Differ Equ Dyn SystDOI 10.1007/s12591-014-0217-7

ORIGINAL RESEARCH

Existence and Controllability Results for FractionalStochastic Semilinear Differential Inclusions

Guendouzi Toufik

© Foundation for Scientific Research and Technological Innovation 2014

Abstract In this paper, we prove the existence and controllability results for fractionalstochastic semilinear differential inclusions involving the Caputo derivative in Hilbertspaces. The results are obtained by using fractional calculation, operator semigroups andBohnenblust-Karlin fixed point theorem. An example is provided to illustrate the theory.

Keywords Hilbert space · Controllability · Fractional stochastic differential inclusions ·Caputo derivative · Semigroup · Bohnenblust-Karlin fixed point theorem

Mathematical Subject Classifications 34G20 · 34G60 · 34A37 · 60H40

Introduction

Controllability is one of the fundamental concepts in mathematical control theory and playsa vital role in both deterministic and stochastic control systems. Roughly speaking, thisproperty means that it is possible to steer a dynamical control system from an arbitrary initialstate to an arbitrary final state by using the set of admissible controls. In recent years, variouscontrollability problems for different kinds of dynamical systems have been studied; cf. [8,9,19,21]. The controllability of nonlinear stochastic systems in infinite dimensional spaces hasbeen extensively investigated by several authors; see [20] and the references therein. Amongthem, Balachandran and Dauer [3] investigated the controllability of nonlinear systems inBanach spaces. Further, Karthikeyan and Balachandran [12] extended the result of [3] to thecase of systems with impulsive effects. In particular, fixed point techniques are effectivelyused to obtain the controllability of evolution systems [13]. Ji et al. [11] established a set ofsufficient conditions for the controllability of impulsive functional differential equations withnonlocal conditions by using the measure of noncompactness combined with Monch fixed

G. Toufik (B)Laboratory of Stochastic Models, Statistic and Applications, Tahar Moulay University, PO.Box 138,En-Nasr, 20000 Saida, Algeriae-mail: tf.guendouzi@gmail.com

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Differ Equ Dyn Syst

point theorem. Very recently, Debbouche and Baleanu [6] derived a set conditions for thecontrollability of a class of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems by using the theory of fractional calculus and fixed point technique.The approximate controllability problems for infinite-dimensional dynamical systems hasbeen studied in [9,18–20]. Sakthivel et al. [18] studied the approximate controllability ofnonlinear deterministic and stochastic evolution systems with unbounded delay in abstractspaces. More recently, Sakthivel et al. [19] investigated the approximate controllability ofnonlinear fractional stochastic control system under the assumptions that the correspondinglinear system is approximately controllable by using the Banach contraction principle.

The subject of fractional calculus and its applications has gained a lot of importanceduring the past three decades, mainly because it has become a powerful tool in modelingseveral complex phenomena in numerous seemingly diverse and widespread fields of scienceand engineering [14,17]. Recently, there has been a significant development in the existenceand uniqueness of solutions of initial and boundary value problem for fractional differentialequations [24].

On the other hand, the theory of differential inclusions has become an active area ofinvestigation due to their applications in the fields such as mechanics, electrical engineering,medicine biology, ecology and so on. One can see [1,5,16]. Recently, Agarwal et al. [2]studied the partial functional differential inclusions involving the Riemann–Liouville deriv-ative. However, the question of fractional stochastic differential inclusions with unboundedoperators involving the Caputo derivative in Hilbert spaces has not been studied extensively.

In this paper, we are interested in the fractional stochastic semilinear differential inclusionsin Hilbert spaces of the type

cDqt x(t) ∈ Ax(t)+ f (t, x(t))+�(t, x(t))

dw(t)

dt, t ∈ J = [0, b], 0 < q < 1,

x(0) = x0,(1)

where cDqt is the Caputo fractional derivative of order q; b > 0; A is the infinitesimal

generator of a strongly continuous semigroup {T (t), t ≥ 0} in H; the state x(·) takes valuesin a Hilbert space H; f : J ×H → H;� : J ×H → P(H) is a nonempty, bounded, closed,and convex multivalued map; {w(t) : t ≥ 0} is a given K-valued Brownian motion or Wienerprocess with a finite trace nuclear covariance operator Q ≥ 0.

Based on our earlier works [22–24], a suitable definition of mild solutions for system (1)is introduced. By using fractional calculation, multivalued mapping and Bohnenblust-Karlinfixed point theorem, an existence result of mild solutions for stochastic system (1) is obtainedunder the assumption of compact semigroup. Further, controllability problem is discussedfor system (1).

Preliminaries

Let (�,F,P) be a complete probability space equipped with a normal filtration Ft , t ∈ J =[0, b] satisfying the usual conditions (i.e., right continuous and F0 containing all P-null sets).We consider three real separable spaces H, K and U , and Q-Wiener process on (�,Fb,P)

with the linear bounded covariance operator Q such that tr Q < ∞. We assume that thereexists a complete orthonormal system {en}n≥1 on K, a bounded sequence of non-negative realnumbers {λn} such that Qen = λnen, n = 1, 2, . . . and a sequence {βn}n≥1 of independentBrownian motions such that

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Differ Equ Dyn Syst

〈w(t), e〉 =∞∑

n=1

√λn〈en, e〉βn(t), e ∈ E, t ∈ [0, b],

and Ft = Fwt , where Fw

t is the sigma algebra generated by {w(s) : 0 ≤ s ≤ t}. LetL0

2 = L2(Q1/2K; H)

be the space of all Hilbert-Schmidt operators from Q1/2K to Hwith the inner product 〈ψ,π〉L0

2= tr [ψQπ�]. Let L2(Fb,K) be the Banach space of all

Fb-measurable square integrable random variables with values in the Hilbert space H. LetE(·) denotes the expectation with respect to the measure P. Let C(

J ; L2(F,H)) be theBanach space of continuous maps from J into L2(F,H) satisfying supt∈J E‖x(t)‖2 < ∞.Let H2(J ; H) be a closed subspace of C(

J ; L2(F,H)) consisting of measurable and Ft -adapted H-valued process x ∈ C(

J ; L2(F,H)) endowed with the norm

‖x‖H2 =(

supt∈J

E‖x(t)‖2H

)1/2

.

Let us recall some known definitions. For more details, we refer to [14]

Definition 2.1 The Riemann–Liouville derivative of order β with lower limit zero for afunction f : [0,∞) → R can be written as

LDα f (t) = 1

(n − α)

dn

dtn

∫ t

0

f (s)

(t − s)α+1−nds, t > 0, n − 1 < α < n, (2)

where is the (Euler) gamma function.

Definition 2.2 The Caputo derivative of order α for a function f : [0,∞) → R can bewritten as

cDα f (t) =L Dα

(f (t)−

n−1∑

k=0

tk

k! f k(0)

), t > 0, n − 1 < α < n. (3)

If f (t) ∈ Cn[0,∞), then

cDα f (t) = 1

(n − α)

∫ t

0(t − s)n−α−1 f n(s)ds = I n−α f n(s), t > 0, n − 1 < α < n

Definition 2.3 The fractional integral of order β with the lower limit 0 for a function f isdefined as

I β f (t) = 1

(β)

∫ t

0

f (s)

(t − s)1−β ds, t > 0, β > 0

whenever the right-hand side is pointwise defined on [0,∞).

We also introduce some basic definitions and results of multivalued maps. For more detailson multivalued maps, see the books of Deimling [7].

In a Hilbert space H, a multivalued map G : H → P(H) is convex (closed) valued, ifG(x) is convex (closed) for all x ∈ H. G is bounded on bounded sets if G(V ) = ∪x∈V G(x)is bounded in H, for any bounded set V of H, i.e., supx∈V

{sup{‖y‖ : y ∈ G(x)}} < ∞.

Definition 2.4 G is called upper semicontinuous (u.s.c.) on H, if for each x∗ ∈ H, the setG(x∗) is a nonempty, closed subset of H, and if for each open set V of H containing G(x∗),the exists an open neighborhood N of x∗ such that G(N ) ⊆ V .

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Differ Equ Dyn Syst

Definition 2.5 G is said to be completely continuous if G(V ) is relatively compact, for everybounded subset V ⊆ H.

If the multivalued map G is completely continuous with nonempty values, then G is u.s.c.,if and only if G has a closed graph (i.e., xn → x∗, yn → y∗, yn ∈ Gxn imply y∗ ∈ Gx∗).

G has a fixed point if there is x ∈ H such that x ∈ Gx .In the following Pbd,cl,cv(H) denotes the set of all nonempty bounded, closed and convex

subset of H.A multivalued map G : J → Pbd,cl,cv(H) is said to be measurable if for each x ∈ H the

mean-square distance between x and G(t) is measurable function on J .For each x ∈ L0

2 define the set of selections of � by

σ ∈ S�,x = {σ ∈ L0

2 : σ(t) ∈ �(t, x(t)) for a.e. t ∈ J}.

Definition 2.6 The multi-valued map �:J × H → Pbd,cl,cv(H) is said to be L2-Carathéodory if

(i) t → �(t, v) is measurable for each v ∈ H;(ii) v → �(t, v) is u.s.c for almost all t ∈ J ;

(iii) for each r > 0, there exists Lσ,r ∈ L1(J,R+) such that

‖�(t, v)‖2 := supσ∈�(t,v)

E‖σ‖2 ≤ Lσ,r (t), for all ‖v‖2 ≤ r and for a.e. t ∈ J.

The following lemma is crucial in the proof of our mains results.

Lemma 2.7 ([15]) Let J be a compact interval and H be a Hilbert space. Let � be anL2-Carathéodory multi-valued map with S�,x �= ∅ and letϒ be a linear continuous mappingfrom L2(J ; H) to H2(J ; H). Then the operator

ϒ ◦ S� : H2 → Pbd,cl,cv(H2), x → (ϒ ◦ S�)(x) = ϒ(S�,x ),is a closed graph operator in H2 × H2.

Now, we give the fixed point theorem due to Bohnenblust and Karlin [4], which is ourmain tool.

Theorem 2.8 Let H be a Hilbert space and K ∈ Pbd,cl,cv(H). Suppose that the operatorG : K → Pbd,cl,cv(K ) is u.s.c. and the set G(K ) is relatively compact in H. Then G has afixed point in K .

Existence of Mild Solutions

Before starting and proving the main result of this section, we present the definition of themild solution to the system (1).

Definition 3.1 A stochastic process x ∈ H2(J,H) is said to be a mild solution of system (1)if x(0) = x0 and there exists a selection σ ∈ S�,x of �(t, x(t)) such that

x(t) = ψ(t)x0 +∫ t

0(t − s)q−1ϕ(t − s) f (s, x(s))ds +

∫ t

0(t − s)q−1ϕ(t − s)σ (s)dw(s),

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Differ Equ Dyn Syst

where

ψ(t) =∫ ∞

0ξq(θ)T

(tqθ

)dθ; ϕ(t) = q

∫ ∞

0θξq(θ)T

(tqθ

)dθ;

T (t) is a C0-semigroup generated by a linear operator A on H; ξq is a probability densityfunction defined on (0,∞), that is ξq(θ) ≥ 0, θ ∈ (0,∞) and ∫∞

0 ξq(θ)dθ = 1.

Lemma 3.2 ([19]) The operators ψ and ϕ have the following properties:

(i) For any fixed t ≥ 0, ψ(t) and ϕ(t) are linear and bounded operators, i.e., for anyx ∈ H,

‖ψ(t)x‖ ≤ M1‖x‖ and ‖ϕ(t)x‖ ≤ q M1

(1 + q)‖x‖.

(ii) {ψ(t), t ≥ 0} and {ϕ(t), t ≥ 0} are strongly continuous, i.e., for x ∈ H and 0 ≤ t1 <t2 ≤ b, we have

‖ψ(t2)x − ψ(t1)x‖ → 0 and ‖ϕ(t2)x − ϕ(t1)x‖ → 0 as t1 → t2.

(iii) For every t > 0, ψ(t) and ϕ(t) are also compact operators if T (t), t ≥ 0 is compact.

We impose the following conditions on data of the problem:(H1) The operator A generates a strongly continuous semigroup {T (t), t ≥ 0} in H, andthere exists a constant M1 ≥ 1 such that supt∈J ‖T (t)‖ ≤ M1.(H2) The function f : J × H → H is continuous and there exists two positive constantsc1, c2 such that

‖ f (t, x)‖2 ≤ c1‖x‖2 + c2, for each x ∈ H, t ∈ J.

(H3) The multi-valued map � : J × H → Pbd,cl,cv(H) is an L2-Carathéodory functionsatisfies the following conditions:

(i) For each t ∈ J , the function �(t, .) : H → Pbd,cl,cv(H) is u.s.c., and for each x ∈H, �(., x) is measurable. Further, for each x ∈ H2, the set

S�,x = {σ ∈ L0

2 : σ(t) ∈ �(t, x(t)) for a.e. t ∈ J}

is nonempty.(ii) For each positive number r and x ∈ H2 with‖x‖H2 ≤ r , there exists a constantβ ∈ (0, q)

and a positive function Lσ,r (·) ∈ L1β (J,R+) such that

‖�(t, x(t))‖2 = sup{‖σ‖2

L02

: σ ∈ �(t, x(t))} ≤ Lσ,r , for each t ∈ J.

(H4) The function s → Lσ,r (s) ∈ L1([0, t],R+)

and there exists a γ > 0 such that

limr→∞ inf

∫ b0 Lσ,r (s)ds

r2 = γ < +∞.

(H5) The semigroup {T (t), t > 0} is compact.

Theorem 3.3 Assume that (H1)–(H5) hold. Then system (1) is solvable on J , provided that

3

(M1

(q)

)2 b2q−1

2q − 1

[bc1 + tr(Q)γ

]< 1. (4)

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Differ Equ Dyn Syst

Proof Transform the system (1) into a fixed point problem. Consider the multivalued operatorA : H2 → Pmes(H2) defined by

A(x) ={

x ∈ H2 : φ(t) = ψ(t)x0 +∫ t

0(t − s)q−1ϕ(t − s) f (s, x(s))ds

+∫ t

0(t − s)q−1ϕ(t − s)σ (s)dw(s); σ ∈ Sσ,x

},

where Pmes(H2) is the family of all nonempty measurable subsets of H2. ��It is clear that the fixed points of A are mild solutions of system (1). Hence we have to

find solutions of the inclusion x ∈ A(x). We show that the multivalued operator A satisfiesthe conditions of Theorem 2.8. The proof will be given in several steps.

Step 1 A(x) is convex for each x ∈ H2. Since � has convex values it follows that S�,x isconvex; so that if σ1, σ2 ∈ S�,x then aσ1 + (1 − a)σ2 ∈ S�,x , for all a ∈ [0, 1], whichimplies clearly that A(x) is convex.

Step 2 The operator A is bounded on bounded subsets of H2. For r > 0 let Br = {x ∈ H2 :‖x‖H2 ≤ r}. Obviously, Br is a bounded, closed and convex set of H2. We claim that thereexists a positive number r such that A(Br ) ⊆ Br .

If this is not true, then for each positive number r , there exists a function xr ∈ B(r), butA(xr ) does not belong to Br , i.e.,

‖A(xr )‖H2 = sup{‖φr‖H2 : φr ∈ (Axr )

}> r,

and

φr (t) = ψ(t)x0 +∫ t

0(t − s)q−1ϕ(t − s) f

(s, xr (s)

)ds

+∫ t

0(t − s)q−1ϕ(t − s)σ r (s)dw(s),

for some σ r ∈ S�,xr .By conditions (H1)–(H3) and Lemma 3.2, we have

r2 < ‖φr (t)‖2

≤ 3‖ψ(t)x0‖2 + 3∫ t

0(t − s)q−1

∥∥ϕ(t − s) f(s, xr (s)

)∥∥2ds

+3tr(Q)∫ t

0(t − s)q−1

∥∥ϕ(t − s)σ r (s)∥∥2

ds

≤ 3M21 ‖x0‖2 + 3

(M1q

(q + 1)

)2 ∫ b

0(t − s)q−1

∥∥ f(s, xr (s)

)∥∥2ds

+3tr(Q)

(M1q

(q + 1)

)2 ∫ b

0(t − s)q−1Lσ,r (s)ds

≤ 3M21 ‖x0‖2 + 3

(M1q

(q + 1)

)2 b2q−1

2q − 1

∫ b

0

(c1‖xr (s)‖2 + c2

)ds

+3tr(Q)

(M1q

(q + 1)

)2 b2q−1

2q − 1

∫ b

0Lσ,r (s)ds

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Differ Equ Dyn Syst

≤ 3M21 ‖x0‖2 + 3

(M1q

(q + 1)

)2 b2q−1

2q − 1

(bc1r2 + bc2

)

+3tr(Q)

(M1q

(q + 1)

)2 b2q−1

2q − 1

∫ b

0Lσ,r (s)ds.

Hence for each φr ∈ A(Br ), we get

r2 < ‖φr‖2H2

= supt∈[0,b]

IE‖φr (t)‖2

≤ 3bM21 ‖x0‖2 + 3

(M1q

(q + 1)

)2 b2q

2q − 1

(c1r2 + c2

)

+3tr(Q)

(M1q

(q + 1)

)2 b2q−1

2q − 1

∫ b

0Lσ,r (s)ds.

Dividing both sides of the above inequality by r2 and taking the limit as r → ∞, using (H3),we get

3

(M1q

(q + 1)

)2 b2q−1

2q − 1

[bc1 + tr(Q)γ

]≥ 1.

This contradicts with the condition (4). Hence, for some r > 0, A(Br ) ⊆ Br .Step 3 A sends bounded sets into equicontinuous sets in H2. For each x ∈ Br let φ ∈ A(x)such that

φ(t) = ψ(t)x0 +∫ t

0(t − s)q−1ϕ(t − s) f (s, x(s))ds +

∫ t

0(t − s)q−1ϕ(t − s)σ (s)dw(s),

for some σ ∈ S�,x . Let 0 ≤ τ1 < τ2 ≤ b. Then we have

‖φ(τ2)− φ(τ1)‖2 ≤ 3[‖(ψ(τ2)− ψ(τ1))x0‖2 +

2∑

i=1

‖�i (τ2)−�i (τ1)‖2].

From the strong continuity of ψ(t), the first term on the R.H.S goes to zero as τ2 − τ1 → 0.Next, it follows from assumptions on the theorem that

‖�1(τ2)−�1(τ1)‖2 ≤ 3∫ τ1

0(τ1 − s)q−1

∥∥(ϕ(τ2 − s)− ϕ(τ1 − s)) f (s, x(s))∥∥2

ds

+3∫ τ1

0

((τ2 − s)q−1 − (τ1 − s)q−1)∥∥ϕ(τ2 − s) f (s, x(s))

∥∥2ds

+3∫ τ2

τ1

(τ2 − s)q−1∥∥ϕ(τ2 − s) f (s, x(s))

∥∥2ds

≤ 3τ

2q−11

2q − 1

∫ τ1

0

∥∥ϕ(τ2 − s)− ϕ(τ1 − s)∥∥2

(c1

∥∥x(s)∥∥2 + c2

)ds

+3

(M1q

(1 + q)

)2( ∫ τ1

0

((τ2 − s)q−1 − (τ1 − s)q−1)ds

)

×( ∫ τ1

0

(c1‖x(s)‖2 + c2

)ds

)

+3(τ2 − τ1)

2q−1

1 − 2q

(M1q

(1 + q)

)2 ∫ τ2

τ1

(c1‖x(s)‖2 + c2

)ds.

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Differ Equ Dyn Syst

Further, we obtain

‖�2(τ2)−�2(τ1)∥∥2 ≤ 3tr(Q)

∫ τ1

0(τ1 − s)q−1

∥∥(ϕ(τ2 − s)− ϕ(τ1 − s))σ (s)∥∥2

ds

+ 3tr(Q)∫ τ1

0

((τ2 − s)q−1 − (τ1 − s)q−1)∥∥ϕ(τ2 − s)σ (s)

∥∥2ds

+ 3tr(Q)∫ τ2

τ1

(τ2 − s)q−1∥∥ϕ(τ2 − s)σ (s)

∥∥2ds

≤ 3tr(Q)τ

2q−11

2q−1

( ∫ τ1

0

∥∥ϕ(τ2−s)−ϕ(τ1 − s)∥∥2

ds

)∥∥Lσ,r∥∥

L1β (J,IR+)

+ 3tr(Q)

(M1q

(1+q)

)2([(τ2−s)q−1−(τ1−s)q−1

] 11−β

ds

)1−β

×∥∥Lσ,r∥∥

L1β (J,IR+)

+ 3tr(Q)(τ2 − τ1)

2q−1

1 − 2q

(M1q

(1 + q)

)2∥∥Lσ,r∥∥

L1β (J,IR+)

.

Since there is δ > 0 such that

‖ϕ(τ2)− ϕ(τ1)‖ ≤ δ√τ1

√τ2 − τ1,

(see [10] proposition 1) and the compactness of ϕ(t), t > 0, from the compactness ofT (t), t > 0 by Lemma 3.2 and assumption (H5), implies the continuity in the uniformoperator topology, we have

‖ϕ(τ2)− ϕ(τ1)‖2 → 0,∥∥ϕ(τ2 − s)− ϕ(τ1 − s)

∥∥2 → 0 as τ2 − τ1 → 0.

Therefore

E‖φ(τ2)− φ(τ1)‖2 → 0 as τ2 − τ1 → 0.

Thus

‖φ(τ2)− φ(τ1)‖2H2

→ 0 as τ2 − τ1 → 0.

Step 4 The set �(t) = {φ(t) : φ ∈ A(Br )

}is relatively compact in K.

Clearly, �(0) = {(Aφ : φ ∈ Br )(0)} = {x0

}is compact; so, it remains to discuss the case

of t > 0. For each ε ∈ (0, t), t ∈ (0, T ], x ∈ Br , arbitrary δ > 0, we define

�ε,δ(t) = {(Aε,δφ)(t) : φ ∈ Br

}

where

(Aε,δφ)(t) =∫ ∞

δ

ξq(θ)T(tqθ

)x0dθ

+ q∫ t−ε

0

∫ ∞

δ

θ(t − s)q−1ξq(θ)T((t − s)qθ

)f (s, x(s))dθds

+ q∫ t−ε

0

∫ ∞

δ

θ(t − s)q−1ξq(θ)T((t − s)qθ

)σ(s)dθds.

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Differ Equ Dyn Syst

Then the sets{(Aε,δφ)(t) : φ ∈ Br

}are relatively compact in H since the operator

T (εqδ), εqδ > 0 is compact in H. We have

‖(Aφ)(t)− (Aε,δφ)(t)‖2

≤ 5

∥∥∥∥∫ δ

0ξq(θ)T

(tqθ

)x0dθ

∥∥∥∥2

+ 5q

∥∥∥∥∫ t

0

∫ δ

0θ(t − s)q−1ξq(θ)T

((t − s)qθ

)f (s, x(s))dθds

∥∥∥∥2

+ q5

∥∥∥∥∫ t

0

∫ ∞

δ

θ(t − s)q−1ξq(θ)T((t − s)qθ

)f (s, x(s))dθds

−∫ t−ε

0

∫ ∞

δ

θ(t − s)q−1ξq(θ)T((t − s)qθ

)f (s, x(s))dθds

∥∥∥∥2

+ q5

∥∥∥∥∫ t

0

∫ δ

0θ(t − s)q−1ξq(θ)T

((t − s)qθ

)σ(s)dθdw(s)

∥∥∥∥2

+ q5

∥∥∥∥∫ t

0

∫ ∞

δ

θ(t − s)q−1ξq(θ)T((t − s)qθ

)σ(s))dθdw(s)

−∫ t−ε

0

∫ ∞

δ

θ(t − s)q−1ξq(θ)T((t − s)qθ

)σ(s)dθdw(s)

∥∥∥∥2

≤ 5M21 ‖x0‖2

∫ δ

0ξq(θ)dθ + 5M2

1 q∫ t

0

∫ δ

0θ(t − s)q−1ξq(θ)

(c1‖x(s)‖2 + c2

)dθds

+ 5M21 q

∫ t

t−ε

∫ ∞

δ

θ(t − s)q−1ξq(θ)(

c1‖x(s)‖2 + c2

)dθds

+ 5q M21 tr(Q)

∫ t

0

∫ δ

0θ(t − s)q−1ξq(θ)Lσ,r (s)dθds

+ 5q M21 tr(Q)

∫ t

t−ε

∫ ∞

δ

θ(t − s)q−1ξq(θ)Lσ,r (s)dθds

≤ 5M21 ‖x0‖2

∫ δ

0ξq(θ)dθ+5M2

1 q

(∫ t

0(t−s)q−1

(c1‖x(s)‖2 + c2

)ds

) ∫ δ

0θξq(θ)dθ

+ 5M21 q

(∫ t

t−ε(t − s)q−1

(c1‖x(s)‖2 + c2

)ds

) ∫ ∞

0θξq(θ)dθ

+ 5q M21 tr(Q)

( ∫ t

0(t − s)q−1Lσ,r (s)ds

) ∫ δ

0θξq(θ)dθ

+ 5q M21 tr(Q)

( ∫ t

t−ε(t − s)q−1Lσ,r (s)ds

) ∫ ∞

0θξq(θ)dθ

≤ 5M21 ‖x0‖2

∫ δ

0ξq(θ)dθ + 5M2

1 qb2q

2q − 1

(c1r2 + c2

) ∫ δ

0θξq(θ)dθ + 5M2

1(2)

(1 + q)

× qε2q−1

2q − 1

(bc1r2 + bc2

)

+ 5q M21 tr(Q)

[(1 − β

q − β

)b

q−β1−β

]1−β‖Lσ,r‖

L1β (J,IR+)

∫ δ

0θξq(θ)dθ

+ 5M21

(2)

(1 + q)qε2q−1

2q − 1‖Lσ,r‖

L1β (J,IR+)

.

Hence, there exist relatively compact sets that can be arbitrarily approximated to the set�(t).Then�(t) is relatively compact in H for all t ∈ (0, T ]. Since it is compact at t = 0, we havethe relative compactness in H for all t ∈ J .

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Differ Equ Dyn Syst

Step 5 A has closed graph. Let xn → x∗ and φn ∈ A(xn) with φn ∈ A(xn). We shall showthat φ∗ ∈ A(x∗).

There exists σn ∈ S�,xn such that

φn(t)=ψ(t)x0+∫ t

0(t − s)q−1ϕ(t − s) f (s, xn(s))ds+

∫ t

0(t − s)q−1ϕ(t − s)σn(s)dw(s).

We must prove that there exists σ∗ ∈ S�,xn such that

φ∗(t)=ψ(t)x0+∫ t

0(t − s)q−1ϕ(t − s) f (s, x∗(s))ds +

∫ t

0(t − s)q−1ϕ(t − s)σ∗(s)dw(s).

Consider the linear continuous operator � : L02 → H2 defined by

(�σ)(t) =∫ t

0(t − s)q−1ϕ(t − s)σ (s)dw(s).

Clearly, � is linear and continuous. Indeed, one has

‖�(σ)(t)‖2 ≤(

M1q

(1 + q)

)2

tr(Q)∫ t

0(t − s)q−1‖σ(s)‖2ds

≤(

M1q

(1 + q)

)2

tr(Q)

[(1 − β

q − β

)b

q−β1−β

]1−β‖Lσ,r‖

L1β (J,H)

.

Let

Zn(t) = φn(t)− ψ(t)x0 −∫ t

0(t − s)q−1ϕ(t − s) f (s, xn(s))ds

Z∗(t) = φ∗(t)− ψ(t)x0 −∫ t

0(t − s)q−1ϕ(t − s) f (s, x∗(s))ds.

We have

Zn(t) ∈ � ◦ S�,xn .

Since f is continuous (see (H2))∥∥Zn(t)− Z∗(t)

∥∥2 → 0 as n → ∞.

Lemma 2.7 implies that � ◦ S� has closed graph. Hence there exists σ∗ ∈ S�,x∗ such that

Z∗(t) =∫ t

0(t − s)q−1ϕ(t − s)σ∗(s)dw(s).

Hence φ∗ ∈ A(x∗), which shows that graph A is closed.As a consequence of Step 1 to Step 5 with the Arzela-Ascoli theorem, we conclude

that A is a compact multivalued map, u.s.c. with convex closed values. As a consequenceof Theorem 2.8, we can deduce that A has a fixed point x which is a mild solution ofsystem (1).

Controllability Results

In the present section, we shall formulate and prove sufficient conditions for the controllabilityof fractional semilinear stochastic differential inclusions. More precisely, we consider thefollowing problem:

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Differ Equ Dyn Syst

{cDq

t x(t) ∈ Ax(t)+ Bu(t)+ f (t, x(t))+�(t, x(t))dw(t)

dt, t ∈ J =[0, b], 0<q<1,

x(0) = x0,

(5)where A, f and � are the same as in above sections, the control function u(t) is given inL2

F (J,U) of admissible control functions, U is a Hilbert space. B is a bounded linear operatorfrom U into H.

Definition 4.1 A stochastic process x ∈ H2(J,H) is a mild solution of (5) if x(0) = x0 andthere exists a selection σ ∈ S�,x of �(t, x(t)) and for each u ∈ L2

F (J,U), it satisfies thefollowing integral equation,

x(t) = ψ(t)x0 +∫ t

0(t − s)q−1ϕ(t − s)[Bu(s)+ f (s, x(s))]ds

+∫ t

0(t − s)q−1ϕ(t − s)σ (s)dw(s),

where ψ(t) and ϕ(t) are given as in Definition 3.1.

Definition 4.2 System (5) is said to be controllable on the interval J if, for every x0, x1 ∈ H,there exists a stochastic control u ∈ L2

F (J,U), which is adapted to the filtration {Ft }t≥0 suchthat the mild solution x of system (5) satisfies x(b) = x1.

We need the following additional hypotheses for the controllability result.(H6) f : J × H → H and σ : J → L0

2 are compact(H7) The linear operator W from L2

F (J,U) into L2(�,H) defined by

Wu =∫ b

0(b − s)q−1ϕ(b − s)Bu(s)ds

has an induced inverse operator W−1 which takes values in L2F (J,U)/K erW , where the

kernel space of W is defined by K erW = {x ∈ L2

F (J,U) : Wu = 0}

and there existpositive constants M2,M3 such that

‖B‖2 ≤ M2 and∥∥W−1

∥∥2 ≤ M3.

The following theorem is our main result in this section.

Theorem 4.3 Assume that (H1)–(H4), (H6) and (H7) are satisfied. Then the system (5) iscontrollable on J provided that

(M1

(q)

)2 b2q−1

2q − 1

[3bc1 + 3tr(Q)γ + M2 M3

(M1

(q)

)2 b2q

2q − 1c1

+M3tr(Q)

(M1

(q)

)2 bq

qγ

]< 1. (6)

Proof For an arbitrary process x(·), define the control process

ubx (t) = W−1

{x1 − ψ(b)x0 −

∫ b

0(b − s)q−1ϕ(b − s) f (s, x(s))ds

−∫ b

0(b − s)q−1ϕ(b − s)σ (s)dw(s)

}(t), t ∈ J,

(7)

where σ ∈ Sσ,x .

123

Differ Equ Dyn Syst

First, we show that the operator A : H2 → P(H2), defined by

A(x) ={

x ∈ H2 : φ(t) = ψ(t)x0 +∫ t

0(t − s)q−1ϕ(t − s)

[f (s, x(s))+ Bub

x (s)]ds

+∫ t

0(t − s)q−1ϕ(t − s)σ (s)dw(s); σ ∈ Sσ,x

}

has a fixed point x , which is a mild solution of system (5).Observe that x1 ∈ (Ax)(b) which means that ub

x steers system (5) from x0 to x1 in finitetime b. This implies that the system (5) is controllable on J .

We note that

‖ubx (t)‖2 ≤ M3‖x1‖2 + M3 M1‖x0‖2 + M3

(M1q

(1 + q)

)2 b2q

2q − 1

(c1r2 + c2

)

+M3tr(Q)

(M1q

(1 + q)

)2[(1 − β

q − β

)b

q−β1−β

]1−β‖Lσ,r‖

L1β (J,H)

.

Similar to the discussion in Section 3, one can check Step 1 to Step 3. Here, we only give themain different steps. Since (H5) is replaced by (H6), there exist some techniques in orderto derive that the set

�(t) = A(Br )(t) = {φ(t) : φ ∈ A(Br )

} ⊂ His relatively compact for any t ∈ J . In fact, by (H6) and Lemma 3.2 the sets

� = {(t − s)q−1ϕ(t − s)σ (s) : t ∈ J, s ∈ [0, t]} ⊂ H

and

� = {(t − s)q−1ϕ(t − s) f (s, x(s)) : t ∈ J, s ∈ [0, t], x ∈ H} ⊂ H

are relatively compact. So for any t ∈ J

�′t =

{∫ t

0(t − s)q−1ϕ(t − s)σ (s)dw(s)

}⊂ tconv�

and

�′t =

{ ∫ t

0(t − s)q−1ϕ(t − s) f (s, x(s))ds

}⊂ tconv�

are relatively compact in H, where conv� means the closure of the convex hull of � in H.By (H7), we obtain that

�′′ ={

ubx = W−1

[x1 − ψ(b)x0 −

∫ b

0(b − s)q−1ϕ(b − s) f (s, x(s))ds

−∫ b

0(b − s)q−1ϕ(b − s)σ (s)dw(s)

]: x ∈ Br

}

is relatively compact in L2F (J,U). As B is bounded, it implies that B�′′ is relatively compact.

So

�′′′ ={ ∫ t

0(t − s)q−1ϕ(t − s)By(s)ds : y ∈ �′′

}⊂ H

123

Differ Equ Dyn Syst

is relatively compact as the map

g →∫ t

0(t − s)q−1ϕ(t − s)g(s)ds

is continuous in H. By Lemma 3.2, we have for any t ∈ J

A(Br )(t) ⊂ ψ(t)x0 +�′t +�′′′,

we have that A(Br )(t) is relatively compact in H for every t ∈ J . Thus, �(t) is relativelycompact in H for each t ∈ J .

Let xn → x∗ as n → ∞, φn ∈ A(xn) and φn → φ∗ as n → ∞. We will show that A hasa closed graph.

Since φn ∈ A(xn), there exists a σn ∈ S�,xn such that

φn(t) = ψ(t)x0 +∫ t

0(t − s)q−1ϕ(t − s) f (s, xn(s))ds

+∫ t

0(t − s)q−1ϕ(t − s)σn(s)dw(s)

+∫ t

0(t − s)q−1ϕ(t − s)BW−1

×[

x1 − ψ(t)x0 −∫ b

0(b − τ)q−1ϕ(b − τ) f (τ, xn(τ ))dτ

−∫ b

0(b − τ)q−1ϕ(b − τ)σn(τ )dw(τ)

](s)ds.

As in Step 5, we must prove that there exists σ∗ ∈ S�,x∗ such that

φ∗(t) = ψ(t)x0 +∫ t

0(t − s)q−1ϕ(t − s) f (s, x∗(s))ds

+∫ t

0(t − s)q−1ϕ(t − s)σ∗(s)dw(s)

+∫ t

0(t − s)q−1ϕ(t − s)BW−1

×[

x1 − ψ(t)x0 −∫ b

0(b − τ)q−1ϕ(b − τ) f (τ, x∗(τ ))dτ

−∫ b

0(b − τ)q−1ϕ(b − τ)σ∗(τ )dw(τ)

](s)ds.

Set

ubx = W−1

{x1 − ψ(t)x0 −

∫ b

0(b − τ)q−1ϕ(b − τ) f (τ, x(τ ))dτ

}(t).

Since f, W−1 are continuous, then ubxn(t) → ub

x∗(t), for t ∈ J , as n → ∞. Clearly, we have∥∥∥∥

(φn(t)− ψ(t)x0 −

∫ t

0(t − s)q−1ϕ(t − s) f (s, xn(s))ds

−∫ t

0(t − s)q−1ϕ(t − s)Bub

xn(s)ds

)

−(φ∗(t)− ψ(t)x0 −

∫ t

0(t − s)q−1ϕ(t − s) f (s, x∗(s))ds

123

Differ Equ Dyn Syst

−∫ t

0(t − s)q−1ϕ(t − s)Bub

x∗(s)ds

)∥∥∥∥2

≤ 3‖φn(t)− φ∗(t)‖2 + 3

(M1q

(1 + q)

)2 b2q

2q − 1supt∈J

‖ f (t, xn(t)− f (t, x∗(t))‖2

+3M2

(M1q

(1 + q)

)2 b2q

2q − 1‖ub

xn− ub

x∗‖2

→ 0 as n → ∞.

Consider the linear continuous operator � : L02 → H2 defined by

(�σ )(t) =∫ t

0(t − s)q−1ϕ(t − s)

[σ(s)dw(s)

−BW−1( ∫ b

0(b − τ)q−1ϕ(b − τ)σ (τ)dw(τ)

)(s)ds

].

Clearly, the operator � is linear and continuous. Using Lemma 2.7, one can deduce that� ◦ S� is closed graph operator. Moreover,

φn(t)− ψ(t)x0 −∫ t

0(t − s)q−1ϕ(t − s) f (s, xn(s))ds

−∫ t

0(t − s)q−1ϕ(t − s)Bub

xn(s)ds ∈ �(S�,xn ).

Since xn → x∗ as n → ∞, it follows from Lemma 2.7 that

φ∗(t)− ψ(t)x0 −∫ t

0(t − s)q−1ϕ(t − s) f (s, x∗(s))ds

−∫ t

0(t − s)q−1ϕ(t − s)Bub

x∗(s)ds

=∫ t

0(t − s)q−1ϕ(t − s)

[σ∗(s)dw(s)− BW−1

×( ∫ b

0(b − τ)q−1ϕ(b − τ)σ∗(τ )dw(τ)

)(s)ds

]

for some σ∗ ∈ S�,x∗ . Therefore φ∗ ∈ A(x∗).As a consequence of Theorem 2.8, we can deduce that A has a fixed point x which is a

mild solution of system (5). Hence (5) is controllable on J . ��

Example

As an application of our results we consider the following fractional stochastic differentialinclusion of the form

⎧⎪⎪⎨

⎪⎪⎩

cDqt x(t, z) ∈ ∂2x(t, z)

∂z2 + μ(t, z)+ f (t, x(t, z))+ �(t, x(t, z))dw

dt,

x(t, 0) = x(t, 1) = 0 t ∈ [0, b],x(0, z) = x0(z) z ∈ [0, 1]

(8)

123

Differ Equ Dyn Syst

where b > 0, 0 < q < 1; (w)(t) is a two sided and standard one dimensional Brownianmotion defined on the filtered probability space (�,F,P); � : J×H → P(H) is a nonempty,bounded, closed and convex multi-valued map, satisfies the assumptions (H3) and (H4);f : J × H → H satisfies (H2). To write the above system into the abstract form of (5), letH = L2[0, 1]. Define the operator A : H → H by Aω = ω′′ with domain

D(A) = {ω ∈ H;ω,ω′ are absolutely continuous, ω′′ ∈ H and ω(0) = ω(1) = 0

}.

Aω =∞∑

n=1

n2(ω, ωn)ωn, ω ∈ D(A),

where ωn(s) = √2 sin(ns), n = 1, 2, . . . is the orthogonal set of eigenvectors in A. It is well

known that A generates a compact, analytic semigroup {T (t), t ≥ 0} in H and

T (t)ω =∞∑

n=1

e−n2(t)(ω, ωn)ωn, ω ∈ H,

where T (t) satisfies the hypotheses (H1) and (H5).Define x(t)(z) = x(t, z), f (t, x(t))(z) = f (t, x(t, z)), �(t, x(t))(z) = �(t, x(t, z)).

Define the bounded linear operator B : U → H by Bu(t)(z) = μ(t, z), 0 ≤ z ≤ 1 andu ∈ U . Therefore, with the above choices, the system (8) can be written to the abstractform (5). Assume that � satisfies (H3) and (H4). Thus all the conditions of Theorem 4.3are satisfied. Hence, system (8) has at least a mild solution on J while the Eq. (6) hold forq ∈ (0, 1).

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