Approximate Controllability of Fractional Sobolev-Type Evolution Equations in Banach Spaces

Abstract

We discuss the approximate controllability of semilinear fractional Sobolev-type differential system under the assumption that the corresponding linear system is approximately controllable. Using Schauder fixed point theorem, fractional calculus and methods of controllability theory, a new set of sufficient conditions for approximate controllability of fractional Sobolev-type differential equations, are formulated and proved. We show that our result has no analogue for the concept of complete controllability. The results of the paper are generalization and continuation of the recent results on this issue.

1. Introduction

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 Samko et al. [1] and the references therein) have attracted many scientists, and notable contributions have been made to both theory and applications of fractional differential equations.

Recently, the existence of mild solutions and stability and (approximate) controllability of (fractional) semilinear evolution system in Banach spaces have been reported by many researchers; see [2–36]. We refer the reader to El-Borai [3, 4], Balachandran and Park [5], Zhou and Jiao [6, 7] Hernández et al. [8], Wang and Zhou [9], Sakthivel et al. [12, 13], Debbouche and Baleanu [14], Wang et al. [15–21], Kumar and Sukavanam [22], Li and Yong [37], Dauer and Mahmudov [28], Mahmudov [27, 29], and the references therein. Complete controllability of evolution systems of Sobolev type in Banach spaces has been studied by Balachandran and Dauer [23], Ahmed [24], and Feckan et al. [2]. However, the approximate controllability of fractional evolution equations of Sobolev type has not been studied.

Motivated by the above-mentioned papers, we study the approximate controllability of a class of fractional evolution equations of Sobolev type: ^𝑐𝐷^𝛼_𝑡((𝑡(𝑡)(𝑡)((𝑡[],𝑥(0)𝐸𝑥))=𝐴𝑥+𝐵𝑢+𝑓𝑡,𝑥)),𝑡∈0,𝑏=𝑥₀, (1) where 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 and 𝐸∶𝐷(𝐸)⊂𝑋→𝑋 are linear operators from a Banach space 𝑋 to 𝑋. The control function 𝑢 takes values in a Hilbert space 𝑈 and 𝑢∈𝐿²([0,𝑏],𝑈). 𝐵∶𝑈→𝑋 is a linear bounded operator. The function 𝑓∈𝐶([0,𝑏]×𝑋,𝑋) will be specified in the sequel. The fractional derivative ^𝑐𝐷^𝛼_𝑡, 0<𝛼<1, is understood in the Caputo sense.

Our aim in this paper is to provide a sufficient condition for the approximate controllability for a class of fractional evolution equations of Sobolev type. It is assumed that 𝐸⁻¹ is compact, and, consequently, the associated linear control system (35) is not exactly controllable. Therefore, our approximate controllability results have no analogue for the concept of complete controllability. In Section 5, we give an example of the system which is not completely controllable, but approximately controllable.

2. Preliminaries

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 a renorming theorem (see, e.g., [37, 38]). In particular, this implies that the duality mapping 𝐽 of 𝑋 into 𝑋^∗ given by the relations ‖𝐽(𝑧)‖_∗=‖𝑧‖,⟨𝐽(𝑧),𝑧⟩=‖𝑧‖²,∀𝑧∈𝑋 (2) is bijective, homogeneous, 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.

The operators 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 and 𝐸∶𝐷(𝐸)⊂𝑋→𝑋 satisfy the following hypotheses:(S1)𝐴 and 𝐸 are linear operators, and 𝐴 is closed;(S2)𝐷(𝐸)⊂𝐷(𝐴) and 𝐸 is bijective;(S3)𝐸⁻¹∶𝑋→𝐷(𝐸) is compact.

The hypotheses (S1)–(S3) and the closed graph theorem imply the boundedness of the linear operator 𝐴𝐸⁻¹∶𝑋→𝑋. Consequently, −𝐴𝐸⁻¹ generates a semigroup {𝑆(𝑡);𝑡≥0} in 𝑋. Assume that max_{0≤𝑡≤𝑏}‖𝑆(𝑡)‖=∶𝑀.

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

Definition 1. 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, (3) provided the right-hand side is pointwise defined on [0,∞), where Γ is the gamma function.

Definition 2. The Caputo derivative of order 𝛼 for a function 𝑓 can be written as ^𝑐𝐷^𝛼𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡₀𝑓^(𝑛)(𝑠)(𝑡−𝑠)^{1+𝛼−𝑛}𝑑𝑠=𝐼^𝑛−𝛼𝑓^(𝑛)(𝑡),𝑡>0,𝑛−1≤𝛼<𝑛. (4)
If 𝑓 is an abstract function with values in 𝑋, then integrals which appear in the above definitions are taken in Bochner’s sense.
For 𝑥∈𝑋 and 0<𝛼<1, we define two families {𝒮_𝐸(𝑡)∶𝑡≥0} and {𝒯_𝐸(𝑡)∶𝑡≥0} of operators by 𝒮_𝛼(𝑡)=^∞₀Ψ_𝛼(𝜃)𝑆𝑡^𝛼𝜃𝑑𝜃,𝒯_𝛼(𝑡)=𝛼^∞₀𝜃Ψ_𝛼(𝜃)𝑆𝑡^𝛼𝜃𝒮𝑑𝜃,_𝐸(𝑡)=𝐸⁻¹𝒮_𝛼(𝑡),𝒯_𝐸(𝑡)=𝐸⁻¹𝒯_𝛼(𝑡), (5) where Ψ_𝛼(𝜃)=1𝜋𝛼^∞_𝑛=1(−1)^𝑛−1Γ(𝑛𝛼+1)()(),𝑛!sin𝑛𝜋𝛼,𝜃∈0,∞ (6) 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,∞ (7)

Lemma 3 (see [2]). The operators 𝒮_𝐸 and 𝒯_𝐸 have the following properties. (i)For any fixed 𝑡≥0, 𝑆_𝐸(𝑡) and 𝑇_𝐸(𝑡) are linear and bounded operators, and 𝒮_𝐸(𝑡)𝑥≤𝑀𝐸⁻¹𝑥‖,𝒯_𝐸‖(𝑡)𝑥≤𝑀𝐸⁻¹Γ(𝛼)‖𝑥‖. (8)(ii){𝒮_𝐸(𝑡)∶𝑡≥0} and {𝒯_𝐸(𝑡)∶𝑡≥0} are compact.

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

Definition 4. A solution 𝑥(⋅;𝑢)∈𝐶([0,𝑏],𝑋) is said to be a mild solution of (1) if for any 𝑢∈𝐿²([0,𝑏],𝑈), the integral equation 𝑥(𝑡)=𝒮_𝐸(𝑡)𝐸𝑥₀+^𝑡₀(𝑡−𝑠)^𝛼−1𝒯_𝐸(𝑡−𝑠)𝐵𝑢(𝑠)+𝑓(𝑠,𝑥(𝑠))𝑑𝑠 (9) is satisfied.

Let 𝑥(𝑏;𝑢) be the state value of (9) at terminal time 𝑏 corresponding to the control 𝑢. Introduce the set ℜ(𝑏)={𝑥(𝑏;𝑢)∶𝑢∈𝐿₂([0,𝑏],𝑈)}, which is called the reachable set of system (9) at terminal time 𝑏; its closure in 𝑋 is denoted by ℜ(𝑏).

Definition 5. System (1) is said to be approximately controllable on [0,𝑏] if ℜ(𝑏)=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 𝑏.

To investigate the approximate controllability of system (9), we assume the following conditions.(H4)The function 𝑓∶[0,𝑏]×𝑋→𝑋 satisfies the following:(a)𝑓(𝑡,⋅)∶𝑋→𝑋 is continuous for each 𝑡∈[0,𝑏] and for each 𝑥∈𝑋,  𝑓(⋅,𝑥)∶[0,𝑏]→𝑋 is strongly measurable;(b)there is a positive integrable function 𝑛∈𝐿¹([0,𝑏],[0,+∞)) and a continuous nondecreasing function Λ_𝑓∶[0,∞)→(0,∞) such that for every (𝑡,𝑥)∈[0,𝑏]×𝑋, we have 𝑓(𝑡,𝑥)≤𝑛(𝑡)Λ_𝑓(‖𝑥‖),liminf_𝑟→∞Λ_𝑓(𝑟)𝑟=𝜎_𝑓<∞. (10)(H5) The following relationship holds: 11+𝜀𝑀²_𝐵𝑀²_𝒯𝑏^2𝛼−1𝑀𝐸2𝛼−1Γ(𝛼)⁻¹𝑏^𝛼𝛼×𝜎_𝑓sup_{[]𝑠∈0,𝑏}𝑛(𝑠)<1, (11) here 𝑀_𝐵∶=‖𝐵‖,𝑀_𝒯∶=‖𝒯_𝐸‖.(H6) For every ℎ∈𝑋,𝑧_𝛼(ℎ)=𝜀(𝜀𝐼+Γ^𝑏₀𝐽)⁻¹(ℎ) converges to zero as 𝜀→0⁺ in strong topology, where Γ^𝑏₀∶=^𝑏₀(𝑏−𝑠)^2(𝛼−1)𝒯_𝐸(𝑏−𝑠)𝐵𝐵^∗𝒯^∗_𝐸(𝑏−𝑠)𝑑𝑠, (12) and 𝑧_𝜀(ℎ) is a solution of 𝜀𝑧_𝜀+Γ^𝑏₀𝐽𝑧_𝜀=𝜀ℎ. (13)

3. 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 system (1) is approximately controllable if for all 𝜀>0, there exists a continuous function 𝑥∈𝐶([0,𝑏],𝑋) such that 𝑢_𝜀(𝑡,𝑥)=(𝑏−𝑡)^𝛼−1𝐵^∗𝒯^∗_𝐸(𝑏−𝑡)𝐽𝜀𝐼+Γ^𝑏₀𝐽⁻¹𝑝(𝑥),𝑥(𝑡)0≤𝑡<𝑏,=𝒮_𝐸(𝑡)𝐸𝑥₀+^𝑡₀(𝑡−𝑠)^𝛼−1𝒯_𝐸(𝑡−𝑠)𝐵𝑢(𝑠)+𝑓(𝑠,𝑥(𝑠))𝑑𝑠, (14) where 𝑝(𝑥)=ℎ−𝒮_𝐸(𝑏)𝐸𝑥₀−^𝑏₀(𝑏−𝑠)^𝛼−1𝒯_𝐸(𝑏−𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠. (15) Having noticed this fact, our goal, in this section, is to find conditions for the solvability of (14). It will be shown that the control in (14) drives system (1) from 𝑥₀ to ℎ−𝜀𝐽𝜀𝐼+Γ^𝑏₀𝐽⁻¹𝑝(𝑥), (16) provided that system (14) has a solution.

Theorem 6. Assume that assumptions (S1)–(S3), (H4), (H5) hold and 1/2<𝛼≤1. Then there exists a solution to (14).

Proof. The proof of Theorem 6 follows from Lemmas 7–9, infinite dimensional analogue of Arzela-Ascoli theorem, and the Schauder fixed point theorem.

For all 𝜀>0, consider the operator Φ_𝜀∶𝐶([0,𝑏],𝑋)→𝐶([0,𝑏],𝑋) defined as follows: Φ_𝜀𝑥(𝑡)∶=𝒮_𝐸(𝑡)𝐸𝑥₀+^𝑡₀(𝑡−𝑠)^𝛼−1𝒯_𝐸()×𝑡−𝑠𝐵𝑢_𝜀(𝑠,𝑥)+𝑓(𝑠,𝑥(𝑠))𝑑𝑠, (17) where 𝑢_𝜀(𝑡,𝑥)=(𝑏−𝑡)^𝛼−1𝐵^∗𝒯^∗_𝐸(𝑏−𝑡)𝐽𝜀𝐼+Γ^𝑏₀𝐽⁻¹𝑝(𝑥),𝑝(𝑥)=ℎ−𝒮_𝐸(𝑏)𝐸𝑥₀−^𝑏₀(𝑏−𝑠)^𝛼−1𝒯_𝐸(𝑏−𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠. (18)

It will be shown that for all 𝜀>0, the operator Φ_𝜀∶𝐶([0,𝑏],𝑋)→𝐶([0,𝑏],𝑋) has a fixed point. To prove this we will employ the Schauder fixed point theorem.

Lemma 7. Under assumptions (S1)–(S3), (H4), (H5), for any 𝜀>0 there exists a positive number 𝑟∶=𝑟(𝜀) such that Φ_𝜀(𝐵_𝑟)⊂𝐵_𝑟.

Proof. Let 𝜀>0 be fixed. If it is not true, then for each 𝑟>0, there exists a function 𝑧_𝑟∈𝐵_𝑟, but Φ_𝜀(𝑧_𝑟)∉𝐵_𝑟. So for some 𝑡=𝑡(𝑟)∈[0,𝑏], one can show that 𝑟≤Φ_𝜀𝑧_𝑟(𝑡)≤𝒮_𝐸(𝑡)𝐸𝑥₀+^𝑡₀(𝑡−𝑠)^𝛼−1𝒯_𝐸()𝑓((𝑠+𝑡−𝑠𝑠,𝑥))𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝒯_𝐸(𝑡−𝑠)𝐵𝑢_𝜀(𝑠,𝑥)𝑑𝑠=∶𝐼₁+𝐼₂+𝐼₃. (19) Let us estimate 𝐼_𝑖,𝑖=1,2,3. By the assumption (H4), we have 𝐼₁≤𝒮_𝐸(𝑡)𝐸𝑥₀≤𝑀𝐸⁻¹𝐸𝑥₀,𝐼₂≤^𝑡₀(𝑡−𝑠)^𝛼−1𝒯_𝐸()𝑓((𝑠≤𝑀𝐸𝑡−𝑠)(𝑡−𝑠𝑠,𝑥))𝑑𝑠₋₁Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓((𝑠≤𝑀𝐸𝑠,𝑥))𝑑𝑠₋₁Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑛(𝑠)Λ_𝑓𝑥(𝑠≤𝑀𝐸(‖)‖)𝑑𝑠₋₁Γ(𝛼)𝑏^𝛼𝛼Λ_𝑓(𝑟)sup_𝑠∈𝐽𝑛(𝑠). (20)(21) Combining the estimates (19)–(21) yields 𝐼₁+𝐼₂<𝑀𝐸⁻¹𝐸𝑥₀+𝑀𝐸⁻¹Γ(𝛼)𝑏^𝛼𝛼Λ_𝑓(𝑟)sup_{[]𝑠∈0,𝑏}𝑛(𝑠)∶=Δ. (22) On the other hand, 𝐼₃≤^𝑡₀(𝑡−𝑠)^𝛼−1𝒯_𝐸(𝑡−𝑠)𝐵𝑢_𝜀(=𝑠,𝑥)𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1(𝑏−𝑠)^𝛼−1𝒯_𝐸(𝑡−𝑠)×𝐵𝐵^∗𝒯^∗_𝐸(𝑏−𝑡)𝐽𝜀𝐼+Γ^𝑏₀𝐽⁻¹𝑝(𝑥)≤𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1(𝑏−𝑠)^𝛼−1𝒯_𝐸(𝑡−𝑠)𝐵𝐵^∗𝒯^∗_𝐸(×𝐽𝑏−𝑡)𝑑𝑠𝜀𝐼+Γ^𝑏₀𝐽⁻¹𝑝(𝑥)≤𝑀²_𝐵𝑀²_𝒯𝑏^2𝛼−1𝐽2𝛼−1𝜀𝐼+Γ^𝑏₀𝐽⁻¹𝑝(𝑥)=𝑀²_𝐵𝑀²_𝒯𝑏^2𝛼−12𝛼−1𝜀𝐼+Γ^𝑏₀𝐽⁻¹𝑝(𝑥≤1)𝜀𝑀²_𝐵𝑀²_𝒯𝑏^2𝛼−1𝑝(𝑥≤12𝛼−1)𝜀𝑀²_𝐵𝑀²_𝒯𝑏^2𝛼−12𝛼−1Δ. (23) Thus, 𝑟≤Φ_𝜀𝑧_𝑟(𝑡1)≤Δ+𝜀𝑀²_𝐵𝑀²_𝒯𝑏^2𝛼−1Δ=12𝛼−11+𝜀𝑀²_𝐵𝑀²_𝒯𝑏^2𝛼−12𝛼−1Δ. (24) Dividing both sides by 𝑟 and taking 𝑟→∞, we obtain that 11+𝜀𝑀²_𝐵𝑀²_𝒯𝑏^2𝛼−1𝑀𝐸2𝛼−1Γ(𝛼)⁻¹𝑏^𝛼𝛼𝜎_𝑓sup_{[]𝑠∈0,𝑏}𝑛(𝑠)≥1, (25) which is a contradiction by assumption (H5). Thus, Φ_𝜀(𝐵_𝑟)⊂𝐵_𝑟 for some 𝑟>0.

Lemma 8. Let assumptions (S1)–(S3), (H4), (H5) hold. Then the set {Φ_𝜀𝑧∶𝑧∈𝐵_𝑟} is an equicontinuous family of functions on [0,𝑏].

Proof. Let 0<𝜂<𝑡<𝑏 and 𝛿>0 such that 𝒯_𝐸𝑠₁−𝒯_𝐸𝑠₂<𝜂 (26) for every 𝑠₁,𝑠₂∈[0,𝑏] with |𝑠₁−𝑠₂|<𝛿. For 𝑧∈𝐵_𝑟, 0<|ℎ|<𝛿,𝑡+ℎ∈[0,𝑏], we have Φ_𝜀𝑧(𝑡+ℎ)−Φ_𝜀𝑧(𝑡≤_𝑡0)(𝑡+ℎ−𝑠)^𝛼−1−(𝑡−𝑠)^𝛼−1×𝒯_𝐸(𝑡+ℎ−𝑠)𝐵𝑢_𝜀()((𝑠+𝑠,𝑧+𝑓𝑠,𝑧))𝑑𝑠_𝑡𝑡+ℎ(𝑡+ℎ−𝑠)^𝛼−1𝒯_𝐸()×𝑡+ℎ−𝑠𝐵𝑢_𝜀()((𝑠+𝑠,𝑧+𝑓𝑠,𝑧))𝑑𝑠_𝑡0(𝑡−𝑠)^𝛼−1𝒯_𝐸(𝑡+ℎ−𝑠)−𝒯_𝐸()×𝑡−𝑠𝐵𝑢_𝜀(𝑠,𝑧)+𝑓(𝑠,𝑧(𝑠))𝑑𝑠. (27) Applying Lemma 3 and the Holder inequality, we obtain Φ_𝜀𝑧(𝑡+ℎ)−Φ_𝜀𝑧(𝑡≤𝑀𝐸₋₁)Γ(𝛼)Λ_𝑓(𝑟)×^𝑡₀(𝑡+ℎ−𝑠)^𝛼−1−(𝑡−𝑠)^𝛼−1𝑛(𝑠)+𝑀𝐸𝑑𝑠₋₁Γ(𝛼)1𝜀𝑀_𝐵𝑀_𝒯Δ×^𝑡₀(𝑡+ℎ−𝑠)^𝛼−1−(𝑡−𝑠)^𝛼−1(𝑏−𝑠)^𝛼−1+𝑀𝐸𝑑𝑠₋₁Γ(𝛼)Λ_𝑓(𝑟)_𝑡^𝑡+ℎ(𝑡+ℎ−𝑠)^𝛼−1𝑛(𝑠)+𝑀𝐸𝑑𝑠₋₁Γ(𝛼)1𝜀𝑀_𝐵𝑀_𝒯Δ×_𝑡^𝑡+ℎ(𝑡+ℎ−𝑠)^𝛼−1(𝑏−𝑠)^𝛼−1+𝑑𝑠𝜂𝑇^𝛼𝛼Λ_𝑓(𝑟)^𝑡₀(𝑡−𝑠)^𝛼−1𝑛(𝑠)𝑑𝑠+𝜂𝑇^𝛼𝛼1𝜀𝑀_𝐵𝑀_𝒯Δ×^𝑡₀(𝑡−𝑠)^𝛼−1(𝑏−𝑠)^𝛼−1𝑑𝑠. (28) Therefore, for 𝜀 sufficiently small, the right-hand side of (28) tends to zero as ℎ→0. On the other hand, the compactness of 𝒯_𝐸(𝑡),𝑡>0, implies the continuity in the uniform operator topology. Thus, the set {Φ_𝜀𝑧∶𝑧∈𝐵_𝑟} is equicontinuous.

Lemma 9. Let assumptions (S1)–(S3), (H4), (H5) hold. Then Φ_𝜀 maps 𝐵_𝑟 onto a precompact set in 𝐵_𝑟.

Proof. Let 0<𝑡≤𝑏 be fixed and let 𝜆 be a real number satisfying 0<𝜆<𝑡. For 𝛿>0, define an operator Φ_𝜀^𝜆,𝛿 on 𝐵_𝑟 by Φ_𝜀^𝜆,𝛿𝑧(𝑡)=𝛼₀^𝑡−𝜆𝐸^−1∞_𝛿𝜃(𝑡−𝑠)^𝛼−1Ψ_𝛼(𝜃)𝑆(𝑡−𝑠)^𝛼𝜃×𝐵𝑢_𝜀(𝑠,𝑧)+𝑓(𝑠,𝑧(𝑠))𝑑𝑠=𝛼𝐸⁻¹𝑆𝜆^𝛼𝛿×₀^{𝑡−𝜆∞}_𝛿𝜃(𝑡−𝑠)^𝛼−1Ψ_𝛼(𝜃)𝑆(𝑡−𝑠)^𝛼𝜃−𝜆^𝛼𝛿×𝐵𝑢_𝜀(𝑠,𝑧)+𝑓(𝑠,𝑧(𝑠))𝑑𝑠. (29) Since 𝐸⁻¹ is a compact operator, the set {(Φ_𝜀^𝜆,𝛿𝑧)(𝑡)∶𝑧∈𝐵_𝑟} is precompact in 𝑋 for every 0<𝜆<𝑡, 𝛿>0. Moreover, for each 𝑧∈𝐵_𝑟, we have Φ_𝜀𝑧(𝑡)−Φ_𝜀^𝜆,𝛿𝑧(𝑡)≤𝛼^𝑡₀^𝛿₀𝜃(𝑡−𝑠)^𝛼−1Ψ_𝛼(𝜃)𝑆(𝑡−𝑠)^𝛼𝜃×𝐵𝑢_𝜀(𝑠,𝑧)+𝑓(𝑠,𝑧(𝑠))𝑑𝜃𝑑𝑠+𝛼^𝑡_𝑡−𝜆^∞_𝛿𝜃(𝑡−𝑠)^𝛼−1Ψ_𝛼(𝜃)𝑆(𝑡−𝑠)^𝛼𝜃×𝐵𝑢_𝜀(𝑠,𝑧)+𝑓(𝑠,𝑧(𝑠))𝑑𝜃𝑑𝑠=∶𝐽₁+𝐽₂. (30) A similar argument, as before, is as follows: 𝐽₁≤𝛼𝑀^𝑡₀(𝑡−𝑠)^𝛼−1𝐵𝑢_𝜀(+𝑓((𝑠×𝑠,𝑧)𝑠,𝑧))𝑑𝑠_𝛿0𝜃Ψ_𝛼(𝜃)1𝑑𝜃≤𝛼𝑀𝜀𝑀_𝐵𝑀_𝒯Δ^𝑡₀(𝑡−𝑠)^𝛼−1(𝑏−𝑠)^𝛼−1𝑑𝑠+Λ_𝑓(𝑟)^𝑡₀(𝑡−𝑠)^𝛼−1𝑛(𝑠)𝑑𝑠^𝛿₀𝜃Ψ_𝛼(𝜃)𝑑𝜃,𝐽₂≤𝛼𝑀^𝑡_𝑡−𝜆(𝑡−𝑠)^𝛼−1𝐵𝑢_𝜀(+𝑓((𝑠×𝑠,𝑧)𝑠,𝑧))𝑑𝑠_∞𝛿𝜃𝜂_𝛼(𝜃)≤𝑑𝜃𝛼𝑀Γ()11+𝛼𝜀𝑀_𝐵𝑀_𝒯Δ^𝑡_𝑡−𝜆(𝑡−𝑠)^𝛼−1(𝑏−𝑠)^𝛼−1𝑑𝑠+Λ_𝑓(𝑟)^𝑡_𝑡−𝜆(𝑡−𝑠)^𝛼−1𝑛(𝑠)𝑑𝑠, (31)(32) where we have used the equality ^∞₀𝜃^𝛽_𝛼Ψ_𝛼(𝜃)Γ𝑑𝜃=1+𝛽Γ.1+𝛼𝛽 (33) From (30)–(32), one can see that for each 𝑧∈𝐵_𝑟, Φ_𝜀𝑧(𝑡)−Φ_𝜀^𝜆,𝛿𝑧(𝑡)⟶0as𝜆⟶0⁺,𝛿⟶0⁺. (34) Therefore, there are relatively compact sets arbitrary close to the set {(Φ_𝜀𝑧)(𝑡)∶𝑧∈𝐵_𝑟}; hence, the set {(Φ_𝜀𝑧)(𝑡)∶𝑧∈𝐵_𝑟} is also precompact in 𝑋.

4. Main Results

Consider the following linear fractional differential system: 𝐷^𝛼_𝑡𝐸𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡),𝑡∈(0,𝑏],𝑥(0)=𝑥₀. (35)(36) It is convenient at this point to introduce the controllability and resolvent operators associated with (35) as 𝐿^𝑏₀=^𝑏₀(𝑏−𝑠)^𝛼−1𝒯_𝐸(𝑏−𝑠)𝐵𝑢(𝑠)𝑑𝑠∶𝐿²])Γ([0,𝑏,𝑈⟶𝑋,_𝑏0=𝐿^𝑏₀𝐿^𝑏₀^∗=^𝑏₀(𝑏−𝑠)^2(𝛼−1)𝒯_𝐸(𝑏−𝑠)×𝐵𝐵^∗𝒯^∗_𝐸(𝑏−𝑠)𝑑𝑠∶𝑋⟶𝑋, (37) respectively, where 𝐵^∗ denotes the adjoint of 𝐵 and 𝒯^∗_𝛼(𝑡) is the adjoint of 𝒯_𝛼(𝑡). It is straightforward that the operator 𝐿^𝑏₀ is a linear bounded operator for 1/2<𝛼≤1.

Theorem 10 (see [27]). The following three conditions are equivalent.(i)Γ is positive; that is, ⟨𝑧^∗,Γ𝑧^∗⟩>0 for all nonzero 𝑧^∗∈𝑋^∗.(ii)For all ℎ∈𝑋,𝐽(𝑧_𝜀(ℎ)) converges to zero as 𝜀→0⁺ in the weak topology, where 𝑧_𝜀(ℎ)=𝜀(𝜀𝐼+Γ𝐽)⁻¹(ℎ) is a solution of (13). (iii)For all ℎ∈𝑋,𝑧_𝜀(ℎ)=𝜀(𝜀𝐼+Γ𝐽)⁻¹(ℎ) converges to zero as 𝜀→0⁺ in the strong topology.

Remark 11. It is known that Theorem 10 (i) holds if and only if Im𝐿^𝑏₀=𝑋. In other words, Theorem 10 (i) holds if and only if the corresponding linear system is approximately controllable on [0,𝑏]. Consequently, assumption (H6) is equivalent to the approximate controllability of the linear system (35).

Theorem 12 (see [27]). Let 𝑝∶𝑋→𝑋  be a nonlinear operator. Assume that 𝑧_𝜀 is a solution of the following equation: 𝜀𝑧_𝜀+Γ^𝑇₀𝐽𝑧_𝜀=𝜀𝑝𝑧_𝜀,𝑝𝑧_𝜀−𝑞⟶0as𝜀⟶0⁺,𝑞∈𝑋. (38) 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 13. Let 1/2<𝛼≤1. Suppose that conditions (S1)–(S3), (H4)–(H5) are satisfied. Besides, assume additionally that there exists 𝑁∈𝐿^∞([0,𝑏],[0,+∞)) such that sup_𝑥∈𝑋𝑓(𝑡,𝑥)≤𝑁(𝑡),fora.e.𝑡∈[0,𝑏]. (39)
Then system (1) is approximately controllable on [0,𝑏].

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,𝑏] under the control 𝑢_𝜀𝑡,𝑥^𝜀=(𝑏−𝑡)^𝛼−1𝐵^∗𝒯^∗_𝐸(𝑏−𝑡)𝐽𝜀𝐼+Γ^𝑏₀𝐽⁻¹𝑝𝑥^𝜀,𝑝𝑥^𝜀=ℎ−𝒮_𝐸(𝑏)𝑥₀−^𝑏₀(𝑏−𝑠)^𝛼−1𝒯_𝐸(𝑏−𝑠)𝑓𝑠,𝑥^𝜀(𝑠)𝑑𝑠 (40) and satisfies the following equality: 𝑥^𝜀(𝑏)=𝒮_𝐸(𝑏)𝑥₀+^𝑏₀(𝑏−𝑠)^𝛼−1𝒯_𝐸()×𝑏−𝑠𝐵𝑢_𝜀𝑠,𝑥^𝜀+𝑓𝑠,𝑥^𝜀(𝑠)𝑑𝑠=𝒮_𝐸(𝑏)𝑥₀+−𝜀𝐼+𝜀𝐼+Γ^𝑏₀𝐽𝜀𝐼+Γ^𝑏₀𝐽⁻¹𝑝𝑥^𝜀+^𝑏₀(𝑏−𝑠)^𝛼−1𝒯_𝐸(𝑏−𝑠)𝑓𝑠,𝑥^𝜀(𝑠)𝑑𝑠=ℎ−𝜀𝜀𝐼+Γ^𝑏₀𝐽⁻¹𝑝𝑥^𝜀. (41) In other words, 𝑧_𝜀=ℎ−𝑥^𝜀(𝑏) is a solution of 𝜀𝐼+Γ^𝑏₀𝐽𝑧_𝜀=𝜀𝑝𝑥^𝜀. (42) By our assumption, ^𝑏₀𝑓𝑠,𝑥^𝜀(𝑠)²𝑑𝑠≤^𝑇₀𝑁²(𝑠)𝑑𝑠. (43) Consequently, the sequence {𝑓(⋅,𝑥^𝜀(⋅))} is bounded. Then there is a subsequence still denoted by {𝑓(⋅,𝑥^𝜀(⋅))} and weakly converges to, say, 𝑓(⋅) in 𝐿²([0,𝑏],𝑋). Then 𝑝𝑥^𝜀=−𝑞_𝑏0(𝑏−𝑠)^𝛼−1𝒯_𝐸(𝑏−𝑠)𝑓𝑠,𝑥^𝜀(𝑠)−𝑓(𝑠)𝑑𝑠≤sup_{0≤𝑡≤𝑏}^𝑡₀(𝑡−𝑠)^𝛼−1𝒯_𝐸()×𝑓𝑡−𝑠𝑠,𝑥^𝜀(𝑠)−𝑓(𝑠)𝑑𝑠⟶0, (44) where 𝑞=ℎ−𝒮_𝐸𝐸𝑥₀−^𝑏₀(𝑏−𝑠)^𝛼−1𝒯_𝐸(𝑏−𝑠)𝑓(𝑠)𝑑𝑠 (45) as 𝜀→0⁺ because of the compactness of an operator 𝑓(⋅)  →  ∫^⋅₀(⋅−𝑠)^𝛼−1𝒯_𝐸(⋅−𝑠)𝑓(𝑠)𝑑𝑠∶𝐿₂([0,𝑏],𝑋)  →  𝐶([0,𝑏],𝑋). Then by Theorem 12 for any ℎ∈𝑋, 𝑥^𝜀(𝑏)−ℎ=𝑧_𝜀⟶0 (46) as 𝜀→0⁺. This gives the approximate controllability. The theorem is proved.

Remark 14. Theorem 13 assumes that the operator 𝐸⁻¹ is compact and, consequently, the associated linear control system (35) is not exactly controllable. Therefore, Theorem 13 has no analogue for the concept of exact controllability.

Remark 15. In order to describe various real-world problems in physical and engineering sciences subject to abrupt changes at certain instants during the evolution process, fractional impulsive differential equations have been used for the system model. Our result can be extended to study the complete and approximate controllability of nonlinear fractional impulsive differential equations of Sobolev type; see [35, 36].

5. Applications

Example 16. Let 𝑋=𝑈=𝐿²[0,𝜋]. Consider the following fractional partial differential equation with control: ^𝑐𝐷_𝑡^3/4𝑥(𝑡,𝜃)−𝑥_𝜃𝜃(𝑡,𝜃)=𝑥_𝜃𝜃()(((),𝑥()()𝑥()(𝜃)𝑡,𝜃+𝑔𝑡,𝑥𝑡,𝜃))+𝑢𝑡,𝜃𝑡,0=𝑥𝑡,𝜋=0,0,𝜃=𝜙,∶0≤𝑡≤𝑏,∶0≤𝜃≤𝜋. (47)
Define 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 by 𝐴∶=𝑥_𝜃𝜃 and 𝐸∶𝐷(𝐸)⊂𝑋→𝑋 by 𝐸𝑥∶=𝑥−𝑥_𝜃𝜃, where each domain, 𝐷(𝐴)and𝐷(𝐸), is given by 𝑥∈𝑋∶𝑥,𝑥_𝜃areabsolutelycontinuous,𝑥_𝜃𝜃∈𝑋,𝑥(𝑡,0)=𝑥(𝑡,𝜋)=0. (48)𝐴 and 𝐸 can be written as follows: 𝐴𝑥∶=^∞_𝑛=1−𝑛²𝑥,𝑒_𝑛𝑒_𝑛,𝑥∈𝐷(𝐴),𝐸𝑥=^∞_𝑛=11+𝑛²𝑥,𝑒_𝑛𝑒_𝑛,𝑥∈𝐷(𝐸), (49) respectively, where 𝑒_𝑛(𝜃)∶=2/𝜋sin𝑛𝜃, 𝑛=1,2,…, is the orthonormal set of eigenvalues of 𝐴. Moreover, for any 𝑥∈𝑋, we have 𝐸⁻¹𝑥=^∞_𝑛=111+𝑛²𝑥,𝑒_𝑛𝑒_𝑛,𝐴𝐸⁻¹𝑥=^∞_𝑛=1−𝑛²1+𝑛²𝑥,𝑒_𝑛𝑒_𝑛,𝑆(𝑡)𝑥=^∞_𝑛=1exp−𝑛²1+𝑛²𝑡𝑥,𝑒_𝑛𝑒_𝑛,𝒯_𝐸(𝑡)=34^∞₀𝐸⁻¹𝜃𝜉_3/4(𝜃)𝑆𝑡^3/4𝜃𝒯𝑑𝜃,_𝐸(𝑡)3𝑥=4^∞_𝑛=111+𝑛²×^∞₀𝜃𝜉_3/4(𝜃)exp−𝑛²1+𝑛²𝑡^3/4𝜃×𝑑𝜃𝑥,𝑒_𝑛𝑒_𝑛=−^∞_𝑛=11𝑛²𝑡^1/4×^∞₀𝜉_3/4(𝜃)𝑑𝑑𝑡exp−𝑛²1+𝑛²𝑡^3/4𝜃×𝑑𝜃𝑥,𝑒_𝑛𝑒_𝑛,Γ^𝑏₀=^𝑏₀(𝑏−𝑠)^2(𝛼−1)𝒯()𝒯()=𝑏−𝑠𝑏−𝑠𝑑𝑠_𝑏0𝑠^2(𝛼−1)𝒯(𝑠)𝒯(𝑠)=𝑑𝑠_𝑏0𝑠^2(𝛼−1)34^∞₀𝐸⁻¹𝜃𝜉_3/4(𝜃)𝑆𝑠^3/4𝜃×3𝑑𝜃4^∞₀𝐸⁻¹𝜃𝜉_3/4(𝜃)𝑆𝑠^3/4𝜃𝑑𝜃𝑑𝑠. (50) It is clear that 𝐸⁻¹ is compact. The linear system corresponding to (47) is completely controllable if and only if there exists 𝛾>0 such that ⟨Γ^𝑏₀𝑥,𝑥⟩≥𝛾‖𝑥‖² for all 𝑥∈𝑋. Assume Γ^𝑏₀=9𝑥,𝑥16^∞_𝑛=111+𝑛²²×^𝑏₀𝑠^{2(𝛼−1)∞}₀𝜃𝜉_3/4(𝜃)exp−𝑛²1+𝑛²𝑠^3/4𝜃𝑑𝜃²×𝑑𝑠𝑥,𝑒_𝑛²⩾𝛾^∞_𝑛=1𝑥,𝑒_𝑛². (51) Then 91161+𝑛²²×^𝑏₀𝑠^{2(𝛼−1)∞}₀𝜃𝜉_3/4(𝜃)exp−𝑛²1+𝑛²𝑠^3/4𝜃𝑑𝜃²9𝑑𝑠⩾𝛾,𝛾≤1161+𝑛²²×^𝑏₀𝑠^−1/2∞₀𝜃𝜉_3/4(𝜃)exp−𝑛²1+𝑛²𝑠^3/4𝜃𝑑𝜃²≤9𝑑𝑠1161+𝑛²²×^𝑏₀𝑠^−1/2∞₀𝜃𝜉_3/4(𝜃)𝑑𝜃²𝑑𝑠⟶0,as𝑛⟶∞,⟹𝛾=0(contradiction), (52) and no such 𝛾>0 exists which satisfies (51), and hence the linear system corresponding to (47) is never completely controllable. We show that the associated linear system is approximately controllable on [0,𝑏]. We need to show that (𝑏−𝑠)^𝛼−1𝐵^∗𝒯^∗_𝐸(𝑏−𝑠)𝑥=0,0≤𝑠<𝑏⇒𝑥=0. Indeed, (𝑏−𝑠)^𝛼−1𝐵^∗𝒯^∗_𝐸()𝑥=()𝑏−𝑠𝑏−𝑠^𝛼−134^∞₀𝜃𝜉_3/4(𝜃)^∞_𝑛=111+𝑛²exp−𝑛²1+𝑛²𝑠^3/4𝜃×𝑥,𝑒_𝑛𝑒_𝑛3=0,4^∞₀𝜃𝜉_3/4(𝜃)^∞_𝑛=111+𝑛²exp−𝑛²1+𝑛²𝑠^3/4𝜃𝑥,𝑒_𝑛𝑒_𝑛=0,𝑥,𝑒_𝑛=0⟹𝑥=0. (53)
Next, we suppose
(H6) 𝑔∶[0,𝑏]×𝑅→𝑅. For each 𝑥∈𝑅, 𝑔(⋅,𝑥) is measurable and for each 𝑡∈[0,𝑏], 𝑔(𝑡,⋅) is continuous. Moreover, sup_𝑥∈𝑅‖𝑔(𝑡,𝑥)‖≤𝑁(𝑡), for a.e. 𝑡∈[0,𝑏].
Define 𝑓∶[0,𝑏]×𝑋→𝑋 by 𝑓(𝑡,𝑥)(𝜃)=𝑔(𝑡,𝑥(𝑡,𝜃)). Now, system (47) can be written in the abstract form (1). Clearly, all the assumptions in Theorem 13 are satisfied if (H6) holds. Then system (47) is approximately controllable on [0,𝑏].