Approximate Controllability of Sobolev Type Nonlocal Fractional Stochastic Dynamic Systems in Hilbert Spaces

Abstract

We study a class of fractional stochastic dynamic control systems of Sobolev type in Hilbert spaces. We use fixed point technique, fractional calculus, stochastic analysis, and methods adopted directly from deterministic control problems for the main results. A new set of sufficient conditions for approximate controllability is formulated and proved. An example is also given to provide the obtained theory.

1. Introduction

We are concerned with the following nonlocal fractional stochastic system of Sobolev type: ^𝐶𝐷^𝑞_𝑡[(𝑡(𝑡)(𝑡)((𝑡((𝑡(𝑡)𝐿𝑥)]=𝑀𝑥+𝐵𝑢+𝑓𝑡,𝑥))+𝜎𝑡,𝑥))𝑑𝑤,𝑥(0)(𝑥(𝑡𝑑𝑡+𝑔))=𝑥₀, (1) where ^𝐶𝐷^𝑞_𝑡 is the Caputo fractional derivative of order 𝑞, 0<𝑞≤1, and 𝑡∈𝐽=[0,𝑏]. Let 𝑋 and 𝑌 be two Hilbert spaces, and the state 𝑥(⋅) takes its values in 𝑋. We assume that the operators 𝐿 and 𝑀 are defined on domains contained in 𝑋 and ranges contained in 𝑌, the control function 𝑢(⋅) belongs to the space 𝐿²_Γ(𝐽,𝑈), a Hilbert space of admissible control functions with 𝑈 as a Hilbert space, and 𝐵 is a bounded linear operator from 𝑈 into 𝑌. It is also assumed that 𝑓∶𝐽×𝑋→𝑌, 𝑔∶𝐶(𝐽∶𝑋)→𝑌 and 𝜎∶𝐽×𝑋→𝐿⁰₂ are appropriate functions; 𝑥₀ is Γ₀ measurable 𝑋-valued random variables independent of 𝑤. Here Γ, Γ₀, 𝐿⁰₂, and 𝑤 will be specified later.

The field of fractional differential equations and its applications has gained a lot of importance during the past three decades, mainly because it has become a powerful tool in modeling several complex phenomena in numerous seemingly diverse and widespread fields of science and engineering [1–8] Recently, there has been a significant development in the existence and uniqueness of solutions of initial and boundary value problem for fractional evolution systems [9].

Controllability is one of the important fundamental concepts in mathematical control theory and plays a vital role in both deterministic and stochastic control systems. Since the controllability notion has extensive industrial and biological applications, in the literature, there are many different notions of controllability, both for linear and nonlinear dynamical systems. Controllability of the deterministic and stochastic dynamical control systems in infinite dimensional spaces is well developed using different kinds of approaches. It should be mentioned that the theory of controllability for nonlinear fractional dynamical systems is still in the initial stage. There are few works in controllability problems for different kinds of systems described by fractional differential equations [10, 11].

The exact controllability for semilinear fractional order system, when the nonlinear term is independent of the control function, is proved by many authors [12–15]. In these papers, the authors have proved the exact controllability by assuming that the controllability operator has an induced inverse on a quotient space. However, if the semigroup associated with the system is compact, then the controllability operator is also compact and hence the induced inverse does not exist because the state space is infinite dimensional [16]. Thus, the concept of exact controllability is too strong and has limited applicability, and the approximate controllability is a weaker concept than complete controllability and it is completely adequate in applications for these control systems.

In [17, 18] the approximate controllability of first order delay control systems has been proved when nonlinear term is a function of both state function and control function by assuming that the corresponding linear system is approximately controllable. To prove the approximate controllability of first order system, with or without delay, a relation between the reachable set of a semilinear system and that of the corresponding linear system is proved in [19–23]. There are several papers devoted to the approximate controllability for semilinear control systems, when the nonlinear term is independent of control function [24–27].

Stochastic differential equations have attracted great interest due to their applications in various fields of science and engineering. There are many interesting results on the theory and applications of stochastic differential equations (see [12, 28–32] and the references therein). To build more realistic models in economics, social sciences, chemistry, finance, physics, and other areas, stochastic effects need to be taken into account. Therefore, many real world problems can be modeled by stochastic differential equations. The deterministic models often fluctuate due to noise, so we must move from deterministic control to stochastic control problems.

In the present literature there are only a limited number of papers that deal with the approximate controllability of fractional stochastic systems [33], as well as with the existence and controllability results of fractional evolution equations of Sobolev type [34].

Sakthivel et al. [35] studied the approximate controllability of a class of dynamic control systems described by nonlinear fractional stochastic differential equations in Hilbert spaces. More recent works can be found in [10, 11]. Debbouche et al. [4] established a class of fractional nonlocal nonlinear integrodifferential equations of Sobolev type using new solution operators. Feckan et al. [36] presented the controllability results corresponding to two admissible control sets for fractional functional evolution equations of Sobolev type in Banach spaces with the help of two new characteristic solution operators and their properties, such as boundedness and compactness.

It should be mentioned that there is no work yet reported on the approximate controllability of Sobolev type fractional deterministic stochastic control systems. Motivated by the above facts, in this paper we establish the approximate controllability for a class of fractional stochastic dynamic systems of Sobolev Type with nonlocal conditions in Hilbert spaces.

The paper is organized as follows: in Section 2, we present some essential facts in fractional calculus, semigroup theory, stochastic analysis, and control theory that will be used to obtain our main results. In Section 3, we state and prove existence and approximate controllability results for Sobolev type fractional stochastic system (1). The last sections deal with an illustrative example and a discussion for possible future work in this direction.

2. Preliminaries

In this section we give some basic definitions, notations, properties, and lemmas, which will be used throughout the work. In particular, we state main properties of fractional calculus [37–40], well-known facts in semigroup theory [41–43], and elementary principles of stochastic analysis [31, 44].

Definition 1. The fractional integral of order 𝛼>0 of a function 𝑓∈𝐿¹([𝑎,𝑏],ℝ⁺) is given by 𝐼^𝛼_𝑎𝑓(𝑡)=1Γ(𝛼)^𝑡_𝑎(𝑡−𝑠)^𝛼−1𝑓(𝑠)𝑑𝑠, (2) where Γ is the gamma function. If 𝑎=0, we can write 𝐼^𝛼𝑓(𝑡)=(𝑔_𝛼∗𝑓)(𝑡), where 𝑔_𝛼(𝑡)1∶=Γ(𝛼)𝑡^𝛼−1,𝑡>0,0,𝑡≤0, (3) and as usual, ∗ denotes the convolution of functions. Moreover, lim_𝛼→0𝑔_𝛼(𝑡)=𝛿(𝑡), with 𝛿 the delta Dirac function.

Definition 2. The Riemann-Liouville derivative of order 𝑛−1<𝛼<𝑛, 𝑛∈ℕ, for a function 𝑓∈𝐶([0,∞)) is given by ^𝐿𝐷^𝛼𝑓(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑛𝑡₀𝑓(𝑠)(𝑡−𝑠)^{𝛼+1−𝑛}𝑑𝑠,𝑡>0. (4)

Definition 3. The Caputo derivative of order 𝑛−1<𝛼<𝑛, 𝑛∈ℕ, for a function 𝑓∈𝐶([0,∞)) is given by ^𝐶𝐷^𝛼𝑓(𝑡)=^𝐿𝐷^𝛼𝑓(𝑡)−^𝑛−1_𝑘=0𝑡^𝑘𝑓𝑘!^(𝑘)(0),𝑡>0. (5)

Remark 4. The following properties hold (see, e.g., [45]).(i)If 𝑓∈𝐶^𝑛([0,∞)), then ^𝐶𝐷^𝛼𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡₀𝑓^(𝑛)(𝑠)(𝑡−𝑠)^{𝛼+1−𝑛}𝑑𝑠=𝐼^𝑛−𝛼𝑓^𝑛(𝑡),𝑡>0,𝑛−1<𝛼<𝑛,𝑛∈ℕ. (6)(ii)The Caputo derivative of a constant is equal to zero.(iii)If 𝑓 is an abstract function with values in 𝑋, then the integrals which appear in Definitions 1–3 are taken in Bochner’s sense.

We introduce the following assumptions on the operators 𝐿 and 𝑀.(H₁)𝐿 and 𝑀 are linear operators, and 𝑀 is closed.(H₂)𝐷(𝐿)⊂𝐷(𝑀) and 𝐿 is bijective.(H₃)𝐿⁻¹∶𝑌→𝐷(𝐿)⊂𝑋 is a linear compact operator.

Remark 5. From (H₃), we deduce that 𝐿⁻¹ is a bounded operator; for short, we denote 𝐶=‖𝐿⁻¹‖. Note (H₃) also implies that 𝐿 is closed since 𝐿⁻¹ is closed and injective; then its inverse is also closed. It comes from (H₁)–(H₃) and the closed graph theorem; we obtain the boundedness of the linear operator 𝑀𝐿⁻¹∶𝑌→𝑌. Consequently, 𝑀𝐿⁻¹ generates a semigroup {𝑆(𝑡)∶=𝑒^{𝑀𝐿−1𝑡},𝑡≥0}. We suppose that 𝑀₀∶=sup_𝑡≥0‖𝑆(𝑡)‖<∞. According to previous definitions, it is suitable to rewrite problem (1) as the equivalent integral equation 𝐿𝑥(𝑡)=𝐿𝑥₀(𝑥+1−𝑔)Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1×(𝑠)(𝑠)((𝑠+1𝑀𝑥+𝐵𝑢+𝑓𝑠,𝑥))𝑑𝑠Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝜎(𝑠,𝑥(𝑠))𝑑𝑤(𝑠), (7) provided the integral in (7) exists. Before formulating the definition of mild solution of (1), we first give the following definitions, corollaries, lemmas, and notations.

Let (Ω,Γ,𝑃) be a complete probability space equipped with a normal filtration Γ_𝑡, 𝑡∈𝐽 satisfying the usual conditions (i.e., right continuous and Γ₀ containing all 𝑃-null sets). We consider four real separable spaces 𝑋, 𝑌, 𝐸, and 𝑈 and 𝑄-Wiener process on (Ω,Γ_𝑏,𝑃) with the linear bounded covariance operator 𝑄 such that tr𝑄<∞. We assume that there exists a complete orthonormal system {𝑒_𝑛}_𝑛≥1 in 𝐸, a bounded sequence of nonnegative real numbers {𝜆_𝑛} such that 𝑄𝑒_𝑛=𝜆_𝑛𝑒_𝑛, 𝑛=1,2,…, and a sequence {𝛽_𝑛}_𝑛≥1 of independent Brownian motions such that ⟨𝑤(𝑡),𝑒⟩=^∞_𝑛=1𝜆_𝑛𝑒_𝑛,𝑒𝛽_𝑛(𝑡),𝑒∈𝐸,𝑡∈𝐽. (8) and Γ_𝑡=Γ^𝑤_𝑡, where Γ^𝑤_𝑡 is the sigma algebra generated by {𝑤(𝑠)∶0≤𝑠≤𝑡}. Let 𝐿⁰₂=𝐿₂(𝑄^1/2𝐸;𝑋) be the space of all Hilbert-Schmidt operators from 𝑄^1/2𝐸 to 𝑋 with the inner product ⟨𝜓,𝜋⟩𝐿⁰₂=tr[𝜓𝑄𝜋^∗]. Let 𝐿²(Γ_𝑏,𝑋) be the Banach space of all Γ_𝑏-measurable square integrable random variables with values in the Hilbert space 𝑋. Let 𝐸(⋅) denote the expectation with respect to the measure 𝑃. Let 𝐶(𝐽;𝐿²(Γ,𝑋)) be the Hilbert space of continuous maps from 𝐽 into 𝐿²(Γ,𝑋) satisfying sup_𝑡∈𝐽𝐸‖𝑥(𝑡)‖²<∞. Let 𝐻₂(𝐽;𝑋) be a closed subspace of 𝐶(𝐽;𝐿²(Γ,𝑋)) consisting of measurable and Γ_𝑡-adapted 𝑋-valued process 𝑥∈𝐶(𝐽;𝐿²(Γ,𝑋)) endowed with the norm ‖𝑥‖_𝐻2=(sup_𝑡∈𝐽𝐸‖𝑥(𝑡)‖²_𝑋)^1/2. For details, we refer the reader to [35, 44] and references therein.

The following results will be used through out this paper.

Lemma 6 (see [33]). Let 𝐺∶𝐽×Ω→𝐿⁰₂ be a strongly measurable mapping such that ∫^𝑏₀𝐸‖𝐺(𝑡)‖^𝑝_𝐿02𝑑𝑡<∞. Then 𝐸^𝑡₀𝐺(𝑠)𝑑𝑤(𝑠)^𝑝≤𝐿_𝐺^𝑡₀𝐸‖𝐺(𝑠)‖^𝑝_𝐿02𝑑𝑠 (9) for all 0≤𝑡≤𝑏 and 𝑝≥2, where 𝐿_𝐺 is the constant involving 𝑝 and 𝑏.

Now, we present the mild solution of the problem (1).

Definition 7 (compare with [46, 47] and [36, 45]). A stochastic process 𝑥∈𝐻₂(𝐽,𝑋) is a mild solution of (1) if, for each control 𝑢∈𝐿²_Γ(𝐽,𝑈), it satisfies the following integral equation: 𝑥(𝑡)=𝒮(𝑡)𝐿𝑥₀(𝑥+−𝑔)_𝑡0(𝑡−𝑠)^𝑞−1𝒯((𝑠)((𝑠+𝑡−𝑠)𝐵𝑢+𝑓𝑠,𝑥))𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝒯(𝑡−𝑠)𝜎(𝑠,𝑥(𝑠))𝑑𝑤(𝑠), (10) where 𝒮(𝑡) and 𝒯(𝑡) are characteristic operators given by 𝒮(𝑡)=^∞₀𝐿⁻¹𝜉_𝑞(𝜃)𝑆𝑡^𝑞𝜃𝒯(𝑡)𝑑𝜃,=𝑞^∞₀𝐿⁻¹𝜃𝜉_𝑞(𝜃)𝑆𝑡^𝑞𝜃𝑑𝜃. (11)

Here, 𝑆(𝑡) is a 𝐶₀-semigroup generated by the linear operator 𝑀𝐿⁻¹∶𝑌→𝑌; 𝜉_𝑞 is a probability density function defined on (0,∞); that is, 𝜉_𝑞(𝜃)≥0, 𝜃∈(0,∞) and ∫^∞₀𝜉_𝑞(𝜃)𝑑𝜃=1.

Lemma 8 (see [45, 48, 49]). The operators {𝒮(𝑡)}_𝑡≥0 and {𝒯(𝑡)}_𝑡≥0 are strongly continuous; that is, for 𝑥∈𝑋 and 0≤𝑡₁<𝑡₂≤𝑏, one has ‖𝒮(𝑡₂)𝑥−𝒮(𝑡₁)𝑥‖→0 and ‖𝒯(𝑡₂)𝑥−𝒯(𝑡₁)𝑥‖→0 as 𝑡₂→𝑡₁.

We impose the following conditions on data of the problem.(i)For any fixed 𝑡≥0, 𝒯(𝑡) and 𝒮(𝑡) are bounded linear operators; that is, for any 𝑥∈𝑋, ‖𝒯(𝑡)𝑥‖≤𝐶𝑀₀‖𝑥‖,‖𝒮(𝑡)𝑥‖≤𝐶𝑀₀Γ𝑞‖𝑥‖. (12)(ii)The functions 𝑓∶𝐽×𝑋→𝑌, 𝜎∶𝐽×𝑋→𝐿⁰₂ and 𝑔∶𝐶(𝐽∶𝑋)→𝑌 satisfy linear growth and Lipschitz conditions. Moreover, there exist positive constants 𝑁₁,𝑁₂>0,??𝐿₁,𝐿₂>0, and 𝑘₁,𝑘₂>0 such that 𝑓(𝑡,𝑥)−𝑓(𝑡,𝑦)²≤𝑁₁𝑥−𝑦²,𝑓(𝑡,𝑥)²≤𝑁₂1+‖𝑥‖²,𝜎(𝑡,𝑥)−𝜎(𝑡,𝑦)²_𝐿02≤𝐿₁𝑥−𝑦²,‖𝜎(𝑡,𝑥)‖²_𝐿02≤𝐿₂1+‖𝑥‖²,𝑔(𝑥)−𝑔(𝑦)²≤𝑘₁𝑥−𝑦²,𝑔(𝑥)²≤𝑘₂1+‖𝑥‖². (13)(iii)The linear stochastic system is approximately controllable on 𝐽.

For each 0≤𝑡<𝑏, the operator 𝛼(𝛼𝐼+Ψ^𝑏₀)⁻¹→0 in the strong operator topology as 𝛼→0⁺, where Ψ^𝑏₀=∫^𝑏₀(𝑏−𝑠)^2(𝑞−1)𝒯(𝑏−𝑠)𝐵𝐵^∗𝒯^∗(𝑏−𝑠)𝑑𝑠 is the controllability Gramian. Here 𝐵^∗ denotes the adjoint of 𝐵, and 𝒯^∗(𝑡) is the adjoint of 𝒯(𝑡).

Observe that Sobolev type nonlocal linear fractional deterministic control system ^𝐶𝐷^𝑞_𝑡[(𝑡(𝑡)(𝑡)𝑥(0)(𝑥(𝑡𝐿𝑥)]=𝑀𝑥+𝐵𝑢,𝑡∈𝐽,+𝑔))=𝑥₀, (14) corresponding to (1) is approximately controllable on 𝐽 iff the operator 𝛼(𝛼𝐼+Ψ^𝑏₀)⁻¹→0 strongly as 𝛼→0⁺. The approximate controllability for linear fractional deterministic control system (14) is a natural generalization of approximate controllability of linear first order control system (𝑞=1, 𝑔=0, and 𝐿 is the identity) [50].

Definition 9. The system (1) is approximately controllable on 𝐽 if ℜ(𝑏)=𝐿²(Ω,Γ_𝑏,𝑋), where ℜ(𝑏)=𝑥(𝑏)=𝑥(𝑏,𝑢)∶𝑢∈𝐿²_Γ(𝐽,𝑈). (15) Here 𝐿²_Γ(𝐽,𝑈) is the closed subspace of 𝐿²_Γ(𝐽×Ω;𝑈), consisting of all Γ_𝑡-adapted, 𝑈-valued stochastic processes.

The following lemma is required to define the control function [35].

Lemma 10. For any 𝑥_𝑏∈𝐿²(Γ_𝑏,𝑋), there exists 𝜑∈𝐿²_Γ(Ω;𝐿²(0,𝑏;𝐿⁰₂)) such that 𝑥_𝑏=𝐸𝑥_𝑏+∫^𝑏₀𝜑(𝑠)𝑑𝑤(𝑠).
Now for any 𝛼>0 and 𝑥_𝑏∈𝐿²(Γ_𝑏,𝑋), one defines the control function in the following form: 𝑢^𝛼(𝑡,𝑥)=𝐵^∗(𝑏−𝑡)^𝑞−1𝒯^∗()×𝑏−𝑡𝛼𝐼+Ψ^𝑏₀⁻¹𝐸𝑥_𝑏−𝒮(𝑏)𝐿𝑥₀(𝑥+−𝑔)_𝑡0𝛼𝐼+Ψ^𝑏₀⁻¹𝜑(𝑠)𝑑𝑤(𝑠)−𝐵^∗(𝑏−𝑡)^𝑞−1𝒯^∗()×𝑏−𝑡_𝑡0𝛼𝐼+Ψ^𝑏₀⁻¹(𝑏−𝑠)^𝑞−1𝒯(𝑏−𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠−𝐵^∗(𝑏−𝑡)^𝑞−1𝒯^∗()×𝑏−𝑡_𝑡0𝛼𝐼+Ψ^𝑏₀⁻¹(𝑏−𝑠)^𝑞−1𝒯()((𝑠(𝑠).𝑏−𝑠×𝜎𝑠,𝑥))𝑑𝑤 (16)

Lemma 11. There exist positive real constants 𝑀, 𝑁 such that, for all 𝑥,𝑦∈𝐻₂, one has 𝐸𝑢^𝛼(𝑡,𝑥)−𝑢^𝛼(𝑡,𝑦)²≤𝑀𝐸𝑥(𝑡)−𝑦(𝑡)²,𝐸𝑢^𝛼(𝑡,𝑥)²≤𝑁1𝑏+𝐸‖𝑥(𝑡)‖². (17)(18)

Proof. We start to prove (17). Let 𝑥,𝑦∈𝐻₂; from Hölder’s inequality, Lemma 6, and the assumption on the data, we obtain 𝐸𝑢^𝛼(𝑡,𝑥)−𝑢^𝛼(𝑡,𝑦)²≤3𝐸𝐵^∗(𝑏−𝑡)^𝑞−1𝒯^∗(𝑏−𝑡)𝛼𝐼+Ψ^𝑏₀⁻¹×𝒮(𝑏)𝐿𝑔(𝑥(𝑡))−𝑔𝑦(𝑡)²+3𝐸𝐵^∗(𝑏−𝑡)^𝑞−1𝒯^∗()×𝑏−𝑡_𝑡0𝛼𝐼+Ψ^𝑏₀⁻¹(𝑏−𝑠)^𝑞−1𝒯()×𝑓((𝑠(𝑠)𝑏−𝑠𝑠,𝑥))−𝑓𝑠,𝑦𝑑𝑠²+3𝐸𝐵^∗(𝑏−𝑡)^𝑞−1𝒯^∗()×𝑏−𝑡_𝑡0𝛼𝐼+Ψ^𝑏₀⁻¹(𝑏−𝑠)^𝑞−1𝒯()×𝜎((𝑠(𝑠)(𝑠𝑏−𝑠𝑠,𝑥))−𝜎𝑠,𝑦𝑑𝑤)²≤3𝛼²‖𝐵‖²(𝑏)^2𝑞−2𝐶𝑀₀²𝐶𝑀₀Γ(𝑞)²‖𝐿‖²𝑘₁𝐸𝑥(𝑡)−𝑦(𝑡)²+3𝛼²‖𝐵‖²(𝑏)^2𝑞−2𝐶𝑀₀Γ𝑞⁴𝑏^2𝑞−1𝑁2𝑞−1¹×^𝑡₀𝐸𝑥(𝑠)−𝑦(𝑠)²+3𝑑𝑠𝛼²‖𝐵‖²(𝑏)^2𝑞−2𝐶𝑀₀Γ𝑞⁴𝑏^2𝑞−1𝐿2𝑞−1¹×^𝑡₀𝐸𝑥(𝑠)−𝑦(𝑠)²≤𝑑𝑠𝑀𝐸𝑥(𝑡)−𝑦(𝑡)², (19) where 𝑀=(3/𝛼²)‖𝐵‖²(𝑏)^2𝑞−2{(𝐶𝑀₀)²(𝐶𝑀₀/Γ(𝑞))²‖𝐿‖²𝑘₁+(𝐶𝑀₀/Γ(𝑞))⁴(𝑏^2𝑞−1/(2𝑞−1))𝑏[𝑁₁+𝐿₁]}. The proof of the inequality (18) can be established in a similar way to that of (17).

3. Approximate Controllability

In this section, we formulate and prove conditions for the existence and approximate controllability results of the nonlocal fractional stochastic dynamic control system of Sobolev type (1) using the contraction mapping principle. For any 𝛼>0, define the operator 𝐹_𝛼∶𝐻₂→𝐻₂ by 𝐹_𝛼𝑥(𝑡)=𝒮(𝑡)𝐿𝑥₀(𝑥+−𝑔)_𝑡0(𝑡−𝑠)^𝑞−1𝒯(𝑡−𝑠)𝐵𝑢^𝛼()((𝑠+𝑠,𝑥+𝑓𝑠,𝑥))𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝒯(𝑡−𝑠)𝜎(𝑠,𝑥(𝑠))𝑑𝑤(𝑠). (20)

We state and prove the following lemma, which will be used for the main results.

Lemma 12. For any 𝑥∈𝐻₂, 𝐹_𝛼(𝑥)(𝑡) is continuous on 𝐽 in 𝐿²-sense.

Proof. Let 0≤𝑡₁<𝑡₂≤𝑏. Then for any fixed 𝑥∈𝐻₂, from (20), we have 𝐸(𝐹_𝛼𝑥)(𝑡₂)−(𝐹_𝛼𝑥)(𝑡₁)²≤4⁴_𝑖=1𝐸Π^𝑥_𝑖(𝑡₂)−Π^𝑥_𝑖(𝑡₁)². (21)
We begin with the first term and get 𝐸Π^𝑥₁(𝑡₂)−Π^𝑥₁(𝑡₁)²=𝐸(𝒮(𝑡₂)−𝒮(𝑡₁))𝐿[𝑥₀−𝑔(𝑥)]²≤‖𝐿‖²𝑥₀²+𝑘₂1+‖𝑥‖²×𝐸𝒮(𝑡₂)−𝒮(𝑡₁)². (22)
The strong continuity of 𝒮(𝑡) implies that the right-hand side of the last inequality tends to zero as 𝑡₂−𝑡₁→0.
Next, it follows from Hölder’s inequality and assumptions on the data that 𝐸Π^𝑥₂(𝑡₂)−Π^𝑥₂(𝑡₁)²=𝐸^𝑡2₀𝑡₂−𝑠^𝑞−1𝒯𝑡₂−𝑠𝐵𝑢^𝛼()−𝑠,𝑥𝑑𝑠_𝑡10𝑡₁−𝑠^𝑞−1𝒯(𝑡₁−𝑠)𝐵𝑢^𝛼(𝑠,𝑥)𝑑𝑠²≤𝐸^𝑡1₀𝑡₁−𝑠^𝑞−1𝒯𝑡₂−𝑠−𝒯𝑡₁−𝑠×𝐵𝑢^𝛼(𝑠,𝑥)𝑑𝑠²+𝐸^𝑡1₀𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^𝑞−1𝒯𝑡₂−𝑠×𝐵𝑢^𝛼(𝑠,𝑥)𝑑𝑠²+𝐸^𝑡2_𝑡1𝑡₂−𝑠^𝑞−1𝒯(𝑡₂−𝑠)𝐵𝑢^𝛼(𝑠,𝑥)𝑑𝑠²≤𝑡₁^2𝑞−12𝑞−1×^𝑡1₀𝐸(𝒯(𝑡₂−𝑠)−𝒯(𝑡₁−𝑠))𝐵𝑢^𝛼(𝑠,𝑥)𝑑𝑠²+𝐶𝑀₀Γ(𝑞)²‖𝐵‖^2𝑡1₀𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^𝑞−12×𝑑𝑠_𝑡10𝐸𝑢^𝛼(𝑠,𝑥)²+𝑡𝑑𝑠₂−𝑡₁^2𝑞−11−2𝑞𝐶𝑀₀Γ(𝑞)²‖𝐵‖^2𝑡2_𝑡1𝐸𝑢^𝛼(𝑠,𝑥)²𝑑𝑠. (23) Also, we have 𝐸Π^𝑥₃(𝑡₂)−Π^𝑥₃(𝑡₁)²=𝐸^𝑡2₀𝑡₂−𝑠^𝑞−1𝒯𝑡₂𝑓((𝑠−−𝑠𝑠,𝑥))𝑑𝑠_𝑡10𝑡₁−𝑠^𝑞−1𝒯(𝑡₁−𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠²≤𝐸^𝑡1₀𝑡₁−𝑠^𝑞−1𝒯𝑡₂−𝑠−𝒯𝑡₁((𝑠−𝑠×𝑓𝑠,𝑥))𝑑𝑠²+𝐸^𝑡1₀𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^𝑞−1×𝒯𝑡₂−𝑠𝑓(𝑠,𝑥(𝑠))𝑑𝑠²+𝐸^𝑡2_𝑡1𝑡₂−𝑠^𝑞−1𝒯𝑡₂−𝑠𝑓(𝑠,𝑥(𝑠))𝑑𝑠²≤𝑡₁^2𝑞−1×2𝑞−1_𝑡10𝐸𝒯𝑡₂−𝑠−𝒯𝑡₁−𝑠𝑓(𝑠,𝑥(𝑠))𝑑𝑠²+𝐶𝑀₀Γ(𝑞)^2𝑡1₀𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^𝑞−12×𝑑𝑠_𝑡10𝐸𝑓(𝑠,𝑥(𝑠))²+𝑡𝑑𝑠₂−𝑡₁^2𝑞−11−2𝑞𝐶𝑀₀Γ(𝑞)^2𝑡2_𝑡1𝐸𝑓(𝑠,𝑥(𝑠))²𝑑𝑠. (24)
Furthermore, we use Lemma 6 and previous assumptions; we obtain 𝐸Π^𝑥₄(𝑡₂)−Π^𝑥₄(𝑡₁)²=𝐸^𝑡2₀𝑡₂−𝑠^𝑞−1𝒯𝑡₂𝜎((𝑠(𝑠)−−𝑠𝑠,𝑥))𝑑𝑤_𝑡10𝑡₁−𝑠^𝑞−1𝒯𝑡₁−𝑠𝜎(𝑠,𝑥(𝑠))𝑑𝑤(𝑠)²≤𝐸^𝑡1₀𝑡₁−𝑠^𝑞−1𝒯𝑡₂−𝑠−𝒯𝑡₁((𝑠(𝑠−𝑠×𝜎𝑠,𝑥))𝑑𝑤)²+𝐸^𝑡1₀𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^𝑞−1𝒯𝑡₂((𝑠(𝑠−𝑠×𝜎𝑠,𝑥))𝑑𝑤)²+𝐸^𝑡2_𝑡1𝑡₂−𝑠^𝑞−1𝒯(𝑡₂−𝑠)𝜎(𝑠,𝑥(𝑠))𝑑𝑤(𝑠)²≤𝐿_𝜎𝑡₁^2𝑞−12𝑞−1×^𝑡1₀𝐸(𝒯(𝑡₂−𝑠)−𝒯(𝑡₁−𝑠))𝜎(𝑠,𝑥(𝑠))𝑑𝑠²+𝐿_𝜎^𝑡1₀𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^𝑞−12×𝑑𝑠_𝑡10𝐸𝒯𝑡₂−𝑠𝜎(𝑠,𝑥(𝑠))²𝑑𝑠+𝐿_𝜎𝑡₂−𝑡₁^2𝑞−11−2𝑞𝐶𝑀₀Γ(𝑞)²×^𝑡2_𝑡1𝐸𝒯(𝑡₂−𝑠)𝜎(𝑠,𝑥(𝑠))²𝑑𝑠. (25)
Hence using the strong continuity of 𝒯(𝑡) and Lebesgue’s dominated convergence theorem, we conclude that the right-hand side of the previous inequalities tends to zero as 𝑡₂−𝑡₁→0. Thus, we conclude 𝐹_𝛼(𝑥)(𝑡) is continuous from the right of [0,𝑏). A similar argument shows that it is also continuous from the left of (0,𝑏].

Theorem 13. Assume hypotheses (i) and (ii) are satisfied. Then the system (1) has a mild solution on 𝐽.

Proof. We prove the existence of a fixed point of the operator 𝐹_𝛼 by using the contraction mapping principle. First, we show that 𝐹_𝛼(𝐻₂)⊂𝐻₂. Let 𝑥∈𝐻₂. From (20), we obtain 𝐸𝐹_𝛼𝑥(𝑡)²≤4sup⁴_{𝑡∈𝐽𝑖=1}𝐸Π^𝑥_𝑖(𝑡)². (26)
Using assumptions (i)-(ii), Lemma 11, and standard computations yields sup_𝑡∈𝐽𝐸Π^𝑥₁(𝑡)²≤𝐶²𝑀²₀‖𝐿‖²𝑥₀²+𝑘₂1+‖𝑥‖²,sup⁴_{𝑡∈𝐽𝑖=2}𝐸Π^𝑥_𝑖(𝑡)²≤𝐶𝑀₀Γ(𝑞)²𝑏^2𝑞−1‖𝐵‖2𝑞−1²𝑁1𝑏+‖𝑥‖²_𝐻2+𝐶𝑀₀Γ𝑞²𝑏^2𝑞−1𝑁2𝑞−1²−𝑏^2𝑞−1𝐿2𝑞−1²𝐿_𝜎1+‖𝑥‖²_𝐻2. (27)(28)
Hence (26)–(28) imply that 𝐸‖𝐹_𝛼𝑥‖²_𝐻2<∞. By Lemma 12, 𝐹_𝛼𝑥∈𝐻₂. Thus for each 𝛼>0, the operator 𝐹_𝛼 maps 𝐻₂ into itself. Next, we use the Banach fixed point theorem to prove that 𝐹_𝛼 has a unique fixed point in 𝐻₂. We claim that there exists a natural 𝑛 such that 𝐹^𝑛_𝛼 is a contraction on 𝐻₂. Indeed, let 𝑥,𝑦∈𝐻₂; we have 𝐸𝐹_𝛼𝑥(𝑡)−𝐹_𝛼𝑦(𝑡)²≤4⁴_𝑖=1𝐸Π^𝑥_𝑖(𝑡)−Π^𝑦_𝑖(𝑡)²≤4𝑘₁𝐶²𝑀²₀‖𝐿‖²𝐸𝑥(𝑡)−𝑦(𝑡)²+4𝐶𝑀₀Γ𝑞²×𝑀‖𝐵‖²𝑏^2𝑞−1+𝑏2𝑞−1^2𝑞−1𝑁2𝑞−1¹+𝑏^2𝑞−1𝐿2𝑞−1¹𝐿_𝜎×𝐸𝑥(𝑡)−𝑦(𝑡)². (29)
Hence, we obtain a positive real constant 𝛾(𝛼) such that 𝐸𝐹_𝛼𝑥(𝑡)−𝐹_𝛼𝑦(𝑡)²≤𝛾(𝛼)𝐸𝑥(𝑡)−𝑦(𝑡)², (30) for all 𝑡∈𝐽 and all 𝑥,𝑦∈𝐻₂. For any natural number 𝑛, it follows from the successive iteration of the previous inequality (30) that, by taking the supremum over 𝐽, (𝐹^𝑛_𝛼𝑥)(𝑡)−(𝐹^𝑛_𝛼𝑦)(𝑡)²_𝐻2≤𝛾^𝑛(𝛼)𝑛!𝑥−𝑦²_𝐻2. (31)
For any fixed 𝛼>0, for sufficiently large 𝑛,𝛾^𝑛(𝛼)/𝑛!<1. It follows from (31) that 𝐹^𝑛_𝛼 is a contraction mapping, so that the contraction principle ensures that the operator 𝐹_𝛼 has a unique fixed point 𝑥_𝛼 in 𝐻₂, which is a mild solution of (1).

Theorem 14. Assume that the assumptions (i)–(iii) hold. Further, if the functions 𝑓 and 𝜎 are uniformly bounded and {𝒯(𝑡)∶𝑡≥0} is compact, then the system (1) is approximately controllable on 𝐽.

Proof. Let 𝑥_𝛼 be a fixed point of 𝐹_𝛼. By using the stochastic Fubini theorem, it can be easily seen that 𝑥_𝛼(𝑏)=𝑥_𝑏−𝛼(𝛼𝐼+Ψ)⁻¹𝐸𝑥_𝑏−𝒮(𝑏)𝐿𝑥₀−𝑔(𝑥)+𝛼^𝑏₀𝛼𝐼+Ψ^𝑏_𝑠⁻¹(𝑏−𝑠)^𝑞−1𝒯(𝑏−𝑠)𝑓𝑠,𝑥_𝛼(𝑠)𝑑𝑠+𝛼^𝑏₀𝛼𝐼+Ψ^𝑏_𝑠⁻¹(𝑏−𝑠)^𝑞−1𝒯()𝜎×𝑏−𝑠𝑠,𝑥_𝛼(𝑠)−𝜑(𝑠)𝑑𝑤(𝑠). (32)
It follows from the assumption on 𝑓, 𝑔, and 𝜎 that there exists 𝐷>0 such that 𝑓(𝑠,𝑥_𝛼(𝑠))²+𝑔(𝑥_𝛼(𝑠))²+𝜎(𝑠,𝑥_𝛼(𝑠))²≤𝐷 (33) for all 𝑠∈𝐽. Then there is a subsequence still denoted by {𝑓(𝑠,𝑥_𝛼(𝑠)),𝑔(𝑥_𝛼(𝑠)),𝜎(𝑠,𝑥_𝛼(𝑠))} which converges weakly to some {𝑓(𝑠),𝑔(𝑠),𝜎(𝑠)} in 𝑌²×𝐿⁰₂.
From the previous equation, we have 𝐸𝑥_𝛼(𝑏)−𝑥_𝑏²≤8𝐸𝛼𝛼𝐼+Ψ^𝑏₀⁻¹(𝐸𝑥_𝑏−𝒮(𝑏)𝐿𝑥₀)²+8𝐸𝛼𝛼𝐼+Ψ^𝑏₀⁻¹²𝒮(𝑏)𝐿𝑔𝑥_𝛼(𝑠)−𝑔(𝑠)²+8𝐸𝛼(𝛼𝐼+Ψ^𝑏₀)⁻¹²𝒮(𝑏)𝐿𝑔(𝑠)²+8𝐸^𝑏₀(𝑏−𝑠)^𝑞−1𝛼𝛼𝐼+Ψ^𝑏_𝑠⁻¹𝜑(𝑠)²_𝐿02𝑑𝑠+8𝐸^𝑏₀(𝑏−𝑠)^𝑞−1𝛼𝛼𝐼+Ψ^𝑏_𝑠⁻¹×𝒯(𝑏−𝑠)𝑓𝑠,𝑥_𝛼(𝑠)−𝑓(𝑠)𝑑𝑠²+8𝐸^𝑏₀(𝑏−𝑠)^𝑞−1𝛼𝛼𝐼+Ψ^𝑏_𝑠⁻¹𝒯(𝑏−𝑠)𝑓(𝑠)𝑑𝑠²+8𝐸^𝑏₀(𝑏−𝑠)^𝑞−1𝛼𝛼𝐼+Ψ^𝑏_𝑠⁻¹×𝒯(𝑏−𝑠)(𝜎(𝑠,𝑥_𝛼(𝑠))−𝜎(𝑠))²_𝐿02𝑑𝑠+8𝐸^𝑏₀(𝑏−𝑠)^𝑞−1×𝛼𝛼𝐼+Ψ^𝑏_𝑠⁻¹𝒯(𝑏−𝑠)𝜎(𝑠)²_𝐿02𝑑𝑠. (34)
On the other hand, by assumption (iii), for all 0≤𝑠<𝑏, the operator 𝛼(𝛼𝐼+Ψ^𝑏_𝑠)⁻¹→0 strongly as 𝛼→0⁺ and moreover ‖𝛼(𝛼𝐼+Ψ^𝑏_𝑠)⁻¹‖≤1. Thus, by the Lebesgue dominated convergence theorem and the compactness of both 𝒮(𝑡) and 𝒯(𝑡) it is implied that 𝐸‖𝑥_𝛼(𝑏)−𝑥_𝑏‖²→0 as 𝛼→0⁺. Hence, we conclude the approximate controllability of (1).

In order to illustrate the abstract results of this work, we give the following example.

4. Example

Consider the fractional stochastic system with nonlocal condition of Sobolev type 𝜕^𝑞𝜕𝑡^𝑞𝑥(𝑧,𝑡)−𝑥_𝑧𝑧(−𝜕𝑧,𝑡)²𝜕𝑧²𝑥()()+𝑓((+𝜎((𝑑𝑤(𝑡)𝑧,𝑡=𝜇𝑧,𝑡𝑡,𝑥𝑧,𝑡))𝑡,𝑥𝑧,𝑡)),𝑥()+𝑑𝑡𝑧,0^𝑚_𝑘=1𝑐_𝑘𝑥𝑧,𝑡_𝑘=𝑥₀(𝑧)[],𝑥()(),𝑧∈0,10,𝑡=𝑥1,𝑡=0,𝑡∈𝐽, (35) where 0<𝑞≤1, 0<𝑡₁<⋯<𝑡_𝑚<𝑏 and 𝑐_𝑘 are positive constants, 𝑘=1,…,𝑚; the functions 𝑥(𝑡)(𝑧)=𝑥(𝑡,𝑧), 𝑓(𝑡,𝑥(𝑡))(𝑧)=𝑓(𝑡,𝑥(𝑡,𝑧)), 𝜎(𝑡,𝑥(𝑡))(𝑧)=𝜎(𝑡,𝑥(𝑡,𝑧)), and 𝑔(𝑥(𝑡))(𝑧)=∑^𝑚_𝑘=1𝑐_𝑘𝑥(𝑧,𝑡_𝑘). The bounded linear operator 𝐵∶𝑈→𝑋 is defined by 𝐵𝑢(𝑡)(𝑧)=𝜇(𝑧,𝑡), 0≤𝑧≤1, 𝑢∈𝑈; 𝑤(𝑡) is a two-sided and standard one-dimensional Brownian motion defined on the filtered probability space (Ω,Γ,𝑃).

Let 𝑋=𝐸=𝑈=𝐿²[0,1]; define the operators 𝐿∶𝐷(𝐿)⊂𝑋→𝑌 and 𝑀∶𝐷(𝑀)⊂𝑋→𝑌 by 𝐿𝑥=𝑥−𝑥^′′ and 𝑀𝑥=−𝑥^′′, where domains 𝐷(𝐿) and 𝐷(𝑀) are given by 𝑥∈𝑋∶𝑥,𝑥^′areabsolutelycontinuous,𝑥^′′𝑥(0)(1).∈𝑋,=𝑥=0 (36) Then 𝐿 and 𝑀 can be written, respectively, as 𝐿𝑥=^∞_𝑛=11+𝑛²𝑥,𝑥_𝑛𝑥_𝑛,𝑥∈𝐷(𝐿),𝑀𝑥=^∞_𝑛=1−𝑛²𝑥,𝑥_𝑛𝑥_𝑛,𝑥∈𝐷(𝑀), (37) where 𝑥_𝑛(𝑧)=(2/𝜋)sin𝑛𝑧, 𝑛=1,2,… is the orthogonal set of eigen functions of 𝑀. Further, for any 𝑥∈𝑋 we have 𝐿⁻¹𝑥=^∞_𝑛=111+𝑛²𝑥,𝑥_𝑛𝑥_𝑛,𝑀𝐿⁻¹𝑥=^∞_𝑛=1−𝑛²1+𝑛²𝑥,𝑥_𝑛𝑥_𝑛,𝑆(𝑡)𝑥=^∞_𝑛=1exp−𝑛²𝑡1+𝑛²𝑥,𝑥_𝑛𝑥_𝑛. (38)

It is easy to see that 𝐿⁻¹ is compact and bounded with ‖𝐿⁻¹‖≤1 and 𝑀𝐿⁻¹ generates the above strongly continuous semigroup 𝑆(𝑡) on 𝑌 with ‖𝑆(𝑡)‖≤𝑒^−𝑡≤1. Therefore, with the above choices, the system (35) can be written as an abstract formulation of (1) and thus Theorem 13 can be applied to guarantee the existence of mild solution of (35). Moreover, it can be easily seen that Sobolev type deterministic linear fractional control system corresponding to (35) is approximately controllable on 𝐽, which means that all conditions of Theorem 14 are satisfied. Thus, fractional stochastic control system of Sobolev type (35) is approximately controllable on 𝐽.

5. Conclusion

Sufficient conditions for the approximate controllability of a class of dynamic control systems described by Sobolev type nonlocal fractional stochastic differential equations in Hilbert spaces are considered. Using fixed point technique, fractional calculations, stochastic analysis, and methods adopted directly from deterministic control problems. In particular, conditions are formulated and proved under the assumption that the approximate controllability of the stochastic control nonlinear dynamical system is implied by the approximate controllability of its corresponding linear part. More precisely, the controllability problem is transformed into a fixed point problem for an appropriate nonlinear operator in a function space. The main used tools are the above required conditions, we guarantee the existence of a fixed point of this operator and study controllability of the considered systems.

Degenerate stochastic differential equations model the phenomenon of convection-diffusion of ideal fluids and therefore arise in a wide variety of important applications, including, for instance, two or three phase flows in porous media or sedimentation-consolidation processes. However, to the best of our knowledge, no results yet exist on approximate controllability for fractional stochastic degenerate systems. Upon making some appropriate assumptions, by employing the ideas and techniques as in this paper, one can establish the approximate controllability results for a class of fractional stochastic degenerate differential equations.

Conflict of Interests

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