Approximate Controllability of Fractional Neutral Evolution Equations in Banach Spaces

Abstract

We discuss the approximate controllability of semilinear fractional neutral differential systems with infinite delay under the assumptions that the corresponding linear system is approximately controllable. Using Krasnoselkii's fixed-point theorem, fractional calculus, and methods of controllability theory, a new set of sufficient conditions for approximate controllability of fractional neutral differential equations with infinite delay are formulated and proved. The results of the paper are generalization and continuation of the recent results on this issue.

1. Introduction and Preliminaries

Many social, physical, biological and engineering problems can be described by fractional partial differential equations. In fact, fractional differential equations are considered as an alternative model to nonlinear differential equations. In the last two decades, fractional differential equations (see, e.g., Samko et al. [1] and references therein) have attracted many scientists, and notable contributions have been made to both theory and applications of fractional differential equations.

Nowadays, controllability theory for linear systems has already been well established, for finite and infinite dimensional systems; see, for instance, [2]. Several authors have extended these concepts to infinite-dimensional systems represented by nonlinear evolution equations in infinite-dimensional spaces, see [3–30]. On the other hand, approximate controllability problems for fractional evolution equations in Hilbert spaces are not yet sufficiently investigated, and there are only few works on it [13–21, 28–30]. So far, the overwhelming majority of the approximate controllability results have only been available for semilinear evolution differential systems in Hilbert spaces, with the exception of the case of [11]. Motivated by the fact that many partial fractional differential equations can be converted into fractional PDE in some Banach spaces, we consider that there is a realistic need to discuss the approximate controllability problem of fractional-order differential systems in Banach spaces. Note that our results are new even for the approximate controllability of fractional neutral differential equations with infinite delay in Hilbert spaces.

Consider the following fractional neutral evolution differential system with infinite delay: 𝑑^𝛼𝑑𝑡^𝛼𝑥(𝑡)−ℎ𝑡,𝑥_𝑡=−𝐴𝑥(𝑡)+𝐵𝑢(𝑡)+𝑓𝑡,𝑥_𝑡,[],𝑥(𝑡)(𝑡)𝑡∈0,𝑇=𝜙∈𝐵_𝑙,(1) where the state 𝑥 takes values in a Banach space 𝑋 and the control function takes values in a Hilbert space 𝑈. The functions ℎ,𝑓 will be specified in the sequel. Let 𝑥_𝑡(⋅) denote 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃), 𝜃∈(−∞,0]. Assume that 𝑙∶(−∞,0]→(−∞,0) is a continuous function satisfying 𝑙=∫⁰_−∞𝑙(𝑡)𝑑𝑡<∞. The Banach space (𝐵_𝑙,‖⋅‖_𝑙) induced by function ℎ is defined as follows: 𝐵_𝑙(]𝜑(𝜃)[],∶=𝜑∶−∞,0⟶𝑋∶forany𝑐>0,isaboundedandmeasurablefunctionon−𝑐,0and∫⁰_−∞𝑙(𝑡)sup_{𝑡≤𝜃≤0}𝜑(𝜃)𝑑𝑡<∞(2) endowed with the norm ‖𝜑‖_𝐵𝑙∶=∫⁰_−∞ℎ(𝑡)sup_{𝑡≤𝜃≤0}‖𝜑(𝜃)‖𝑑𝑡. It should be mentioned that (approximate) controllability results for first- and second-order partial neutral functional differential equations with infinite delay were considered by Sakthivel et al. [18], Chalishajar [8], and Chalishajar and Acharya [9].

Throughout this paper, unless otherwise specified, the following notations will be used. Let (𝑋,‖⋅‖) be a separable reflexive Banach space, and let (𝑋^∗,‖⋅‖_∗) stand for its dual space with respect to the continuous pairing ⟨⋅,⋅⟩. We may assume, without loss of generality, that 𝑋 and 𝑋^∗ are smooth and strictly convex, by virtue of renorming theorem (see, e.g., [10]). In particular, this implies that the duality mapping 𝐽 of 𝑋 into 𝑋^∗ given by the following relations: ‖𝐽(𝑧)‖_∗=‖𝑧‖,⟨𝐽(𝑧),𝑧⟩=‖𝑧‖²,∀𝑧∈𝑋(3) is bijective, homogeneous, and demicontinuous, that is, continuous from 𝑋 with a strong topology into 𝑋^∗ with weak topology and strictly monotonic. Moreover, 𝐽⁻¹∶𝑋^∗→𝑋 is also duality mapping.

In this paper, we also assume that −𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is the infinitesimal generator of a compact analytic semigroup 𝑆(𝑡), 𝑡>0, of uniformly bounded linear operator in 𝑋, that is, there exists 𝑀>1 such that ‖𝑆(𝑡)‖_𝐿(𝑋)≤𝑀 for all 𝑡≥0. Without loss of generality, let 0∈𝜌(𝐴), where 𝜌(𝐴) is the resolvent set of 𝐴. Then, for any 𝛽>0, we can define 𝐴^−𝛽 by 𝐴^−𝛽1∶=Γ𝛽^∞₀𝑡^𝛽−1𝑆(𝑡)𝑑𝑡.(4) It follows that each 𝐴^−𝛽 is an injective continuous endomorphism of 𝑋. Hence, we can define 𝐴^𝛽∶=(𝐴^−𝛽)⁻¹, which is a closed bijective linear operator in 𝑋. It can be shown that each 𝐴^𝛽 has dense domain and that 𝐷(𝐴^𝛾)⊂𝐷(𝐴^𝛽) for 0≤𝛽≤𝛾. Moreover, 𝐴^𝛽+𝛾𝑥=𝐴^𝛽𝐴^𝛾𝑥=𝐴^𝛾𝐴^𝛽𝑥 for every 𝛽, 𝛾∈𝑅 and 𝑥∈𝐷(𝐴^𝜇) with 𝜇∶=max(𝛽,𝛾,𝛽+𝛾), where 𝐴⁰=𝐼 and 𝐼 is the identity in 𝑋.

We denote by 𝑋_𝛽 the Banach space of 𝐷(𝐴^𝛽) equipped with norm ‖𝑥‖_𝛽∶=‖𝐴^𝛽𝑥‖ for 𝑥∈𝐷(𝐴^𝛽), which is equivalent to the graph norm of 𝐴^𝛽. Then, we have 𝑋_𝛾↪𝑋_𝛽, for 0≤𝛽≤𝛾 (with 𝑋₀=𝑋 ), and the embedding is continuous. Moreover, 𝐴^𝛽 has the following basic properties.

Lemma 1 (see [31]). 𝐴^𝛽 has the following properties:(i)𝑆(𝑡)∶𝑋→𝑋_𝛽 for each 𝑡>0 and 𝛽≥0.(ii)𝐴^𝛽𝑆(𝑡)𝑥=𝑆(𝑡)𝐴^𝛽𝑥 for each 𝑥∈𝐷(𝐴^𝛽) and 𝑡≥0.(iii)For every 𝑡>0, 𝐴^𝛽𝑆(𝑡) is bounded in 𝑋, and there exists 𝑀_𝛽>0 such that 𝐴^𝛽𝑆(𝑡)≤𝑀_𝛽𝑡^−𝛽.(5)(iv)𝐴^−𝛽 is a bounded linear operator for 0≤𝛽≤1 in 𝑋.

From Lemma 1(iv), since 𝐴^−𝛽 is a bounded linear operator for 0≤𝛽≤1, there exists a constant 𝐶_𝛽 such that ‖𝐴^−𝛽‖≤𝐶_𝛽 for 0≤𝛽≤1.

Let us recall the following known definitions in fractional calculus. For more details, see [1].

Definition 2. The fractional integral of order 𝛼>0 with the lower limit 0 for a function 𝑓 is defined as 𝐼^𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡₀𝑓(𝑠)(𝑡−𝑠)^1−𝛼𝑑𝑠,𝑡>0,𝛼>0,(6) provided that the right-hand side is pointwise defined on [0,∞), where Γ is the gamma function.

Definition 3. Riemann-Liouville derivative of order 𝛼 with the lower limit 0 for a function 𝑓∶[0,∞)→𝑅 can be written as ^𝐿𝐷^𝛼𝑓(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑛𝑡₀𝑓(𝑠)(𝑡−𝑠)^{𝛼+1−𝑛}𝑑𝑠,𝑡>0,𝑛−1<𝛼<𝑛,(7)

Definition 4. The Caputo derivative of order 𝛼 for a function 𝑓∶[0,∞)→𝑅 can be written as ^𝑐𝐷^𝛼𝑓(𝑡)=^𝐿𝐷^𝛼𝑓(𝑡)−^𝑛−1_𝑘=0𝑡^𝑘𝑓𝑘!^(𝑘)(0),𝑡>0,𝑛−1<𝛼<𝑛.(8)

For 𝑥∈𝑋, we define two families {𝔖_𝛼(𝑡)∶𝑡≥0} and {𝔄_𝛼(𝑡)∶𝑡≥0} of operators by 𝔖_𝛼(𝑡)=^∞₀Ψ_𝛼(𝜃)𝑆𝑡^𝛼𝜃𝔄𝑑𝜃,_𝛼(𝑡)=𝛼^∞₀𝜃Ψ_𝛼(𝜃)𝑆𝑡^𝛼𝜃𝑑𝜃,(9) where Ψ_𝛼(𝜃)=1𝜋𝛼^∞_𝑛=1(−1)^𝑛−1Γ(𝑛𝛼+1)()(),𝑛!sin𝑛𝜋𝛼,𝜃∈0,∞(10) is the function of Wright type defined on (0,∞) which satisfies Ψ_𝛼(𝜃)≥0,^∞₀Ψ_𝛼(𝜃)𝑑𝜃=1,_∞0𝜃^𝜁Ψ_𝛼(𝜃)Γ()𝑑𝜃=1+𝜁Γ()().1+𝛼𝜁,𝜁∈−1,∞(11)

The following lemma follows from the results in [32–34].

Lemma 5. The operators 𝔖_𝛼 and 𝔄_𝛼 have the following properties:(i)for any fixed 𝑡≥0 and any 𝑥∈𝑋_𝛽, one has the operators 𝔖_𝛼(𝑡) and 𝔄_𝛼(𝑡) which are linear and bounded operators; that is, for any 𝑥∈𝑋_𝛽,𝔖_𝛼(𝑡)𝑥_𝛽≤𝑀‖𝑥‖_𝛽,𝔄_𝛼(𝑡)𝑥_𝛽≤𝑀Γ(𝛼)‖𝑥‖_𝛽;(12)(ii)the operators 𝔖_𝛼(𝑡) and 𝔄_𝛼(𝑡) are strongly continuous for all 𝑡≥0;(iii)𝔖_𝛼(𝑡) and 𝔄_𝛼(𝑡) are norm continuous in 𝑋 for 𝑡>0;(iv)𝔖_𝛼(𝑡) and 𝔄_𝛼(𝑡) are compact operators in 𝑋 for 𝑡>0;(v)for every 𝑡>0, the restriction of 𝔖_𝛼(𝑡) to 𝑋_𝛽 and the restriction of 𝔄_𝛼(𝑡) to 𝑋_𝛽 are norm continuous;(vi)for every 𝑡>0, the restriction of 𝔖_𝛼(𝑡) to 𝑋_𝛽 and the restriction of 𝔄_𝛼(𝑡) to 𝑋_𝛽 are compact operators in 𝑋_𝛽;(vii)for all 𝑥∈𝑋 and 𝑡∈[0,𝑇], 𝐴^𝛽𝔄_𝛼(𝑡)𝑥≤𝐶_𝛽𝑡^−𝛼𝛽‖𝑥‖,𝐶_𝛽𝑀∶=^𝛽𝛼Γ2−𝛽Γ.1+𝛼1−𝛽(13)

In this paper, we adopt the following definition of mild solution of (1).

Definition 6. A solution 𝑥(⋅;𝑢)∈𝐶([0,𝑇],𝑋) is said to be a mild solution of (1) if for any 𝑢∈𝐿₂([0,𝑇],𝑈) the integral equation 𝑥(𝑡)=𝔖_𝛼(𝑡)𝜙(0)+𝑔0,𝜙−𝑔𝑡,𝑥_𝑡−^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝔄_𝛼(𝑡−𝑠)𝑔𝑠,𝑥_𝑠+𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝔄_𝛼(𝑡−𝑠)𝐵𝑢(𝑠)+𝑓𝑠,𝑥_𝑠𝑑𝑠(14) is satisfied.

Let 𝑥(𝑇;𝑢) be the state value of (14) at terminal time 𝑇 corresponding to the control 𝑢. Introduce the set ℜ(𝑇)={𝑥(𝑇;𝑢)∶𝑢∈𝐿₂([0,𝑇],𝑈)}, which is called the reachable set of the system (14) at terminal time 𝑇, and its closure in 𝑋 is denoted by ℜ(𝑇).

Definition 7. The system (1) is said to be approximately controllable on [0,𝑇] if ℜ(𝑇)=𝑋; that is, given an arbitrary 𝜀>0, it is possible to steer from the point 𝑥₀ to within a distance 𝜀 from all points in the state space 𝑋 at time 𝑇.

To investigate the approximate controllability of the system (14), we assume the following conditions.(H₁)𝐴 is the infinitesimal generator of an analytic semigroup of bounded linear operators 𝑆(𝑡) in 𝑋, 0∈𝜌(𝐴), 𝑆(𝑡) is compact for 𝑡>0, and there exists a positive constant 𝑀 such that ‖𝑆(𝑡)‖≤𝑀.(H₂)The function 𝑔∶[0,𝑇]×𝐵_𝑙→𝑋 is continuous, and there exists some constant 𝑀_𝑔>0,0<𝛽<1, such that 𝑔 is 𝑋_𝛽-valued and 𝐴^𝛽𝑔(𝑡,𝑥)−𝐴^𝛽𝑔𝑡,𝑦≤𝑀_𝑔𝑥−𝑦_𝐵𝑙,𝑥,𝑦∈𝐵_𝑙[],𝐴,𝑡∈0,𝑇_𝛽𝑔(𝑡,𝑥)≤𝑀_𝑔1+‖𝑥‖_𝐵𝑙.(15)(H₃)The function 𝑓∶[0,𝑇]×𝐵_𝑙→𝑋 satisfies the following:(a)𝑓(𝑡,⋅)∶𝐵_𝑙→𝑋 is continuous for each 𝑡∈[0,𝑇] and for each 𝑥∈𝐵_𝑙,𝑓(⋅,𝑥)∶[0,𝑇]→𝑋 is strongly measurable;(b) there is a positive integrable function 𝑛∈𝐿^∞([0,𝑇],[0,+∞)) and a continuous nondecreasing function Λ_𝑓∶[0,∞)→(0,∞) such that for every (𝑡,𝑥)∈[0,𝑇]×𝐵_𝑙, we have 𝑓(𝑡,𝑥)≤𝑛(𝑡)Λ_𝑓‖𝑥‖_𝐵𝑙,liminf_𝑟→∞Λ_𝑓(𝑟)𝑟=𝜎_𝑓<∞.(16)(H₄)The following relationship holds: 11+𝜀𝑀²_𝐵𝑀²_𝔄𝑇^2𝛼−1×𝑀2𝛼−1_𝑔𝐴^−𝛽𝑙+𝐶_1−𝛽𝑀_𝑔𝑇^𝛼𝛽𝑀𝛼𝛽𝑙+Γ(𝛼)𝑇^𝛼𝛼𝜎_𝑓sup_𝑠∈𝐽𝑛(𝑠)<1.(17) Here, 𝑀_𝐵∶=‖𝐵‖,𝑀_𝔄∶=‖𝔄_𝛼‖,and𝐶_1−𝛽=𝛼𝑀_1−𝛽Γ(1+𝛽)/Γ(1+𝛼𝛽).(H_ac)For every ℎ∈𝑋𝑧_𝛼(ℎ)=𝜀(𝜀𝐼+Γ^𝑇₀𝐽)⁻¹(ℎ) converges to zero as 𝜀→0⁺ in strong topology, where Γ^𝑇₀∶=^𝑇₀(𝑇−𝑠)^2(𝛼−1)𝔄_𝛼(𝑇−𝑠)𝐵𝐵^∗𝔄^∗_𝛼(𝑇−𝑠)𝑑𝑠(18) and 𝑧_𝜀(ℎ) is a solution of the equation 𝜀𝑧_𝜀+Γ^𝑇₀𝐽𝑧_𝜀=𝜀ℎ.(19)

Let 𝐶_𝑇=𝑥∶𝑥∈𝐶((−∞,𝑇],𝑋),𝑥₀=𝜙∈𝐵_𝑙.(20) set ‖⋅‖_𝑇 be a seminorm defined by ‖𝑥‖_𝑇=𝑥_0𝐵𝑙+sup_{0≤𝑠≤𝑇}‖𝑥(𝑠)‖,𝑥∈𝐶_𝑇.(21)

Lemma 8 (see [8]). Assume that 𝑥∈𝐶_𝑇, then for all 𝑡∈[0,𝑇],𝑥_𝑡∈𝐵_𝑙 and 𝑙‖𝑥(𝑡)‖≤𝑥_𝑡𝐵𝑙≤𝑙sup_{0≤𝑠≤𝑡}‖𝑥(𝑠)‖+𝑥_0𝐵𝑙.(22)

2. Existence Theorem

In order to formulate the controllability problem in the form suitable for application of fixed-point theorem, it is assumed that the corresponding linear system is approximately controllable. Then, it will be shown that the system (1) is approximately controllable if for all 𝜀>0 there exists a continuous function 𝑥∈𝐶([0,𝑇],𝑋) such that 𝑢_𝜀(𝑡,𝑥)=(𝑇−𝑡)^𝛼−1𝐵^∗𝔄^∗_𝛼(𝑇−𝑡)𝐽𝜀𝐼+Γ^𝑇₀𝐽⁻¹𝑝(𝑥),𝑥(𝑡)=𝔖_𝛼(𝑡)𝜙(0)+𝑔0,𝜙−𝑔𝑡,𝑥_𝑡−^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝔄_𝛼(𝑡−𝑠)𝑔𝑠,𝑥_𝑠+𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝔄_𝛼(𝑡−𝑠)𝐵𝑢_𝜀(𝑠,𝑥)+𝑓𝑠,𝑥_𝑠𝑑𝑠,(23) where 𝑝(𝑥)=ℎ−𝔖_𝛼(𝑇)𝜙(0)+𝑔0,𝜙+𝑔𝑇,𝑥_𝑇+^𝑇₀(𝑇−𝑠)^𝛼−1𝐴𝔄_𝛼(𝑇−𝑠)𝑔𝑠,𝑥_𝑠−𝑑𝑠_𝑇0(𝑇−𝑠)^𝛼−1𝔄_𝛼(𝑇−𝑠)𝑓𝑠,𝑥_𝑠𝑑𝑠,ℎ∈𝑋.(24) Having noticed this fact, our goal in this section is to find conditions for solvability of (23). Note that it will be shown that the control in (23) drives the system (1) from 𝜙(0) to ℎ−𝜀𝐽𝜀𝐼+Γ^𝑇₀𝐽⁻¹𝑝(𝑥)(25) provided that the system (23) has a solution.

Theorem 9. Assume that assumptions (H₁)–(H₄) hold and 1/2<𝛼<1. Then, there exists a solution to (23).

Proof. The proof of Theorem 9 follows from Lemmas 10–14 and infinite dimensional analogue of Arzela-Ascoli theorem.

For all 𝜀>0, consider the operator Φ_𝜀∶𝐶_𝑇→𝐶_𝑇 defined as follows: Φ_𝜀𝑥(𝑡)𝜙(𝑡)(]𝔖∶=𝑡∈−∞,0_𝛼(𝑡)𝜙(0)+𝑔0,𝜙−𝑔𝑡,𝑥_𝑡−∫^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝔄_𝛼(𝑡−𝑠)𝑔𝑠,𝑥_𝑠+∫𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝔄_𝛼()×𝑡−𝑠𝐵𝑢_𝜀(𝑠,𝑥)+𝑓𝑠,𝑥_𝑠𝑑𝑠,𝑡∈[0,𝑇].(26) It will be shown that for all 𝜀>0, the operator Φ_𝜀∶𝐶_𝑇→𝐶_𝑇 has a fixed point.

Suppose that 𝑥(𝑡)=𝜙(𝑡)+𝑧(𝑡),𝑡∈(−∞,𝑇], where 𝜙(𝑡) is taken as 𝜙(𝑡) for 𝑡∈(−∞,0], while for 𝑡∈[0,𝑇], it is defined as 𝔖_𝛼(𝑡)𝜙(0). Set 𝐶⁰_𝑇=𝑧∈𝐶_𝑇∶𝑧₀=0∈𝐵_𝑙.(27) For any 𝑧∈𝐶⁰_𝑇, we have ‖𝑧‖_𝑇=𝑧_0𝐵𝑙+sup_{0≤𝑠≤𝑇}‖𝑧(𝑠)‖=sup_{0≤𝑠≤𝑇}‖𝑧(𝑠)‖.(28) Thus, (𝐶⁰_𝑇,‖⋅‖_𝑇) is a Banach space. For each positive number 𝑟>0, set 𝐁_𝑟∶=𝑧∈𝐶⁰_𝑇∶‖𝑧‖_𝑇≤𝑟.(29) It is clear that 𝐵_𝑟 is bounded closed convex set in 𝐶⁰_𝑇. For any 𝑧∈𝐵_𝑟, we see that 𝜙_𝑡+𝑧_𝑡𝐵𝑙≤𝜙_𝑡𝐵𝑙+𝑧_𝑡𝐵𝑙≤𝑙sup_{0≤𝑠≤𝑡}𝜙(𝑠)+𝜙_0𝐵𝑙+𝑙sup_{0≤𝑠≤𝑡}‖𝑧(𝑠)‖+𝑧_0𝐵𝑙≤𝑙𝑀𝜙(0)+𝑟+𝜙_𝐵𝑙∶=𝑅(𝑟).(30) Consider the maps Π_𝜀,Θ_𝜀∶𝐶⁰_𝑇→𝐶⁰_𝑇 defined by Π_𝜀𝑧(𝑡)(],∶=𝔖0𝑡∈−∞,0_𝛼(𝑡)𝑔0,𝜙−𝑔𝑡,𝜙_𝑡+𝑧_𝑡−∫^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝔄_𝛼(𝑡−𝑠)𝑔𝑠,𝜙_𝑠+𝑧_𝑠[],Θ𝑑𝑠𝑡∈0,𝑇_𝜀𝑧(𝑡)(],∶=∫0𝑡∈−∞,0_𝑡0(𝑡−𝑠)^𝛼−1𝔄_𝛼()×𝑡−𝑠𝐵𝑢_𝜀𝑠,𝜙+𝑧+𝑓𝑠,𝜙_𝑠+𝑧_𝑠𝑑𝑠,𝑡∈[0,𝑇].(31) Obviously, the operator Φ_𝜀 has a fixed point if and only if operator Π_𝜀+Θ_𝜀 has a fixed point. In order to prove that Π_𝜀+Θ_𝜀 has a fixed point we will employ the Krasnoselskii’s fixed-point theorem.

Lemma 10. Under assumptions (H₁)–(H₄), for any 𝜀>0, there exists a positive number 𝑟∶=𝑟(𝜀) such that Π_𝜀𝑧_𝑟+Θ_𝜀𝑦_𝑟⊂𝐁_𝑟 for all 𝑧,𝑦∈𝐁_𝑟𝑟.

Proof. Let 𝜀>0 be fixed. If it is not true, then for each 𝑟>0, there are functions 𝑧_𝑟,𝑦_𝑟∈𝐵_𝑟, but Π_𝜀(𝑧_𝑟)+Θ_𝜀(𝑦_𝑟)∉𝐁_𝑟. So for some 𝑡=𝑡(𝑟)∈[0,𝑇], one can show that 𝑟≤Π_𝜀𝑧_𝑟(𝑡)+Θ_𝜀𝑦_𝑟(𝑡≤𝔖_𝛼)(𝑡)𝑔0,𝜙+𝑔𝑡,𝜙_𝑡+𝑧_𝑡+^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝔄_𝛼(𝑡−𝑠)𝑔𝑠,𝜙_𝑠+𝑧_𝑠+𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝔄_𝛼(𝑡−𝑠)𝑓𝑠,𝜙_𝑠+𝑦_𝑠+𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝔄_𝛼(𝑡−𝑠)𝐵𝑢_𝜀𝑠,𝜙+𝑦𝑑𝑠=∶𝐼₁+𝐼₂+𝐼₃+𝐼₄+𝐼₅.(32) Let us estimate 𝐼_𝑖,𝑖=1,…,4. By the assumption (H₂), we have 𝐼₁≤𝑀𝐴^−𝛽𝐴^𝛽𝑔0,𝜙≤𝑀𝑀_𝑔𝐴^−𝛽1+𝜙_𝐵𝑙,𝐼₂≤𝐴^−𝛽𝐴^𝛽𝑔𝑡,𝜙_𝑡+𝑧_𝑡≤𝑀_𝑔𝐴^−𝛽1+𝜙_𝑡+𝑧_𝑡𝐵𝑙(33)≤𝑀_𝑔𝐴^−𝛽(1+𝑅(𝑟)).(34) Using Lemma 5 and the Hölder inequality, one can deduce that 𝐼₃≤^𝑡₀(𝑡−𝑠)^𝛼−1𝐴^1−𝛽𝔄_𝛼(𝑡−𝑠)𝐴^𝛽𝑔𝑠,𝜙_𝑠+𝑧_𝑠≤𝑀𝑑𝑠_1−𝛽𝛼Γ1+𝛽Γ1+𝛼𝛽^𝑡₀(𝑡−𝑠)^𝛼𝛽−1𝐴^𝛽𝑔𝑠,𝜙_𝑠+𝑧_𝑠𝑑𝑠≤𝐾𝛼,𝛽^𝑡₀(𝑡−𝑠)^𝛼𝛽−1𝑀_𝑔1+𝜙_𝑠+𝑧_𝑠𝐵𝑙𝑀𝑑𝑠≤𝐾𝛼,𝛽_𝑔𝑇^𝛼𝛽((𝑟.𝛼𝛽1+𝑅))(35) Using the assumption (H₃), one has 𝐼₄≤^𝑡₀(𝑡−𝑠)^𝛼−1𝔄_𝛼(𝑡−𝑠)𝑓𝑠,𝜙_𝑠+𝑦_𝑠≤𝑀𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝜙_𝑠+𝑦_𝑠≤𝑀𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑛(𝑠)Λ_𝑓𝜙_𝑠+𝑦_𝑠≤𝑀𝑑𝑠Γ(𝛼)𝑇^𝛼𝛼Λ_𝑓(𝑅(𝑟))sup_𝑠∈𝐽𝑛(𝑠).(36) Combining the estimates (32)–(36) yields 𝐼₁+𝐼₂+𝐼₃+𝐼₄<𝑀𝑀_𝑔𝐴^−𝛽1+𝜙_𝐵𝑙+𝑀_𝑔𝐴^−𝛽(1+𝑅(𝑟))+𝐶_1−𝛽𝑀_𝑔𝑇^𝛼𝛽((𝑟+𝑀𝛼𝛽1+𝑅))Γ(𝛼)𝑇^𝛼𝛼Λ_𝑓(𝑅(𝑟))sup_𝑠∈𝐽𝑛(𝑠)∶=Δ.(37) On the other hand, 𝐼₅≤^𝑡₀(𝑡−𝑠)^𝛼−1𝔄_𝛼(𝑡−𝑠)𝐵𝑢_𝜀=𝑠,𝜙+𝑦𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1(𝑇−𝑠)^𝛼−1𝔄_𝛼(𝑡−𝑠)𝐵𝐵^∗𝔄^∗_𝛼×(𝑇−𝑡)𝐽𝜀𝐼+Γ^𝑇₀𝐽⁻¹𝑝≤𝜙+𝑦𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1(𝑇−𝑠)^𝛼−1𝔄_𝛼(𝑡−𝑠)𝐵𝐵^∗𝔄^∗_𝛼(×𝐽𝑇−𝑡)𝑑𝑠𝜀𝐼+Γ^𝑇₀𝐽⁻¹𝑝𝜙+𝑦≤𝑀²_𝐵𝑀²_𝔄𝑇^2𝛼−1𝐽2𝛼−1𝜀𝐼+Γ^𝑇₀𝐽⁻¹𝑝𝜙+𝑦=𝑀²_𝐵𝑀²_𝔄𝑇^2𝛼−12𝛼−1𝜀𝐼+Γ^𝑇₀𝐽⁻¹𝑝≤1𝜙+𝑦𝜀𝑀²_𝐵𝑀²_𝔄𝑇^2𝛼−1𝑝≤12𝛼−1𝜙+𝑦𝜀𝑀²_𝐵𝑀²_𝔄𝑇^2𝛼−12𝛼−1Δ.(38) Thus, 𝑟≤Π_𝜀𝑧_𝑟(𝑡)+Θ_𝜀𝑦_𝑟(𝑡1)≤Δ+𝜀𝑀²_𝐵𝑀²_𝔄𝑇^2𝛼−1Δ=12𝛼−11+𝜀𝑀²_𝐵𝑀²_𝔄𝑇^2𝛼−12𝛼−1Δ.(39) Dividing both sides of (39) by 𝑟 and taking 𝑟→∞, we obtain that 11+𝜀𝑀²_𝐵𝑀²_𝔄𝑇^2𝛼−1×𝑀2𝛼−1_𝑔𝐴^−𝛽𝑙+𝑀_𝑔𝐶_1−𝛽𝑇^𝛼𝛽𝑀𝛼𝛽𝑙+Γ(𝛼)𝑇^𝛼𝛼𝜎_𝑓sup_𝑠∈𝐽𝑛(𝑠)≥1,(40) which is a contradiction by assumption (H₄). Thus, Π_𝜀𝑧_𝑟+Θ_𝜀𝑦_𝑟⊂𝐁_𝑟 for some 𝑟>0.

Lemma 11. Let assumptions (H₁)–(H₄) hold. Then, Θ₁ is contractive.

Proof. Let 𝑥,𝑦∈𝐁_𝑟. Then, Π_𝜀𝑥(𝑡)−Π_𝜀𝑦(𝑡≤𝑔𝜙)𝑡,_𝑡+𝑥_𝑡−𝑔𝑡,𝜙_𝑡+𝑦_𝑡+^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝔄_𝛼()×𝑔𝜙𝑡−𝑠𝑠,_𝑠+𝑥_𝑠−𝑔𝑠,𝜙_𝑠+𝑦_𝑠≤𝐴𝑑𝑠_−𝛽𝑀_𝑔𝑥_𝑡−𝑦_𝑡𝐵𝑙+𝐶_1−𝛽×^𝑡₀(𝑡−𝑠)^𝛼𝛽−1𝐴^𝛽𝑔𝑠,𝜙_𝑠+𝑥_𝑠−𝑔𝑠,𝜙_𝑠+𝑦_𝑠≤𝐴𝑑𝑠_−𝛽𝑀_𝑔𝑥_𝑡−𝑦_𝑡𝐵𝑙+𝐶_1−𝛽𝑀_𝑔×^𝑡₀(𝑡−𝑠)^𝛼𝛽−1𝑥_𝑠−𝑦_𝑠𝐵𝑙𝑑𝑠.(41) Hence, Π_𝜀𝑥(𝑡)−Π_𝜀𝑦(𝑡)≤𝑀_𝑔𝑙𝐴^−𝛽+𝐶_1−𝛽𝑇^𝛼𝛽𝛼𝛽sup_{0≤𝑠≤𝑡}𝑥(𝑠)−𝑦(𝑠),(42) where we have used the fact that 𝑥₀=𝑦₀=0. Thus, sup_{0≤𝑡≤𝑇}Π_𝜀𝑥(𝑡)−Π_𝜀𝑦(𝑡)≤𝑀_𝑔𝑙𝐴^−𝛽+𝐶_1−𝛽𝑇^𝛼𝛽𝛼𝛽sup_{0≤𝑠≤𝑇}𝑥(𝑠)−𝑦(𝑠),(43) so Θ₁ is a contraction by assumption (H₄).

Lemma 12. Let assumptions (H₁)–(H₄) hold. Then, Θ_𝜀 maps bounded sets to bounded sets in 𝐵_𝑟.

Proof. By the similar argument as Lemma 10, we obtain Θ_𝜀𝑧(𝑡<1)1+𝜀𝑀²_𝐵𝑀²_𝔄𝑇^2𝛼−1×𝑀2𝛼−1Γ(𝛼)𝑇^𝛼𝛼Λ_𝑓(𝑅(𝑟))sup_𝑠∈𝐽𝑛(𝑠)∶=𝑟₁(𝜀)(44) which implies that Θ_𝜀𝑧∈𝐁_𝑟1(𝜀).

Lemma 13. Let assumptions (H₁)–(H₄) hold. Then, the set {Θ_𝜀𝑧∶𝑧∈𝐁_𝑟} is an equicontinuous family of functions on [0,𝑇].

Proof. Let 0<𝜂<𝑡<𝑇 and 𝛿>0 such that 𝔄_𝛼𝑠₁−𝔄_𝛼𝑠₂<𝜂(45) for every 𝑠₁,𝑠₂∈[0,𝑇] with |𝑠₁−𝑠₂|<𝛿. For 𝑧∈𝐁_𝑟, 0<|ℎ|<𝛿,𝑡+ℎ∈[0,𝑇], we have Θ_𝜀𝑧(𝑡+ℎ)−Θ_𝜀𝑧(𝑡≤_𝑡0)(𝑡+ℎ−𝑠)^𝛼−1−(𝑡−𝑠)^𝛼−1𝔄_𝛼()×)𝑡+ℎ−𝑠(𝑇−𝑠^𝛼−1𝐵𝑣_𝜀𝑠,𝜙+𝑧+𝑓𝑠,𝜙_𝑠+𝑧_𝑠+𝑑𝑠_𝑡𝑡+ℎ(𝑡+ℎ−𝑠)^𝛼−1𝔄_𝛼()×)𝑡+ℎ−𝑠(𝑇−𝑠^𝛼−1𝐵𝑣_𝜀𝑠,𝜙+𝑧+𝑓𝑠,𝜙_𝑠+𝑧_𝑠+𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝔄_𝛼(𝑡+ℎ−𝑠)−𝔄_𝛼()×)𝑡−𝑠(𝑇−𝑠^𝛼−1𝐵𝑣_𝜀𝑠,𝜙+𝑧+𝑓𝑠,𝜙_𝑠+𝑧_𝑠𝑑𝑠.(46) Applying (38) and the Hölder inequality, we obtain Θ_𝜀𝑧(𝑡+ℎ)−Θ_𝜀𝑧(𝑡≤𝑀)Γ(𝛼)Λ_𝑓(𝑅(𝑟))^𝑡₀(𝑡+ℎ−𝑠)^𝛼−1−(𝑡−𝑠)^𝛼−1𝑛(𝑠)+𝑀𝑑𝑠Γ(𝛼)1𝜀𝑀_𝐵𝑀_𝔄Δ×^𝑡₀(𝑡+ℎ−𝑠)^𝛼−1−(𝑡−𝑠)^𝛼−1(𝑇−𝑠)^𝛼−1+𝑀𝑑𝑠Γ(𝛼)Λ_𝑓(𝑅(𝑟))_𝑡^𝑡+ℎ(𝑡+ℎ−𝑠)^𝛼−1𝑛(𝑠)+𝑀𝑑𝑠Γ(𝛼)1𝜀𝑀_𝐵𝑀_𝔄Δ_𝑡^𝑡+ℎ(𝑡+ℎ−𝑠)^𝛼−1(𝑇−𝑠)^𝛼−1+𝑑𝑠𝜂𝑇^𝛼𝛼Λ_𝑓(𝑅(𝑟))^𝑡₀(𝑡−𝑠)^𝛼−1𝑛(𝑠)+𝑑𝑠𝜂𝑇^𝛼𝛼1𝜀𝑀_𝐵𝑀_𝔄Δ^𝑡₀(𝑡−𝑠)^𝛼−1(𝑇−𝑠)^𝛼−1𝑑𝑠.(47) Therefore, for 𝜀 sufficiently small, the right-hand side of (47) tends to zero as ℎ→0. On the other hand, the compactness of 𝔄_𝛼(𝑡), 𝑡>0 implies the continuity in the uniform operator topology. Thus, the set {Θ_𝜀𝑧∶𝑧∈𝐁_𝑟} is equicontinuous.

Lemma 14. Let assumptions (H₁)–(H₄) hold. Then, Θ_𝜀 maps 𝐁_𝑟 onto a precompact set in 𝐁_𝑟.

Proof. Let 0<𝑡≤𝑇 be fixed and 𝜀 be a real number satisfying 0<𝜆<𝑡. For 𝛿>0, define an operator Θ_𝜀^𝜆,𝛿 on 𝐵_𝑟 by Θ_𝜀^𝜆,𝛿𝑧(𝑡)=𝛼₀^{𝑡−𝜆∞}_𝛿𝜃(𝑡−𝑠)^𝛼−1𝜂_𝛼(𝜃)𝑆(𝑡−𝑠)^𝛼𝜃×𝐵𝑢_𝜀𝑠,𝜙+𝑧+𝑓𝑠,𝜙_𝑠+𝑧_𝑠𝜆𝑑𝑠=𝛼𝑆^𝛼𝛿₀^{𝑡−𝜆∞}_𝛿𝜃(𝑡−𝑠)^𝛼−1𝜂_𝛼(𝜃)𝑆(𝑡−𝑠)^𝛼𝜃−𝜆^𝛼𝛿×𝐵𝑢_𝜀𝑠,𝜙+𝑧+𝑓𝑠,𝜙_𝑠+𝑧_𝑠𝑑𝑠.(48) Since 𝑆(𝑡), 𝑡>0 is a compact operator, the set {(Θ_𝜀^𝜆,𝛿𝑧)(𝑡)∶𝑧∈𝐁_𝑟} is precompact in 𝐻 for every 0<𝜆<𝑡, 𝛿>0. Moreover, for each 𝑧∈𝐁_𝑟, we have Θ_𝜀𝑧(𝑡)−Θ_𝜀^𝜆,𝛿𝑧(𝑡)≤𝛼𝐸^𝑡₀^𝛿₀𝜃(𝑡−𝑠)^𝛼−1𝜂_𝛼(𝜃)𝑆(𝑡−𝑠)^𝛼𝜃×𝐵𝑢_𝜀𝑠,𝜙+𝑧+𝑓𝑠,𝜙_𝑠+𝑧_𝑠𝑑𝜃𝑑𝑠+𝛼𝐸^𝑡_𝑡−𝜆^∞_𝛿𝜃(𝑡−𝑠)^𝛼−1𝜂_𝛼(𝜃)𝑆(𝑡−𝑠)^𝛼𝜃×𝐵𝑢_𝜀𝑠,𝜙+𝑧+𝑓𝑠,𝜙_𝑠+𝑧_𝑠𝑑𝜃𝑑𝑠∶=𝐽₁+𝐽₂.(49) A similar argument as before 𝐽₁≤𝛼𝑀^𝑡₀(𝑡−𝑠)^𝛼−1𝐵𝑢_𝜀𝑠,𝜙+𝑧+𝑓𝑠,𝜙_𝑠+𝑧_𝑠×𝑑𝑠_𝛿0𝜃𝜂_𝛼(𝜃)1𝑑𝜃≤𝛼𝑀𝜀𝑀_𝐵𝑀_𝔄Δ^𝑡₀(𝑡−𝑠)^𝛼−1(𝑇−𝑠)^𝛼−1𝑑𝑠+Λ_𝑓(𝑅(𝑟))^𝑡₀(𝑡−𝑠)^𝛼−1𝑛(𝑠)𝑑𝑠^𝛿₀𝜃𝜂_𝛼(𝜃),𝐽𝑑𝜃₂≤𝛼𝑀^𝑡_𝑡−𝜆(𝑡−𝑠)^𝛼−1𝐵𝑢_𝜀𝑠,𝜙+𝑧+𝑓𝑠,𝜙_𝑠+𝑧_𝑠×𝑑𝑠_∞𝛿𝜃𝜂_𝛼(𝜃)≤𝑑𝜃𝛼𝑀Γ()11+𝛼𝜀𝑀_𝐵𝑀_𝔄Δ^𝑡_𝑡−𝜆(𝑡−𝑠)^𝛼−1(𝑇−𝑠)^𝛼−1𝑑𝑠+Λ_𝑓(𝑅(𝑟))^𝑡_𝑡−𝜆(𝑡−𝑠)^𝛼−1𝑛(𝑠)𝑑𝑠,(50) where we have used the equality ^∞₀𝜃^𝛽𝜂_𝛼(𝜃)Γ𝑑𝜃=1+𝛽Γ1+𝛼𝛽.(51) Form (49) to (50), one can see that for each 𝑧∈𝐁_𝑟, Θ_𝜀𝑧(𝑡)−Θ_𝜀^𝜆,𝛿𝑧(𝑡)⟶0as𝜆⟶0⁺,𝛿⟶0⁺.(52) Therefore, there are relatively compact sets arbitrary close to the set {(Θ_𝜀𝑧)(𝑡)∶𝑧∈𝐁_𝑟}. Hence, the set {(Θ_𝜀𝑧)(𝑡)∶𝑧∈𝐁_𝑟} is also precompact in 𝐁_𝑟.

3. Main Results

Consider the following linear fractional differential system: 𝐷^𝛼_𝑡𝑥(𝑡)(𝑡)(𝑡)[],𝑥(0)(0).=𝐴𝑥+𝐵𝑢,𝑡∈0,𝑇(53)=𝜙(54) The approximate controllability for the linear fractional system (53) is a natural generalization of approximate controllability of linear first-order control system. It is convenient at this point to introduce the controllability operator associated with (53) as 𝐿^𝑇₀=^𝑇₀(𝑇−𝑠)^𝛼−1𝔄_𝛼()(𝑠)Γ𝑇−𝑠𝐵𝑢𝑑𝑠,_𝑇0=𝐿^𝑇₀𝐿^𝑇₀^∗=^𝑇₀(𝑇−𝑠)^2(𝛼−1)𝔄_𝛼(𝑇−𝑠)𝐵𝐵^∗𝔄^∗_𝛼(𝑇−𝑠)𝑑𝑠,(55) respectively, where 𝐵^∗ denotes the adjoint of 𝐵 and 𝔄^∗_𝛼(𝑡) is the adjoint of 𝔄_𝛼(𝑡). It is straightforward that the operator 𝐿^𝑇₀ is a linear-bounded operator for 1/2<𝛼≤1.

Theorem 15 (see [11]). The following three conditions are equivalent: (i)Γ^𝑇₀ is positive, that is, ⟨𝑧^∗,Γ^𝑇₀𝑧^∗⟩>0 for all nonzero 𝑧^∗∈𝑋^∗.(ii)For all ℎ∈𝑋,𝐽(𝑧_𝜀(ℎ)) converges to the zero as 𝜀→0⁺ in the weak topology, where 𝑧_𝜀(ℎ)=𝜀(𝜀𝐼+Γ^𝑇₀𝐽)⁻¹(ℎ) is a solution of the equation 𝜀𝑧_𝜀+Γ^𝑇₀𝐽𝑧_𝜀=𝛼ℎ.(56)(iii)For all ℎ∈𝑋,𝑧_𝜀(ℎ)=𝜀(𝜀𝐼+Γ^𝑇₀𝐽)⁻¹(ℎ) converges to the zero as 𝜀→0⁺ in the strong topology.

Remark 16. It is known that Theorem 15(i) holds if and only if Im𝐿^𝑇₀=𝑋. In other words, Theorem 15(i) holds if and only if the corresponding linear system is approximately controllable on [0,𝑇].

Theorem 17 (see [11]). Let 𝑝∶𝑋→𝑋 be a nonlinear operator. Assume that 𝑧_𝜀 is a solution of the following equation: 𝜀𝑧_𝜀+Γ^𝑇₀𝐽𝑧_𝜀=𝛼𝑝𝑧_𝜀,𝑝𝑧(57)_𝜀−𝑝⟶0as𝜀⟶0⁺,𝑝∈𝑋.(58) Then, there exists a subsequence of the sequence {𝑧_𝜀} strongly converging to zero as 𝜀→0⁺.

We are now in a position to state and prove the main result of the paper.

Theorem 18. Let 1/2<𝛼<1. Suppose that conditions (H₁)–(H₄) and (H_ac) are satisfied. Besides, assume additionally that(H_gc)𝑔∶[0,𝑇]×𝑋→𝑋 and 𝐴^𝛽𝑔(𝑇,⋅) is continuous from the weak topology of 𝑋 to the strong topology of 𝑋.(H_ub) There exists 𝑁∈𝐿^∞([0,𝑇],[0,+∞)) such that sup_{𝑥∈𝐵𝑙}𝑓(𝑡,𝑥)+sup_𝑦∈𝑋𝐴^𝛽𝑔𝑡,𝑦≤𝑁(𝑡),fora.e.𝑡∈[0,𝑇].(59)
Then, the system (1) is approximately controllable on [0,𝑇].

Proof. Let 𝑥^𝜀 be a fixed point of Φ_𝜀 in 𝐵_𝑟(𝜀). Then, 𝑥^𝜀 is a mild solution of (1) on [0,𝑇] under the control 𝑢_𝜀𝑡,𝑥^𝜀=(𝑇−𝑡)^𝛼−1𝐵^∗𝑆^∗(𝑇−𝑡)𝐽𝜀𝐼+Γ^𝑇₀𝐽⁻¹𝑝𝑥^𝜀,𝑝𝑥^𝜀=ℎ−𝔖_𝛼(𝑇)𝜙(0)+𝑔0,𝜙(0)+𝑔𝑇,𝑥^𝜀(𝑇)+^𝑇₀(𝑇−𝑠)^𝛼−1𝐴𝔄_𝛼(𝑇−𝑠)𝑔𝑠,𝑥^𝜀_𝑠𝑑𝑠−^𝑇₀(𝑇−𝑠)^𝛼−1𝔄_𝛼(𝑇−𝑠)𝑓𝑠,𝑥^𝜀_𝑠𝑑𝑠(60) and satisfies the following equality: 𝑥^𝜀(𝑇)=𝔖_𝛼(𝑇)𝜙(0)+𝑔0,𝜙(0)−𝑔𝑇,𝑥^𝜀(𝑇−_𝑇0)(𝑇−𝑠)^𝛼−1𝐴𝔄_𝛼(𝑇−𝑠)𝑔𝑠,𝑥^𝜀_𝑠+𝑑𝑠_𝑇0(𝑇−𝑠)^𝛼−1𝔄_𝛼(𝑡−𝑠)𝐵𝑢_𝜀(𝑠,𝑥)+𝑓𝑠,𝑥^𝜀_𝑠𝑑𝑠=𝔖_𝛼(𝑇)𝜙(0)+𝑔0,𝜙(0)−𝑔𝑇,𝑥^𝜀(𝑇−_𝑇0)(𝑇−𝑠)^𝛼−1𝐴𝔄_𝛼(𝑇−𝑠)ℎ𝑠,𝑥^𝜀_𝑠+𝑑𝑠−𝜀𝐼+𝜀𝐼+Γ^𝑇₀𝐽𝜀𝐼+Γ^𝑇₀𝐽⁻¹𝑝𝑥^𝜀+^𝑇₀(𝑇−𝑠)^𝛼−1𝔄_𝛼(𝑡−𝑠)𝑓𝑠,𝑥^𝜀_𝑠𝑑𝑠=ℎ−𝜀𝜀𝐼+Γ^𝑇₀𝐽⁻¹𝑝𝑥^𝜀.(61) In other words, 𝑧_𝜀=ℎ−𝑥^𝜀(𝑇) is a solution of the equation 𝜀𝐼+Γ^𝑇₀𝐽𝑧_𝜀=𝜀𝑝𝑥^𝜀.(62) It follows that 𝜀𝐽𝑧_𝜀,𝑧_𝜀+𝐽𝑧_𝜀,Γ^𝑇₀𝐽𝑧_𝜀=𝜀𝐽𝑧_𝜀,𝑝𝑥^𝜀,𝜀𝑧_𝜀²+𝐽𝑧_𝜀,Γ^𝑇₀𝐽𝑧_𝜀=𝜀𝐽𝑧_𝜀,𝑝𝑥^𝜀,𝜀𝑧_𝜀²≤𝜀𝐽𝑧_𝜀,𝑝𝑥^𝜀≤𝜀𝑧_𝜀𝑝𝑥^𝜀,𝑧_𝜀=𝐽𝑧_𝜀≤𝑝𝑥^𝜀.(63) On the other hand, by (H_ub), 𝑝𝑥^𝜀≤‖ℎ‖𝜙(0(𝑇)+𝑀+𝑀)+𝑁_1−𝛽𝛼Γ1+𝛽Γ1+𝛼𝛽^𝑇₀(𝑇−𝑠)^𝛼𝛽−1𝑁(𝑠)+𝑀𝑑𝑠Γ(𝛼)^𝑇₀(𝑇−𝑠)^𝛼−1𝑁(𝑠)𝑑𝑠.(64) From (63) and (64), it follows that 𝑥^𝜀(𝑇)⇀𝑥 weakly as 𝜀→0⁺ and by the assumption (H_gc)  𝐴^𝛽𝑔(𝑇,𝑥^𝜀(𝑇))→𝐴^𝛽𝑔(𝑇,𝑥) strongly as 𝜀→0⁺. Moreover, because of assumption (H_ub), ^𝑇₀𝑓𝑠,𝑥^𝜀_𝑠²𝑑𝑠+^𝑇₀𝐴^𝛽𝑔𝑠,𝑥^𝜀_𝑠²𝑑𝑠≤^𝑇₀𝑁(𝑠)𝑑𝑠.(65) Consequently, the sequences {𝑓(⋅,𝑥^𝜀_⋅)},{𝐴^𝛽𝑔(⋅,𝑥^𝜀_⋅)} are bounded. Then, there is a subsequence still denoted by {𝑓(⋅,𝑥^𝜀_⋅),𝐴^𝛽𝑔(⋅,𝑥^𝜀_⋅)} which weakly converges to, say, (𝑓(⋅),𝑔(⋅)) in 𝐿₂([0,𝑇],𝑋). Then, 𝑝𝑥^𝜀=𝑔−𝑝𝑇,𝑥^𝜀(𝑇)(𝑥+−𝑔𝑇,)_𝑇0(𝑇−𝑠)^𝛼−1𝐴^1−𝛽𝔄_𝛼(𝑇−𝑠)𝐴^𝛽𝑔𝑠,𝑥^𝜀_𝑠(𝑠+−𝑔)𝑑𝑠_𝑇0(𝑇−𝑠)^𝛼−1𝔄_𝛼(𝑇−𝑠)𝑓𝑠,𝑥^𝜀_𝑠(𝑠≤𝐴−𝑓)𝑑𝑠_−𝛽𝐴^𝛽𝑔𝑇,𝑥^𝜀(𝑇)−𝐴^𝛽𝑔(𝑇,𝑥)+sup_{0≤𝑡≤𝑇}^𝑡₀(𝑡−𝑠)^𝛼−1𝐴^1−𝛽𝔄_𝛼(𝑡−𝑠)𝐴^𝛽𝑔𝑠,𝑥^𝜀_𝑠−𝑔(𝑠)𝑑𝑠+sup_{0≤𝑡≤𝑇}^𝑡₀(𝑡−𝑠)^𝛼−1𝔄_𝛼(𝑡−𝑠)𝑓𝑠,𝑥^𝜀_𝑠−𝑓(𝑠)𝑑𝑠⟶0,(66) where 𝑝=ℎ−𝔖_𝛼(𝑇)𝜙(0)(0)(𝑥)++𝑔0,𝜙+𝑔𝑇,_𝑇0(𝑇−𝑠)^𝛼−1𝐴^1−𝛽𝔄_𝛼()𝑔(𝑠)−𝑇−𝑠𝑑𝑠_𝑇0(𝑇−𝑠)^𝛼−1𝔄_𝛼(𝑇−𝑠)𝑓(𝑠)𝑑𝑠(67) as 𝜀→0⁺ because of compactness of an operator 𝑓(⋅)→∫^⋅₀(⋅−𝑠)^𝛼−1𝔄_𝛼(⋅−𝑠)𝑓(𝑠)𝑑𝑠∶𝐿₂([0,𝑇],𝑋)→𝐶([0,𝑇],𝑋). Then, by Theorem 17𝑥^𝜀(𝑇)−ℎ=𝑧_𝜀⟶0(68) as 𝜀→0⁺. This gives the approximate controllability. The theorem is proved.

Remark 19. Theorem 18 assumes that the operator 𝐴 generates a compact semigroup. If the compactness condition holds on the bounded operator that maps the control function or the generated 𝐶₀-semigroup, then the controllability operator 𝐿^𝑇₀ is also compact, and its inverse does not exist if the state space is infinite dimensional, and, consequently, the associated linear control system (53) is not exactly controllable. Therefore, the concept of complete controllability is too strong in infinite dimensional spaces, and the approximate controllability notion is more appropriate. Thus, Theorem 18 has no analogue for the concept of complete controllability.

4. Applications

In this section, we illustrate the obtained result. Let 𝑋=𝐿₂[0,𝜋], and let 𝐴 be defined as follows: 𝐴𝑧=−𝑧^′′(69) with domain 𝐷(𝐴)=𝑧∈𝑋∣𝑧,𝑑𝑧𝑑𝑑𝜉areabsolutelycontinuous,₂𝑧𝑑𝜉²∈𝑋and𝑧(0)=𝑧(𝜋)=0.(70) Recall that 𝐴 is the infinitesimal generator of a strongly continuous semigroup 𝑆(𝑡),𝑡>0, on 𝑋 which is analytic compact and self-adjoint, and the eigenvalues are −𝑛², 𝑛∈𝑁, with corresponding normalized eigenvectors 𝑒_𝑛(𝜉)∶=(2/𝜋)^1/2sin(𝑛𝜉) and 𝑆(𝑡)𝑒_𝑛=𝑒^−𝑛2𝑡𝑒_𝑛,𝑛=1,2,….(71) Moreover, the following hold.(a){𝑒_𝑛∶𝑛∈𝑁} is an orthonormal basis of 𝑋.(b)If 𝑧∈𝐷(𝐴), then 𝐴(𝑧)=∑^∞_𝑛=1𝑛²⟨𝑧,𝑒_𝑛⟩𝑒_𝑛.(c)For 𝑧∈ℍ, 𝐴^−1/2𝑧=∑^∞_𝑛=1(1/𝑛)⟨𝑧,𝑒_𝑛⟩𝑒_𝑛.(d)The operator 𝐴^1/2 is given as 𝐴^1/2𝑧=∑^∞_𝑛=1𝑛⟨𝑧,𝑒_𝑛⟩𝑒_𝑛 on the space 𝐷[𝐴^1/2]={𝑧∈𝑋∶∑^∞_𝑛=1𝑛⟨𝑧,𝑒_𝑛⟩𝑒_𝑛∈𝑋}.

For 1/2<𝛼<1, consider the neutral system 𝜕^𝛼𝜕𝑡^𝛼𝑥(𝑡,𝜉)+^𝜋₀𝑏()𝑥()=𝜕𝜃,𝜉𝑡,𝜃d𝜃₂𝜕𝜉²𝑥()(((),𝑥()()𝑥()(𝜉)𝑡,𝜉+𝑝𝑡,𝑥𝑡,𝜉))+𝐵𝑢𝑡,𝜉𝑡,0=𝑥𝑡,𝜋=0,𝑡≥0,0,𝜉=𝜑,0≤𝜉≤𝜋,(72) where 𝑝∶[0,𝑇]×𝑅→𝑅 is continuous functions. 𝐵 is a linear continuous mapping from 𝑈={𝑢=∑^∞_𝑛=2𝑢_𝑛𝑒_𝑛∣‖𝑢‖²_𝑈∶=∑^∞_𝑛=2𝑢²_𝑛<∞} to 𝑋 as follows: 𝐵𝑢=2𝑢₂+^∞_𝑛=2𝑢_𝑛𝑒_𝑛.(73)

To write the initial-boundary value problem (72) in the abstract form, we assume the following.(A1)The function 𝑏 is measurable and ^𝜋₀^𝜋₀𝑏²(𝜃,𝜉)d𝜃d𝜉<∞.(74)(A2)The function (𝜕/𝜕𝜉)𝑏(𝜃,𝜉) is measurable, 𝑏(𝜃,0)=𝑏(𝜃,𝜋)=0, and let 𝐿₁=^𝜋₀^𝜋₀𝜕𝑏()𝜕𝜉𝜃,𝜉²d𝜃d𝜉^1/2.(75)

Define 𝑓,𝑔∶[0,𝑇]×𝑋→𝑋 by 𝑔(𝑥)(𝜉)=^𝜋₀𝑏()𝑥(𝜃)𝑓(𝜉)((𝜉.𝜃,𝜉d𝜃,𝑡,𝑥)(=𝑝𝑡,𝑥))(76)

From (A1), it is clear that 𝑔 is bounded linear operator on 𝑋. Furthermore, 𝑔(𝑥)∈𝐷[𝐴^1/2], and ‖𝐴^1/2𝑔‖≤𝐿₁. In fact, from the definition of 𝑔 and (A2), it follows that 𝑔(𝑥),𝑒_𝑛=^𝜋₀^𝜋₀𝑏(𝜃,𝜉)𝑥(𝜃)d𝜃𝑒_𝑛(𝜉)=1d𝜉𝑛2𝜋^1/2𝜋₀𝜕𝑏()𝑥(𝜃)(=1𝜕𝜉𝜃,𝜉d𝜃,cos𝑛𝜉)𝑛2𝜋^1/2𝑔₁(𝑥),cos(𝑛𝜉),(77) where 𝑔₁(𝑥)=∫^𝜋₀(𝜕/𝜕𝜉)𝑏(𝜃,𝜉)𝑥(𝜃)d𝜃. From (A2), we know that 𝑔₁∶𝑋→𝑋 is a bounded linear operator with ‖𝑔₁‖≤𝐿₁. Hence, ‖𝐴^1/2𝑔(𝑥)‖=‖𝑔₁(𝑥)‖, which implies the assertion. Moreover, assume that 𝑓 and 𝑔 satisfy conditions of Theorem 18. Thus, the problem (72) can be written in the abstract form 𝑑^𝛼𝑑𝑡^𝛼𝑥(𝑡)((𝑡(𝑡)((𝑡(𝑡),𝑥(0)+𝑔𝑡,𝑥))=𝐴𝑥+𝑓𝑡,𝑥))+𝐵𝑢=𝑥₀,𝑡∈[0,𝑇].(78) Now, consider the associated linear system 𝑑^𝛼𝑑𝑡^𝛼𝑥(𝑡)(𝑡)(𝑡),𝑥(0)=𝐴𝑥+𝐵𝑢=𝑥₀,𝑡∈[0,𝑇].(79) Show that it is approximately controllable on [0,𝑇] for 1/2<𝛼<1. It is easy to see that if 𝑧=∑^∞_𝑛=1⟨𝑧,𝑒_𝑛⟩𝑒_𝑛, then 𝐵^∗𝑣=2𝑣₁+𝑣₂𝑒₂+^∞_𝑛=3𝑣_𝑛𝑒_𝑛,𝐵^∗𝔄^∗_𝛼(𝑇−𝑠)𝑧=𝐵^∗𝛼^∞₀𝜃Ψ_𝛼(𝜃)𝑆^∗(𝑇−𝑠)^𝛼𝜃𝑧𝑑𝜃=𝛼^∞₀𝜃Ψ_𝛼(𝜃)2𝑧,𝑒₁𝑒^{−(𝑇−𝑠)𝛼𝜃}𝑒₁+𝑧,𝑒₂𝑒^{−4(𝑇−𝑠)𝛼𝜃}𝑒₂+^∞_𝑛=3𝑒^{−𝑛2(𝑇−𝑠)𝛼𝜃}𝑧,𝑒_𝑛𝑒_𝑛=2𝑑𝜃𝑧,𝑒₁𝛼^∞₀𝜃Ψ_𝛼(𝜃)𝑒^{−(𝑇−𝑠)𝛼𝜃}+𝑑𝜃𝑧,𝑒₂𝛼^∞₀𝜃Ψ_𝛼(𝜃)𝑒^{−4(𝑇−𝑠)𝛼𝜃}𝑑𝜃𝑒₂+𝛼^∞_𝑛=3^∞₀𝜃Ψ_𝛼(𝜃)𝑒^{−𝑛2(𝑇−𝑠)𝛼𝜃}𝑑𝜃𝑧,𝑒_𝑛𝑒_𝑛(𝑇−𝑠)^𝛼−1𝐵^∗𝔄^∗_𝛼(𝑇−𝑠)𝑧²=(𝑇−𝑠)^2(𝛼−1)2𝛼^∞₀𝜃Ψ_𝛼(𝜃)𝑒^{−(𝑇−𝑠)𝛼𝜃}𝑑𝜃𝑧,𝑒₁+𝛼^∞₀𝜃Ψ_𝛼(𝜃)𝑒^{−4(𝑇−𝑠)𝛼𝜃}𝑑𝜃𝑧,𝑒₂²+(𝑇−𝑠)^{∞2(𝛼−1)}_𝑛=3𝛼^∞_𝑛=3^∞₀𝜃Ψ_𝛼(𝜃)𝑒^{−𝑛2(𝑇−𝑠)𝛼𝜃}𝑑𝜃²×𝑧,𝑒_𝑛²=0.(80) It follows that ⟨𝑧,𝑒₁⟩=⟨𝑧,𝑒₂⟩=⋯=⟨𝑧,𝑒_𝑛⟩=⋯=0, and consequently, 𝑧=0, which means that (79) is approximately controllable on [0,𝑇]. Therefore, from Theorem 18, the system (72) is approximately controllable [0,𝑇].

5. Conclusion

In this paper, abstract results concerning the approximate controllability of fractional semilinear evolution systems with infinite delay in a separable reflexive Banach space are obtained. Approximate controllability result for semilinear systems is obtained by means of the Krasnoselskii’s fixed-point theorem under the compactness assumption. It is also proven that the controllability of the semilinear system is implied by the approximate controllability of the associated linear system under some natural conditions. Upon making some appropriate assumptions, by employing the ideas and techniques as in this paper, one can establish the approximate controllability results for a wide class of fractional deterministic and stochastic evolution equations.

Acknowledgment

N. I. Mahmudov is thankful to the reviewers for making valuable suggestions leading to the better presentation of this paper.