Complete Controllability for Fractional Evolution Equations

Abstract

The paper is concerned with the complete controllability of fractional evolution equation with nonlocal condition by using a more general concept for mild solution. By contraction fixed point theorem and Krasnoselskii's fixed point theorem, we obtain some sufficient conditions to ensure the complete controllability. Our obtained results are more general to known results.

1. Introduction

Fractional differential equations have recently been proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. It draws a great application in nonlinear oscillations of earthquakes and many physical phenomena such as seepage flow in porous media and in fluid dynamic traffic model. There has been a significant development in fractional differential equations in recent years, see the monographs of Kilbas et al. [1], Miller and Ross [2], Podlubny [3], Lakshmikantham et al. [4], and the papers [5–14] and the references therein.

Some recent papers investigated the problem of the existence of a mild solution for abstract differential equation with fractional derivative [15–23]. However, the results in [15, 16, 18, 19] are incorrect since the considered variation of constant formulas is not appropriate [17]. Zhou and Jiao [22, 23] introduced two characteristic solution operators and gave a suitable concept on a mild solution by applying Laplace transform and probability density functions. But the condition that the analytic semigroup {𝑇(𝑡)}_𝑡≥0 was uniformly bounded was too strong. Shu et al. [20] researched the existence of mild solutions for impulsive fractional partial differential equation. But, Fec̆kan et al. [24] had pointed out that the definition of solution of impulsive fractional differential equation was not correct. By using Laplace transform, Shu and Wang [25] gave a definition of mild solution for fractional differential equation with order 1<𝛼<2 and investigated its existence. Agarwal et al. [26] studied the existence and dimension of the set for mild solutions of semilinear fractional differential equations inclusions.

In 1960, Kalman first introduced the concept of controllability which leads to some very important results regarding the behavior of linear and nonlinear dynamical systems. There are various works of complete controllability of systems represented by differential equations, integrodifferential equations, differential inclusions, neutral functional differential equations, and impulsive differential inclusions in Banach spaces (see [8, 27–29] and the references therein). Recently, more and more researchers also pay attention to study the controllability of fractional order evolution systems (see [21, 30, 31] and the references and therein). Unfortunately, the concept of mild solutions used in [30, 31] was not suitable for fractional evolution systems at all and the corresponding definition of mild solutions is only a simple extension of the mild solutions of integer order systems. Wang and Zhou [21] investigated the complete controllability of fractional evolution systems with two characteristic solution operators introduced by them.

The nonlocal condition can be applied in physics with better effect than the classical initial condition 𝑥(0)=𝑥₀. Nonlocal condition was initiated by Byszewski [32] when he proved the existence and uniqueness of mild and classical solutions of nonlocal Cauchy problems. As remarked by Byszewski and Lakshmikantham [33], the nonlocal condition can be more useful than the standard initial condition to describe some physical phenomena.

Inspired by the above discussions, in this paper, we consider a class of fractional evolution equations. By using a more general definition of mild solution, we obtain some sufficient conditions to ensure the complete controllability.

We consider the following fractional evolution equations: ^𝐶𝐷^𝛼_𝑡𝑥(𝑡)(𝑡)((𝑡(𝑡)[],𝑥(0)(𝑥)=𝐴𝑥+𝑓𝑡,𝑥))+𝐵𝑢,𝑡∈𝐽=0,𝑏+𝑔=𝑥₀, (1) where ^𝐶𝐷^𝛼_𝑡 is the Caputo fractional derivative of order 𝛼∈(0,1], the state 𝑥(⋅) takes value in a Banach space 𝑋, the control function 𝑢(⋅) is given in 𝐿²(𝐽,𝑈), with 𝑈 as a Banach space, 𝐵 is a bounded linear operator from 𝑈 into 𝑋, 𝐴 is a sectorial operator on 𝑋, 𝑓∶𝐽×𝑋→𝑋 and 𝑔∶𝑋→𝑋 are given functions satisfying some assumptions, and 𝑥₀∈𝑋.

The rest of this paper is organized as follows. In Section 2, some notations and preparations are given. A suitable concept on a mild solution for our problem is introduced. In Section 3, the complete controllability results are obtained by using fixed point theorems. Some conclusions are given in Section 4.

2. Preliminaries

In this section, we will firstly introduce fractional integral and derivative, some notations about sectorial operators, solution operators, and analytic solution operators and then give the definition of a mild solution of system (1).

Throughout this paper, ℝ, ℂ denote the sets of real and complex numbers, respectively, and ℝ₊=[0,∞). By 𝐶(𝐽,𝑋), we denote the space of all continuous functions from 𝐽 to 𝑋. ℒ(𝑋) is the space of all bounded linear operators from 𝑋 to 𝑋. 𝐷(𝐴) denotes domain of 𝐴, while 𝜌(𝐴) means resolvent set of 𝐴 and 𝑅(𝜆,𝐴)=(𝜆𝐼−𝐴)⁻¹ stands for the resolvent operator of 𝐴.

Definition 1 (see [3]). The fractional integral of order 𝑝 with the lower limit 𝑎 for a function 𝑓∶[𝑎,∞)→ℝ is defined as _𝑎𝐼^𝑝_𝑡𝑓(𝑡)=1Γ𝑝^𝑡_𝑎𝑓(𝑠)(𝑡−𝑠)^1−𝑝𝑑𝑠,𝑡>𝑎,𝑝>0, (2) provided that the right side is point-wise defined on [𝑎,∞), where Γ(⋅) is the gamma function.

Definition 2 (see [3]). The Riemann-Liouville derivative of order 𝑝>0 for a function 𝑓∶[𝑎,∞)→ℝ is defined as ^𝐿_𝑎𝐷^𝑝_𝑡𝑓(𝑡)=1Γ𝑑𝑛−𝑝𝑑𝑡^𝑛𝑛𝑡_𝑎𝑓(𝑠)(𝑡−𝑠)^{𝑝+1−𝑛}𝑑𝑠,𝑡>𝑎,𝑛−1<𝑝<𝑛. (3)

Definition 3 (see [3]). The Caputo derivative of order 𝑝>0 for a function 𝑓∶[𝑎,∞)→ℝ is defined as ^𝐶_𝑎𝐷^𝑝_𝑡𝑓(𝑡)=^𝐿_𝑎𝐷^𝑝_𝑡𝑓(𝑡)−^𝑛−1_𝑘=0𝑡^𝑘𝑓𝑘!^(𝑘)(0),𝑡>𝑎,𝑛−1<𝑝<𝑛. (4) Let ^𝐶𝐷^𝑝_𝑡𝑓(𝑡)=^𝐶₀𝐷^𝑝_𝑡𝑓(𝑡).

Remark 4. (i) If 𝑓∈𝐶^𝑛[𝑎,∞), then ^𝐶_𝑎𝐷^𝑝_𝑡𝑓(𝑡)=1Γ𝑛−𝑝^𝑡_𝑎𝑓^(𝑛)(𝑠)(𝑡−𝑠)^{𝑝+1−𝑛}𝑑𝑠,𝑡>𝑎,𝑛−1<𝑝<𝑛. (5)
The Caputo derivative of a constant is equal to zero.
If 𝑓 is an abstract function with values in 𝑋, then the integrals which appear in Definitions 1 and 2 are taken in Bochner’s sense.

Definition 5. An operator 𝐴 is said to be sectorial if there are constants 𝜔∈ℝ, 𝜃∈(0,𝜋/2), and 𝑀>0 such that the resolvent of 𝐴 exists outside the sector Σ_𝜃(𝜔)=𝜆∈ℂ∶𝜆≠𝜔,arg(𝜆−𝜔)<𝜃 (6) with ‖𝑅(𝜆,𝐴)‖_ℒ(𝑋)≤𝑀|𝜆−𝜔|,𝜆∉Σ_𝜃(𝜔). (7)

Consider the following Cauchy problem for the Caputo derivative evolution equation of order 𝛼(0<𝛼≤1): ^𝐶𝐷^𝛼_𝑡𝑢(𝑡)(𝑡)[],𝑢(0)=𝐴𝑢,𝑡∈𝐽=0,𝑏=𝑢₀. (8)

Definition 6 (see [7]). A family {𝑆_𝛼(𝑡)}_𝑡≥0⊆ℒ(𝑋) is called a solution operator for system (8), if the following conditions are satisfied:(a){𝑆_𝛼(𝑡)}_𝑡≥0 is strongly continuous, for 𝑡≥0 and 𝑆_𝛼(0)=𝐼.(b)𝑆_𝛼(𝑡)𝐷(𝐴)⊆𝐷(𝐴) and 𝐴𝑆_𝛼(𝑡)𝑥=𝑆_𝛼(𝑡)𝐴𝑥, for all 𝑥∈𝐷(𝐴) and 𝑡≥0.(c)𝑆_𝛼(𝑡)𝑥 is a solution of the following integral equation: 𝑢(𝑡)=𝑢₀+1Γ(𝛼)^𝑡₀𝐴𝑢(𝑡)(𝑡−𝑠)^1−𝛼𝑑𝑠, (9)for all 𝑥∈𝐷(𝐴) and 𝑡≥0.

Remark 7 (see [7]). If {𝑆_𝛼(𝑡)}_𝑡≥0 is the solution operator of system (8), then 𝐴𝑥=Γ(𝛼+1)lim_𝑡→0𝑆_𝛼(𝑡)𝑥−𝑥𝑡^𝛼, (10) where 𝐷(𝐴) consists of those 𝑥∈𝑋 for which this limit exists. We call 𝐴 the infinitesimal generator of {𝑆_𝛼(𝑡)}_𝑡≥0 or say that 𝐴 generates {𝑆_𝛼(𝑡)}_𝑡≥0.

Remark 8 (see [7]). The solution operator {𝑆_𝛼(𝑡)}_𝑡≥0 of system (8) is defined as follows: 𝑆_𝛼(𝑡)=12𝜋𝑖_𝛾𝑒^𝜆𝑡𝜆^𝛼−1𝑅𝜆^𝛼,𝐴𝑑𝜆, (11) where 𝛾 is a suitable path such that 𝜆^𝛼∉Σ_𝜃(𝜔), for 𝜆∈𝛾.

A operator 𝐴 is said to belong to ℭ^𝛼(𝑋,𝑀,𝜔) or ℭ^𝛼(𝑀,𝜔), if system (8) has solution operator {𝑆_𝛼(𝑡)}_𝑡≥0 satisfying |𝑆_𝛼(𝑡)|≤𝑀𝑒^𝜔𝑡, 𝑡≥0. Denote ℭ^𝛼(𝜔)=⋃{ℭ^𝛼(𝑀,𝜔),𝑀≥1} and ℭ^𝛼=⋃{ℭ^𝛼(𝜔),𝜔≥0}.

Definition 9 (see [7]). A solution operator {𝑆_𝛼(𝑡)}_𝑡≥0 of system (8) is called analytic, if {𝑆_𝛼(𝑡)}_𝑡≥0 admits an analytic extension to a sectorial ∑_𝜃0={𝜆∈ℂ⧵{0}∶|arg𝜆|<𝜃₀} for some 𝜃₀∈(0,𝜋/2]. An analytic solution operator is said to be of analyticity type (𝜃₀,𝜔₀), if, for each 𝜃<𝜃₀ and 𝜔>𝜔₀, there is an 𝑀=𝑀(𝜃,𝜔) such that |𝑆_𝛼(𝑡)|≤𝑒^{𝜔𝑅𝑒𝑡}, 𝑡∈∑_𝜃={𝑡∈ℂ⧵{0}∶|arg𝑡|<𝜃}. Denote 𝒜^𝛼𝜃₀,𝜔₀=𝐴∈ℭ^𝛼𝑆∶𝐴generatesanalyticsolution_𝛼(𝑡)oftype𝜃₀,𝜔₀. (12)

Lemma 10 (see [7]). Let 𝛼∈(0,2); a linear closed densely defined operator 𝐴 belongs to 𝒜^𝛼(𝜃₀,𝜔₀), if 𝜆^𝛼∈𝜌(𝐴), for each 𝜆∈Σ_𝜃0+𝜋/2 and, for any 𝜔>𝜔₀, 𝜃<𝜃₀, there is a constant 𝐶=𝐶(𝜃,𝜔) such that 𝜆^𝛼−1𝑅𝜆^𝛼≤𝐶,𝐴|𝜆−𝜔|𝑓𝑜𝑟𝜆∈Σ_𝜃+𝜋/2(𝜔). (13) Next, we consider the definition of the mild solution of system (1).

According to Definitions 1 and 2, it is suitable to rewrite the nonlocal Cauchy problem (1) in the equivalent integral equation 𝑥(𝑡)=𝑥₀(𝑥)+1−𝑔Γ(𝛼)×^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝑥(𝑠)+𝑓(𝑠,𝑥(𝑠))+𝐵𝑢(𝑠)𝑑𝑠, (14) provided that the integral in (14) exists.

The following Lemma 11 is discussed in [20]; for the sake of completeness, we outline its proof here.

Lemma 11. If (14) holds and 𝐴 is a sectorial operator, then we have 𝑥(𝑡)=𝑆_𝛼(𝑡)𝑥₀−𝑔(𝑥)+^𝑡₀𝑇_𝛼()𝑓((𝑠+𝑡−𝑠𝑠,𝑥))𝑑𝑠_𝑡0𝑇_𝛼(𝑡−𝑠)𝐵𝑢(𝑠)𝑑𝑠, (15) where𝑆_𝛼(𝑡)=12𝜋𝑖_𝛾𝑒^𝜆𝑡𝜆^𝛼−1𝑅𝜆^𝛼𝑇,𝐴𝑑𝜆,_𝛼(𝑡)=12𝜋𝑖_𝛾𝑒^𝜆𝑡𝑅𝜆^𝛼,𝐴𝑑𝜆, (16) and 𝛾 is a suitable path such that 𝜆^𝛼∉Σ_𝜃(𝜔), for 𝜆∈𝛾.

Proof. By applying the Laplace transform to (14), we have ℒ(𝑥𝜆)=𝑥)(⁰−𝑔(𝑥)𝜆+𝐴ℒ(𝑥)(𝜆)+ℒ𝑓+𝐵𝑢(𝜆)𝜆^𝛼. (17) Since (𝜆^𝛼𝐼−𝐴)⁻¹ exists, that is, 𝜆^𝛼∈𝜌(𝐴), from the above equation, we obtain ℒ(𝑥)(𝜆)=𝜆^𝛼−1𝜆^𝛼𝐼−𝐴⁻¹𝑥₀(𝑥)+𝜆−𝑔_𝛼𝐼−𝐴⁻¹ℒ𝑓+𝐵𝑢(𝜆). (18) Therefore, by the Laplace inverse transform, we have 𝑥(𝑡)=𝑆_𝛼(𝑡)𝑥₀−𝑔(𝑥)+^𝑡₀𝑇_𝛼()𝑓((𝑠+𝑡−𝑠𝑠,𝑥))𝑑𝑠_𝑡0𝑇_𝛼(𝑡−𝑠)𝐵𝑢(𝑠)𝑑𝑠. (19)

Lemma 12. If 𝛼∈(0,1] and 𝐴∈𝒜^𝛼(𝜃₀,𝜔₀), then the operators 𝑆_𝛼(𝑡) and 𝑇_𝛼(𝑡) are continuous on 𝑡∈ℝ⁺.

Proof. For 0≤𝑡^′<𝑡^′′, by Lemma 10, we have 𝑆_𝛼𝑡^′′−𝑆_𝛼𝑡^′=_𝛾𝑒^{𝜆𝑡′′}−𝑒^𝜆𝑡′𝜆^𝛼−1𝑅𝜆^𝛼,𝐴𝑑𝜆≤𝐶_𝛾𝑒^{𝜆𝑡′′}−𝑒^𝜆𝑡′|𝑑𝜆|||;𝜆−𝜔 (20) choose the integration path 𝛾 as follows: 𝛾=𝜔+𝑟𝑒^{−𝑖((𝜋/2)+𝛿)}∶𝜚≤𝑟<∞𝜔+𝜚𝑒^𝑖𝜑∶𝜑≤𝜋2+𝛿𝜔+𝑟𝑒^{𝑖((𝜋/2)+𝛿)}∶𝜚≤𝑟<∞, (21) such that 𝛾 is oriented counterclockwise, where 𝛿∈(0,𝛿₀), 𝜔>𝜔₀, and 𝜚>0.
From (20), we have 𝑆_𝛼𝑡^′′−𝑆_𝛼𝑡^′≤2𝐶^∞_𝜚𝑒^{𝜆𝑡′′}−𝑒^𝜆𝑡′𝑑𝑟𝑟+𝐶₍₎^{(𝜋/2)+𝛿}_{−𝜋/2−𝛿}𝑒^{𝜆𝑡′′}−𝑒^𝜆𝑡′≤𝑑𝜑2𝐶𝜚^∞_𝜚𝑒^{(𝜔+𝑟cos((𝜋/2)+𝛿)−𝑖𝑟sin((𝜋/2)+𝛿))𝑡′′}−𝑒^{(𝜔+𝑟cos((𝜋/2)+𝛿)−𝑖𝑟sin((𝜋/2)+𝛿))𝑡′}𝑑𝑟+𝐶₍₎^{(𝜋/2)+𝛿}_{−𝜋/2−𝛿}𝑒^{(𝜔+𝜚cos𝜑+𝑖𝜚sin𝜑)𝑡′′}−𝑒^{(𝜔+𝜚cos𝜑+𝑖𝜚sin𝜑)𝑡′}𝑑𝜑. (22) By noticing that cos((𝜋/2)+𝛿)<0, by the dominated convergence theorem, we have |𝑆_𝛼(𝑡^′′)−𝑆_𝛼(𝑡^′)|→0 as 𝑡^′′→𝑡^′, which implies that 𝑆_𝛼(𝑡) is continuous on 𝑡. For the same reason, 𝑇_𝛼(𝑡) is too continuous on 𝑡. The proof is complete.

Lemma 13 (see [20]). If 𝛼∈(0,1] and 𝐴∈𝒜^𝛼(𝜃₀,𝜔₀), then, for any 𝑡>0, we have 𝑇_𝛼(𝑡)≤𝐶𝑒^𝜔𝑡1+𝑡^𝛼−1,𝜔>𝜔₀. (23)

If 𝐴∈𝒜^𝛼(𝜃₀,𝜔₀), then |𝑆_𝛼(𝑡)|≤𝑀𝑒^𝜔𝑡 and |𝑇_𝛼(𝑡)|≤𝐶𝑒^𝜔𝑡(1+𝑡^𝛼−1), for all 𝑡∈(0,+∞). Let 𝑀_𝑆=sup_{0≤𝑡≤𝑏}𝑆_𝛼(𝑡),𝑀_𝑇=sup_{0≤𝑡≤𝑏}𝐶𝑒^𝜔𝑡1+𝑡^1−𝛼; (24) we have 𝑆_𝛼(𝑡)≤𝑀_𝑆,𝑇_𝛼(𝑡)≤𝑡^𝛼−1𝑀_𝑇∀𝑡∈(0,+∞). (25)

Lemma 14 (Krasnoselskii’s fixed point theorem). Let 𝐸 be a Banach space, let 𝐵 be a bounded closed and convex subset of 𝐸, and let 𝐹₁ and 𝐹₂ be maps of 𝐵 into 𝐸 such that 𝐹₁𝑥+𝐹₂𝑦∈𝐵 for every pair 𝑥,𝑦∈𝐵. If 𝐹₁ is a contraction and 𝐹₂ is completely continuous, then the equation 𝐹₁𝑥+𝐹₂𝑥=𝑥 has a solution on 𝐵.

In [34], Reich gave a general fixed point theorem which contained Krasnoselskii’s fixed point theorem, for more details we can see the reference.

Definition 15. A function 𝑥∶𝐽→𝑋 is called a mild solution of system (1), if 𝑥 satisfies the following equation 𝑥(𝑡)=𝑆_𝛼(𝑡)𝑥₀−𝑔(𝑥)+^𝑡₀𝑇_𝛼()𝑓((𝑠+𝑡−𝑠𝑠,𝑥))𝑑𝑠_𝑡0𝑇_𝛼(𝑡−𝑠)𝐵𝑢(𝑠)𝑑𝑠, (26) where 𝑆_𝛼(𝑡)=12𝜋𝑖_𝛾𝑒^𝜆𝑡𝜆^𝛼−1𝑅𝜆^𝛼𝑇,𝐴𝑑𝜆,_𝛼(𝑡)=12𝜋𝑖_𝛾𝑒^𝜆𝑡𝑅𝜆^𝛼,𝐴𝑑𝜆 (27) and 𝛾 is a suitable path such that 𝜆^𝛼∉Σ_𝜃(𝜔), for 𝜆∈𝛾.

Remark 16. When 𝛼=1, 𝑆_𝛼(𝑡)=𝑇_𝛼(𝑡) is a 𝐶₀-semigroup and system (1) degenerates into 1 order evolution equation. However, the limits of 𝑆_𝛼(𝑡) and 𝑇_𝛼(𝑡) in [21–23] did not exist as 𝛼→1⁻.

Remark 17. When 𝐴 generates a 𝐶₀-semigroup {𝑇(𝑡)}_𝑡≥0 in system (1), we have 𝑆_𝛼(𝑡)=^∞₀𝜙_𝛼(𝜃)𝑇𝑡^𝛼𝜃𝑇𝑑𝜃,_𝛼(𝑡)=𝛼𝑡^𝛼−1∞₀𝜃𝜙_𝛼(𝜃)𝑇𝑡^𝛼𝜃𝑑𝜃, (28) where 0<𝛼<1 and 𝜙(𝜃) is a probability density function defined on (0,∞) in [21–23]. So, this definition is more general to that in [21–23].

Remark 18. It is easy to verify that a classical solution of system (1) is a mild solution of the same system.

Definition 19. The system (1) is said to be completely controllable on 𝐽, if, for every 𝑥₀,𝑥₁∈𝑋, there exists a control 𝑢∈𝐿²(𝐽,𝑈), such that a mild solution 𝑥 of system (1) satisfies 𝑥(𝑏)=𝑥₁.
In this paper, we assume the following.(H₁)𝑓∶𝐽×𝑋→𝑋 is continuous and there exist constant 𝑞∈(0,𝛼) and function 𝑚∈𝐿^1/𝑞(𝐽,ℝ⁺) such that 𝑓(𝑡,𝑥(𝑡))−𝑓𝑡,𝑦(𝑡)≤𝑚(𝑡)𝑥−𝑦, (29) for all 𝑡∈𝐽 and 𝑥,𝑦∈𝑋;(H₂)𝑔∶𝑋→𝑋 is continuous and there exists a constant 𝐿_𝑔>0 such that 𝑔(𝑥)−𝑔𝑦≤𝐿_𝑔𝑥−𝑦, (30) for all 𝑥,𝑦∈𝑋.
It is easy to see that if (H₂) holds, then the following assumption holds: (H^′₂) 𝑔∶𝑋→𝑋 is continuous and there exist positive constants 𝐾_𝑔 and 𝑑 such that 𝑔(𝑥)≤𝐾_𝑔‖𝑥‖+𝑑, (31) for all 𝑥∈𝑋; (H₃)the operator family {𝑆_𝛼(𝑡)}_𝑡≥0 is compact;(H₄)the linear operator 𝐵∶𝐿²(𝐽,𝑈)→𝐿(𝐽,𝑋) is bounded; 𝑊∶𝐿²(𝐽,𝑈)→𝑋 defined by 𝑊𝑢=^𝑏₀𝑇_𝛼(𝑏−𝑠)𝐵𝑢(𝑠)𝑑𝑠 (32) has an inverse operator 𝑊⁻¹ which takes values in 𝐿²(𝐽,𝑈)/Ker𝑊 and there exist two positive constants 𝑀₂,𝑀₃>0 such that ‖𝐵‖≤𝑀₂,𝑊⁻¹≤𝑀₃. (33)

3. Complete Controllability Results

Theorem 20. Suppose that (𝐻₁), (𝐻₂), and (𝐻₄) are satisfied; then system (1) is completely controllable on 𝐽, provided that 𝐴∈𝒜^𝛼(𝜃₀,𝜔₀) and Θ=𝑀_𝑆𝐿_𝑔+𝑀_𝑇𝑏^𝛼−𝑞1−𝑞𝛼−𝑞^1−𝑞‖𝑚‖_𝐿^1/𝑞×1+𝑀₂𝑀₃𝑀_𝑇𝑏^𝛼𝛼<1. (34)

Proof. Using hypothesis (H₄) for an arbitrary function 𝑥∈𝐶(𝐽,𝑋), we defined the control function 𝑢_𝑥(𝑡) by 𝑢_𝑥(𝑡)=𝑊⁻¹𝑥₁−𝑆_𝛼(𝑏)𝑥₀(𝑥)−−𝑔_𝑏0𝑇_𝛼(𝑏−𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠. (35) We show that using this control, the operator 𝐹 on 𝐶(𝐽,𝑋) by (𝐹𝑥)(𝑡)=𝑆_𝛼(𝑡)𝑥₀−𝑔(𝑥)+^𝑡₀𝑇_𝛼()𝑓((𝑠+𝑡−𝑠𝑠,𝑥))𝑑𝑠_𝑡0𝑇_𝛼(𝑡−𝑠)𝐵𝑢_𝑥(𝑠)𝑑𝑠 (36) has a fixed point 𝑥, which is a mild solution of system (1).
It is obvious that (𝐹𝑥)(𝑏)=𝑥₁, which means that 𝑢_𝑥 steers the mild 𝑥 from 𝑥₀ to 𝑥₁ in finite time 𝑏. This implies that system (1) is completely controllable on 𝐽. Next, we will prove that 𝐹 has a fixed point on 𝐶(𝐽,𝑋).
Taking 𝑡∈[0,𝑏] and, for all 𝑥,𝑦∈𝐶(𝐽,𝑋), we have, from (H₁), (H₄), and (35), 𝑢_𝑥(𝑡)−𝑢_𝑦(𝑡=𝑊₋₁)𝑆_𝛼(𝑏)𝑔(𝑥)𝑦+−𝑔_𝑏0𝑇_𝛼(𝑏−𝑠)𝑓(𝑠,𝑥(𝑠))−𝑓𝑠,𝑦(𝑠)𝑑𝑠≤𝑀₃𝑀_𝑇^𝑏₀(𝑏−𝑠)^𝛼−1𝑚(𝑠)𝑑𝑠𝑥−𝑦+𝑀₃𝑀_𝑆𝐿_𝑔≤𝑀𝑥−𝑦₃𝑀_𝑇𝑏^𝛼−𝑞1−𝑞𝛼−𝑞^1−𝑞‖𝑚‖_𝐿^1/𝑞+𝑀₃𝑀_𝑆𝐿_𝑔𝑥−𝑦; (37) by (25), we have 𝑡)−(𝑡=𝑆(𝐹𝑥)(𝐹𝑦)_𝛼(𝑡)𝑔𝑦(𝑥)+−𝑔_𝑡0𝑇_𝛼(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠))−𝑓𝑠,𝑦(𝑠)𝑑𝑠+^𝑡₀𝑇_𝛼(𝑡−𝑠)𝐵𝑢_𝑥(𝑠)−𝑢_𝑦(𝑠)≤𝑆𝑑𝑠_𝛼(𝑡𝑔𝑦(𝑥+)−𝑔)_𝑡0𝑇_𝛼()𝑓((𝑠(𝑠)+𝑡−𝑠𝑠,𝑥))−𝑓𝑠,𝑦𝑑𝑠_𝑡0𝑇_𝛼(𝑡−𝑠)𝐵𝑢_𝑥(𝑠)−𝑢_𝑦(𝑠)≤𝑀𝑑𝑠_𝑆𝐿_𝑔𝑦−𝑥+𝑀_𝑇^𝑡₀(𝑡−𝑠)^𝛼−1𝑚(𝑠+𝑀)𝑦−𝑥𝑑𝑠_𝑇𝑡0(𝑡−𝑠)^𝛼−1𝐵𝑢_𝑥(𝑠)−𝑢_𝑦(𝑠)≤𝑀𝑑𝑠_𝑆𝐿_𝑔𝑦−𝑥+𝑀_𝑇^𝑡₀(𝑡−𝑠)^{(𝛼−1)/(1−𝑞)}𝑑𝑠^1−𝑞×^𝑡₀(𝑚(𝑠))^1/𝑞𝑑𝑠^𝑞+𝑏𝑦−𝑥^𝛼𝛼𝑀₂𝑀₃𝑀_𝑇×𝑀_𝑇𝑏^2𝛼−𝑞𝛼1−𝑞𝛼−𝑞^1−𝑞‖𝑚‖_𝐿^1/𝑞+𝑀_𝑆𝐿_𝑔≤𝑀𝑦−𝑥_𝑆𝐿_𝑔+𝑀_𝑇𝑏^𝛼−𝑞1−𝑞𝛼−𝑞^1−𝑞‖𝑚‖_𝐿^1/𝑞×1+𝑀₂𝑀₃𝑀_𝑇𝑏^𝛼𝛼𝑦−𝑥=Θ𝑦−𝑥. (38) Hence, 𝐹 is a contraction mapping and has a unique fixed point 𝑥^∗∈𝐶(𝐽,𝑋). Therefore, this 𝑥^∗ is a mild solution of system (1). The proof is complete.

Theorem 21. Suppose that (𝐻₁), (H^′₂), (𝐻₃), and (𝐻₄) are satisfied; then system (1) is completely controllable on 𝐽 provided that 𝐴∈𝒜^𝛼(𝜃₀,𝜔₀) and 𝑀₄=𝑀₂𝑀₃𝑀_𝑇𝑏^𝛼𝛼×𝑀+1_𝑆𝐾_𝑔+𝑀_𝑇1−𝑞𝛼−𝑞^1−𝑞𝑏^𝛼−𝑞‖𝑚‖_𝐿^1/𝑞<1. (39)

Proof. Define 𝐹₁𝑥(𝑡)=^𝑡₀𝑇_𝛼()𝑓((𝑠+𝑡−𝑠𝑠,𝑥))𝑑𝑠_𝑡0𝑇_𝛼(𝑡−𝑠)𝐵𝑢_𝑥(𝑠)𝐹𝑑𝑠,₂𝑥(𝑡)=𝑆_𝛼(𝑡)𝑥₀−𝑔(𝑥), (40) for 𝑡∈[0,𝑏] and any 𝑥∈𝐶(𝐽,𝑋). Taking into account (35), by (H₁), (H^′₂), and (H₄), we have 𝑢_𝑥(𝑡)=𝑊⁻¹𝑥₁−𝑆_𝛼(𝑏)𝑥₀(𝑥)−−𝑔_𝑏0𝑇_𝛼()𝑓((𝑠≤𝑊𝑏−𝑠𝑠,𝑥))𝑑𝑠₋₁𝑥₁+𝑆_𝛼(𝑏)𝑥₀(𝑥)+−𝑔_𝑏0𝑇_𝛼(𝑏−𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠≤𝑀₃𝑥₁+𝑀_𝑆𝑥₀(𝑥+𝑀−𝑔)_𝑇𝑏0(𝑏−𝑠)^𝛼−1𝑚(𝑠)‖𝑥‖+𝑓(𝑠,0)𝑑𝑠≤𝑀₃𝑥₁+𝑀_𝑆𝑥₀+𝐾_𝑔‖𝑥‖+𝑀+𝑑_𝑇1−𝑞𝛼−𝑞^1−𝑞𝑏^𝛼−𝑞‖𝑚‖_𝐿^1/𝑞×‖𝑥‖+𝑏^𝛼𝛼sup_𝑠∈𝐽𝑓(𝑠,0). (41) In order to make the following process clear, we divide it into several steps.
𝑆𝑡𝑒𝑝 I. For 𝑡∈[0,𝑏] and any 𝑥,𝑦∈𝐶(𝐽,𝑋), we have 𝐹₁𝑥(𝑡=_𝑡0)𝑇_𝛼()𝑓((𝑠+𝑡−𝑠𝑠,𝑥))𝑑𝑠_𝑡0𝑇_𝛼(𝑡−𝑠)𝐵𝑢_𝑥(𝑠)≤𝑑𝑠_𝑡0𝑇_𝛼()𝑓((𝑠()+𝑡−𝑠𝑠,𝑥))−𝑓𝑠,0𝑑𝑠_𝑡0𝑇_𝛼(𝑡−𝑠)𝑓(𝑠,0)𝑑𝑠+^𝑡₀𝑇_𝛼(𝑡−𝑠)𝐵𝑢_𝑥(𝑠)≤𝑀𝑑𝑠_𝑇𝑡0(𝑡−𝑠)^𝛼−1𝑚(𝑠𝑥‖+𝑀)‖𝑑𝑠_𝑇sup_𝑠∈𝐽𝑓(𝑠,0)^𝑡₀(𝑡−𝑠)^𝛼−1+𝑀𝑑𝑠_𝑇𝑡0(𝑡−𝑠)^𝛼−1𝑀₂𝑢_𝑥(𝑠≤𝑀)𝑑𝑠_𝑇𝑡0(𝑡−𝑠)^{(𝛼−1)/(1−𝑞)}𝑑𝑠^1−𝑞‖𝑚‖_𝐿^1/𝑞‖𝑥‖+𝑏^𝛼𝛼𝑀_𝑇sup_{[]𝑠∈0,𝑏}𝑓(+𝑏𝑠,0)^𝛼𝛼×𝑀_𝑇𝑀₂𝑀₃𝑥₁+𝑀_𝑆𝑥₀+𝐾_𝑔‖𝑥‖+𝑀+𝑑_𝑇1−𝑞𝛼−𝑞^1−𝑞𝑏^𝛼−𝑞‖𝑚‖_𝐿^1/𝑞‖𝑥‖+𝑏^𝛼𝛼sup_𝑠∈𝐽𝑓(≤𝑏𝑠,0)1+^𝛼𝛼𝑀_𝑇𝑀₂𝑀₃𝑀_𝑇1−𝑞𝛼−𝑞^1−𝑞𝑏^𝛼−𝑞‖𝑚‖_𝐿^1/𝑞+𝑀₂𝑀₃𝑀_𝑇𝑏^𝛼𝛼𝑀_𝑆𝐾_𝑔‖𝑥‖+𝑀₂𝑀₃𝑀_𝑇𝑏^𝛼𝛼𝑥₁+𝑀_𝑆𝑥₀+𝑀+𝑑₂𝑀₃𝑀_𝑇𝑏^𝛼𝛼𝑏+1^𝛼𝛼𝑀_𝑇sup_𝑠∈𝐽𝑓(,𝐹𝑠,0)₂𝑦(𝑡)=𝑆_𝛼(𝑡)𝑥₀𝑦≤𝑀−𝑔_𝑆𝑥₀+𝐾_𝑔𝑦+𝑑. (42) By the condition 𝑀₄<1, we can find 𝑘₀>0 such that, for 𝑥,𝑦∈𝐵_𝑘0={𝑥∈𝐶(𝐽,𝑋)∶‖𝑥‖≤𝑘₀}, 𝐹₁𝑥+𝐹₂𝑦≤𝑘₀i.e.𝐹₁𝑥+𝐹₂𝑦∈𝐵_𝑘0. (43)
𝑆𝑡𝑒𝑝 II. 𝐹₁ is a contraction mapping on 𝐵_𝑘0.
For any 𝑥,𝑦∈𝐵_𝑘0 and 𝑡∈[0,𝑏], we have 𝐹₁𝑥(𝑡)−𝐹₁𝑦(𝑡≤_𝑡0)𝑇_𝛼()𝑓((𝑠(𝑠)+𝑡−𝑠𝑠,𝑥))−𝑓𝑠,𝑦𝑑𝑠_𝑡0𝑇_𝛼(𝑡−𝑠)𝐵𝑢_𝑥(𝑠)−𝑢_𝑦(𝑠)≤𝑀𝑑𝑠_𝑇𝑡0(𝑡−𝑠)^𝛼−1𝑚(𝑠+𝑀)𝑥−𝑦𝑑𝑠_𝑇𝑀₂^𝑡₀(𝑡−𝑠)^𝛼−1𝑢_𝑥(𝑠)−𝑢_𝑦(𝑠≤𝑀)𝑑𝑠_𝑇𝑡0(𝑡−𝑠)^{(𝛼−1)/(1−𝑞)}𝑑𝑠^1−𝑞‖𝑚‖_𝐿^1/𝑞𝑥−𝑦+𝑀₂𝑀_𝑇𝑏^𝛼𝛼𝑀₃𝑀_𝑇𝑏^𝛼−𝑞1−𝑞𝛼−𝑞^1−𝑞‖𝑚‖_𝐿^1/𝑞+𝑀₃𝑀_𝑆𝐾_𝑔≤𝑀𝑥−𝑦_𝑇1−𝑞𝛼−𝑞^1−𝑞𝑏^𝛼−𝑞‖𝑚‖_𝐿^1/𝑞𝑏1+^𝛼𝛼𝑀_𝑇𝑀₂𝑀₃+𝑏^𝛼𝛼𝑀_𝑇𝑀₂𝑀₃𝑀_𝑆𝐾_𝑔𝑥−𝑦=∶Υ𝑥−𝑦. (44) From the condition 𝑀₄<1, we obtain Υ<1, which implies that 𝐹₁ is a contraction mapping.
𝑆𝑡𝑒𝑝 III. 𝐹₂ is a completely continuous operator.
First, we will prove that 𝐹₂ is continuous on 𝐵_𝑘0. Let {𝑥_𝑛}⊆𝐵_𝑘0 with 𝑥_𝑛→𝑥∈𝐵_𝑘0. By (H^′₂), we have 𝑔𝑥_𝑛⟶𝑔(𝑥)as𝑛⟶∞. (45) So, we have 𝐹₂𝑥_𝑛(𝑡)−𝐹₂𝑥(𝑡=𝑆_𝛼)(𝑡)𝑥₀−𝑔𝑥_𝑛−𝑆_𝛼(𝑡)𝑥₀(𝑥)=𝑆−𝑔_𝛼(𝑡)𝑔(𝑥)−𝑔𝑥_𝑛≤𝑀_𝑆𝑔(𝑥)−𝑔𝑥_𝑛⟶0as𝑛⟶∞, (46) which implies that 𝐹₂ is continuous.
Next, we will show that {𝐹₂𝑥,𝑥∈𝐵_𝑘0} is relatively compact. It suffices to show that the family of function {𝐹₂𝑥,𝑥∈𝐵_𝑘0} is uniformly bounded and equicontinuous and, for any 𝑡∈[0,𝑏], {(𝐹₂𝑥)(𝑡),𝑥∈𝐵_𝑘0} is relatively compact.
For any 𝑥∈𝐵_𝑘0, we have ‖𝐹₂𝑥‖≤𝑘₀ which implies that {𝐹₂𝑥,𝑥∈𝐵_𝑘0} is uniformly bounded. In the following, we will show that {𝐹₂𝑥,𝑥∈𝐵_𝑘0} is a family of equicontinuous functions.
For any 𝑥∈𝐵_𝑘0 and 0≤𝑡^′<𝑡^′′≤𝑏, we have 𝐹₂𝑥𝑡^′′−𝐹₂𝑥𝑡^′=𝑆_𝛼𝑡^′′𝑥₀−𝑔(𝑥)−𝑆_𝛼𝑡^′𝑥₀(𝑥)≤𝑆−𝑔_𝛼𝑡^′′−𝑆_𝛼𝑡^′𝑥₀(𝑥≤𝑆−𝑔)_𝛼𝑡^′′−𝑆_𝛼𝑡^′𝑥₀+𝐾_𝑔𝑘₀+𝑑. (47) From Lemma 12, we have |(𝐹₂𝑥)(𝑡^′′)−(𝐹₂𝑥)(𝑡^′)|→0 independently of 𝑥∈𝐵_𝑘0 as 𝑡^′′−𝑡^′→0, which means that {𝐹₂𝑥,𝑥∈𝐵_𝑘0} is equicontinuous.
By the compactness of {𝑆_𝛼(𝑡)}_𝑡≥0, we know that {(𝐹₂𝑥)(𝑡),𝑥∈𝐵_𝑘0} is relatively compact. Therefore, {𝐹₂𝑥,𝑥∈𝐵_𝑘0} is relatively compact by Arzela-Ascoli theorem. The continuity of 𝐹₂ and relative compactness of {𝐹₂𝑥,𝑥∈𝐵_𝑘0} imply that 𝐹₂ is a completely continuous operator. By using Krasnoselskii’s fixed point theorem, we obtain that 𝐹₁+𝐹₂ has a fixed point on 𝐵_𝑘0. Therefore, the nonlocal Cauchy problem (1) has at least one mild solution. The proof is complete.

When there is no control term, system (1) degenerates to the following system: ^𝐶𝐷^𝛼_𝑡𝑥(𝑡)(𝑡)((𝑡[],𝑥(0)(𝑥)=𝐴𝑥+𝑓𝑡,𝑥)),𝑡∈𝐽=0,𝑏+𝑔=𝑥₀. (48) As a direct result of Theorems 20 and 21, we have the following corollaries.

Corollary 22. Suppose that (𝐻₁), (𝐻₂), and (𝐻₄) are satisfied; then system (48) has a unique mild solution, if 𝐴∈𝒜^𝛼(𝜃₀,𝜔₀) and Θ=𝑀_𝑆𝐿_𝑔+𝑀_𝑇𝑏^𝛼−𝑞1−𝑞𝛼−𝑞^1−𝑞‖𝑚‖_𝐿^1/𝑞<1. (49)

Corollary 23. Suppose that (𝐻₁), (𝐻^′₂), (𝐻₃), and (𝐻₄) are satisfied; then system (48) has at least one mild solution, if 𝐴∈𝒜^𝛼(𝜃₀,𝜔₀) and 𝑀₄=𝑀_𝑆𝐾_𝑔+𝑀_𝑇1−𝑞𝛼−𝑞^1−𝑞𝑏^𝛼−𝑞‖𝑚‖_𝐿^1/𝑞<1. (50)

4. Conclusions

In this paper, we introduce a more general definition for mild solution of fractional evolution equation with nonlocal condition based on solution operator. By contraction fixed point theorem and Krasnoselskii’s fixed point theorem, we obtain some sufficient conditions to ensure the complete controllability for system (1). Here, we do not require the operator 𝐴 to be the infinitesimal generator of an analytic semigroup {𝑇(𝑡)}_𝑡≥0 of uniform boundedness. So, the results we obtained are more general. For fractional evolution equation with Riemann-Liouville derivative, since it is equipped with a singular initial, it will be a difficult problem.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publishing of this paper.

Acknowledgments

The authors would like to thank the Editor and the anonymous reviewers for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (11271309 and 11301451), the Specialized Research Fund for the Doctoral Program of Higher Education (20114301110001), and the Key Projects of Hunan Provincial Natural Science Foundation of China (12JJ2001).