Approximate Controllability of Fractional Integrodifferential Evolution Equations

Abstract

This paper addresses the issue of approximate controllability for a class of control system which is represented by nonlinear fractional integrodifferential equations with nonlocal conditions. By using semigroup theory, p-mean continuity and fractional calculations, a set of sufficient conditions, are formulated and proved for the nonlinear fractional control systems. More precisely, the results are established under the assumption that the corresponding linear system is approximately controllable and functions satisfy non-Lipschitz conditions. The results generalize and improve some known results.

1. Introduction

In recent years, use theory of fractional calculus and fractional differential equations has gained importance and popularity due to its applications in various fields of science and engineering. Various physical phenomena in science and engineering can be successfully modeled by using fractional calculus theory. Due to its tremendous scope and applications, several papers have been devoted to study the existence of mild solutions of fractional differential equations (see [1–4] and references therein). On the other hand, controllability is an important property of a control system which plays an important role in the analysis and design of control systems [5–8]. Most literatures in this direction so far have been concerned with controllability of nonlinear differential equations in infinite-dimensional spaces without fractional derivatives (see [9] and references therein). Using generalized open mapping theorem, a set of sufficient conditions for constrained local relative controllability near the origin are formulated and proved for the semilinear systems with delayed controls in [10, 11].

Recently, only few papers deal with the controllability of fractional dynamical systems [12–14]. Klamka [15, 16] derived a set of sufficient conditions for the local controllability of finite-dimensional fractional discrete-time semilinear systems. However, the problem of controllability for fractional systems has not been fully investigated, and there is still room open for further research in this area [17]. Moreover, the approximate controllable systems are more prevalent, and very often, approximate controllability is completely adequate in applications (see [18–21] and references therein). Therefore, it is important, in fact necessary, to study the weaker concept of controllability, namely, approximate controllability for nonlinear fractional integrodifferential systems. Motivated by this fact, in this paper, we consider the approximate controllability of the fractional nonlinear integrodifferential evolution equations with nonlocal initial condition in the following form: ^𝐶𝐷^𝑞𝑥(𝑡)=−𝐴𝑥(𝑡)+𝐵𝑢(𝑡)+𝑡^𝑛𝑓((𝑡),(𝑡,[]𝑡,𝑥𝐻𝑥)())𝑡∈𝐽=0,𝑏,𝑛∈𝑍⁺(),𝑥(0)(𝑥),𝑞∈0,1=𝑔+𝑥₀∈𝑋_𝛾, (1) where the state 𝑥(⋅) takes the values in a Hilbert space 𝑋, ^𝐶𝐷^𝑞 denotes Caputo derivative, −𝐴∶𝐷(𝐴)→𝑋 is the infinitesimal generator of an analytic semigroup {𝑇(𝑡),𝑡≥0} on 𝑋; the control function 𝑢(⋅) is given in 𝐿²(𝐽,𝑈); 𝑈 is a Hilbert space; 𝐵 is a bounded linear operator from 𝑈 into 𝑋; the operator 𝐻 is defined by (𝐻𝑥)(𝑡)=∫^𝑏₀ℎ(𝑡,𝑠,𝑥(𝑠))𝑑𝑠; the nonlinear term 𝑓∶𝐽×𝑋_𝛾×𝑋_𝛾→𝑋(or𝑋_𝛾) is a given function, where here 𝐽=[0,𝑏]; and 𝑋_𝛾=𝐷(𝐴^𝛾)(0<𝛾<1) is a Hilbert space with the norm ‖𝑥‖_𝛾=‖𝐴^𝛾𝑥‖ for 𝑥∈𝑋_𝛾. The functions 𝑓, ℎ, and 𝑔 will be specified later. In fact, our results in this paper are motivated by the recent work of [20], and the fractional integrodifferential equations are studied in [4]. The main objective of this paper is to derive conditions for the approximate controllability of (1) with non-Lipschitz coefficients, and the associated linear system is approximately controllable.

2. Preliminaries

In this section, we provide definitions, lemmas, and notations necessary to establish our main results [4]. Let ‖𝑓‖_{𝐿𝑝(𝐽,𝑅+)} denote the 𝐿^𝑝(𝐽,𝑅⁺) norm of 𝑓 whenever 𝑓∈𝐿^𝑝(𝐽,𝑅⁺) for some 𝑝 with 1<𝑝<∞. Let 𝒞_𝛾 denote the Banach space 𝐶(𝐽,𝑋_𝛾) endowed with sup norm given by ‖𝑥‖_∞≡sup_𝑡∈𝐽‖𝑥‖_𝛾, for 𝑥∈𝒞_𝛾, 𝛾∈(0,1).

Let us recall the following known results.

The fractional integral of order 𝛽 with the lower limit 0 for a function 𝑓 is defined as 𝐼^𝛽𝑓(𝑡)=1Γ𝛽^𝑡₀𝑓(𝑠)(𝑡−𝑠)^1−𝛽𝑑𝑠,𝑡>0,𝛽>0, (2) provided the right-hand side is pointwise defined on [0,∞), where Γ(⋅) is the gamma function.

Riemann-Liouville derivative of order 𝛽 with lower limit zero for a function 𝑓∶[0,∞)→𝑅 can be written as ^𝐿𝐷^𝛽𝑓(𝑡)=1Γ𝑑𝑛−𝛽𝑑𝑡^𝑛𝑛𝑡₀𝑓(𝑠)(𝑡−𝑠)^{𝛽+1−𝑛}𝑑𝑠,𝑡>0,𝑛−1<𝛽<𝑛. (3)

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

Remark 1 (see [4]). (i) If 𝑓(𝑡)∈𝐶^𝑛[0,∞), then 𝐷^𝛽𝑓(𝑡)=1Γ𝑛−𝛽^𝑡₀𝑓^𝑛(𝑠)(𝑡−𝑠)^{𝛽+1−𝑛}𝑑𝑠=𝐼^𝑛−𝛽𝑓^𝑛(𝑠),𝑡>0,0≤𝑛−1<𝛽<𝑛. (5)
(ii) The Caputo derivative of a constant is equal to zero.
(iii) If 𝑓 is an abstract function with values in 𝑋, then integrals which appear in the above results are taken in Bochner’s sense.
For additional details concerning the fractional derivative, we refer the reader to [3].
To define the mild solution for the control system (1), we associate problem (1) to the following integral equation [4]: 𝑥(𝑡)=𝒯(𝑡)𝑥₀+𝑔(𝑥)+^𝑡₀(𝑡−𝑠)^𝑞−1𝒮()(𝑠)+𝑡−𝑠𝐵𝑢𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝑠^𝑛𝒮(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠),(𝐻𝑥)(𝑠))𝑑𝑠,𝑡∈𝐽, (6) where 𝒯(𝑡)=∫^∞₀𝜉_𝑞(𝜃)𝑇(𝑡^𝑞𝜃)𝑑𝜃, 𝒮(𝑡)=𝑞∫^∞₀𝜃𝜉_𝑞(𝜃)𝑇(𝑡^𝑞𝜃)𝑑𝜃, 𝜉_𝑞(𝜃)=(1/𝑞)𝜃^{−1−1/𝑞}𝑤_𝑞(𝜃^−1/𝑞)≥0, 𝑤_𝑞(𝜃)=(1/𝜋)∑^∞_𝑛=1(−1)^𝑛−1𝜃^{−𝑞𝑛−1}(Γ(𝑛𝑞+1)/𝑛!)sin(𝑛𝜋𝑞), 𝜃∈(0,∞),  and 𝜉_𝑞 is a probability density function defined on (0,∞); that is 𝜉_𝑞(𝜃)≥0, 𝜃∈(0,∞), and ∫^∞₀𝜉_𝑞(𝜃)𝑑𝜃=1.

Definition 2. A function 𝑥(⋅;𝑥₀,𝑢)∈𝐶(𝐽,𝑋_𝛾) is said to be a mild solution of (1) if for any 𝑢(⋅)∈𝐿²(𝐽,𝑈) the integral equation (6) is satisfied.

Definition 3. The system (1) is said to be approximately controllable on the interval 𝐽 if ℜ(𝑏,𝑥₀)=𝑋, where ℜ𝑏,𝑥₀=𝑥_𝑏𝑥₀;𝑢(0)∶𝑢(⋅)∈𝐿²(𝐽,𝑈) (7) is called the reachable set of system (1) at terminal time 𝑏, and its closure in 𝑋 is denoted by ℜ(𝑏,𝑥₀); let 𝑥_𝑏(𝑥₀;𝑢) be the state value of (1) at terminal time 𝑏 corresponding to the control 𝑢 and the initial value 𝑥₀∈𝑋.

Consider the following linear fractional differential control system 𝐷^𝑞_𝑡𝑥(𝑡)(𝑡)(𝑡)[]𝑥(0)=𝐴𝑥+𝐵𝑢,𝑡∈𝐽=0,𝑏,0<𝑞<1,=𝑥₀. (8) The approximate controllability for the system (8) is a natural generalization of approximate controllability for the linear first order control system (see [18]). It is convenient at this point to introduce the controllability operator associated with the linear system Γ^𝑏₀=^𝑏₀(𝑏−𝑠)^𝑞−1𝒮(𝑏−𝜏)𝐵𝐵^∗𝒮^∗(𝑏−𝜏)𝑑𝜏, (9) where 𝐵^∗ denotes the adjoint of 𝐵 and 𝒮^∗(𝑡) is the adjoint of 𝒮(𝑡). It is straightforward that the operator Γ^𝑏₀ is a linear bounded operator. Let 𝑅(𝛼,Γ^𝑏₀)=(𝛼𝐼+Γ^𝑏₀)⁻¹ for 𝛼>0.

Lemma 4. The linear fractional control system (8) is approximately controllable on 𝐽 if and only if 𝛼𝑅(𝛼,Γ^𝑏₀)→0 as 𝛼→0⁺ in the strong operator topology.

The proof of this lemma is a straightforward adaptation of the proof of Theorem 2 of [18].

Lemma 5 (see [4]). For each 𝜓∈𝐿^𝑝(𝐽,𝑋) with 1≤𝑝<+∞, one has lim_ℎ→0∫^𝑏₀‖𝜓(𝑡+ℎ)−𝜓(𝑡+ℎ)‖^𝑝𝑑𝑡=0, where 𝜓(𝑠)=0 for 𝑠 does not belong to 𝐽.

Lemma 6 (see [4]). The operators 𝒯 and 𝒮 have the following properties.(i) For fixed 𝑡≥0, 𝒯(𝑡) and 𝒮(𝑡) are linear and bounded operators. For any 𝑥∈𝑋, ‖𝒯(𝑡)𝑥‖≤𝑀‖𝑥‖,‖𝒮(𝑡)𝑥‖≤𝑀𝑞Γ‖𝑥‖.𝑞+1 (10)(ii){𝒯(𝑡),𝑡≥0} and {𝒮(𝑡),≥0} are strongly continuous.(iii) For every 𝑡>0, 𝒯(𝑡) and 𝒮(𝑡) are also compact operators.(iv) For any 𝑥∈𝑋, 𝛽∈(0,1), and 𝛾∈(0,1) one has 𝐴𝒮(𝑡)𝑥=𝐴^1−𝛽𝒮(𝑡)𝐴^𝛽𝐴𝑥,𝑡∈𝐽,_𝛾𝒮(𝑡≤𝑀)^𝛾𝑞Γ2−𝛾Γ𝑡1+𝑞1−𝛾^−𝛾𝑞,0<𝑡≤𝑏. (11)(v) For fixed 𝑡≥0 and any 𝑥∈𝑋_𝛾, one has ‖𝒯(𝑡)𝑥‖_𝛾≤𝑀‖𝑥‖_𝛾,‖𝒮(𝑡)𝑥‖_𝛾≤𝑀𝑞Γ‖𝑥‖𝑞+1^𝛾. (12)(vi)𝒯_𝛾(𝑡) and 𝒮_𝛾(𝑡), 𝑡>0 are uniformly continuous; that is, for each fixed 𝑡>0 and 𝜖>0, there exists ℎ>0 such that 𝒯_𝛾(𝑡+𝜖)−𝒯_𝛾(𝑡)_𝛾<𝜖,for|𝜖|𝒮𝑡+𝜖≥0,<ℎ,_𝛾(𝑡+𝜖)−𝒮_𝛾(𝑡)_𝛾<𝜖,for𝑡+𝜖≥0,|𝜖|<ℎ, (13)
 where 𝒯_𝛾(𝑡)=∫^∞₀𝜉_𝑞(𝜃)𝑇_𝛾(𝑡^𝑞𝜃)𝑑𝜃, 𝒮_𝛾(𝑡)=𝑞∫^∞₀𝜃𝜉_𝑞(𝜃)𝑇_𝛾(𝑡^𝑞𝜃)𝑑𝜃.

3. Main Result

In this section, we present our main result on approximate controllability of control system (1). We prove that under certain conditions, approximate controllability of the linear system (8) implies the approximate controllability of nonlinear fractional system (1). In order to establish the result, we require the following assumptions.(𝐻₁) The function 𝑓∶𝐽×𝑋_𝛾×𝑋_𝛾→𝑋_𝛾 is Carathéodory, and there exists a positive function 𝜌∈𝐿^𝑝(𝐽,𝑅⁺) for some 𝑝 with 1<𝑝<∞ such that ‖𝑓(𝑡,𝑥,𝑦)‖_𝛾≤𝜌(𝑡) for all 𝑥,𝑦∈𝑋_𝛾 and 𝑡∈𝐽.(𝐻₂) The function 𝑔∶𝒞_𝛾→𝑋_𝛾 is completely continuous, and there exist 𝛽₁,𝛽₂>0 such that 𝑔(𝑥)_𝛾≤𝛽₁‖𝑥‖_∞+𝛽₂. (14)(𝐻₃) The function ℎ∶𝐷_ℎ×𝑋_𝛾→𝑋_𝛾 is continuous, and there exist 𝐿₁,𝐿₂>0 such that ‖ℎ(𝑡,𝑠,𝑥)−ℎ(𝑡,𝑠,𝑦)‖_𝛾≤𝐿₁‖𝑥−𝑦‖_𝛾+𝐿₂ for each (𝑡,𝑠)∈𝐷_ℎ={(𝑡,𝑠)∈𝑅²,0≤𝑠,𝑡≤𝑏} and 𝑥,𝑦∈𝑋_𝛾.(𝐻₄) The linear fractional control system (8) is approximately controllable. (𝐻₅) The function 𝑓∶𝐽×𝑋_𝛾×𝑋_𝛾→𝑋_𝛾 is continuous and uniformly bounded, and there exists 𝑁>0 such that ‖𝑓(𝑡,𝑥,𝑦)‖≤𝑁 for all (𝑡,𝑥,𝑦)∈𝐽×𝑋_𝛾×𝑋_𝛾.(𝐻₆) The semigroup 𝑇(𝑡) is compact.

In order to prove the required result, for 𝛼>0, we define the operator 𝐹_𝛼 on 𝐶(𝐽,𝑋_𝛾) as 𝐹_𝛼𝑥(𝑡)=𝑧(𝑡), (15) where 𝑧(𝑡)=𝒯(𝑡)𝑥₀+𝑔(𝑥)+^𝑡₀(𝑡−𝑠)^𝑞−1𝒮()(𝑠)+𝑡−𝑠𝐵𝑣𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝑠^𝑛𝒮(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠),(𝐻𝑥)(𝑠))𝑑𝑠,𝑡∈𝐽,𝑣(𝑡)=𝐵^∗𝒮^∗(𝑏−𝑡)𝑅𝛼,Γ^𝑏₀𝑝(𝑥(⋅,𝑝(𝑥(⋅))))=𝑥_𝑏−𝒯(𝑏)𝑥₀(𝑥−+𝑔)_𝑏0(𝑏−𝑠)^𝑞−1𝑠^𝑛𝒮()((𝑠),(𝑠𝑏−𝑠×𝑓𝑠,𝑥𝐻𝑥)())𝑑𝑠. (16)(17)

Theorem 7. Under the conditions (𝐻₁), (𝐻₂), (𝐻₃), and (𝐻₆), the control system (1) admits a mild solution on [0,𝑏], and here 𝑀_𝐵=‖𝐵‖ and 𝑀₁=‖𝜌‖_𝐿𝑝²_(𝐽,𝑅+).

Proof. The main aim in this section is to find conditions for solvability of systems (16) and (17) for 𝛼>0. In the Banach space 𝐶(𝐽,𝑋_𝛾), consider a set 𝐵_𝑟0=𝑥(⋅)∈𝐶𝐽,𝑋_𝛾∶‖𝑥‖_𝛾≤𝑟, (18) where 𝑟 is the positive constant. Now, it will be shown that, using Schauder’s fixed point theorem, for all 𝛼>0, the operator 𝐹_𝛼∶𝐶(𝐽,𝑋_𝛾)→𝐶(𝐽,𝑋_𝛾) has a fixed point.
First, we prove that for an arbitrary 𝛼>0 and there is a positive constant 𝑟₀=𝑟₀(𝛼) such that 𝐹_𝛼∶𝐵_𝑟0→𝐵_𝑟0.
Let 𝑥∈𝐵_𝑟0, and then, for 𝑡∈𝐽 and 1/𝑝<𝑞(⇒(𝑝𝑞−1))>0, using Holders inequality, Lemma 6, and conditions (𝐻₁) and (𝐻₂) in (17), we obtain ‖𝑣(𝑡)‖_𝛾≤𝐵^∗𝒮^∗(𝑏−𝑡)𝑅𝛼,Γ^𝑏₀×𝑥_𝑏𝛾−𝒯(𝑏)𝑥₀+𝑔(𝑥)_𝛾+^𝑏₀(𝑏−𝑠)^𝑞−1𝑠^𝑛×𝒮(𝑏−𝑠)𝑓(𝑠,𝑥(𝑠),(𝐻𝑥)(𝑠))_𝛾≤𝑀𝑑𝑠_𝐵𝑀𝑞𝑥𝛼Γ𝑞+1^𝑏𝛾+𝑀𝑥_0𝛾+𝛽₁‖𝑥‖_∞+𝛽₂+𝑀₁𝑀𝑞Γ𝑞+1𝑝−1𝑝𝑞−1^{(𝑝−1)/𝑝}×𝑝−1𝑛𝑝²+𝑝−1^{(𝑝−1)/𝑝2}𝑏^{𝑛+𝑞−1/𝑝2}. (19) It follows from (16) that ‖𝑧(𝑡)‖_𝛾≤𝒯(𝑡)𝑥₀+𝑔(𝑥)_𝛾+^𝑡₀(𝑡−𝑠)^𝑞−1‖𝒮(𝑡−𝑠)𝐵𝑣(𝑠)‖_𝛾+𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝑠^𝑛𝒮(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠),(𝐻𝑥)(𝑠))_𝛾𝑥𝑑𝑠≤𝑀_0𝛾+𝛽₁‖𝑥‖_∞+𝛽₂+𝑀_𝐵𝑀𝑞Γ𝑞+1𝑝−1𝑞𝑝−1^{(𝑝−1)/𝑝}𝑏^{((𝑞−1)𝑝+𝑝−1)/𝑝}𝑏‖𝑣(𝑠)‖_𝛾+𝑀₁𝑀𝑞Γ𝑞+1𝑝−1𝑝𝑞−1^{(𝑝−1)/𝑝}𝑝−1𝑛𝑝²+𝑝−1^{(𝑝−1)/𝑝2}×𝑏^{𝑛+𝑞−1/𝑝2}. (20) From the above two inequalities, we get that ‖(𝐹_𝛼𝑥)(𝑡)‖_𝛾≤𝑟₀. This follows that 𝐹_𝛼 maps 𝐵_𝑟0 into itself.
For each 𝛼>0, we prove the operator 𝐹_𝛼 maps 𝐵_𝑟0 into a relatively compact subset of 𝐵_𝑟0. First, we show that 𝑉(𝑡)={(𝐹_𝛼𝑥)(𝑡)∶𝑥∈𝐵_𝑟0} is relatively compact in 𝑋_𝛾 for every 𝑡∈𝐽. The case 𝑡=0 is obvious. Let 𝑡∈(0,𝑏] be fixed, and for each 𝜀∈(0,𝑡), arbitrary 𝛿>0, and 𝑥∈𝐵_𝑟0, we define the operator 𝐹_𝛼^𝜀,𝛿 by 𝐹_𝛼^𝜀,𝛿𝑥(𝑡)=𝒯𝑥₀+𝑔(𝑥)+𝑞₀^{𝑡−𝜀∞}_𝛿𝜃(𝑡−𝑠)^𝑞−1𝜉_𝑞(𝜃)𝑇(𝑡−𝑠)^𝑞𝜃×𝐵𝑣(𝑠)𝑑𝜃𝑑𝑠+𝑞₀^{𝑡−𝜀∞}_𝛿𝜃(𝑡−𝑠)^𝑞−1𝑠^𝑛𝜉_𝑞(𝜃)𝑇(𝑡−𝑠)^𝑞𝜃((𝑠),(𝑠𝜀×𝑓𝑠,𝑥𝐻𝑥)())𝑑𝜃𝑑𝑠=𝑇^𝑞𝛿×^∞_𝛿𝜉_𝑞(𝜃)𝑇𝑡^𝑞𝜃−𝑇𝜀^𝑞𝛿𝑑𝜃𝑥₀(𝑥+𝑇𝜀+𝑔)_𝑞𝛿𝑞×₀^{𝑡−𝜀∞}_𝛿𝜃(𝑡−𝑠)^𝑞−1𝜉_𝑞(𝜃)×𝑇(𝑡−𝑠)^𝑞𝜃−𝑇𝜀^𝑞𝛿(𝑠)𝜀𝐵𝑣𝑑𝜃𝑑𝑠+𝑇^𝑞𝛿𝑞₀^{𝑡−𝜀∞}_𝛿𝜃(𝑡−𝑠)^𝑞−1𝑠^𝑛𝜉_𝑞(𝜃)×𝑇(𝑡−𝑠)^𝑞𝜃−𝑇𝜀^𝑞𝛿((𝑠),(𝑠𝜀×𝑓𝑠,𝑥𝐻𝑥)())𝑑𝜃𝑑𝑠=𝑇^𝑞𝛿𝑦(𝑡,𝜀). (21) Since 𝑇(𝜀^𝑞𝛿) is compact in 𝑋_𝛾 and 𝑦(𝑡,𝜀) is bounded in 𝐵_𝑟0, the set 𝑉_𝜀(𝑡)={(𝐹_𝛼^𝜀,𝛿𝑥)(𝑡)∶𝑥∈𝐵_𝑟0} is relatively compact in 𝑋_𝛾 [4]. On the other hand, using (𝐻₁) and Holder’s inequality, we have(𝐹_𝛼𝑥)(𝑡)−(𝐹_𝛼^𝜀,𝛿𝑥)(𝑡)_𝛾≤𝑞^𝑡₀^𝛿₀𝜃(𝑡−𝑠)^𝑞−1𝜉_𝑞(𝜃)×𝑇((𝑡−𝑠)^𝑞𝜃)[𝐵𝑣(𝑠)+𝑠^𝑛𝑓(𝑠,𝑥(𝑠),(𝐻𝑥)(𝑠))]_𝛾𝑑𝜃𝑑𝑠+𝑞^𝑡_𝑡−𝜀^∞_𝛿𝜃(𝑡−𝑠)^𝑞−1𝜉_𝑞(𝜃)×𝑇((𝑡−𝑠)^𝑞𝜃)[𝐵𝑣(𝑠)+𝑠^𝑛𝑓(𝑠,𝑥(𝑠),(𝐻𝑥)(𝑠))]_𝛾𝑑𝜃𝑑𝑠≤𝑀_𝐵𝑀𝑞𝑝−1𝑞𝑝−1^{(𝑝−1)/𝑝}𝑡^{1+𝑞−1/𝑝}‖𝑣(𝑠)‖_𝛾^𝛿₀𝜃𝜉_𝑞(𝜃)𝑑𝜃+𝑀₁𝑀𝑞𝑝−1𝑝𝑞−1^{(𝑝−1)/𝑝}𝑝−1𝑛𝑝²+𝑝−1^{(𝑝−1)/𝑝2}𝑏^{𝑛+𝑞−1/𝑝2}×^𝛿₀𝜃𝜉_𝑞(𝜃)+𝑀𝑑𝜃_𝐵𝑀𝑞Γ𝑞+1𝑝−1𝑞𝑝−1^{(𝑝−1)/𝑝}𝜀^{1+𝑞−1/𝑝}‖𝑣(𝑠)‖_𝛾+𝑀₁𝑀𝑞Γ𝑞+1𝑝−1𝑝𝑞−1^{(𝑝−1)/𝑝}𝑝−1𝑛𝑝²+𝑝−1^{(𝑝−1)/𝑝2}𝜀^{𝑛+𝑞−1/𝑝2}. (22)This implies that there are relatively compact sets 𝑉_𝜀(𝑡) arbitrarily close to the set 𝑉(𝑡) for each 𝑡∈(0,𝑏]. Thus, 𝑉(𝑡) is relatively compact in 𝑋_𝛾 for all 𝑡∈(0,𝑏] since it is compact at 𝑡=0. Therefore, we have the relatively compactness in 𝑋_𝛾 for all 𝑡∈[0,𝑏].
Next we show that 𝑉(𝑡)={(𝐹_𝛼𝑥)(𝑡)∶𝑥∈𝐵_𝑟0} is an equicontinuous family of functions on [0,𝑏]. By the compactness of the set, we can prove that the functions 𝐹_𝛼𝑥,𝑥∈𝐵_𝑟0 are equicontinuous at 𝑡=0. For any 𝑥∈𝐵_𝑟0 and 0<𝑡₁<𝑡₂≤𝑏, we have 𝑧(𝑡₂)−𝑧(𝑡₁)_𝛾≤(𝒯(𝑡₂)−𝒯(𝑡₁))[𝑥₀+𝑔(𝑥)]_𝛾+^𝑡1₀𝑡₂𝑠^𝑞−1[𝒮(𝑡₂−𝑠)−𝒮(𝑡₁−𝑠)]𝐵𝑣(𝑠)𝑑𝑠_𝛾+^𝑡1₀𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^𝑞−1𝒮(𝑡₁−𝑠)𝐵𝑣(𝑠)𝑑𝑠_𝛾+^𝑡2_𝑡1𝑡₂−𝑠^𝑞−1𝒮(𝑡₂−𝑠)𝐵𝑣(𝑠)𝑑𝑠_𝛾+^𝑡1₀𝑡₂−𝑠^𝑞−1𝑠^𝑛𝒮𝑡₂−𝑠−𝒮𝑡₁((𝑠),(𝑠−𝑠×𝑓𝑠,𝑥𝐻𝑥)())𝑑𝑠_𝛾+^𝑡1₀𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^𝑞−1𝑠^𝑛𝒮𝑡₂((𝑠),(𝑠−𝑠×𝑓𝑠,𝑥𝐻𝑥)())𝑑𝑠_𝛾+^𝑡2_𝑡1𝑡₂−𝑠^𝑞−1𝑠^𝑛𝒮(𝑡₂−𝑠)𝑓(𝑠,𝑥(𝑠),(𝐻𝑥)(𝑠))𝑑𝑠_𝛾≤𝐼₁+𝐼₂+𝐼₃+𝐼₄+𝐼₅+𝐼₆+𝐼₇. (23) Now, we have to prove that 𝐼₁, 𝐼₂, 𝐼₃, 𝐼₄, 𝐼₅, 𝐼₆, and 𝐼₇ tend to 0 independently of 𝑥∈𝐵_𝑟0 as 𝑡₂−𝑡₁→0. By Lemma 6, one can easily show that lim_{𝑡1→𝑡2}𝐼₁=0. By using Lemmas 5 and 6 and the Lagrange mean value theorem and following the similar procedure as in the proof of Theorem 3.2 of [4], one can deduce that lim_{𝑡1→𝑡2}𝐼₅,𝐼₆,𝐼₇=0. In a similar way, we can obtain 𝐼₂≤Γ(2)Γ𝑀𝑞+1𝛾^1+𝛾𝑝−1²(𝑞−1)𝑝²+𝑝−1^{2(𝑝−1)2/𝑝2}×𝑏^{((𝑞−1)𝑝2+(𝑝−1)2)/𝑝2}×^𝑏₀𝑡₂−𝑠^−𝑞𝛾−𝑡₁−𝑠^{−𝑞𝛾𝑝2/(𝑝−1)}𝑑𝑠^{(𝑝−1)/𝑝2}×𝑏𝑀_𝐵‖𝑣(𝑠,𝐼)‖₃≤𝑀𝑞Γ𝑞+1^𝑏₀𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^{𝑞−1𝑝/(𝑝−1)}𝑑𝑠^{(𝑝−1)/𝑝}×𝑏𝑀_𝐵‖𝑣(𝑠,𝐼)‖₄≤𝑀𝑞Γ𝑞+1𝑝−1𝑝𝑞−1^{(𝑝−1)/𝑝}𝑡₂−𝑡₁^{((𝑞−1)𝑝+𝑝−1)/𝑝}×𝑀_𝐵‖𝑣(𝑠)‖𝑡₂−𝑡₁. (24) By the Lagrange mean value theorem and Lemmas 5 and 6, it can be easily seen that 𝐼₂, 𝐼₃, and 𝐼₄ tend to 0 as 𝑡₂−𝑡₁→0. Thus, the right-hand side of (23) tends to 0 as 𝑡₂−𝑡₁→0 which means that {(𝐹_𝛼𝑥)∶𝑥∈𝐵_𝑟0} is a family of equicontinuous functions. It can be easily seen that for all 𝛼>0, 𝐹_𝛼 is continuous on 𝐶(𝐽,𝑋_𝛾). Hence, by Arzela-Ascoli’s theorem, 𝐹_𝛼 is compact. By Schauder’s fixed point theorem, 𝐹_𝛼 has a fixed point 𝑥∈𝐵_𝑟0. Thus, the control system (1) has at least one mild solution on [0,𝑏].

Theorem 8. Assume that (𝐻₁)--(𝐻₆) hold. Then, system (1) is approximately controllable on [0,𝑏].

Proof. Linear system (8) is approximately controllable, by Lemma 4, and it can be seen that approximate controllability of (8) is equivalent to convergence of the operator 𝛼𝑅(𝛼,Γ^𝑏₀) to zero operator in the strong operator topology as 𝛼→0⁺. Let 𝑥_𝛼(⋅) be a fixed point of 𝐹_𝛼 in 𝐵_𝑟. Any fixed point of 𝐹_𝛼 is a mild solution of (1) under the control 𝑢_𝛼(𝑡)=𝐵^∗𝒮^∗(𝑏−𝑡)𝑅(𝛼,Γ^𝑏₀)𝑝(𝑥_𝛼) and satisfies 𝑥_𝛼(𝑏)=𝑥_𝑏−𝛼𝑅(𝛼,Γ^𝑏₀)𝑝(𝑥_𝛼). By the condition (𝐻₅), we have ∫^𝑏₀‖𝑓(𝑠,𝑥_𝛼(𝑠),(𝐻𝑥_𝛼)(𝑠))‖²𝑑𝑠≤𝑏𝑁², and consequently, the sequence {𝑓(𝑠,𝑥_𝛼(𝑠),(𝐻𝑥_𝛼)(𝑠))} is bounded in 𝐿²(𝐽,𝑋_𝛾). Then there is a subsequence denoted by {𝑓(𝑠,𝑥_𝛼(𝑠),(𝐻𝑥_𝛼)(𝑠))} that converges weakly to say 𝑓(𝑠) in 𝐿²(𝐽,𝑋_𝛾). Now, the compactness of 𝒮(𝑡) implies that 𝒮(𝑏−𝑠)𝑓(𝑠,𝑥_𝛼(𝑠),(𝐻𝑥_𝛼)(𝑠))→𝒮(𝑏−𝑠)𝑓(𝑠) in 𝐿²(𝐽,𝑋_𝛾), and we obtain 𝑝𝑥_𝛼=−𝑤_𝑏0(𝑏−𝑠)^𝑞−1𝑠^𝑛𝒮()×𝑓𝑥𝑏−𝑠𝑠,_𝛼(𝑠),𝐻𝑥_𝛼(𝑠)−𝑓(𝑠)𝑑𝑠≤sup_{0≤𝑡≤𝑏}^𝑡₀(𝑡−𝑠)^𝑞−1𝑠^𝑛𝒮()×𝑓𝑥𝑡−𝑠𝑠,_𝛼(𝑠),𝐻𝑥_𝛼(𝑠)−𝑓(𝑠)𝑑𝑠⟶0as𝛼⟶0⁺, (25) where 𝑤=𝑥_𝑏−𝒯(𝑏)𝑥₀+𝑔𝑥_𝛼−^𝑏₀(𝑏−𝑠)^𝑞−1𝒮(𝑏−𝑠)𝑠^𝑛𝑓(𝑠)𝑑𝑠. (26) Then, we obtain 𝑥_𝛼(𝑏)−𝑥_𝑏≤𝛼𝑅𝛼,Γ^𝑏₀(𝑤+)𝛼𝑅𝛼,Γ^𝑏₀𝑝𝑥_𝛼≤−𝑤𝛼𝑅𝛼,Γ^𝑏₀(𝑤)+𝑝𝑥_𝛼−𝑤. (27) It follows from Lemma 4 and estimation (25) that ‖𝑥_𝛼(𝑏)−𝑥_𝑏‖→0as𝛼→0⁺. This proves the approximate controllability of (1). The proof is completed.

Example 9. As an application of the obtained theory, we consider a control system which is represented by fractional partial differential equation. Let 𝑋=𝑈=𝐿²([0,1]), and let 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 be the operator defined by 𝐴𝑧=−𝑧^′′ with domain 𝐷(𝐴)=𝑧∈𝐿²([0,1])∶𝑧,𝑧^′areabsolutelycontinuous,𝑧^′′∈𝐿²([0,1]),𝑧(0)=𝑧(1)=0. (28) Let 𝐷(𝐴^1/2)={𝑧(⋅)∈𝑋∶∑^∞_𝑛=1(𝑧,𝑒_𝑛)𝑒_𝑛∈𝑋}, and let 𝑋_1/2=𝐷(𝐴^1/2,‖⋅‖_1/2), where ‖⋅‖_1/2=‖𝐴^1/2‖₂. For each 𝑧∈𝐷(𝐴^1/2) and ‖𝐴^−1/2‖=1, the operator 𝐴^1/2 is given by 𝐴^1/2𝑧=∑^∞_𝑛=1𝑛²(𝑧,𝑒_𝑛)𝑒_𝑛, where 𝑒_𝑛(𝑦)=√2sin(𝑛𝑦) and 0≤𝑦≤1, 𝑛=1,2,… is the orthogonal set of eigenvectors of 𝐴. It is well known that −𝐴 generates a compact analytic semigroup of uniformly bounded linear operator.
Consider the fractional partial differential equation with control in the following form: 𝜕^𝑞𝜕𝑡^𝑞𝑥=𝜕𝑡,𝑦₂𝜕𝑦²𝑥+𝑒𝑡,𝑦+𝜇𝑡,𝑦_−𝑡𝑒^𝑡+𝑒^−𝑡cos𝑥𝑡,𝑦+^𝑏₀cos(𝑡𝑠)𝑥𝑠,𝑦𝑑𝑠+𝑒^−𝑡,[][],𝑥()()𝑥=𝑡∈0,1,𝑦∈0,1𝑡,0=𝑥𝑡,1=0,𝑡>0,0,𝑦^𝜎_𝑖=0¹₀𝑘₁𝑦,𝜏𝑥𝑡_𝑖+,𝜏𝑑𝜏_𝜎𝑖=010𝑘₂𝜕𝑦,𝜏𝑥𝑡𝜕𝜏^𝑖,𝜏𝑑𝜏, (29) where 0≤𝑞≤1; 𝜇∶[0,1]×[0,1]→[0,1] is continuous; 𝑘₁,𝑘₂∈𝐿²([0,1]×[0,1]); 𝜎∈𝑁.
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≤1. Let 𝐶([0,1],𝑋_1/2) be a Banach space equipped with sup norm. Further, we define the operators 𝑓∶[0,1]×𝑋_1/2→𝑋_1/2 and 𝑔∶𝐶([0,1],𝑋_1/2)→𝑋_1/2 by 𝑓((𝑡),(𝑡𝑦=𝑒𝑡,𝑥𝐻𝑥)())_−𝑡𝑒^𝑡+𝑒^−𝑡cos𝑥(𝑡)+^𝑏₀cos(𝑡𝑠)𝑥(𝑠)𝑑𝑠𝑦+𝑒^−𝑡,𝑔(𝑥)𝑦=^𝜎_𝑖=0(𝒦𝑥)𝑡_𝑖𝑦,for𝑥∈𝐶𝐽,𝑋_1/2, (30) where (𝐻𝑥)(𝑡)(𝑦)=∫^𝑏₀cos(𝑡𝑠)𝑥(𝑠,𝑦)𝑑𝑠,(𝒦𝜙)(𝜏)=∫¹₀𝑘₁(𝑦,𝜏)𝜙(𝜏)𝑑𝜏+∫¹₀𝑘₂(𝑦,𝜏)(𝜕/𝜕𝜏)𝜙(𝜏)𝑑𝜏 for all 𝜙∈𝑋_1/2. Moreover, the linear fractional control system corresponding to (29) is approximately controllable. Thus, with the above choices of 𝐴, 𝐵, 𝑓, and 𝑔, the system (29) can be written to the abstract form of (1). Therefore, all the conditions of Theorem 8 are satisfied. Hence, by Theorem 8, the fractional nonlinear integrodifferential system (29) is approximately controllable on [0,1].

Note. The considered system (1) is of the more general form, and in particular, if functions 𝑓 and ℎ have various physical meanings, it is important to note that (1) has a great diversity. The result in this paper assumes that the linear system has a compact semigroup and consequently is not completely controllable. Moreover, the functions with Lipschitz condition are considerably strong when one discusses various applications in the real-world problems. Such an assumption is removed from this paper.