# Controllability of Fractional Neutral Stochastic Integro-Differential Systems with Infinite Delay

Authors: Xichao Sun; Litan Yan; Jing Cui
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper is concerned with the controllability of a class of fractional neutral stochastic integro-differential systems with infinite delay in an abstract space. By employing fractional calculus and Sadovskii's fixed point principle without assuming severe compactness condition on the semigroup, a set of sufficient conditions are derived for achieving the controllability result.


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## Body

## 1. Introduction

It is well known that the fractional calculus is a classical mathematical notion and is a generalization of ordinary differentiation and integration to arbitrary (noninteger) order. Nowadays, studying fractional-order calculus has become an active research field [1–7]. Much effort has been devoted to apply the fractional calculus to networks control. For example, Chen et al. [8], Delshad et al. [9], and Wang and Zhang [10] studied the synchronization for fractional-order complex dynamical networks; Zhang et al. [11] investigated a fractional order three-dimensional Hopfield neural network and pointed out that chaotic behaviors can emerge in a fractional network; Kaslik and Sivasundaram [12] discussed the local stability for fractional-order neural networks of Hopfield type by applying the linear stability theory of fractional-order system.One of the emerging branches of this study is the theory of fractional evolution equations, say, evolution equations, where the integer derivative with respect to time is replaced by a derivative of fractional order. The increasing interest in this class of equations is motivated both by their application to problems from fluid dynamic traffic model, viscoelasticity, heat conduction in materials with memory, electrodynamics with memory, and also because they can be employed to approach nonlinear conservation laws (see [13] and references therein). In addition, neutral stochastic differential equations with infinite delay have become important in recent years as mathematical models of phenomena in both science and engineering, for instance, in the theory development in Gurtin and Pipkin [14] and Nunziato [15] for the description of heat conduction in materials with fading memory. It should be pointed out that the deterministic models often fluctuate due to noise, which is random or at least appears to be so. Therefore, we must move from deterministic problems to stochastic ones. We mention here the recent papers [16, 17] concerning the existence of mild solutions of fractional stochastic systems.As one of the fundamental concepts in mathematical control theory, controllability plays an important role both in deterministic and stochastic control problems such as stabilization of unstable systems by feedback control. Roughly speaking, controllability generally means that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. Controllability problems for different nonlinear stochastic systems in infinite dimensional spaces have been extensively studied in many papers; see [18–22] and references therein. We would also like to mention that the controllability for stochastic systems with infinite delay has been investigated by Balasubramaniam  et al. [23, 24] and Ren  et al. [25] using some abstract spaces. Nevertheless, to the best of our knowledge, it seems that little is known about the controllability of fractional neutral stochastic differential equations with infinite delay, and the aim of this paper is to close this gap.In this paper, we are interested in the controllability of a class of fractional neutral stochastic integro-differential systems with infinite delay of the followin form:(1)Dcαdtα[x(t)+G(t,xt)]=-Ax(t)+Bu(t)+∫0tσ(t,s,xs)dW(s),t∈J,x(t)=ϕ(t),t∈(-∞,0].
Here, J:=[0,b], b>0. x(·) takes value in a real separable Hilbert space ℍ with inner product (·,·) and norm |·|ℍ. The fractional derivative Dcα, α∈(1/2,1] is understood in the Caputo sense. -A:𝒟(-A)⊂ℍ→ℍ is the infinitesimal generator of a analytic semigroup of a bounded linear operator S(t), t≥0, on ℍ. Let 𝕂 be another separable Hilbert space with inner product (·,·)𝕂 and norm ∥·∥𝕂. W is a given 𝕂-valued Wiener process with a finite trace nuclear covariance operator Q≥0 defined on a filtered complete probability space (Ω,ℱ,{ℱt}t≥0,P). The control function u(·) takes value in L2(J,𝕌) of admissible control functions for a separable Hilbert space 𝕌, and B is a bounded linear operator from 𝕌 into ℍ. The histories xt:Ω→Ch defined by xt={x(t+θ),θ∈(-∞,0]} belong to the phase space Ch, which will be defined in Section 2. The initial data ϕ={ϕ(t),t∈(-∞,0]} is an ℱ0-measurable, Ch-valued random variable independent of W with finite second moments, and G:J×Ch→ℍ, σ:J×J×ℍ→ℒ20(𝕂,ℍ) are appropriate mappings specified later (here, ℒ20(𝕂,ℍ) denotes the space of all Q-Hilbert-Schmidt operators from 𝕂 into ℍ, which is going to be defined later).The structure of this paper is as follows. In Section2, we briefly present some basic notations and preliminaries. The controllability result of system (1) is investigated by means of Sadovskii’s fixed point theorem and operator theory in Section 3. Conclusion is given in Section 4.


## 2. Preliminaries

For more details in this section, we refer the reader to  Pazy [26], Da Prato and Zabczyk [27], and Samko  et al. [28]. Throughout this paper, (ℍ,|·|ℍ) and (𝕂,∥·∥𝕂) denote two real separable Hilbert spaces. We denote by ℒ(𝕂,ℍ) the set of all linear bounded operators from 𝕂 into ℍ, equipped with the usual operator norm ∥·∥. In this paper, we use the symbol ∥·∥ to denote norms of operators regardless of the spaces potentially involved when no confusion possibly arises.Let(Ω,ℱ,{ℱt}t≥0,P) be a filtered complete probability space satisfying the usual condition, which means that the filtration is a right continuous increasing family and ℱ0 contains all P-null sets. W=(Wt)t≥0 is a Q-Wiener process defined on (Ω,ℱ,{ℱt}t≥0,P) with covariance operator Q such that TrQ<∞. We assume that there exist a complete orthonormal system {ek}k≥1 in 𝕂, a bounded sequence of nonnegative real numbers λk such that Qek=λkek, k=1,2,…, and a sequence of independent Brownian motions {βk}k≥1 such that
(2)(W(t),e)𝕂=∑k=1∞‍λk(ek,e)𝕂βk(t),e∈𝕂,t≥0.
Let ℒ20=ℒ2(Q1/2𝕂,ℍ) be the space of all Hilbert-Schmidt operators from Q1/2𝕂 to ℍ with the inner product 〈φ,ϕ〉ℒ20=Tr[φQϕ*].Suppose that0∈ρ(-A), where ρ(-A) is the resolvent set of -A, then the semigroup S(·) is uniformly bounded. That is to say, ∥S(t)∥≤M, t≥0, for some constant M>0. Then, for α∈(0,1], it is possible to define the fractional power operator Aα as a closed linear operator on its domain 𝒟(Aα). Furthermore, the subspace 𝒟(Aα) is dense in ℍ, and the expression
(3)∥x∥α=∥Aαx∥,x∈𝒟(Aα),
defines a norm on ℍα:=𝒟(Aα). The following properties are well known.Lemma 1 (Pazy [26]).
Suppose that the preceding conditions are satisfied.(a)
If0<β<α≤1, then ℍα⊂ℍβ and the embedding is compact whenever the resolvent operator of A is compact.(b)
For everyα∈(0,1], there exists a positive constant Cα such that
(4)∥AαS(t)∥≤Cαtα,t>0.Let now us recall some basic definitions and results of fractional calculus.Definition 2.
The fractional integral of orderα with the lower limit 0 for a function f is defined as
(5)Iαf=1Γ(α)∫0t‍f(s)(t-s)1-αds,t>0,α>0,
provided the right-hand side is pointwise defined on [0,∞), where Γ(·) is the gamma function.Definition 3.
The Caputo derivative of orderα with the lower limit 0 for a function f can be written as
(6)cDαf(t)=1Γ(n-α)∫0t‍fn(s)(t-s)α+1-nds=In-αfn(t),t>0,0≤n-1<α<n.Iff is an abstract function with values in ℍ, then the integrals that appear in the previous definitions are taken in Bochner’s sense.Assume thath:(-∞,0]→(0,+∞) with l=∫-∞0‍h(t)dt<+∞ is a continuous function. Recall that the abstract phase space 𝒞h is defined by
(7)𝒞h={∫-∞0‍sups≤θ≤0(E|φ(θ)|2)1/2φ:(-∞,0]⟶ℍ,foranya>0,(E|φ(θ)|2)1/2isaboundedandmeasurablefunctionon[-a,0],∫-∞0‍h(s)sups≤θ≤0(E|φ(θ)|2)1/2ds<+∞}.
If 𝒞h is endowed with the norm
(8)∥φ∥𝒞h=∫-∞0‍h(s)sups≤θ≤0(E|φ(θ)|2)1/2ds,φ∈𝒞h,
then (𝒞h,∥·∥𝒞h) is a Banach space (see Liu et al. [29]).At the end of this section, we recall the fixed point theorem of Sadovskii [30].Lemma 4.
LetΦ be a condensing operator on a Banach space ℍ; that is, Φ is continuous, and take bounded sets into bounded sets, and μ(Φ(B))≤μ(B) for every bounded set B of ℍ with μ(B)>0. If Φ(N)⊂N for a convex, closed, and bounded set N of ℍ, then Φ has a fixed point in ℍ (where μ(·) denotes Kuratowski's measure of noncompactness.)


## 3. Main Results

In this section, we obtain controllability of system (1). We first present the definition of mild solutions.Definition 5.
Anℍ-valued stochastic process {x(t),t∈(-∞,b]} is said to be a mild solution of system (1) if  (i)
x(t) is ℱt-adapted and measurable for each t≥0;(ii)
x(t) is continuous on [0,b] almost surely and for each s∈[0,t), the function (t-s)α-1ATα(t-s)G(s,xs) is integrable such that the following stochastic integral equation is verified:
(9)x(t)=Sα(t)[ϕ(0)+G(0,ϕ)]-G(t,xt)-∫0t‍(t-s)α-1ATα(t-s)G(s,xs)ds+∫0t‍(t-s)α-1Tα(t-s)Bu(s)ds+∫0t‍(t-s)α-1Tα(t-s)×[∫0s‍σ(s,τ,xτ)dW(τ)]ds;(iii)
x(t)=ϕ(t) on (-∞,0] with ∥ϕ∥𝒞h2<∞, where
(10)Sα(t)x=∫0∞‍ηα(θ)S(tαθ)xdθ,Tα(t)x=α∫0∞‍θηα(θ)S(tαθ)xdθ
with ηα a probability density function defined on (0,∞).Definition 6.
System (1) is said to be controllable on the interval J, if for every initial stochastic process ϕ∈𝒞h defined on (-∞,0], there exists a stochastic control u∈L2(J,𝕌), which is adapted to the filtration {ℱt}t≥0 such that the mild solution x(t) of (1) satisfies x(b)=x*, where x* and b are preassigned terminal state and time, respectively.The following properties ofSα(t) and Tα(t) that appeared in Zhou and Jiao [7] are useful.Lemma 7.
Under previous assumptions onS(t), t≥0, and A, then  (i)
for any fixedt≥0, Sα(t) and Tα(t) are linear and bounded operators such that for any x∈ℍ, |Sα(t)x|ℍ≤M|x|ℍ, |Tα(t)x|ℍ≤(Mα/Γ(1+α))|x|ℍ;(ii)
Sα(t) and Tα(t) are strongly continuous;(iii)
for anyx∈ℍ, β∈(0,1), and θ∈(0,1], one has
(11)ATα(t)x=A1-βTα(t)Aβx,∥AθTα(t)∥≤αCθtαθΓ(2-θ)Γ(1+α(1-θ)),t∈[0,b].In this paper, we will work under the following assumptions.(A0)
-A is the infinitesimal generator of an analytic semigroup of bounded linear operators S(t) in ℍ; S(t), t>0, is continuous in the uniform operator topology and ∥S(t)∥≤M for some M>0.(A1)
The functionG:J×𝒞h→ℍ is continuous, and there exist some constants MG>0, β∈(0,1) with αβ>1/2, such that G is ℍβ-valued and
(12)E|AβG(t,x)-AβG(t,y)|ℍ2≤MG∥x-y∥𝒞h2,x,y∈𝒞h,t∈J,E|AβG(t,x)|ℍ2≤MG(∥x∥𝒞h2+1).(A2)
σ:J×J×𝒞h→L(𝕂,ℍ) satisfies the following:
(2a)
for each(t,s)∈D:=J×J, σ(t,s,·):𝒞h→L(𝕂,ℍ) is continuous and for each x∈𝒞h, σ(·,·,x):D→L(𝕂,ℍ) is strongly measurable;(2b)
there is a positive integrable functionm∈L1([0,b]) and a continuous nondecreasing function Ξ:[0,∞)→(0,∞) such that for every (t,s,x)∈J×J×𝒞h, we have
(13)∫0t‍E|σ(t,s,x)|ℒ202ds≤m(t)Ξ(∥x∥𝒞h2),liminfr→∞Ξ(r)rds=Λ<∞;(2c)
for anyx,y∈𝒞h, t≥0, there exists a positive constant Mσ such that
(14)∫0t‍E|σ(t,s,x)-σ(t,s,y)|ℒ202ds≤Mσ∥x-y∥𝒞h2.(A3)
The linear operatorW from L2(J,𝕌) into ℍ defined by
(15)Wu=∫0b‍(b-s)α-1Tα(b-s)Bu(s)dshas an invertible operatorW-1 defined on L2(J,𝕌)/kerW (see [31]), and there exist a pair of constants N1,N2>0 such that
(16)∥B∥2≤N1,∥W-1∥≤N2.(A4)
There exists a compact set℧⊆ℍ such that S(t)σ(s,τ,ξ)∈℧ for all t∈J, 0≤τ≤s≤b and ξ∈𝒞h with finite second moment.(A5)
Assume that the following relationship holds:(17)20(1+5N1N2N)l2×(M0+MGP+Nb-1TrQΛ∫0b‍m(s)ds)<1, where
(18)N=b2α2α-1(MαΓ(1+α))2,M0=∥A-β∥2MG,(19)P=α2C1-β2Γ2(1+β)Γ2(1+αβ)b2αβ2αβ-1.Denote byC((-∞,b],ℍ) the space of all continuous ℍ-valued stochastic processes {ξ(t),t∈(-∞,b]}. Let
(20)𝒞b={x:x∈C((-∞,b],ℍ),x0=ϕ∈𝒞h}.
Set ∥·∥b to be a seminorm defined by
(21)∥x∥b=∥x0∥𝒞h+sups∈[0,b](E|x(s)|2)1/2,x∈𝒞b.
We have the following useful lemma that appeared in Liu et al.  [29].Lemma 8.
Assume thatx∈𝒞b; then, for all t∈J, xt∈𝒞h. Moreover,
(22)l(E|x(t)|2)1/2≤∥xt∥𝒞h≤lsups∈[0,t](E|x(s)|2)1/2+∥x0∥𝒞h,
where l=∫-∞0‍h(s)ds<∞.The main object of this paper is to explain and prove the following theorem.Theorem 9.
Assume that assumptions(A0)–(A5) hold. Then, system (1) is controllable on J provided that
(23)L≔3l2(1+3N1N2N)(M0+MGP)+9N1N2N2TrQMσl2<1.Proof.
Using assumption(A3), for an arbitrary process x(·), define the control process
(24)ux(t)=W-1{[∫0s‍σ(s,τ,xτ)dW(τ)]x*-Sα(b)[ϕ(0)+G(0,ϕ)]+G(b,xb)+∫0b‍(b-s)α-1ATα(b-s)G(s,xs)ds-∫0b‍(b-s)α-1Tα(b-s)×[∫0s‍σ(s,τ,xτ)dW(τ)]ds}(t).
We transform (1) into a fixed point problem. Consider the operator Q:𝒞b→𝒞b defined by
(25)(Qx)(t)={ϕ(t),t∈(-∞,0];Sα(t)[ϕ(0)+G(0,ϕ)]-G(t,xt)-∫0t‍(t-s)α-1ATα(t-s)G(s,xs)ds+∫0t‍(t-s)α-1Tα(t-s)Bux(s)ds+∫0t‍(t-s)α-1Tα(t-s)×[∫0s‍σ(s,τ,xτ)dW(τ)]ds,t∈J.
It follows from Hölder inequality, Lemma 7, and assumption (A1) that
(26)E|∫0t‍(t-s)α-1ATα(t-s)G(s,xs)ds|ℍ2≤E[∫0t‍|(t-s)α-1A1-βTα×(t-s)AβG(s,xs)|ℍds∫0t‍]2≤PE[∫0t‍|(t-s)α(β-1)(t-s)α-1×AβG(s,xs)|ℍds∫0t‍]2≤P∫0t‍(t-s)2(αβ-1)ds×∫0t‍MG(1+|xs|𝒞h2)ds,
which deduces that (t-s)α-1ATα(t-s)G(s,xs) is integrable on J by Bochner's theorem (see [32]), where P is defined in (19).
In what follows, we will show that using the controlux(·), the operator Q has a fixed point, which is then a mild solution for system (1).
Forϕ∈𝒞h, define
(27)ϕ~(t)={ϕ(t),t∈(-∞,0];Sα(t)ϕ(0),t∈J.
Then, ϕ~∈𝒞b. Set x(t)=ϕ~(t)+z(t), t∈(-∞,b]. It is easy to check that x satisfies (1) if and only if z0=0 and
(28)z(t)=Sα(t)G(0,ϕ)-G(t,ϕ~t+zt)-∫0t‍(t-s)α-1ATα(t-s)G(s,ϕ~s+zs)ds+∫0t‍(t-s)α-1Tα(t-s)Buϕ~+z(s)ds+∫0t‍(t-s)α-1Tα(t-s)×[∫0s‍σ(s,τ,ϕ~τ+zτ)dW(τ)]ds,
where uϕ~+z is obtained by replacing xt by ϕ~t+zt in (24). Let
(29)𝒞b0={z∈𝒞b,z0=0∈𝒞h}.
For each z∈𝒞b0, we have
(30)∥z∥b=∥z0∥𝒞h+sups∈[0,b](E|z(s)|2)1/2=sups∈[0,b](E|z(s)|2)1/2.
Thus, (𝒞b0,∥·∥b) is a Banach space. For q>0, set
(31)Bq={y∈𝒞b0,∥y∥b2≤q};
then, for each q, Bq is clearly a bounded closed convex set in 𝒞b0. For z∈Bq, from Lemma 8, we see that
(32)∥zt+ϕ~t∥𝒞h2≤2(∥zt∥𝒞h2+∥ϕ~t∥𝒞h2)≤4(l2sups∈[0,t]E|z(s)|2+∥z0∥𝒞h2+l2sups∈[0,t]E|ϕ~(s)|2+∥ϕ~0∥𝒞h2)≤4l2(q+M2E|ϕ(0)|ℍ2)+4∥ϕ∥𝒞h2.
Consider the mapΠ:𝒞b0→𝒞b0 defined by
(33)(Πz)(t)={0,t∈(-∞,0];Sα(t)G(0,ϕ)-G(t,ϕ~t+zt)-∫0t‍(t-s)α-1ATα(t-s)×G(s,ϕ~s+zs)ds+∫0t‍(t-s)α-1Tα(t-s)Buϕ~+z(s)ds+∫0t‍(t-s)α-1Tα(t-s)×[∫0s‍σ(s,τ,ϕ~τ+zτ)dW(τ)]ds,t∈J.
A similar argument as (26), we can show that Π is well defined on Bq for each q>0.  Note that the operator Q with a fixed point is equivalent to show that the operator Π has fixed point. To this end, we decompose Π as Π=Π1+Π2, where the operators Π1 and Π2 are defined on Bq, respectively, by
(34)(Π1z)(t)=Sα(t)G(0,ϕ)-G(t,ϕ~t+zt)-∫0t‍(t-s)α-1ATα(t-s)G(s,ϕ~s+zs)ds+∫0t‍(t-s)α-1Tα(t-s)Buϕ~+z(s)ds,(Π2z)(t)=∫0t‍(t-s)α-1Tα(t-s)×[∫0s‍σ(s,τ,ϕ~τ+zτ)dW(τ)]ds.
Thus, Theorem 9 follows from the next theorem.Theorem 10.
Assume that the assumptions of Theorem9 hold. Then, Π=Π1+Π2 satisfies all the conditions in Lemma 4.Proof.
The proof is followed by the several steps.
Step  1. There exists a positive number q such that Π(Bq)⊂Bq. If it is not true, then for each positive number q, there exists a function zq(·)∈Bq, but Π(zq)∉Bq; that is, E|(Πzq)(t)|ℍ2>q for some t=t(q)∈J. An elementary calculation can show that
(35)q≤E|Π(zq)(t)|ℍ2≤5E|Sα(t)G(0,ϕ)|ℍ2+5E|G(t,ϕ~t+ztq)|ℍ2+5E|∫0t‍(t-s)α-1ATα(t-s)G(s,ϕ~s+zsq)ds|ℍ2+5E|∫0t‍(t-s)α-1Tα(t-s)×[∫0s‍σ(s,ϕ~τ+zτq)dW(τ)]ds|ℍ2+5E|∫0t‍(t-s)α-1Tα(t-s)Buϕ~+zq(s)ds|ℍ2=5∑i=15‍Ii.
We estimate each term on the right hand side of (35). By virtue of Lemma 8, assumptions (A0)-(A1), and (32), we obtain
(36)I1+I2≤M2∥A-β∥2E|AβG(0,ϕ)|ℍ2+∥A-β∥2E|AβG(t,ϕ~t+ztq)|ℍ2≤M0[M2(1+∥ϕ∥𝒞h2)+(1+∥ϕ~t+ztq∥𝒞h2)]≤M0[M2(1+∥ϕ∥𝒞h2)+(1+4l2(q+M2E|ϕ(0)|ℍ2)+4∥ϕ∥𝒞h2)];
in view of Lemma 7 and Hölder inequality, we have
(37)I3≤E[∫0t‍|(t-s)α-1A1-βTα(t-s)×AβG(s,ϕ~s+zsq)|ℍds∫0t‍]2≤α2C1-β2Γ2(1+β)Γ2(1+αβ)E×[∫0t‍|(t-s)αβ-1AβG(s,ϕ~s+zsq)|ℍds]2≤α2C1-β2Γ2(1+β)Γ2(1+αβ)∫0t‍(t-s)2(αβ-1)ds×∫0t‍MG(1+|ϕ~s+zsq|𝒞h2)ds≤MGα2C1-β2Γ2(1+β)Γ2(1+αβ)b2αβ-12αβ-1×∫0t‍(1+|ϕ~s+zsq|𝒞h2)ds≤MGP(1+4l2(q+M2E|ϕ(0)|ℍ2)+4∥ϕ∥𝒞h2),
where M0 and P are defined in (18) and (19), respectively.
Applying Burkhölder-Davis-Gundy's inequality and assumptions(A2), we get
(38)I4≤E[∫0t‍|[∫0s‍σ(s,τ,ϕ~τ+zτq)dW(τ)](t-s)α-1Tα(t-s)×[∫0s‍σ(s,τ,ϕ~τ+zτq)dW(τ)]|ℍds]2≤(MαΓ(1+α))2∫0t‍(t-s)2(α-1)ds×∫0t‍E|∫0s‍σ(s,τ,ϕ~τ+zτq)dW(τ)|ℍ2ds≤(MαΓ(1+α))2b2α-12α-1×∫0t‍TrQ∫0s‍E|σ(s,ϕ~τ+zτq)|ℒ202dτds≤(MαΓ(1+α))2b2α-12α-1TrQ×∫0t‍m(s)Ξ(∥ϕ~τ+zτq∥𝒞h2)ds≤(MαΓ(1+α))2TrQb2α-12α-1Ξ×(4l2(q+M2E|ϕ(0)|ℍ2)+4∥ϕ∥𝒞h2)∫0t‍m(s)ds.
On the other hand, in view of (24) and (A3), we have
(39)E|Buzq+ϕ~(t)|ℍ2≤N1N2E|{∫0s‍x*-Sα(b)[ϕ(0)+G(0,ϕ)]+G(b,ϕ~b+zbq)+∫0b‍(b-s)α-1ATα(b-s)G(s,ϕ~s+zsq)ds-∫0b‍(b-s)α-1Tα(b-s)×[∫0s‍σ(s,τ,ϕ~τ+zτq)dW(τ)]ds}(t)|ℍ2;
thus, by the same procedure as (36)–(38), it follows that
(40)I5≤(MαΓ(1+α))2∫0t‍(t-s)2(α-1)ds∫0t‍E|Buzq+ϕ~(s)|ℍ2ds≤N{∫0b‍C0+5N1N2×[∫0b‍4MG∥A-β∥2l2q+4l2MGPq+2Nb-1TrQΞ×(4l2(q+M2E|ϕ(0)|ℍ2)+4∥ϕ∥𝒞h2)×∫0b‍m(s)ds]},
where
(41)C0=5N1N2[∥ϕ∥𝒞h2E|x*|ℍ2+2M2E|ϕ(0)|ℍ2+2M2M0(1+∥ϕ∥𝒞h2)+M0(1+4l2M2E|ϕ(0)|ℍ2+4∥ϕ∥𝒞h2)+MGP(1+4l2M2E|ϕ(0)|ℍ2+4∥ϕ∥𝒞h2)].M0 and P are defined in (18) and (19), respectively.
Combining these estimates (35) to (40) yields
(42)q≤E|Π(zq)(t)|ℍ2≤L0+20M0l2(1+5N1N2N)q+20l2MGP(1+5N1N2N)q+5(MαΓ(1+α))2TrQb2α-12α-1×NΞ(4l2(q+M2E|ϕ(0)|ℍ2)+4∥ϕ∥𝒞h2)∫0b‍m(s)ds,
where
(43)L0=5N+5M2M0(1+∥ϕ∥𝒞h2)C0+5M0(1+4l2M2E|ϕ(0)|ℍ2+4∥ϕ∥𝒞h2)+5MGP(1+4l2M2E|ϕ(0)|ℍ2+4∥ϕ∥𝒞h2).
Dividing both sides of (42) by q and taking q→∞, we obtain that
(44)20(1+5N1N2N)l2×(M0+MGP+Nb-1TrQΛ∫0b‍m(s)ds)≥1,
which is a contradiction by assumption (A5). Thus, for some positive number q, Π(Bq)⊂Bq.
Step  2. Π1 is a contractive mapping. Let x,v∈Bq. From the assumptions on G and σ, it is easy to verify that the following inequality holds:
(45)E|ux+ϕ~(s)-uv+ϕ~(s)|ℍ2≤3N2[∫0b(MαΓ(1+α))2‍∥A-β∥2MG∥xb-vb∥𝒞h2+α2C1-β2Γ2(1+β)Γ2(1+αβ)b2αβ-12αβ-1∫0b‍MG∥xs-vs∥𝒞h2ds+(MαΓ(1+α))2b2α-12α-1∫0b‍MσTrQ∥xs-vs∥𝒞h2ds].
Thus, by the assumptions, we have
(46)E|(Π1x)(t)-(Π1v)(t)|ℍ2≤3E|G(t,ϕ~t+xt)-G(t,ϕ~t+vt)|ℍ2+3E|∫0t‍(t-s)α-1ATα(t-s)×[G(s,ϕ~s+xs)-G(s,ϕ~s+vs)]ds∫0t‍|ℍ2+3E|∫0t‍(t-s)α-1Tα(t-s)B(uϕ~+x-uϕ~+v)ds|ℍ2≤3M0l2(1+3N1N2N)sup0≤s≤bE|x(s)-v(s)|ℍ2+3PMGl2(1+3N1N2N)sup0≤s≤bE|x(s)-v(s)|ℍ2+9N1N2N2TrQMσl2sup0≤s≤bE|x(s)-v(s)|ℍ2,
where we have used the fact that u0=v0=0. Hence,
(47)∥Π1x-Π1v∥b2≤L∥u-v∥b2;
so, Π1 is a contraction by (23).
Step  3. We show that the operator Π2 is compact. Let q>0 be such that Π2(Bq)⊂Bq. The proof will be divided into the following claims.
Claim  1. Π2 maps bounded sets into bounded sets in Bq.
Indeed, it is enough to show that there exists a positive constantΔ such that for each t∈J, z∈Bq, we have ∥Π2z∥b2≤Δ. If z∈Bq, from (32), it follows that
(48)∥zt+ϕ~t∥𝒞h2≤4l2(q+M2E|ϕ(0)|ℍ2)+4∥ϕ∥𝒞h2≔q′.
By the similar argument as before, we get
(49)E|Π2z(t)|ℍ2≤(MαΓ(1+α))2∫0t‍(t-s)2(α-1)ds×∫0t‍E|∫0s‍σ(s,τ,ϕ~τ+zτ)dW(τ)|ℍ2ds≤(MαΓ(1+α))2TrQb2α-12α-1Ξ(q′)∫0t‍m(s)ds≤(MαΓ(1+α))2TrQb2α-12α-1Ξ(q′)∫0b‍m(s)ds≔Δ.
Therefore, for each z∈Bq, it holds that ∥Π2z∥b2≤Δ.
Claim  2. The set {Π2z,z∈Bq} is an equicontinuous family of functions on J. Let 0<ϵ<t<b and δ>0 such that ∥Tα(s1)-Tα(s2)∥<ϵ, for every s1,s2∈J with |s1-s2|<δ. For z∈Bq, 0<|h|<δ, t+h∈J, we have
(50)E|Π2z(t+h)-Π2z(t)|ℍ2=E|∫0t+h‍(t+h-s)α-1Tα(t+h-s)×[∫0s‍σ(s,τ,ϕ~τ+zτ)dW(τ)]ds-∫0t‍(t-s)α-1Tα(t-s)×[∫0s‍σ(s,τ,ϕ~τ+zτ)dW(τ)]ds|ℍ2≤3E|∫tt+h‍(t+h-s)α-1Tα(t+h-s)×[∫0s‍σ(s,τ,ϕ~τ+zτ)dW(τ)]ds|ℍ2+3E|∫0t‍[(t+h-s)α-1-(t-s)α-1]Tα(t+h-s)×[∫0s‍σ(s,τ,ϕ~τ+zτ)dW(τ)]ds|ℍ2+3E|∫0t‍(t-s)α-1[Tα(t+h-s)-Tα(t-s)]×[∫0s‍σ(s,τ,ϕ~τ+zτ)dW(τ)]ds|ℍ2.
Applying assumption (A2), Hölder inequality, together with Lemma 7, we get
(51)E|Π2z(t+h)-Π2z(t)|ℍ2≤3Z(α)∫tt+h‍(t+h-s)2(α-1)ds×∫tt+h‍E|∫0s‍σ(s,τ,ϕ~τ+zτ)dW(τ)|ℍ2ds+3Z(α)∫0t‍[(t+h-s)α-1-(t-s)α-1]2ds×∫0t‍E|∫0s‍σ(s,τ,ϕ~τ+zτ)dW(τ)|ℍ2ds+3ϵ∫0t‍(t-s)2(α-1)ds×∫0t‍E|∫0s‍σ(s,τ,ϕ~τ+zτ)dW(τ)|ℍ2ds≤3Z(α)h2α-12α-1∫tt+h‍TrQMσm(s)Ξ(q′)ds+3Z(α)∫0t‍[(t+h-s)α-1-(t-s)α-1]2ds×∫0t‍TrQMσm(s)Ξ(q′)ds+3ϵ∫0t‍(t-s)2(α-1)ds∫0t‍TrQMσm(s)Ξ(q′)ds,
with Z(α)=(Mα/Γ(1+α))2. Therefore, for ϵ sufficiently small, the right hand side of (51) tends to zero as h→0. Thus, the set {Π2z,z∈Bq} is equicontinuous by assumption (A1).
Claim  3. Π2 maps Bq into a precompact set in Bq. Let 0<t≤b be fixed and ϵ a real number satisfying 0<ϵ<t. For δ>0, define an operator Π2ϵ,δ on Bq by
(52)(Π2ϵ,δz)(t)=α∫0t-ϵ‍∫δ∞‍θ(t-s)α-1ηα(θ)S((t-s)αθ)×[∫0s‍σ(s,τ,ϕ~τ+zτ)dW(τ)]dθds=S(ϵαδ)α∫0t-ϵ‍∫δ∞‍θ(t-s)α-1ηα(θ)S×((t-s)αθ-ϵαδ)×[∫0s‍σ(s,τ,ϕ~τ+zτ)dW(τ)]dθds.
Since S(t)σ(s,τ,ϕ~τ+zτ) belongs to the compact set ℧, the set {Π2ϵ,δz)(t),z∈Bq} is precompact in ℍ for every ϵ∈(0,t), δ>0. Moreover, for each z∈Bq, we have
(53)E|(Π2z)(t)-(Π2ϵ,δz)(t)|ℍ2≤2α2E|∫0t‍∫0δ‍θ(t-s)α-1ηα(θ)S((t-s)αθ)×[∫0s‍σ(s,τ,ϕ~τ+zτ)dW(τ)]dθds|ℍ2+2α2E|∫t-ϵt‍∫δ∞‍θ(t-s)α-1ηα(θ)S((t-s)αθ)×[∫0s‍σ(s,τ,ϕ~τ+zτ)dW(τ)]dθds|ℍ2=J1+J2.
An application of Hölder inequality together with Burkhölder-Davis-Gundy's inequality, it follows that
(54)J1≤2α2M2∫0t‍(t-s)2(α-1)E×|∫0s‍σ(s,τ,ϕ~τ+zτ)dW(τ)|ℍ2ds×∫0t‍|∫0δ‍θηα(θ)dθ|2ds≤2α2M2b∫0t‍(t-s)2(α-1)TrQ×∫0s‍E|σ(s,τ,ϕ~τ+zτ)|ℒ202dτds|∫0δ‍θηα(θ)dθ|2≤2α2M2b∫0t‍TrQΞ(q′)m(s)(t-s)2(α-1)ds×|∫0δ‍θηα(θ)dθ|2,J2≤2α2M2∫t-ϵt‍(t-s)2(α-1)E×|∫0s‍σ(s,τ,ϕ~τ+zτ)dW(τ)|ℍ2ds×∫t-ϵt‍|∫0∞‍θηα(θ)dθ|2ds≤2α2M2ϵΓ2(1+α)∫t-ϵt‍TrQΞ(q′)m(s)(t-s)2(α-1)ds.
From (54), we deduce that for each z∈Bq,
(55)E|(Π2z)(t)-(Π2ϵ,δz)(t)|ℍ2⟶0asϵ⟶0+,δ⟶0+.
Therefore, there are relatively compact sets arbitrary close to the set {Π2(t),z∈Bq}; hence, the set {(Π2z)(t),z∈Bq} is also precompact in Bq.
Thus, from Arzelá-Ascoli theorem together with assumptions onf and σ, we conclude that Π2 is a compact operator. Therefore, Π is a condensing map on Bq. This completes the proof.Remark 11.
In order to describe various real-world problems in physical and engineering sciences subject to abrupt changes at certain instants during the evolution process, impulsive differential equations have been used to model the system. The technique used here can be extended to establish the controllability of neutral fractional stochastic integro-differential systems with impulsive effect and infinite delay. The controllability result can be obtained by suitably introducing the impulsive effects defined in [19].


## 4. Conclusions

In this paper, we have studied the controllability of fractional neutral stochastic integro-differential systems with infinite delay in an abstract space. Through fractional calculus and Sadovskii's fixed point principle, we have investigated the sufficient conditions for the controllability of the system considered.



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*Source: 316906-2013-04-18.xml*