# The Existence, Uniqueness, and Controllability of Neutral Stochastic Delay Partial Differential Equations Driven by Standard Brownian Motion and Fractional Brownian Motion

Authors: Dehao Ruan; Jiaowan Luo
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

We focus on a class of neutral stochastic delay partial differential equations perturbed by a standard Brownian motion and a fractional Brownian motion. Under some suitable assumptions, the existence, uniqueness, and controllability results for these equations are investigated by means of the Banach fixed point method. Moreover, an example is presented to illustrate our main results.


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## Body

## 1. Introduction

Fractional Brownian motion (fBm for short) with Hurst parameterH∈(0,1) is a centered Gaussian process {βH(t),t≥0} which is often used to model many complex phenomena in applications when the systems contain rough external forcing. When H=1/2, the fBm is the standard Brownian motion which is a Markov process and martingale, and we can use the classical Itô theory to construct a stochastic integration with respect to the standard Brownian motion. However, when H≠1/2, the fBm is neither a Markov process nor a semimartingale, and the fBm behaviors become quite different. The classical techniques based on PDE cannot apply to in this context. For more details on fBm, we refer the readers to the articles [1–5].Neutral differential equations (NDEs for short) are a family of differential equations depending on the past as well as the present state which involve derivatives with delays as well as the function itself. This family of equations has great applications in mathematical, electrical engineering, ecology, and other fields of science. In recent years, NDEs have received more and more attention; see, for example, [6–16] and the reference therein.The observations of stock prices processes suggest that they are not self-similar. As a result, the mixed model describes the stock prices behavior in a better way. Since the pioneering paper by Cheridito [17], mixed stochastic models containing both a standard Brownian motion and fBm gained a lot of attention. The main reason for this is that they allow us to model systems driven by white noise and fractional Gaussian noise. For more details on this topic see [18–21].The controllability problem is one of the most attractive and important problems of differential equations because of their great significance and applications in physics, population dynamics, engineering, mathematical biology, and other areas of science. Controllability for stochastic models only containing standard Brownian motion has been investigated very well (see [7, 12, 16, 22]). Recently, controllability for stochastic systems only driven by fBm has gained a lot of attention; we refer to [6, 9, 11]. Up to now, there is no paper which considers the controllability for stochastic systems driven by standard Brownian motion and fBm.In this paper, we study the existence, uniqueness, and controllability results for a class of neutral stochastic delay partial differential equations (NSDPDEs for short) driven by a standard Brownian motion and an fBm in the abstract form(1)dxt+ht,xt-rt=Axt+ft,xt-ρt+Butdt+gt,xt-ηtdWt+σtdBQHt,t≥0,xs=φs∈C-τ,0;L2Ω;X,-τ≤s≤0,where A is the infinitesimal generator of an analytic semigroup {S(t),t≥0} of bounded linear operators in a real separable Hilbert space X. The delay functions r,ρ,η:[0,+∞)→[0,τ](τ≥0) are continuous, and f,h:[0,+∞)×X→X, g:[0,+∞)×X→LQ(1)0(Y1;X), and σ:[0,+∞)→LQ(2)0(Y2;X) are Borel measurable satisfying appropriate conditions. The control mapping u(·) takes values in L2([0,T];U), and the Hilbert space of admissible control mappings for a real separable Hilbert U, B is a bounded linear operator from U to X. {W(t),t≥0} denotes a standard Brownian motion in a real separable Hilbert space Y1. {BQH(t),t≥0} denotes a fractional Brownian motion in a real separable Hilbert space Y2 with Hurst parameter H∈(1/2,1). Let LQ(i)0(Yi;X)(i=1,2) be the space of all Q(i)-Hilbert-Schmidt operators from Yi into X.Recently, Boufoussi and Hajji in [8] considered the existence and uniqueness problems of a class of neutral stochastic delay differential equations; that is, B≡0 and g≡0 in (1) by means of the Banach fixed point theory. Very recently, Liu and Luo in [13] studied the existence and uniqueness problems of a wide class of neutral stochastic delay partial differential equations, that is, B≡0 in (1) by means of the Banach fixed point theory. However, they all required 1/2<β<1 which is a stronger condition than ours (see Section 3).In this paper, based on the above papers, we also use the Banach fixed point theory to consider a class of neutral stochastic delay partial differential equations, which is more general than ones in [8, 13]. Some conditions on the existence, uniqueness, and controllability of (1) are obtained. The related known results in Boufoussi and Hajji [8] and Liu and Luo in [13] are improved and generalized.The contents of this paper are as follows. In Section2, some notions, definitions, and lemmas which will be needed throughout this paper are introduced. In Section 3, the existence and uniqueness of mild solutions of (1) are proved. In Section 4, the controllability of (1) is investigated by means of the Banach fixed point theory. In Section 5, an example is given to illustrate our main results. At last, in Section 6, our conclusion is presented.


## 2. Preliminaries

Let(Ω,F,{Ft}t≥0,P) be a complete probability space equipped with a normal filtration {Ft}t≥0 satisfying standard assumptions; that is, the filtration is right continuous and F0 contains all P-null sets. Let C([-τ,T]; L2(Ω;X)) be the family of all continuous functions from [-τ,T] into L2(Ω;X). Let X and Yi,i=1,2, be three real separable Hilbert spaces. We denote by L(Yi;X) the space of all bounded linear operators from Yi to X, i=1,2. We suppose that {en(i)}n∈N+ is a complete orthonormal basis in Yi. Let Q(i)∈L(Yi;X) be an operator defined by Q(i)en(i)=λn(i)en(i) with finite trace trQ(i)=∑n=1∞λn(i)<∞, where {λn(i)}n∈N+ are nonnegative real numbers. Then there exists a real-valued sequence {ωn(t)}n∈N+ of one-dimensional standard Brownian motions mutually independent on (Ω,F,{Ft}t≥0,P) such that (2)Wt=∑n=1∞λn1ωnten1,t≥0.Consider that a Y2-valued stochastic process BQH(t) is defined by the formal infinite sum (see [2]). (3)BQHt=∑n=1∞λn2βnHten2,t≥0,where the sequence {βnH(t)}n∈N+ is mutually independent scalar fBms with Hurst parameter H∈(1/2,1).LetLQ(i)0(Yi;X) be the space of all Q(i)-Hilbert-Schmidt operators from Yi to X,i=1,2. Now we show the following definitions.Definition 1.
Letξi∈L(Yi;X) and define (4)ξiLQi0≔trξiQiξi∗=∑n=1∞λniξieniX21/2.
IfξiLQ(i)0<∞, then ξi is called a Q(i)-Hilbert-Schmidt operator and the space LQ(i)0≔LQ(i)0(Yi;X) is a real separable Hilbert space equipped with the inner product 〈φ,ψ〉LQi0=∑n=1∞φeni,ψeni,i=1,2.LetT>0, β={β(t);t∈[0,T]} be a Wiener process and {βH(t),t∈[0,T]} be the one-dimensional fBm with Hurst parameter H∈(1/2,1). βH has the following stochastic integral representation:(5)βHt=∫0tKHt,sdβs,KH(t,s) is the kernel defined as (6)KHt,s=cHs1/2-H∫stu-sH-3/2uH-1/2dufor t>s, where cH=H(2H-1)/B(2-2H,H-1/2)1/2, and B(·,·) denotes the Beta function. Set KH(t,s)=0 for t≤s. Let H denote the reproducing kernel Hilbert space of the fBm. In fact H is the closure of the set of indicator functions {I[0;t],t∈[0,T]} with respect to the inner product. (7)I0,t,I0,sH=RHt,s.Then the mapping I[0,t]→βH(t) can be extended to an isometry between H and the first Wiener chaos of the fBm span¯L2(Ω){βH,t∈[0,T]}. The image of an element φ∈H by this isometry is called the Wiener integral of φ with respect to βH (see [23]).We recall that forψ,φ∈H their inner product in H is given by (8)ψ,φH=H2H-1∫0T∫0Tψsφtt-s2H-2dsdt.KH∗ is an operator from H to L2([0,T]) defined as(9)KH∗φs=∫stφr∂KHr,s∂rdr.Then (10)KH∗I0,ts=KHt,sI0,tsand KH∗ is an isometry between the set of indicator functions {I[0;t],t∈[0,T]} and L2([0,T]) that can be extended to H (see [1]).Letψ(s) for s∈[0,T] be a mapping with values in L20(Y2;X). The stochastic integral of ψ with respect to BQH is defined by(11)∫0tψsdBQHs=∑n=1∞∫0tλn2ψsen2dβnHs=∑n=1∞∫0tλn2KH∗ψen2sdβns,where βn is the standard Brownian motion which was introduced in (5).In order to set existence, uniqueness, and controllability problems, we need the following lemmas which are Lemma 2 in [2] and Lemma 7.7 in [24], respectively.Lemma 2 (see [2]).
For anyφ:[0,T]↦LQ(2)0(Y2;X) and 0≤a≤b≤T, if ∑n=1∞λn(2)<∞ and the series ∑n=1∞λn(2)φ(t)en(2) is uniformly convergent for t∈[0,T], then one has (12)E∫abφsdBQHsX2≤cHH2H-1b-a2H-1∫abφsLQ202ds,
whereH∈(1/2,1) and cH=H(2H-1)/B(2-2H,H-1/2).Lemma 3 (see [24]).
For anyr≥1 and for arbitrary predictable process Φ(·)∈LQ(1)0(Y1;X), one has (13)sups∈0,t⁡E∫0sϕσdWσ2r≤r2r-1r∫0tEϕsLQ102r1/rdsrt∈0,T.We assume that0∈ρ(A), where ρ(A) is the resolvent set of A and the analytic semigroup {S(t),t≥0} in X is uniformly bounded; that is, S(t)X≤M for some positive constant M≥1. Then, for α∈(0,1], it is possible under some circumstance to define the fractional power (-A)α as a closed linear operator with domain D((-A)α). Moreover, the subspace D((-A)α) is dense in X and the expression (14)vα=-AαvX,v∈D-Aαdefines a norm on D((-A)α).LetHα denote the Banach space D((-A)α) equipped with the norm ·α, then the following properties are well known (see [25]).Lemma 4 (see [25]).
Suppose that the preceding conditions are satisfied.(1)
If0<β≤α, then the injection Hα↪Hβ is continuous.(2)
For every0<β≤1, there exists Mβ>0 such that (15)-AβSt≤Mβt-β,0≤t≤T.


## 3. Existence and Uniqueness of Mild Solution

In this section, the Banach fixed point theory is used to investigate the existence and uniqueness problems of (1). Now, we introduce the definition of mild solution of (1).Definition 5.
AnX-valued stochastic process x(t) is called a mild solution of (1) if x(·)∈C([-τ,T];L2(Ω;X)) and x(t)=φ(t) for t∈[-τ,0] and for t∈[0,T] satisfies(16)xt=Stφ0+h0,x-r0-ht,xt-rt-∫0tASt-shs,xs-rsds+∫0tSt-sfs,xs-ρs+Busds+∫0tSt-sgs,xs-ηsdWs+∫0tSt-sσsdBQHsP-a.s.In order to set the existence and uniqueness problems, we assume that the following conditions hold:(C.1)
A is the infinitesimal generator of an analytic semigroup {S(t),t≥0} of bounded linear operators in X and there exists a positive constant M≥1 such that S(t)X≤M for 0≤t≤T.(C.2)
The mappingsf:[0,+∞)×X→X and g:[0,+∞)×X→LQ(1)0(Y1;X) satisfy the following conditions: for any x,y∈X and t∈[0,T], there exist nonnegative constants Lf, Cf, Lg, and Cg, such that
(i)
f(t,x)-f(t,y)X≤Lfx-yX,f(t,x)X2≤Cf(1+xX2);(ii)
g(t,x)-g(t,y)LQ(1)0≤Lgx-yX,g(t,x)LQ(1)02≤Cg(1+xX2).(C.3)
The mappingh(t,x)∈D((-A)β) satisfies the following conditions: for any x,y∈X and t≥0, there exist nonnegative constants Lh,Ch≥0 and β∈(0,1) such that
(i)
(-A)βh(t,x)-(-A)βh(t,y)X≤Lhx-yX;(ii)
(-A)βh(t,x)X2≤Ch(1+xX2);(iii)
the constantsLh and β satisfy the following inequality k≔Lh(-A)-β<1.(C.4)
The mapping(-A)βh is continuous in mean square: for all x∈C([0,T];L2(Ω;X)), limt→sE(-A)βh(t,x(t))-(-A)βh(s,x(s))X2=0.(C.5)
The mappingσ:[0,T]→LQ(2)0(Y2;X) satisfies ∫0Tσ(s)LQ(2)02ds<∞, ∀T>0.(C.6)
The mappingu(·) takes values in L2([0,T];U) and B is a bounded linear operator from U to X.Theorem 6.
Let conditions (C.1)–(C.6) be satisfied. Then system (1) has a unique mild solution on [-τ,T].Proof.
DenoteBT≔C([-τ,T];L2(Ω;X)) by subspace of all continuous functions form [-τ,T] into L2(Ω;X) equipped with the norm ζBT=sups∈[-τ,T](Eζ(s)2)1/2. Fix T>0 and consider the space ST≔{x∈BT:x(s)=φ(s), for s∈[-τ,0]}. ST is a closed subspace of BT equipped with the norm ·BT. Obviously, ST and BT are two Banach spaces. Define an operator π:ST→ST by π(x)(t)=φ(t) for t∈[-τ,0] and for t∈[0,T],(17)πxt=Stφ0+h0,x-r0-ht,xt-rt-∫0tASt-shs,xs-rsds+∫0tSt-sfs,xs-ρs+Busds+∫0tSt-sgs,xs-ηsdWs+∫0tSt-sσsdBQHs.
We first verify thatπ is continuous in mean square on [0,T]. Let x∈ST,0≤t<T, and let ε be positive and sufficiently small (similar estimates hold for ε<0), then (18)
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It is easy to know thatEKi(ε)X2→0 as ε→0,i=1,2,4,7.
Further, by using Cauchy-Schwarz inequality, we get(19)EK3εX2≤2E∫0tSε-I-A1-βSt-s-Aβhs,xs-rsdsX2+2E∫tt+ε-A1-βSt+ε-s-Aβhs,xs-rsdsX2≔2EK31εX2+2EK32εX2.
Applying conditions (C.1) and (C.3) toEK31(ε)X2, we can obtain(20)EK31εU2≤Sε-I2M1-β2E∫0t1t-s1-β-Aβhs,xs-rsdsX2≤Sε-I2M1-β2Tββ-1∫0t1t-s1-βE-Aβhs,xs-rsX2ds≤Sε-I2M1-β2T2ββ-2Ch1+sups∈-τ,T⁡ExsX2⟶0
asε→0. Applying condition (C.1) and (C.3) to EI32(ε)X2, we have(21)EK32εU2≤M1-β2E∫tt+ε1t+ε-s1-β-Aβhs,xs-rsdsX2≤M1-β2εββ-1∫tt+ε1t+ε-s1-βE-Aβhs,xs-rsX2ds≤M1-β2ε2ββ-2Ch1+sups∈-τ,T⁡ExsX2⟶0
asε→0. That is to say, EK3(ε)U2→0 as ε→0.
For the fifth term, by using Lemma3, we can obtain (22)EK5εX2≤2E∫0tSε-ISt-sgs,xs-ηsdWsX2+2E∫tt+εSt+ε-sgs,xs-ηsdWsX2≤2Sε-I2∫0tESt-sgs,xs-ηsLQ102ds+2∫tt+εESt+ε-sgs,xs-ηsLQ102ds⟶0
asε→0.
For the sixth term, we get(23)EK6εX2≤2E∫0tSt+ε-s-St-sσsdBQHsX2+2E∫tt+εSt+ε-sσsdBQHsX2≔2EK61εX2+2EK62εX2.
Firstly, apply Lemma2 to EK61(ε)X2, and we have (24)EK61εX2≤cHH2H-1T2H-1∫0tSt-sSε-IσsLQ202ds≤Sε-I2cHH2H-1T2H-1M2∫0tσsLQ202ds⟶0
asε→0. For every s fixed, we get S(ε)σ(s)→σ(s) as ε→0 and S(ε)σ(s)LQ(2)0≤Mσ(s)LQ(2)0.
By Lemma2 and the Lebesgue dominated convergence theorem, we can obtain (25)EK62εX2≤cHH2H-1ε2H-1M2∫tt+εσsLQ202ds⟶0
asε→0. Therefore EK6(ε)X2→0 as ε→0. Thus, π is continuous in mean square on [0,T]. So we conclude that π(ST)⊂ST.
Moreover, we shall show thatπ is contractive in ST1 with some 0<T1≤T. Let x,y∈ST and for any fixed t∈[0,T], we get (26)Eπxt-πytX2≤1kEht,xt-rt-ht,yt-rt2X+31-kE∫0t-ASt-shs,xs-rs-hs,ys-rsdsX2+31-kE∫0tSt-sfs,xs-ρs-fs,ys-ρsdsX2+31-kE∫0tSt-sgs,xs-ηs-gs,ys-ηsdWsX2.
By conditions (C.1)–(C.4) and Lemma3, we have (27)Eπxt-πyt2≤kExt-rt-yt-rtX2+3M1-β2Lh2t2β1-kβ2sups∈-τ,T⁡Exs-ysX2+3M2Lf2t21-ksups∈-τ,T⁡Exs-ysX2+3M2Lg2t1-ksups∈-τ,T⁡Exs-ysX2.
Hence(28)supt∈-τ,TEπxt-πytX2≤γtsupt∈-τ,T⁡Ext-ytX2,
where(29)γt=k+3M1-β2Lh2t2β1-kβ2+3M2Lf2t21-k+3M2Lg2t1-k.
By condition (C.3), we haveγ(0)=k<1, then there exists 0<T1≤T such that 0<γ(T1)<1 which shows that π is a contraction mapping in ST1; therefore π has a unique fixed point in ST1, which is a unique mild solution of (1) on [-τ,T1]. Repeat this procedure to extend the mild solution to [τ,T]. The proof is completed.Remark 7.
Boufoussi and Hajji in [8] and Liu and Luo in [13], respectively, considered the existence and uniqueness of solutions for special cases of (1). In (20) and (21), we use the Hölder inequality and linear growth condition to obtain the continuity of K3(ε) and obtain a weaker result than those of [8, 13]. The necessary conditions both in [8, 13] are (30)1/2<β<1.
However, our condition is(31)0<β<1.
In this sense, this paper improves and generalizes the results in [8, 13].


## 4. Controllability Result

In this section, we focus on the controllability problem of (1). Now we introduce the concept of controllability of neutral stochastic delay partial differential equations.Definition 8.
System (1) is said to be controllable on the finite interval [-τ,T], if, for each initial stochastic process φ defined on [-τ,0] and x1∈X, there exists a stochastic control u∈L2([0,T];U) which is adapted to the filtration {Ft}t≥0 such that the mild solution x(·) of (1) satisfies x(T)=x1, where x1 and T are the preassigned terminal state and time, respectively.In order to set the controllability problem, we assume that the following conditions hold:(C.7)
The linear operatorω from U to X defined by ωu=∫0TS(T-s)Bu(s)ds has an inverse operator ω-1 with values in L2([0,T];U)∖kerω, where kerω={x∈L2([0,T]);U):ωx=0} (see [22]), and there exists a pair of finite positive constants MB and Mω such that B≤MB and ω-1≤Mω.Theorem 9.
Let conditions (C.1)–(C.7) be satisfied. Then, system (1) is controllable on [-τ,T].Proof.
Using the condition (C.7) for an arbitrary mappingx(·), define the stochastic control by(32)ut=ω-1x1-STφ0+h0,x-r0+hT,xT-rT+∫0TAST-shs,xs-rsds-∫0TST-sfs,xs-ρsds-∫0TST-sgs,xs-ηsdWs-∫0TST-sσsdBQHst,t∈0,T.
Applying this control to the operatorπ. Obviously, to find a fixed point for the operator π is equivalent to prove the existence of mild solutions of equation. Clearly, π(T)=x1, which implies that the stochastic control u steers the system from the initial state φ to x1 in time T, provided we can find a fixed point of the operator π which means that the system is controllable on [-τ,T].
First, we shall prove thatπ is continuous in mean square on [0,T]. Let x∈ST, 0≤t<T, and ε be positive and sufficiently small (similar estimates hold for ε<0), then (33)
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From Section3, we can obtain that EKi(ε)X2→0 as ε→0, i=1,2,…,6. Now we estimate the term EK7(ε)X2, and we have (34)EK7εX2≤2E∫tt+εSt+ε-vBω-1x1-STφ0+h0,x-r0+hT,xT-rT+∫0TAST-shs,xs-rsds-∫0TST-sfs,xs-ρsds-∫0TST-sgs,xs-ηsdWs-∫0TST-sσsdBQHsdvX2+2E∫0tSt+ε-s-St-sBω-1x1-STφ0+h0,x-r0+hT,xT-rT+∫0TAST-shs,xs-rsds-∫0TST-sfs,xs-ρsds-∫0TST-sgs,xs-ηsdWs-∫0TST-sσsdBQHsdvX2≔2EK71εX2+2EK72εX2.
Applying conditions (C.1)–(C.7) and Lemmas2–4 to EK71(ε)X2, we get (35)EK71εX2≤7M2MB2Mω2E∫tt+εx1dvX2+7M2MB2Mω2E∫tt+εMx0+h0,x-r0dvX2+7M2MB2Mω2E∫tt+εhT,xT-rTdvX2+7M2MB2Mω2E∫tt+ε∫0T-A1-βST-s-Aβhs,xs-rsdsdvX2+7M2MB2Mω2E∫tt+ε∫0TST-sfs,xs-ρsdsdvX2+7M2MB2Mω2E∫tt+ε∫0TST-sgs,xs-ηsdWsdvX2+7M2MB2Mω2E∫tt+ε∫0TST-sσsdBQHsdvX2≤7M2MB2Mω2ε2Ex1X2+7M4MB2Mω2ε2Ex0+h0,x-r0X2+7M2MB2Mω2Ch-A-β2ε21+sups∈-τ,T⁡ExsX2+7M2MB2Mω2M1-β2Chβ-2T2βε21+sups∈-τ,T⁡ExsX2+7M4MB2Mω2CfT2ε21+sups∈-τ,T⁡ExsX2+7M4MB2Mω2CgTε21+sups∈-τ,TExsX2+7M4MB2Mω2cHH2H-1T2H-1ε2∫0TσsLQ202ds.
ThenEK71(ε)X2→0 as ε→0. Using the similar technique to EK72(ε)X2, we have (36)EK72εX2≤7MB2Mω2E∫0tSt+ε-v-St-vx1dvX2+7MB2Mω2E∫0tSt+ε-v-St-vMx0+h0,x-r0dvX2+7MB2Mω2E∫0tSt+ε-v-St-vhT,xT-rTdvX2+7MB2Mω2E∫0tSt+ε-v-St-v∫0TAST-shs,xs-rsdsdvX2+7MB2Mω2E∫0tSt+ε-v-St-v∫0TST-sfs,xs-ρsdsdvX2+7MB2Mω2E∫0tSt+ε-v-St-v∫0TST-sgs,xs-ηsdWsdvX2+7MB2Mω2E∫0tSt+ε-v-St-v∫0TST-sσsdBQHsdvX2≤7MB2Mω2Ex1X2+7M2MB2Mω2Ex0+h0,x-r0X2+7MB2Mω2Ch-A-β21+sups∈-τ,T⁡ExsX2+7MB2Mω2M1-β2Chβ-2T2β1+sups∈-τ,T⁡ExsX2+7M2MB2Mω2CfT21+sups∈-τ,T⁡ExsX2+7M2MB2Mω2CgT1+sups∈-τ,T⁡ExsX2+7M2MB2Mω2cHH2H-1T2H-1∫0TσsLQ202ds×∫0tSt+ε-v-St-vdv2.
ThenEK72(ε)X2→0 as ε→0.
Thus,π is indeed continuous in mean square on [0,T]. So we conclude that π(ST)⊂ST.
Last, we will show thatπ is a contraction mapping in ST1 with some 0<T1≤T. Let x,y∈ST, as proceeding as we did previously; we can obtain (37)Eπxt-πytX2≤1kEht,xt-rt-ht,yt-rtX2+71-kE∫0t-A1-βSt-s-Aβhs,xs-rs-hs,ys-rsdsX2+E∫0tSt-sfs,xs-ρs-fs,ys-ρsdsX2+E∫0tSt-sgs,xs-ηs-gs,ys-ηsdWsX2+E∫0tSt-vBω-1hT,xT-rT-hT,yT-rTdvX2+E∫0tSt-vBω-1∫0TAST-shs,xs-rs-hs,ys-rsdsdvX2+E∫0tSt-vBω-1∫0TST-sfs,xs-ρs-fs,ys-ρsdsdvX2+E∫0tSt-vBω-1∫0TST-sgs,xs-ηs-gs,ys-ηsdWsdvX2.
Thus(38)Eπxt-πyt2≤kExt-rt-yt-rtX2+7M1-β2Lh2t2β1-kβ2sups∈-τ,T⁡Exs-ysX2+7M2Lf2t21-ksups∈-τ,T⁡Exs-ysX2+7M2Lg2t1-ksups∈-τ,T⁡Exs-ysX2+7M2MB2Mω2Lh2-A-β2t21-kExT-rT-yT-rTX2+7M2M1-β2MB2Mω2Lh2T2βt21-kβ2sups∈-τ,T⁡Exs-ysX2+7M4MB2Mω2Lf2T2t21-ksups∈-τ,T⁡Exs-ysX2+7M4MB2Mω2Lg2Tt21-ksups∈-τ,T⁡Exs-ysX2.
Hence(39)supt∈-τ,T⁡Eπxt-πytX2≤btsupt∈-τ,T⁡Ext-ytX2,
where(40)bt=k+7M1-β2Lh2t2β1-kβ2+7M2Lf2t21-k+7M2Lg2t1-k+7M2MB2Mω2Lh2-A-β2t21-k+7M2M1-β2MB2Mω2Lh2T2βt21-kβ2+7M4MB2Mω2Lf2T2t21-k+7M4MB2Mω2Lg2Tt21-k.
By condition (C.3), we know thatb(0)=k<1 then there exists 0<T1≤T such that 0<γ(T1)<1 which shows that π is a contraction mapping in ST1. Then π has a unique fixed point in ST1, which is a unique mild solution of (1) on [-τ,T1]. Repeating this procedure can be used to extend the mild solution to the entire interval [-τ,T]. Obviously, π(x)(T)=x1 which means that system (1) is controllable on [-τ,T]. This completes the proof.


## 5. An Example

In this section, we provide an example to illustrate our main results. Consider the following NSDPDE with finite delaysτ1, τ2, and τ3(0≤τi≤τ<∞,i=1,2,3):(41)dxt,ξ-Ht,xt-τ1,ξ=∂2∂ξ2xt,ξ+μt,ξ+Ft,xt-τ2,ξdt+Gt,xt-τ3,ξdWt+σtdBQHt,0≤ξ≤π,t∈0,T,xt,0=xt,π=0,t∈0,T,xt,ξ=φt,ξ,t∈-τ,0,0≤ξ≤π,where W(t) is a cylindrical Brownian motion and BQH(t) is a cylindrical fractional Brownian motion in X.LetX=L2([0,π]) and define the operator A:D(A)⊂X→X by A=∂2/∂ξ2 with the domain (42)DA≔z∈X,z,z′are  absolutely  continuous  on,z′′∈X,z0=zπ=0.It is well known that an analytic semigroup{S(t)}t≥0 is generated by the operator A in X satisfying S(t)X≤e-t. Furthermore,(43)Au=∑i=1∞n2u,enen,u∈DA,where en(x)=2/πsin⁡(nx) is a complete orthonormal set of eigenvectors of A. Then, the following properties hold:(i)
Ify∈D(A), then Ay=∑n=1∞n2〈y,en〉en.(ii)
For eachy∈X, A-1/2y=∑n=1∞(1/n)〈y,en〉en. In particular, A-1/2X=1.(iii)
The operatorA1/2 is given by (44)A1/2y=∑n=1∞ny,xnenon the spaceD[A1/2]={y(·)∈X,∑n=1∞n〈y,en〉enΣ}.We assume that the following conditions hold:(1)
LetB:U→X be a bounded linear operator defined by (45)Butξ=μt,ξ,0≤ξ≤π,u∈L20,π,U.(2)
The operatorω:∈L2([0,π],U)→X is given by (46)ωuξ=∫0TST-sμs,ξds,0≤ξ≤π,has a bounded invertible operatorω-1, and satisfies condition (C.7). For the construction of the operator ω and its inverse, see [26] for more details.(3)
There exist nonnegative constantsLh<1, Lf, and Lg such that, for all t∈[-τ,T], x,y∈X, and ξ∈[0,π], (47)Ft,xt,ξ-Ft,yt,ξX≤Lfxt,ξ-yt,ξX,Gt,xt,ξ-Gt,yt,ξX≤Lgxt,ξ-yt,ξX,-A1/2Ht,xt,ξ--A1/2Ht,yt,ξX≤Lhxt,ξ-yt,ξX.(4)
There exist nonnegative constantsCf,Cg, and Ch, such that, for all t∈[-τ,T], x∈X, and ξ∈[0,π], (48)Ft,xt,ξX2≤Cf1+xt,ξX2,Gt,xt,ξX2≤Cg1+xt,ξX2,-A1/2Ht,xt,ξX2≤Ch1+xt,ξX2.(5)
The mapping(-A)1/2H is continuous in mean square on [-τ,T]: for all t∈[-τ,T], x∈X, and ξ∈[0,π], we have (49)limt→s⁡E-A1/2Ht,xt,ξ--A1/2Hs,xs,ξX2=0.(6)
The mappingσ:[0,T]→LQ(2)0(Y2;X) satisfies (50)∫0TσsLQ202ds<∞,∀T>0.Define the operators F, G, and H:R+×L2([0,π])→L2([0,π]) by (51)ht,φξ=Ht,φ-τ1ξforξ∈0,π,φ∈L20,π,ft,φξ=Ft,φ-τ2ξforξ∈0,π,φ∈L20,π,gt,φξ=Gt,φ-τ3ξforξ∈0,π,φ∈L20,π.If we put(52)xtξ=xt,ξ,t∈0,T,ξ∈0,π,xt,ξ=φt,ξ,t∈-τ,0,ξ∈0,π,then problem (41) can be written in the abstract form (53)dxt+ht,xt-rt=Axt+ft,xt-ρt+Butdt+gt,xt-ηtdWt+σtdBQHt,0≤t≤T,xs=φs,-τ≤s≤0.Thus all the conditions of Theorems 6 and 9 are fulfilled. Therefore, we conclude that the system (41) has a unique mild solution which is controllable on [-τ,T].


## 6. Conclusion

In this paper, by using the Banach fixed point theorem, we obtain some sufficient conditions ensuring the existence, uniqueness, and controllability of neutral stochastic delay partial differential equations driven by standard Brownian motion and fractional Brownian motion. In our next paper, we will explore the existence, uniqueness, and controllability problems with non-Lipschitz condition.



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*Source: 7502514-2018-04-02.xml*