it is enough to show that the operator Φ has a fixed point in ℬ_𝑏.
Define the function 𝑦(⋅)∶(−∞,𝑏]→𝑋 by 𝑦(𝑡)=𝜙(𝑡)(],[],,𝑡∈−∞,00,𝑡∈0,𝑏 (23) then 𝑦₀=𝜙. For each 𝑧∈𝐶(𝐽,𝑅) with 𝑧(0)=0, let the function 𝑧 be defined by 𝑧(𝑡)=(],𝑧(𝑡)[].0,𝑡∈−∞,0,𝑡∈0,𝑏 (24) If 𝑥(⋅) satisfies (15), then we can decompose 𝑥(⋅) as 𝑥(𝑡)=𝑦(𝑡)+𝑧(𝑡) for 𝑡∈𝐽, which implies that 𝑥_𝑡=𝑦_𝑡+𝑧_𝑡 for 𝑡∈𝐽 and the function 𝑧(⋅) satisfies𝑧(𝑡)=𝑆_𝑞(𝑡)𝜙(0)+𝑔0,𝜙−𝑔𝑡,𝑦_𝑡+𝑧_𝑡+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝐵𝑢_𝑦+𝑧(𝑠)𝑑𝑠+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)×𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐻𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡∈0,𝑡₁,𝑆_𝑞(𝑡)𝜙(0)+𝑔0,𝜙+𝑆_𝑞𝑡−𝑡₁𝐼₁𝑦𝑡⁻₁+𝑧𝑡⁻₁+𝑆_𝑞𝑡−𝑡₁×𝑔𝑡₁,𝑦_𝑡1+𝑧_𝑡1+𝐼₁𝑦_𝑡−1+𝑧_𝑡⁻₁−𝑔𝑡₁,𝑦_𝑡1+𝑧_𝑡1−𝑔𝑡,𝑦_𝑡+𝑧_𝑡+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝐵𝑢_𝑦+𝑧(𝑠)𝑑𝑠+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐻𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡∈𝑡₁,𝑡₂,⋮𝑆_𝑞(𝑡)𝜙(0)+𝑔0,𝜙+∑^𝑚_𝑖=1𝑆_𝑞𝑡−𝑡_𝑖𝐼_𝑖𝑦𝑡⁻_𝑖+𝑧𝑡⁻_𝑖+∑^𝑚_𝑖=1𝑆_𝑞𝑡−𝑡_𝑖𝑔𝑡_𝑖,𝑦_𝑡𝑖+𝑧_𝑡𝑖+𝐼_𝑖𝑦_𝑡−𝑖+𝑧_𝑡⁻_𝑖−𝑔𝑡_𝑖,𝑦_𝑡𝑖+𝑧_𝑡𝑖−𝑔𝑡,𝑦_𝑡+𝑧_𝑡+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝐵𝑢_𝑦+𝑧(𝑠)𝑑𝑠+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐻𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡∈𝑡_𝑚,𝑏, (25)where𝑢_𝑦+𝑧(𝑡)=𝐵^∗𝑆^∗_𝑞𝑡₁−𝑡Γ^𝑡1₀⁻¹×𝑥_𝑡1−𝑆_𝑞𝑡₁𝜙(0)+𝑔0,𝜙+𝑔𝑡₁,𝑦_𝑡1+𝑧_𝑡1−∫^𝑡1₀𝑆_𝑞𝑡₁−𝑠𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐻𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡∈0,𝑡₁,𝐵^∗𝑆^∗_𝑞𝑡₂−𝑡Γ^𝑡2₀⁻¹×𝑥_𝑡1−𝑆_𝑞𝑡₂𝜙(0)+𝑔0,𝜙−𝑆_𝑞𝑡₂−𝑡₁𝐼₁𝑦𝑡⁻₁+𝑧𝑡⁻₁−𝑆_𝑞𝑡₂−𝑡₁𝑔𝑡₁,𝑦_𝑡1+𝑧_𝑡1+𝐼₁𝑦_𝑡−1+𝑧_𝑡⁻₁−𝑔𝑡₁,𝑦_𝑡1+𝑧_𝑡1+𝑔𝑡₂,𝑦_𝑡2+𝑧_𝑡2−∫^𝑡2₀𝑆_𝑞𝑡₂−𝑠×𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐻𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡∈𝑡₁,𝑡₂,⋮𝐵^∗𝑆^∗_𝑞(𝑏−𝑡)Γ^𝑏₀⁻¹×𝑥_𝑏−𝑆_𝑞(𝑏)𝜙(0)+𝑔0,𝜙−∑^𝑚_𝑖=1𝑆_𝑞𝑏−𝑡_𝑖𝐼_𝑖𝑦𝑡⁻_𝑖+𝑧𝑡⁻_𝑖−∑^𝑚_𝑖=1𝑆_𝑞𝑏−𝑡_𝑖𝑔𝑡_𝑖,𝑦_𝑡𝑖+𝑧_𝑡𝑖+𝐼_𝑖𝑦_𝑡−𝑖+𝑧_𝑡⁻_𝑖−𝑔𝑡_𝑖,𝑦_𝑡𝑖+𝑧_𝑡𝑖+𝑔𝑏,𝑦_𝑏+𝑧_𝑏−∫^𝑏₀𝑆_𝑞(𝑏−𝑠)×𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐻𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡∈𝑡_𝑚,𝑏. (26)
Let ℬ⁰_𝑏={𝑧∈ℬ_𝑏∶𝑧₀=0∈ℬ_ℎ}. For any 𝑧∈ℬ⁰_𝑏, we get ‖𝑧‖_ℬ⁰_𝑏=sup_𝑠∈𝐽‖𝑧(𝑠)‖_𝑋+𝑧_0ℬℎ=sup_𝑠∈𝐽‖𝑧(𝑠)‖_𝑋,𝑧∈ℬ⁰_𝑏. (27) Thus, (ℬ⁰_𝑏,‖⋅‖_ℬ⁰_𝑏) is a Banach space. Define the operator Π∶ℬ⁰_𝑏→ℬ⁰_𝑏 by(𝑡)=𝑆Π𝑧^𝑞(𝑡)𝜙(0)+𝑔0,𝜙−𝑔𝑡,𝑦_𝑡+𝑧_𝑡+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝐵𝑢_𝑦+𝑧(𝑠)𝑑𝑠+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐻𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡∈0,𝑡₁,𝑆_𝑞(𝑡)𝜙(0)+𝑔0,𝜙+𝑆_𝑞𝑡−𝑡₁𝐼₁𝑧𝑡⁻₁+𝑆_𝑞𝑡−𝑡₁×𝑔𝑡₁,𝑦_𝑡1+𝑧_𝑡1+𝐼₁𝑦_𝑡−1+𝑧_𝑡⁻₁−𝑔𝑡₁,𝑦_𝑡1+𝑧_𝑡1−𝑔𝑡,𝑦_𝑡+𝑧_𝑡+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝐵𝑢_𝑦+𝑧(𝑠)𝑑𝑠+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)×𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐻𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡∈𝑡₁,𝑡₂,⋮𝑆_𝑞(𝑡)𝜙(0)+𝑔0,𝜙+∑^𝑚_𝑖=1𝑆_𝑞𝑡−𝑡_𝑖𝐼_𝑖𝑧𝑡⁻_𝑖+∑^𝑚_𝑖=1𝑆_𝑞𝑡−𝑡_𝑖𝑔𝑡_𝑖,𝑦_𝑡𝑖+𝑧_𝑡𝑖+𝐼_𝑖𝑦_𝑡−𝑖+𝑧_𝑡⁻_𝑖−𝑔𝑡_𝑖,𝑦_𝑡𝑖+𝑧_𝑡𝑖−𝑔𝑡,𝑦_𝑡+𝑧_𝑡+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝐵𝑢_𝑦+𝑧(𝑠)𝑑𝑠+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)×𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐻𝑦(𝑠)+𝑧(𝑠)𝑑𝑠,𝑡∈𝑡_𝑚,𝑏. (28)It can be easily seen that the operator Φ has a fixed point if and only if Π has a fixed point. Now, we will prove that Π has a unique fixed point. In order to prove this, we show that Π is a contraction mapping. Let 𝑧,𝑧^∗∈ℬ⁰_𝑏, for all 𝑡∈[0,𝑡₁], we have (Π𝑧)(𝑡)−Π𝑧^∗(𝑡)_𝑋≤𝑔𝑡,𝑦_𝑡+𝑧_𝑡−𝑔𝑡,𝑦_𝑡+𝑧^∗_𝑡𝑋+^𝑡₀𝑆_𝑞𝑡−𝜂𝐵𝐵^∗𝑆^∗_𝑞𝑡₁−𝜂Γ^𝑡1₀⁻¹_𝑋×𝑔𝑡₁,𝑦_𝑡1+𝑧_𝑡1−𝑔𝑡₁,𝑦_𝑡1+𝑧^∗_𝑡1𝑋+^𝑡1₀𝑆_𝑞𝑡₁−𝑠_𝐿(𝑋)×𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐻𝑦(𝑠)+𝑧(𝑠)−𝑓𝑠,𝑦_𝑠+𝑧^∗_𝑠,𝐻𝑦(𝑠)+𝑧^∗(𝑠)_𝑋+𝑑𝑠𝑑𝜂_𝑡0𝑆_𝑞(𝑡−𝑠)_𝐿(𝑋)×𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐻𝑦(𝑠)+𝑧(𝑠)−𝑓𝑠,𝑦_𝑠+𝑧^∗_𝑠,𝐻𝑦(𝑠)+𝑧^∗(𝑠)_𝑋𝑑𝑠≤𝐿_𝑔𝑧_𝑡−𝑧^∗_𝑡ℬℎ+1𝛾𝑀²_𝐵𝑀^2𝑡₀𝐿_𝑔𝑧_𝑡−𝑧^∗_𝑡ℬℎ+𝑀^𝑡1₀𝜇₁𝑧_𝑠−𝑧^∗_𝑠ℬℎ+𝜇₂×𝐻𝑦(𝑠)−𝑧(𝑠)𝑦(𝑠)−−𝐻𝑧^∗(𝑠)_𝑋𝑑𝑠𝑑𝜂+𝑀^𝑡1₀𝜇₁𝑧_𝑠−𝑧^∗_𝑠ℬℎ+𝜇₂𝐻𝑦(𝑠)−𝑧(𝑠)−𝐻(𝑦(𝑠)−𝑧^∗(𝑠))_𝑋≤1𝑑𝑠1+𝛾𝑀²_𝐵𝑀²𝐿_𝑔𝐶^∗₁+𝑀𝑏𝜇₁𝐶^∗₁+𝜇₂𝐻^∗×𝑧−𝑧^∗_ℬ0𝑏. (29) Similarly, for 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], 𝑘=1,2,…,𝑚, we can obtain (Π𝑧)(𝑡)−Π𝑧^∗(𝑡)_𝑋≤^𝑘_𝑖=1𝑆_𝑞𝑡−𝑡_{𝑖𝐿(𝑥)}𝐼_𝑖𝑧𝑡⁻_𝑖−𝐼_𝑖𝑧𝑡⁻_𝑖𝑋+^𝑘_𝑖=1𝑆_𝑞𝑡−𝑡_{𝑖𝐿(𝑋)}×𝑔𝑡_𝑖,𝑦_𝑡𝑖+𝑧_𝑡𝑖+𝐼_𝑖𝑦_𝑡−𝑖+𝑧_𝑡⁻_𝑖−𝑔𝑡_𝑖,𝑦_𝑡𝑖+𝑧^∗_𝑡𝑖+𝐼_𝑖𝑦_𝑡−𝑖+𝑧^∗_𝑡⁻_𝑖𝑋+𝑔𝑡_𝑖,𝑦_𝑡𝑖+𝑧_𝑡𝑖−𝑔𝑡_𝑖,𝑦_𝑡𝑖+𝑧^∗_𝑡𝑖𝑋+𝑔𝑡,𝑦_𝑡+𝑧_𝑡−𝑔𝑡,𝑦_𝑡+𝑧^∗_𝑡𝑋+^𝑡₀𝑆_𝑞𝑡−𝜂𝐵𝐵^∗𝑆^∗_𝑞(𝑡_𝑘+1−𝜂)Γ^𝑡1₀⁻¹_𝑋×^𝑘_𝑖=1𝑆_𝑞(𝑡_𝑘+1−𝑡_𝑖)_𝐿(𝑥)𝐼_𝑖(𝑧(𝑡⁻_𝑖))−𝐼_𝑖(𝑧(𝑡⁻_𝑖))_𝑋+^𝑘_𝑖=1𝑆_𝑞𝑡_𝑘+1−𝑡_{𝑖𝐿(𝑋)}×𝑔𝑡_𝑖,𝑦_𝑡𝑖+𝑧_𝑡𝑖+𝐼_𝑖𝑦_𝑡−𝑖+𝑧_𝑡⁻_𝑖−𝑔𝑡_𝑖,𝑦_𝑡𝑖+𝑧^∗_𝑡𝑖+𝐼_𝑖𝑦_𝑡−𝑖+𝑧^∗_𝑡⁻_𝑖𝑋+𝑔𝑡_𝑖,𝑦_𝑡𝑖+𝑧_𝑡𝑖−𝑔𝑡_𝑖,𝑦_𝑡𝑖+𝑧^∗_𝑡𝑖𝑋+𝑔𝑡_𝑘+1,𝑦_𝑡𝑘+1+𝑧_𝑡𝑘+1−𝑔𝑡_𝑘+1,𝑦_𝑡𝑘+1+𝑧^∗_{𝑡𝑘+1𝑋}+^𝑡𝑘+1₀𝑆_𝑞𝑡_𝑘+1−𝑠_𝐿(𝑋)×𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐻𝑦(𝑠)+𝑧(𝑠)−𝑓(𝑠,𝑦_𝑠+𝑧^∗_𝑠,𝐻(𝑦(𝑠)+𝑧^∗(𝑠)))_𝑋+𝑑𝑠𝑑𝜂_𝑡0𝑆_𝑞(𝑡−𝑠)_𝐿(𝑋)×𝑓𝑠,𝑦_𝑠+𝑧_𝑠,𝐻𝑦(𝑠)+𝑧(𝑠)−𝑓𝑠,𝑦_𝑠+𝑧^∗_𝑠,𝐻𝑦(𝑠)+𝑧^∗(𝑠)_𝑋≤1𝑑𝑠1+𝛾𝑀²_𝐵𝑀²𝑘𝑀𝜌+𝑘𝑀𝐿_𝑔𝐶^∗₁2+𝜌+𝐿_𝑔𝐶^∗₁+𝑀𝑏𝜇₁𝐶^∗₁+𝜇₂𝐻^∗𝑧−𝑧^∗_ℬ0𝑏. (30) Thus, for all 𝑡∈[0,𝑏], we get the estimate (Π𝑧)(𝑡)−Π𝑧^∗(𝑡)_𝑋≤11+𝛾𝑀²_𝐵𝑀²𝑚𝑀𝜌+𝑚𝑀𝐿_𝑔𝐶^∗₁2+𝜌+𝐿_𝑔𝐶^∗₁+𝑀𝑏𝜇₁𝐶^∗₁+𝜇₂𝐻^∗𝑧−𝑧^∗_ℬ0𝑏≤𝐿𝑧−𝑧^∗_ℬ0𝑏. (31) Thus, we have ‖(Π𝑧)(𝑡)−(Π𝑧^∗)(𝑡)‖_𝑋≤𝐿‖𝑧−𝑧^∗‖_ℬ⁰_𝑏 for all 𝑡∈[0,𝑏]. Since 𝐿<1, this implies that Π is a contraction mapping, and hence, Π has a unique fixed point 𝑧∈ℬ⁰_𝑏. Thus, the system (1) is exactly controllable on [0,𝑏]. The proof is complete.

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 𝜙 and every finite time horizon 𝑏>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 [23–25]. 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.

Definition 9. The fractional control system (1) is said to be approximately controllable on [0,𝑏] if the closure of the reachable set ℛ(𝑏,𝜙) is dense in 𝑋; that is, ℛ(𝑏,𝜙)=𝑋.

Remark 10. Assume that the linear fractional control system 𝐷^𝑞_𝑡𝑥(𝑡)(𝑡)+(𝑡)[],𝑥(0)=𝐴𝑥𝐵𝑢)(,𝑡∈0,𝑏=𝜙 (32) is approximate controllable. Let us now introduce the operators associated with (32) as Γ^𝑏₀=∫^𝑏₀𝒮_𝑞(𝑏−𝑠)𝐵𝐵^∗𝒮^∗_𝑞(𝑏−𝑠)𝑑𝑠,  𝑅(𝛼,Γ^𝑏₀)=(𝛼𝐼+Γ^𝑏₀)⁻¹ for 𝛼>0. It should be mentioned that the approximate controllability of (32) is equivalent to the convergence of function 𝛼𝑅(𝛼,Γ^𝑏₀) to zero, as 𝛼→0⁺ in the strong operator topology (see [23, 27] and references therein).

Theorem 11. Assume that conditions (H1)–(H4) hold and that the family {𝑆_𝑞(𝑡)∶𝑡>0} is compact. In addition, assume that the function 𝑓 is uniformly bounded and the linear system associated with the system (1) is approximately controllable, then the nonlinear fractional control system with infinite delay (1) is approximately controllable on [0,𝑏].

Proof. For each 𝛼>0, define the operator Ψ∶ℬ_𝑏→ℬ_𝑏 by Ψ𝑥(𝑡)=𝑧(𝑡), where𝑧(𝑡)=𝜙(𝑡)(],𝑆,𝑡∈−∞,0_𝑞(𝑡)𝜙(0)+𝑔0,𝜙−𝑔𝑡,𝑥_𝑡+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝐵𝑢_𝑥(𝑠)𝑑𝑠+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝐻𝑥(𝑠)𝑑𝑠,𝑡∈0,𝑡₁,𝑆_𝑞(𝑡)𝜙(0)+𝑔0,𝜙+𝑆_𝑞𝑡−𝑡₁𝐼₁𝑥𝑡⁻₁+𝑆_𝑞𝑡−𝑡₁𝑔𝑡₁,𝑥_𝑡1+𝐼₁𝑥_𝑡⁻₁−𝑔𝑡₁,𝑥_𝑡1−𝑔𝑡,𝑥_𝑡+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝐵𝑢_𝑥(𝑠)𝑑𝑠+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝐻𝑥(𝑠)𝑑𝑠,𝑡∈𝑡₁,𝑡₂,⋮𝑆_𝑞(𝑡)𝜙(0)+𝑔0,𝜙+∑^𝑚_𝑖=1𝑆_𝑞𝑡−𝑡_𝑖𝐼_𝑖𝑥𝑡⁻_𝑖+∑^𝑚_𝑖=1𝑆_𝑞𝑡−𝑡_𝑖𝑔𝑡_𝑖,𝑥_𝑡𝑖+𝐼_𝑖𝑥_𝑡⁻_𝑖−𝑔𝑡_𝑖,𝑥_𝑡𝑖−𝑔𝑡,𝑥_𝑡+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝐵𝑢_𝑥(𝑠)𝑑𝑠+∫^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝐻𝑥(𝑠)𝑑𝑠,𝑡∈𝑡_𝑚,𝑢,𝑏_𝑥(𝑡)=𝐵^∗𝑆^∗_𝑞𝑡₁−𝑡𝑅𝛼,Γ^𝑡1₀𝑝(𝑥(⋅)),𝑡∈0,𝑡₁,𝐵^∗𝑆^∗_𝑞𝑡₂−𝑡𝑅𝛼,Γ^𝑡2_𝑡1𝑝(𝑥(⋅)),𝑡∈𝑡₁,𝑡₂,⋮𝐵^∗𝑆^∗_𝑞(𝑏−𝑡)𝑅𝛼,Γ^𝑏_𝑡𝑚𝑝(𝑥(⋅)),𝑡∈𝑡_𝑚,𝑏, (33)where 𝑝(𝑥(⋅=𝑥))_𝑡1−𝑆_𝑞𝑡₁𝜙(0)+𝑔0,𝜙+𝑔𝑡₁,𝑥_𝑡1−∫^𝑡1₀𝑆_𝑞𝑡₁−𝑠𝑓𝑠,𝑥_𝑠,𝐻𝑥(𝑠)𝑑𝑠,𝑡∈0,𝑡₁,𝑥_𝑡1−𝑆_𝑞𝑡₂𝜙(0)+𝑔0,𝜙−𝑆_𝑞𝑡₂−𝑡₁𝐼₁𝑥𝑡⁻₁−𝑆_𝑞𝑡₂−𝑡₁𝑔𝑡₁,𝑥_𝑡1+𝐼₁𝑥_𝑡⁻₁−𝑔𝑡₁,𝑥_𝑡1+𝑔𝑡₂,𝑥_𝑡2−∫^𝑡2₀𝑆_𝑞𝑡₂−𝑠𝑓𝑠,𝑥_𝑠,𝐻𝑥(𝑠)𝑑𝑠,𝑡∈𝑡₁,𝑡₂,⋮𝑥_𝑏−𝑆_𝑞(𝑏𝜙(0)−)+𝑔0,𝜙_𝑚𝑖∑₌₁𝑆_𝑞𝑏−𝑡_𝑖𝐼_𝑖𝑥𝑡⁻_𝑖−∑^𝑚_𝑖=1𝑆_𝑞𝑏−𝑡_𝑖𝑔𝑡_𝑖,𝑥_𝑡𝑖+𝐼_𝑖𝑥_𝑡⁻_𝑖−𝑔𝑡_𝑖,𝑥_𝑡𝑖+𝑔𝑏,𝑥_𝑏−∫^𝑏₀𝑆_𝑞(𝑏−𝑠)𝑓𝑠,𝑥_𝑠,𝐻𝑥(𝑠)𝑑𝑠,𝑡∈𝑡_𝑚,𝑏. (34) One can easily show that for all 𝛼>0, the operator Ψ has a fixed point by employing the technique used in Theorem 8 with some changes.
Let 𝑥^𝛼(⋅) be a fixed point of Ψ. Further, any fixed point of Ψ is a mild solution of (1) under the control 𝑢^𝛼(𝑡)=𝐵^∗𝑆^∗_𝑞𝑡₁−𝑡𝑅𝛼,Γ^𝑡1₀𝑝𝑥^𝛼,𝑡∈0,𝑡₁,𝐵^∗𝑆^∗_𝑞𝑡₂−𝑡𝑅𝛼,Γ^𝑡2₀𝑝𝑥^𝛼,𝑡∈𝑡₁,𝑡₂,⋮𝐵^∗𝑆^∗_𝑞(𝑏−𝑡)𝑅𝛼,Γ^𝑏₀𝑝𝑥^𝛼,𝑡∈𝑡_𝑚,𝑏 (35) and satisfies 𝑥^𝛼𝑡₁=𝑥_𝑡1−𝛼𝑅𝛼,Γ^𝑡1₀𝑝𝑥^𝛼,𝑡∈0,𝑡₁,𝑥^𝛼𝑡₂=𝑥_𝑡2−𝛼𝑅𝛼,Γ^𝑡2₀𝑝𝑥^𝛼,𝑡∈𝑡₁,𝑡₂,⋮𝑥^𝛼(𝑏)=𝑥_𝑏−𝛼𝑅𝛼,Γ^𝑏₀𝑝𝑥^𝛼,𝑡∈𝑡_𝑚,𝑏. (36) Moreover, by the assumption that 𝑓 is uniformly bounded, there exists 𝑁>0 such that 𝑓(𝑠,𝑥^𝑏₀^𝛼_𝑠,𝐻𝑥^𝛼(𝑠))²𝑑𝑠≤𝑏𝑁², (37) and consequently, the sequence {𝑓(𝑠,𝑥^𝛼_𝑠,𝐻𝑥^𝛼(𝑠))} is bounded in 𝐿²(𝐽,𝑋). Then, there is a subsequence still denoted by {𝑓(𝑠,𝑥^𝛼_𝑠,𝐻𝑥^𝛼(𝑠))} that converges weakly to say 𝑓(𝑠) in 𝐿²(𝐽,𝑋). Now, we define 𝑤=𝑥_𝑡1−𝑆_𝑞𝑡₁𝜙(0)𝑡+𝑔0,𝜙+𝑔₁,𝑥^𝛼_𝑡1−∫^𝑡1₀𝑆_𝑞𝑡₁−𝑠𝑓(𝑠)𝑑𝑠,𝑡∈0,𝑡₁,𝑥_𝑡1−𝑆_𝑞𝑡₂𝜙(0)+𝑔0,𝜙−𝑆_𝑞𝑡₂−𝑡₁𝐼₁𝑥^𝛼𝑡⁻₁−𝑆_𝑞𝑡₂−𝑡₁𝑔𝑡₁,𝑥^𝛼_𝑡1+𝐼₁𝑥^𝛼_𝑡⁻₁−𝑔𝑡₁,𝑥^𝛼_𝑡1+𝑔𝑡₂,𝑥^𝛼_𝑡2−∫^𝑡2₀𝑆_𝑞𝑡₂−𝑠𝑓(𝑠)𝑑𝑠,𝑡∈𝑡₁,𝑡₂,⋮𝑥_𝑏−𝑆_𝑞(𝑏𝜙(0)−)+𝑔0,𝜙_𝑚𝑖∑₌₁𝑆_𝑞𝑏−𝑡_𝑖𝐼_𝑖𝑥^𝛼𝑡⁻_𝑖−∑^𝑚_𝑖=1𝑆_𝑞𝑏−𝑡_𝑖𝑔𝑡_𝑖,𝑥^𝛼_𝑡𝑖+𝐼_𝑖𝑥^𝛼_𝑡⁻_𝑖−𝑔𝑡_𝑖,𝑥^𝛼_𝑡𝑖+𝑔𝑏,𝑥^𝛼_𝑏−∫^𝑏₀𝑆_𝑞(𝑏−𝑠)𝑓(𝑠)𝑑𝑠,𝑡∈𝑡_𝑚,𝑏. (38) Now, for 𝑡∈[0,𝑡₁], we have 𝑝𝑥^𝛼=−𝑤_𝑡10𝑆_𝑞𝑡₁−𝑠𝑓𝑠,𝑥^𝛼_𝑠,𝐻𝑥^𝛼(𝑠)−𝑓(𝑠)𝑑𝑠≤sup_{𝑡∈[0,𝑡1]}^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑥^𝛼_𝑠,𝐻𝑥^𝛼(𝑠)−𝑓(𝑠)𝑑𝑠. (39) Also, for 𝑡∈[𝑡_𝑖,𝑡_𝑖+1], 𝑖=1,…,𝑚, we have 𝑝𝑥^𝛼=−𝑤_𝑡𝑖+10𝑆_𝑞𝑡_𝑖+1−𝑠𝑓𝑠,𝑥^𝛼_𝑠,𝐻𝑥^𝛼(𝑠)−𝑓(𝑠)𝑑𝑠≤sup_{𝑡∈(𝑡𝑖,𝑡𝑖+1]}^𝑡₀𝑆_𝑞(𝑡−𝑠)𝑓𝑠,𝑥^𝛼_𝑠,𝐻𝑥^𝛼(𝑠)−𝑓(𝑠)𝑑𝑠. (40) By using infinite-dimensional version of the Ascoli-Arzela theorem, it is easy to show that an operator 𝑙(⋅)→∫^⋅₀𝑆_𝑞(⋅−𝑠)𝑙(𝑠)𝑑𝑠∶𝐿¹(𝐽,𝑋)→𝐶(𝐽,𝑋) is compact. Hence, for all 𝑡∈[0,𝑏], we obtain that ‖𝑝(𝑥^𝛼)−𝑤‖→0 as 𝛼→0⁺. Moreover, from (36) we get for 𝑡∈[0,𝑡₁], 𝑥^𝛼𝑡₁−𝑥_𝑡1≤𝛼𝑅𝛼,Γ^𝑡1₀(𝑤+)𝛼𝑅𝛼,Γ^𝑡1₀𝑝𝑥^𝛼≤−𝑤𝛼𝑅𝛼,Γ^𝑡1₀(𝑤)+𝑝𝑥^𝛼−𝑤. (41) It follows from Remark 10 and (39) that ‖𝑥_𝛼(𝑡₁)−𝑥_𝑡1‖→0 as 𝛼→0⁺. Similarly, in the view of (40), for 𝑡∈(𝑡_𝑖,𝑡_𝑖+1], 𝑖=1,…,𝑚, 𝑥^𝛼𝑡_𝑖+1−𝑥_𝑡𝑖+1→0as𝛼→0⁺. (42) Thus, for all 𝑡∈[0,𝑏], we get ‖𝑥^𝛼(𝑏)−𝑥_𝑏‖→0 as 𝛼→0⁺. This proves the approximate controllability of (1). The proof is completed.

Example 12. Now, we present an example to illustrate the abstract results of this paper which do not aim at generality but indicate how our theorem can be applied to concrete problems. Let 𝑋=𝐿²[0,𝜋]. Define 𝐴∶𝑋→𝑋 by 𝐴𝑧=𝑧^′′ with domain 𝐷(𝐴)={𝑧∈𝑋∶𝑧,𝑧^′ are absolutely continuous, 𝑧^′′∈𝑋,𝑧(0)=𝑧(𝜋)=0}. Then, 𝐴 generates an analytic semigroup {𝑇(𝑡),𝑡>0} in 𝑋, and it is given by [14] 𝑇(𝑡)𝑧=^∞_𝑛=1𝑒^−𝑛2𝑡𝑧,𝑒_𝑛𝑒_𝑛,𝑧∈𝑋, (43) where 𝑒_𝑛(𝑦)=2/𝜋sin𝑛𝑦, 𝑛=1,2,3,… is the orthogonal set of eigenvectors of 𝐴. Also, define an infinite dimensional control space 𝑈 by 𝑈={𝑢∣𝑢=∑^∞_𝑛=2𝑢_𝑛𝑒_𝑛 with ∑^∞_𝑛=2𝑢²_𝑛<∞} with norm defined by ‖𝑢‖_𝑈=(∑^∞_𝑛=2𝑢²_𝑛)^1/2. Define a continuous linear map 𝐵 from 𝑈 to 𝑋 as 𝐵𝑢=2𝑢₂𝑒₁+∑^∞_𝑛=2𝑢_𝑛𝑒_𝑛 for 𝑢=∑^∞_𝑛=2𝑢_𝑛𝑒_𝑛∈𝑈.
Note that the subordination principle of solution operator implies that 𝐴 is the infinitesimal generator of a solution operator {𝑆_𝑞(𝑡)}_𝑡≥0. Since 𝑆_𝑞(𝑡) is strongly continuous on [0,∞), by the uniformly bounded theorem, there exists a constant 𝑀>0 such that ‖𝑆_𝑞(𝑡)‖_𝐿(𝑋)≤𝑀 for 𝑡∈[0,𝑏] [14].
Consider the following fractional partial integrodifferential equation with infinite delay and control in the following form: ^𝑐𝐷^𝑞_𝑡𝑥𝑡,𝑦+^𝑡_−∞𝑎𝑡,𝑦,𝑠−𝑡𝑄₁𝑥=𝜕𝑠,𝑦𝑑𝑠₂𝜕𝑦²𝑥𝑡,𝑦+^𝑡_−∞𝑎𝑡,𝑦,𝑠−𝑡𝑄₁𝑥+1𝑠,𝑦𝑑𝑠Γ1−𝑞^𝑡₀(𝑡−𝑠)^−𝑞×𝜇𝑠,𝑦+^𝑠_−∞𝐾(𝑠,𝑥,𝜉−𝑠)×𝑄₂𝑥+𝜉,𝑦𝑑𝜉_𝑠0𝑔(𝜉,𝑠)𝑒^{−𝑥(𝜉,𝑦)}[][]𝑑𝜉𝑑𝑠,𝑡∈𝐽=0,1,𝑦∈0,𝜋,𝑡≠𝑡_𝑘, (44) with 𝑥(𝑡,0)=𝑥(𝑡,𝜋)=0, 𝑥(𝑡,𝑦)=𝜙(𝑡,𝑦), 𝑡∈(−∞,0], 𝑦∈[0,𝜋], Δ𝑥(𝑡_𝑖)(𝑦)=∫^𝑡𝑖_−∞𝑞_𝑖(𝑡_𝑖−𝑠)𝑥(𝑠,𝑦)𝑑𝑠, and 𝑦∈[0,𝜋], where 𝜕^𝑞_𝑡 is the Caputo fractional partial derivative of order 0<𝑞<1; 𝜇∶[0,1]×[0,𝜋]→[0,𝜋] is continuous and in 𝑡; 𝑞_𝑖∶𝑅→𝑅 are continuous; 0<𝑡₁<𝑡₂<⋯<𝑡_𝑛<𝑏 are prefixed numbers; and 𝜙∈ℬ_ℎ.
Let ℎ(𝑠)=𝑒^2𝑠, 𝑠<0, then 𝑙=∫⁰_∞ℎ(𝑠)𝑑𝑠=1/2. Let ℬ_ℎ be the phase space endowed with the norm ‖𝜑‖_ℬℎ=∫⁰_−∞ℎ(𝑠)sup_{𝜃∈[𝑠,0]}‖𝜑(𝜃)‖_𝐿2𝑑𝑠. Let 𝑥(𝑡)(𝑦)=𝑥(𝑡,𝑦), and define the bounded linear operator 𝐵∶𝑈→𝑋 by (𝐵𝑢)(𝑡)(𝑦)=𝜇(𝑡,𝑦), 0≤𝑦≤𝜋, 𝑔(𝑡,𝜙)(𝑦)=∫⁰_−∞𝑎(𝑡,𝑦,𝜃)𝑄₁(𝜙(𝜃)(𝑦))𝑑𝜃, 𝐼_𝑘(𝑥(𝑡⁻_𝑖))(𝑦)=∫^𝑡𝑖_−∞𝑞_𝑖(𝑡_𝑖−𝑠)𝑥(𝑠,𝑦)𝑑𝑠, and 𝑓(𝑡,𝜙,𝐻𝑥(𝑡))(𝑦)=∫⁰_−∞𝐾(𝑡,𝑦,𝜃)𝑄₂(𝜙(𝜃)(𝑦))𝑑𝜃+𝐻𝑥(𝑡)(𝑦), where 𝐻𝑥(𝑡)(𝑦)=∫^𝑡₀𝑔(𝑠,𝑡)𝑒^{−𝑥(𝑠,𝑦)}𝑑𝑠 and 𝜙(𝜃)(𝑦)=𝜙(𝜃,𝑦), (𝜃,𝑦)∈(−∞,0]×[0,𝜋]. Moreover, the linear fractional control system corresponding to (44) is exactly controllable. Further, if we impose suitable conditions on 𝑎, 𝐾, 𝑄₁, 𝑄₂, 𝑔, 𝑞_𝑖, and 𝐵 to verify assumptions on Theorem 8, then the system (44) can be written in the abstract form of (1). Therefore, all the conditions of Theorem 8 are satisfied, and hence, the nonlinear fractional control system (44) is exactly controllable on [0,b].

Acknowledgment

The work of Yong Ren is supported by the National Natural Science Foundation of China (no. 11371029).