APPROXIMATE CONTROLLABILITY OF IMPULSIVE STOCHASTIC SYSTEMS DRIVEN BY ROSENBLATT PROCESS AND BROWNIAN MOTION Текст научной статьи по специальности «Математика»

URAL MATHEMATICAL JOURNAL, Vol. 8, No. 2, 2022, pp. 59-70

DOI: 10.15826/umj.2022.2.005

APPROXIMATE CONTROLLABILITY OF IMPULSIVE STOCHASTIC SYSTEMS DRIVEN BY ROSENBLATT PROCESS AND BROWNIAN MOTION

Abbes Benchaabane

Laboratory of Analysis and Control of Differential Equations "ACED", Univ. 8 May 1945 Guelma, Algeria benchaabane.abbes@univ-guelma.dz

Abstract: In this paper we consider a class of impulsive stochastic functional differential equations driven simultaneously by a Rosenblatt process and standard Brownian motion in a Hilbert space. We prove an existence and uniqueness result and we establish some conditions ensuring the approximate controllability for the mild solution by means of the Banach fixed point principle. At the end we provide a practical example in order to illustrate the viability of our result.

Keywords: Approximate controllability, Fixed point theorem, Rosenblatt process, Mild solution stochastic impulsive systems.

1. Introduction

It is well known that approximate controllability is one of the fundamental concepts in mathematical control theory for infinite differential systems and plays a significant role in both deterministic and in stochastic dynamical systems. Approximate controllability means that the system can be moved to an arbitrary small neighborhood of the final state. Some recent researches on the existence results of approximate controllability are [8, 9, 14, 25].

Recently, there has been increasing interest in the analysis of control synthesis problems for impulsive systems due to their significance both in theory and applications, for example, in problems of sudden environmental changes, radiation of electromagnetic waves and changes in the interconnections of subsystems. For some recent researches on the existence results for impulsive stochastic differential equations, we refer the reader to monographs [3-5, 10, 23, 24, 29]. In these models, the processes are characterized by the fact that they undergo abrupt changes of state at certain moments of time between intervals of continuous evolution. For basic concepts about the impulsive systems see [12, 17].

In recent years, there has been a growing interest in stochastic functional differential equations driven by the Rosenblatt process [2, 19, 20, 22]. The theory of Rosenblatt process has been developed accordingly due to its nice properties see [13, 16, 27]. Tudor [28] investigated the Rosenblatt process which is Gaussian and the calculus for it is much easier than other processes. However, in concrete situations where the Gaussianity is not plausible for the model, one can employ the Rosenblatt process. There is corresponding literature devoted to various theoretical aspects of impulse systems controlled by Rosenblatt processes [7, 15, 18, 20].

Some dynamical systems of a special kind require a mixed process to model their dynamics [1,

26].

Inspired by the above studies, this article is devoted to demonstrating the approximate controllability of a soft solution for a class of neutral functional-stochastic differential equations controlled

by a Wiener process and a Rosenblatt process independent of the form

dx(t) = Ax(t)dt + Bu(t)dt + f (t, x(t)) dt + g (t, x(t)) dW(t) + a(t)dZH(t), t € [0,T], t = tfc, ( )

Ax(tk) = x(t+) - x(t-) = Ik(x(t-)), k = 1,2,..., m, 1 ' ;

x(0) = x0 € X,

where x(-) takes values in the separable Hilbert space X, A : D(A) c X ^ X is a closed, linear, and densely defined operator on X. Let B be a bounded linear operator from the Hilbert space U into X.

Let the control u € ([0,T],U) which is the Hilbert space of all square integrable and Ft-adapted processes with values in U. Let QK be a positive, self adjoint and trace class operator on K and let L2(K, X) be the space of all QK-Hilbert-Schmidt operators acting between K and X equipped with the Hilbert-Schmidt norm ||.||L2. The W is a QK-Wiener process on Hilbert space K.

Let Q be a positive, self adjoint and trace class operator on Y and let L>(Y, X) be the space of all Q -Hilbert-Schmidt operators acting between Y and X equipped with the Hilbert-Schmidt norm ||.||Lo. Let ZH be a Q-Rosenblatt process on a Hilbert space Y. The process W and ZH are independent. The functions f, g and a will be specified later. Moreover, the fixed moments of times tk satisfy 0 = t0 < ti < ... < tm < tm+1 = T, x(t+) and x(t-) represent the right and left limits of x(t) at t = tk. Here Ax(tk) = x(t+) — x(t-) represents the jump in the state x at time tk, where Ik determines the size of the jump.

Let (Q, FT, P) be the complete probability space with the natural filtration {Ft | t € [0, T]} generated by random variables {ZH(s), W(s), s € [0,T]}. Let x0 be an F0-measurable random variable independent of W and ZH satisfying E ||x0||2 < to. We define the following classes of functions: let L2(Q, Ft, X) be the Hilbert space of all FT—measurable, square integrable variables with values in X, ([0,T],X) is the Hilbert space of all square integrable and Ft—adapted processes with values in X.

The space C ([0,T], L2(Q, FT, X)) is the Banach space of continuous maps except for a finite number of points tk at which x(t-) and x(t+) exists and x(t-) = x(tk) satisfying the condition

supíe[0,T] E ||x(t)|2 <

and AT is the closed subspace of C ([0, T], L2(Q, FT, X)) consisting of measurable and Ft-adapted processes x(t), then AT is a Banach space with the norm defined by

( 2^ 1/2 II^iIaT = (ksuPíe[0,T] E ||x(t)|1 )

Let {ZH(t), t € [0,T]} be the one-dimensional Rosenblatt process with parameter H € (1/2,1), ZH has the following representation (see Tudor [28])

Zh(t) = d(H) / *

00

r* dKH , J)KH

(-u., yi)—-—(u,y2)du

Iy1vy2 du du

where

B(t)tg[o,T] is the Wiener process, B(-, ■) is the Beta function,

¿=n±±, dm = * /=£=:, CB = ' H{2H-1)

H + 1\ 2(2H - 1)' V B(2 - 2H, H - 1/2);

KH(t,s) = 1{i>s}CHs1/2-H /V - s)H-3/2uH-1/2du.

1 J s

Let X and Y be two real separable Hilbert spaces, L(Y; X) be the space of bounded linear operator from Y to X, Q e L(Y; X) be an operator defined by Qen = Anen with finite trace

-n - '»nc-n

<x

^n

tr Q = ^ An < œ, An > 0

n=1

and {en} is a complete orthonormal basis in Y.

We define the infinite dimensional Q-Rosenblatt process on Y as

Zn{t) = \ZKenZn{t),

n=1

where (zn)n>0 is a family of real independent Rosenblatt processes. Consider the following fundamental inequality.

Lemma 1 [21]. If $ : [0,T] ^ L0(Y; X) satisfies

rT

f ||0(s)MLo ds < œ,

J0 2

then we have

rt 2 rt

E

(s)

/0

< 2Ht2H-1 / ||^(s)|Lg ds.

0

Definition 1. For each u e ([0, T],U), a stochastic process x e AT is a mild solution of (1.1) if we have

x(t) = S(t)x0 + / S(t - s) (Bu(s) + f (s, x(s))) ds J0

+ / S(t - s)g(s,x(s))dW(s)+/ S(t - s)a(s)dZn(s)+ V S(t - tfc)/fc(x(t-)). 70 70 0<ifc <t

Let x(T; u) be the state value of system (1.1) at terminal time T corresponding to control u. The set

R(T) = {x(T; u) : u € Lf ([0, T], U)} is called the reachable set of (1.1) at the terminal time T.

Definition 2. The stochastic control system (1.1) is called approximately controllable on the

interval [0, T] if

R(T) = C2{tl,?T,X).

For the proof of the main result, we impose the following conditions on data of the problem.

(Hyp 1) A is the infinitesimal generator of a compact semigroup {S(t), t > 0} on X such that ||S(t)|| < M, for some constant M > 0.

(Hyp 2) 1. The function f : [0,T] x X ^ X is continuous and there exists a constant Cf such that for x, y € X and t € [0, T]

|f(t,x)||2 < Cf(1 + ||x||2), |f(t,x) - f(t,y)||2 < Cf ||x - y||2 .

2. The function g : [0, T] x X — L2(K, X) is continuous and there exists a constant Cg such that for x, y e X and t e [0, T]

||g(t,x)||L2 < Cg(1 + ||x||2), ||g(t,x) - g(t,y)||^ < Cg ||x - y||2 .

(Hyp 3) The function a : [0, T] — is bounded by a positive constant L for all t e [0,T]. (Hyp 4) /k : X — X is continuous and there exist constants dk, > 0 such that, for x,y e X

(i) ||/fc(x) - /fc(y) ||2 < dfc ||x y||2 , k e {1,..., m} ,

(ii) 114(x)||2 < qfc( 1 + ||x||2) , k € {1,..., m}

/ m \

(iii) M2m E dk <

(Hyp 5) For each 0 < t < T, the operator a(a/ + rT) 1 — 0 in the strong operator topology as a — 0+, with rT e L(X, X) and

T

rT

f S(T - t)BB*S*(T - t)dt.

■J s

(Hyp 6) 1. The function f : [0,T] x X — X is continuous and there exists a constant Cf such that for x, y e X and t e [0, T]

||f (t,x) - f (t,y)||2 < Cf ||x - y||2 .

2. The function g : [0, T] x X — L2(K, X) is continuous and there exists a constant Cg such

that for x, y e X and t e [0, T]

||g(t,x) - g(t,y)||^ < Cg ||x - y||2 .

3. The functions f and g are uniformly bounded, then there exists C > 0 such that

||f(s,x(s)||2 + ||g(s,x(s)||L < C.

Lemma 2 [6]. For any xT e L2(Q, FT, X) there exists a unique ^ e ([0,T]; L2(K, X)) such that t

xt = E(xt) + / f(s)dW(s). 0

For any a > 0 and an arbitrary function x(.), we define the control function for system (1.1) in the following form

ua(t, x) = B*S*(T - t)(a/ + rT)-1 (E(xt) - S(T)x0)

+B *S *(T - t) / (a/ + rT )-1f(s)dW (s) - B*S*(T - t) / (a/ + rT )-1S(T - s)a(s))dZn (s) ./0 ./0

-B*S*(T - t) / (a/ + rT)-1S(T - s)f (s, x(s))ds

J0

-B*S*(T - t) / (a/ + rT)-1S(T - s)g(s,x(s))dW(s)

0s

-B*S*(T - t)(a/ + rT)-1 S(t - tfc)/fc(x(t-)),

0<ifc <t

the function ua(t,x) is defined so that the system driven by this command has a unique solution (see Theorem 1) and moreover the system is approximately controllable (see Theorem 2).

Lemma 3. There exists positive real constant Mu such that, for all x, y € A^ we have

B\\ua(t,x)-ua(t,y)\\2<^\\x-y\\2AT,

a

E|K(i,x)||2<^(l + ||x||

iaT

(1.2) (1.3)

Proof. Let x, y € AT, we have

r t

E||ua(t,x) - ua(t,y)||2 < 3E B*S*(T - t) / (a/ + rT)-1S(T - s)[/(s,x(s)) - /(s,y(s))] ds

JQ

+3E +3E

B*S*(T - t) / (a/ + rT)-1S(T - s)[g(s,x(s)) - g(s,y(s))] dW(s)

B*S*(T - t)(a/ + rT)-1 ^ S(T - tk) [/fc(x(t-)) - /k(y(t-))]

k=i

Using the Holder inequality, Ito isometric theorem and the assumptions on the data, we obtain

E |\ua(t, x) - ua(t, y)||2 < ^ \\B\\2 M4TCf f E ||x(s) - y(s)||2 ds

a Jo

3 rt 3 m .

+ \\B\\2 M^Cg / E ||x(s) - y{s)II2 ds + -2\\B\\2 M4m(V dfc)E || [lk(x(t;)) - Ik(y(trk a Jo a \r /

3

^llßfM^T sup E ||x(s) — y(s)||2

a se[Q,T]

+-||5||2M4^T sup E||x(s)-y(s)||2 + m(Vdfc) sup E ||x(s) — y{s)\\

a se[Q,T] ; «c[QT]

< — \\B\\2M4

a

k=i se[Q,T ] m

T 2Cf + TCg + ^^dk

k=i

llx - y^AT

Mu

a2

llx - yllAT

where

Mß = 3 llBll2 M4

T 2Cf + TCg + m PT

k=i

The proof of the second (1.3) is similar.

2. Approximate controllability

For any a > 0, define the operator : A^ ^ A^ by

t

(Fax)(t) = S(t)xo + / S(t - s)(Bua(s,x) + /(s,x(s))) ds

y Q

+ / S(t - s)g(s,x(s))dW(s)+/ S(t - s)a(s)dZn(s)+ V S(t - tk)/k(x(t-)).

QQ

Q<tfc<t

□

The first main result is the following theorem.

2

2

t

2

Q

m

2

m

2

m

Theorem 1. Under assumptions (Hyp 1)-(Hyp 5), the system (1.1) has a mild solution on [0,T].

T

Proof. Step 1. Let 0 < t1 < t2 < T. Then for any fixed x e AT

E ||(Fax)(t2) - (F«x)(t1)|2 < 6E ||(S(t2) - S(t1)) xo|2

+6E

a ft 2

ft 1

+6E

t2

/ S(t2 - s)f (s,x(s))ds - / S(t1 - s)f (s,x(s))ds /o J0

r ti

+6E +6E

+6E

f S(t2 - s)g(s,x(s))dW(s) - S(t1 - s)g(s,x(s))dW(s) 0 Jo

r t2 />t1 2

I S(t2 - s)a(s)dZn(s) - S(t1 - s)a(s)dZn(s) 00

^ S(t2 - tk)/k(x(t—)) - ^ S(t1 - tk)/k(x(t—))

0<tfc <t2 0<tfc <ti

/ 2 S(t2 - s)Bua(s,x(s))ds - / 1 S(t1 - s)Bua(s,x(s))ds 00

= 6 (J1 + J2 + J3 + J4 + J5 + Ja).

Thus we obtain by Holder inequality, Ito isometric theorem and the assumptions (Hyp 1)-(Hyp 5)

J1 < ||S(t2) - S(t1)|2 E

J2 < 2E

t1

0

t1

(S(t2 - s) - S(t1 - s)) f (s, x(s))ds

+ 2E

t2

< 2t1 f1 E ||(S(t2 - s) - S(t1 - s)) f(s,x(s))||2 ds + 2M2(t2 - T E ||f(s,x(s))f ds,

0 t1

S(t2 - s)f (s, x(s))ds

J3 < 2E

(S(t2 - s) - S(t1 - s)) g(s,x(s))dW(s)

0

2+E

t2

S(t2 - s)g(s,x(s))dW(s)

< 2 E 0

t2

(S(t2 - s) - S(t1 - s)) g(s, x(s)) ds + 2M2 / E ||g(s, x(s)) ||L2 ds,

L2

J4 < 2E

t1

(S(t2 - s) - S(t1 - s)) a(s)dZn(s)

+ 2E

t1 t2

S(t2 - s)a(s)dZn(s)

< 4Ht2H-1 J 1 E ||(S(t2 - s) - S(t1 - s)) a(s)||Lo ds + 4M2H (t2H—1 - t?H—^ jf ' ||^(s)|Lg ds,

J5 < 2m ^ E ||S(t2 - s)/k(x(t—))||2 + 2m ^ E ||(S(t2 - s) - S(t1 - s)) /k(x(t—))|'2 t1<tfc <t2 0<tfc <t1

< 2mM2 ^ E ||/k(x(t—))112 + 2m ^ E ||(S(t2 - s) - S(t1 - s)) /k(x(t-))|2 ,

Ja < 2E

t1

0

t1<tfc <t2 J t1

0<tfc <t1

(S(t2 - s) - S(t1 - s)) Bua(s, x)ds

0

+ 2E

t2

S(t2 - s)Bua(s,x)ds

< 2t1 [ 1 E ||(S(t2 - s) - S(t1 - s)) Bua(s,x)||2ds + 2M2 ||B||2 (t2 - / 2 E ||ua(s, x)||2ds. J0 Jt1

Consequently, using the strong continuity of S(t), as well as the Lebesgue's dominated convergence theorem, we conclude that the right side of the above inequality tends to zero when t2 - t1 — 0. Thus we conclude that (Fax)(t) is continuous in [0,T].

2

2

2

2

2

2

0

2

2

2

2

2

0

2

2

Step 2. Let x € A^, then we have

E ||(F«x)(t)||2 < 6E ||S(i)xof + 6E / S(t - s)Bua(s,x)ds

J o

ft 2 rt

t

+6E +6E

S (t — s)f (s, x(s))ds

0

S(t — s)a(s)dZH (s)

+ 6E

+ 6E

S (t2 — s)g(s,x(s))dW (s)

0

S(t — tfc)/fc(x(t-))

o<tfc <t

By Holder inequality, Lemma 3, Ito isometric theorem and the assumptions (Hyp 1)-(Hyp 5), we have

E ||(F«x)(t)||2 < 6E ||S(t)x0||2 + 6M2 ||B||2tW ||ua(s,x)||2ds

0

+6M2TE i ||f (s,x(s))||2 ds + 6M2E / ||g(s,x(s))|L2 ds ./0 ./0 ft m 2 +12M2HT2H-1E / ||a(s)||Lo ds + 6mM^E ||/k(x(t-))||2 .

2 k=1

Hence

E ||(F0.ï)(Î)||2 < 6M2E ||.T0||2 + 6M2 ||B||2T2^ (l + \\x\\2at)

a v 2 /

+6M2T2Cf (i + ||x|£t) + 6M2TCg (i + ||X|^TT)

m

+12M2HT2H-1TL + 6mM2 ( ^ qfc) (i + ||x||Ar)

fc=i

< 6M2 (E ||x01|2 + 2HT2H-1TL

+6M2[ ||B||2 T2

M„

a

+ TCg + m(J]qfc) ) (i + ||x

fc=i

we thus obtain that ||(Fax)||^r < oo. Since (Fax)(t) is continuous on [0,T], therefore maps AT, in itself.

Step 3. Let x,y e AT, then for any fixed t e [0,T] we have

t

||(F«x)(t) — (F«y)(t)||2 < 4E / S(t — s)B (ua(s,x) — ua(s,y)) ds

Jo

ft 2

+4E +4E +4E

S(t — s) (f (s,x(s)) — f (s,y(s))) ds

/0

t2 S(t — s) (g(s, x(s)) — g(s, y(s))) dW(s)

S(t — tfc) (4(x(t-)) — /fc(y(t-)))

0<tfc<t

By assumptions (Hyp 1)-(Hyp 5) combined with Holder's inequality, Lemma 3 and Ito isometric

2

2

t

2

2

0

m

2

2

0

theorem, we get that

||(F«x)(t) - (F«y)(t)||2

f-t r t

r2||T->n2+ / 1l„,a/„ „,ar„ „All2 i at\/t24

< 4M2 ||B||2t / ||ua(s,x) - ua(s,y)||2 ds + 4M2t / ||f(s,x(s)) - f (s,y(s))||2 ds ./0 ./0

/■ t / m \ +4M2 J ||g(s, x(s)) - g(s, y(s))||L2 ds + 4M2m( dfc) ||/fc(x(t-)) - /fc(y(t-)) ||2

fc=i Therefore,

2

||(Fax)(t) - (Fay)(t)| • l.\/2 II / ||x(s)-y(s)||2ds + 4M2iCf [ \\x{s) - y{s)\\2 ds

a2 7o 7o

/" t / m \ +4M2Cfl / ||x(s) - y(s)||2ds + 4M2m( Vdfc) ||x(t-) - y(t-)||2 .

70 Vi 7

Then we have

sup E ||(Fax)(t) - (F«y)(t)||2 se[o,t]

,M„ m

(A/i " \

l|P||2 t + t(tCf + Cfl) + dk) ) sup E||.T(s)-y(s)||i

a k=1 / se[o,t]

= <p(t) sup E ||x(s) - y(s)||2 se[o,t]

where

M m

v?(i) = 4M2 \\B\\2t2-f + AM2t{tCf + Cfl) + 4M2m(^dfc) a fc=i

We have (see (Hyp 4)—(iii))

p(0) = k=1 dfc J < 1-

So there is T1 with 0 < T1 < T such that 0 < ^(T1) < 1 and is a contraction mapping on A^1 and consequently has a unique fixed point. So by repeating the procedure, we extend the solution to the interval [0, T] in several finite steps. □

The second main result is the following theorem.

Theorem 2. Under assumptions (Hyp 1), (Hyp 3), (Hyp 4), (Hyp 5) and (Hyp 6), the system (1.1) is approximately controllable on [0,T].

Proof. Let xa the solution of system (1.1) corresponding to ^(t, x) = ^a(t,x). We obtain by the stochastic Fubini theorem

m

2

xa(T) = xt - a(a/ + rT)-1 (Ext - S(T)xo)

i'T i'T +a / (a/ + rT)-1S(T - s)f (s,x(s)ds + a / (a/ + rT)-1 [S(T - s)g(s,x(s) - tf(s)j dW(s) Jo Jo

| T m

+a / (a/ + rT)-1S(T - s)a(s)dZH(s) + a(a/ + rT)-1V S(T - tfc)/fc(xa(t-)). 7o fc=i

By the hypotheses (Hyp 6-2), there is a subsequence still designated by {f(s,xa(s),g(s,xa(s)} which converges weakly to some {f(s),g(s)} in X x L2 and {/k(xa(t-))} weakly converging to {/k(w)} in X. By the compactness of {S(t) : t > 0}, we have

S(T - s)f (s,x«(s) — S(T - s)f (s),

S(T - s)g(s,x«(s) — S(T - s)g(s),

S(T - tfc)/fc(xa(t-)) — S(T - tfc)/fc(w).

By hypothesis (Hyp 5), we have

( a(a/ + rT)-1 — 0 strongly as a — 0+, for all 0 < s < T,

\ ¡a(a/ + rT)-1! < 1.

So, by the Lebesgue dominated convergence theorem we obtain

i'T

E ||xa(T) — xt||2 < 9E ||a(a1 + rT)-1 (Ext — S(T)xo)||2 + 9E J ||a(a1 + rT)-1^(s) ||L2 ds +18HT2H-1 J ||a(a1 + rT)-1S(T — s)a(s) ||Lg ds + 9e( J ||a(a1 + rT)-1S(T — s)f (s) || ds) +9e( JT ||a(a1 + rT)-11| ||S(T — s) (f(s,xa(s)) — f(s))|| ds)2

/" t

+9^ ||a(a1 + rT)-1S(T — s)g(s)||L ds

Jo 2

+9E / ||a(a1 + rT)-1||2 ||S(T — s)(g(s,xa(s)) — g(s))||L2 ds

0

m

+9e|| ^ a(a1 + rT )-1S(T — tfc )/fc (w)

a(al +rs ) 1( fc=1

mm

2

^ 0 as a ^ 0+.

+9E ||a(a/ + rT)-11| | ^ S(T - tfc)/fc(xa(t-)) - ^ S(T - tfc)/fc(w)

fc=1 fc=1 Then the system (1.1) is approximately controllable. □

3. Example

In this section we present an example. Let X = L2[0,n], U = L2[0,n] and x0 e L2[0,n]. Let A c D(A) : X — X be the linear operator given by Ay = y'', where

D(A) = {y e X / y, y' are absolutely continuous y'' e X, y(0) = y(n) = 0}.

Let B e L(R, X) be defined as

(Bu)(z) = b(x)u, 0 < z < n, u e R, b(x) e L2[0,n].

Here W(t) denotes a one dimensional standard Brownian motion and is a Rosenblatt process, the processes W and are independent.

Consider the control system driven by the process W and to illustrate the obtained theory

1 d2 \

dx(t, z) = z) + b(z)u(t) + /1 (t, x(t, z)) jdt

+g1 (t,x(t,z)) dw(t) + a(t)dZH, t e [0,T] , z e [0,n],

Ax(tk,z) = x(t^,z)-x(t^,z) =-^x(tk,z), t = tk, k = 1,...,m,

x(t, 0) = x(t, n) = 0, t e [0, T], x(0, z)= x0(z), z e [0,n].

Suppose /i,gi: R+ x R ^ R are continuous, satisfy the Lipschitz condition and the linear growth condition and are uniformly bounded.

First of all, note that there exists a complete orthonormal set {en}n>1 of eigenvectors of A with

e-n{z) = sf (2/"7r) sin??,z, 0 < z < 7r, n = 1,2,... and the compact semigroup S(t), t > 0, that is generated by A such that

Ay = - £ n2 (y, en) en(y), y G D(A),

n=1

S(t)y = £ e-nt (x, en) e„(y), y G X.

n=1

Now define the functions: / : [0, T] x X ^ X, g : [0, T] x X ^ L(K; X) as follows

/ (t,x)(z) = /i(t,x(z)),

g(t,x)(z) = gi(t,x(z))

for t G [0,T], x G X and 0 < z < n. Consequently, by [11, Theorem 4.1.7], we have that the deterministic linear system (3.1) is approximately controllable on every [0,t], t > 0, provided that

/ b(z)en (z)dz = 0, for n = 1,2,3,....

Jo

Hence, all conditions of Theorem 2 are satisfied, and consequently system (3.1) is approximately controllable on [0,T].

4. Conclusion

Approximate controllability of a class of impulsive stochastic functional differential equations driven simultaneously by a Rosenblatt process and standard Brownian motion in a Hilbert space are obtained. The controllability problem is transformed into a fixed point problem for an appropriate nonlinear operator in a function space. By using some famous fixed point theorems and the approximating technique some new existence and controllability results are obtained.

We also remark that the same idea can be used to study the controllability and the exponential stability of impulsive stochastic functional differential equations driven simultaneously by a Rosenblatt process and standard Brownian motion under non-Lipschitz condition and with non local conditions.

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