Fractionalcalculusisanemergingfieldintheareaoftheappliedmathemat-icsthatdealswithderivativesandintegralsofarbitraryordersaswellaswiththeirapplications.Duringthehistoryoffractionalcalculusitwasreportedthatthepuremathematicalformulationsoftheinvestigatedprob

doi:10.7151/dmdico.1144

EXISTENCE AND CONTROLLABILITY OF FRACTIONAL-ORDER IMPULSIVE STOCHASTIC SYSTEM WITH INFINITE DELAY

Toufik Guendouzi

Laboratory of Stochastic Models, Statistic and Applications Tahar Moulay University PO. Box 138 En-Nasr, 20000 Saida, Algeria

e-mail: tf.guendouzi@gmail.com

Abstract

This paper is concerned with the existence and approximate controllabil- ity for impulsive fractional-order stochastic infinite delay integro-differential equations in Hilbert space. By using Krasnoselskii’s fixed point theorem with stochastic analysis theory, we derive a new set of sufficient conditions for the approximate controllability of impulsive fractional stochastic system under the assumption that the corresponding linear system is approximately con- trollable. Finally, an example is provided to illustrate the obtained theory.

Keywords: existence result, approximate controllability, fractional stochas- tic differential equations, resolvent operators, infinite delay.

2010 Mathematics Subject Classification: 34G20, 34G60, 34A37, 60H40.

1. Introduction

Fractional calculus is an emerging field in the area of the applied mathemat- ics that deals with derivatives and integrals of arbitrary orders as well as with their applications. During the history of fractional calculus it was reported that the pure mathematical formulations of the investigated problems started to be dressed with more applications in various fields. As a result during the last decade fractional calculus has been applied successfully to almost every field of science and engineering. However, despite of the fact that several fields of application of fractional differentiation and integration are already well established, some others have just started.

Many applications of fractional calculus dynamics can be found in turbulence and fluid dynamics, stochastic dynamical system, plasma physics and controlled

thermonuclear fusion, nonlinear control theory, image processing, nonlinear bio- logical systems, astrophysics, etc. (see for more details Refs. [5, 14, 16] and the references therein).

On the other hand, there is also an increasing interest in the recent issue re- lated to dynamical fractional systems oriented towards the field of control theory concerning heat transfer, lossless transmission lines [18], the use of discretizing devices supported by fractional calculus. In recent years, various controllability problems for different kinds of dynamical systems have been studied in many publications [1, 13, 24]. From the mathematical point of view, the problems of exact and approximate controllability are to be distinguished. However, the con- cept of exact controllability is usually too strong and has limited applicability.

Approximate controllability is a weaker concept than complete controllability and it is completely adequate in applications [4, 12]. In particular, the fixed point techniques are widely used in studying the controllability problems for nonlinear control systems. Klamka studied the practical applicability of the fixed point theorem in solving various controllability problems for different types of dynamical control systems. Wang derived a set of sufficient conditions for the approximate controllability of differential equations with multiple delays by im- plementing some natural conditions such as growth conditions for the nonlinear term and compactness of the semigroup. Sakthivel and Anandhi [19] investigated the problem of approximate controllability for a class of nonlinear impulsive dif- ferential equations with state-dependent delay by using semigroup theory and fixed point technique.

Moreover, the study of stochastic differential equations has attracted great interest due to its applications in characterizing many problems in physics, biol- ogy, chemistry, mechanics, and so on. The deterministic models often fluctuate due to noise, so we must move from deterministic control to stochastic control problems. In the present literature there is only a limited number of papers that deal with the controllability of stochastic systems [10, 20]. Klamka [11] derived a set of sufficient conditions for constrained local relative controllability near the origin for semilinear finite-dimensional dynamical control systems by using the generalized open mapping theorem.

Sakthivel et al. [21] studied the approximate controllability of nonlinear de- terministic and stochastic evolution systems with unbounded delay in abstract spaces. Muthukumar and Balasubramaniam [15] derived a set of sufficient condi- tions for the approximate controllability of mixed stochastic Volterra-Fredholm type integro-differential systems in Hilbert spaces by using the Banach fixed point theorem. More recently, the approximate controllability of fractional stochastic evolution equations has been studied in [22]. The authors obtained a new set of sufficient conditions for the approximate controllability of nonlinear fractional stochastic control system under the assumptions that the corresponding linear

system is approximately controllable. However, to the best of our knowledge, the approximate controllability problem for impulsive fractional stochastic integro- differential system with infinite delay has not been investigated yet. Motivated by this consideration, in this paper we will study the approximate controllability for impulsive fractional-order stochastic infinite delay integro-differential system in Hilbert space under the assumption that the associated linear system is ap- proximately controllable. Our paper is organized as follows. Section 2 is devoted to a review of some essential results in fractional calculus and the resolvent oper- ators that will be used in this work to obtain our main results. In Section 3, we state and prove the existence of mild solution and controllability result. Section 4 deals with an example to illustrate the abstract results.

2. Preliminaries and basic properties

Let H, K be two separable Hilbert spaces and L(K, H) be the space of bounded linear operators from K into H. For convenience, we will use the same notation k.k to denote the norms in H, K and L(K, H), and use (., .) to denote the inner product of H and K. Let (Ω, F , {Ft}_t≥0, IP) be a complete filtered probability space satisfying that F0 contains all IP-null sets of F. ω = (ωt)t≥0be a Q-Wiener process defined on (Ω, F , {F_t}_t≥0, IP) with the covariance operator Q such that T rQ < ∞. We assume that there exists a complete orthonormal system {ek}_k≥1 in K, a bounded sequence of nonnegative real numbers λ_k such that Qe_k = λ_ke_k, k = 1, 2, . . . , and a sequence of independent Brownian motions {β_k}_k≥1such that

(ω(t), e)K=

∞

X

k=1

pλ_k(e_k, e)Kβ_k(t), e ∈ K, t ≥ 0.

Let L⁰₂ = L2(Q¹²K, H) be the space of all Hilbert-Schmidt operators from Q¹²K to H with the inner product hϕ, ψi_L⁰

2 = T r(ϕQψ^?).

The purpose of this paper is to investigate the existence of mild solution and the approximate controllability for the following impulsive fractional stochastic differential equations with infinite delay involving the Caputo derivative in the form

(1)



















cD^α_tx(t) = Ax(t) + Bu(t) + f (t, xt, F x(t)) + σ(t, xt, Gx(t))dω(t) dt , t ∈ J = [0, T ], T > 0, t 6= tk,

∆x(t_k) = I_k(x(t⁻_k)), k = 1, 2, . . . , m, x(t) = φ(t), φ(t) ∈ B_h,

where ^cD^α_t is the Caputo fractional derivative of order α, 0 < α; x(.) takes the value in the separable Hilbert space H; A : D(A) ⊂ H → H is the infinitesimal generator of an α-resolvent family {S_α(t)}_t≥0; the control function u(.) is given in L²(J, U ), U is a Hilbert space; B is a bounded linear operator from U into H.

The history xt : (−∞, 0] → H, xt(θ) = x(t + θ), θ ≤ 0, belongs to an abstract phase space B_h; f : J × B_h× H → H and σ : J × B_h× L⁰₂ → H are appropriate functions to be specified later; I_k : B_h → H, k = 1, 2, . . . , m, are appropriate functions. The terms F x(t) and Gx(t) are given by F x(t) = Rt

0K(t, s)x(s)ds and Gx(t) = Rt

0P (t, s)x(s)ds respectively, where K, P ∈ C(D, IR⁺) are the set of all positive continuous functions on D = {(t, s) ∈ IR² : 0 ≤ s ≤ t ≤ T }.

Here 0 = t₀ ≤ t₁ ≤ · · · ≤ t_m ≤ t_m+1 = T , ∆x(t_k) = x(t⁺_k) − x(t⁻_k), x(t⁺_k) = lim_h→0x(t_k+ h) and x(t⁻_k) = lim_h→0x(t_k− h) represent the right and left limits of x(t) at t = t_k, respectively. The initial data φ = {φ(t), t ∈ (−∞, 0]} is an F₀-measurable, B_h-valued random variable independent of ω with finite second moments.

Now, we assume that h : (−∞, 0] → (0, ∞) with l = R0

−∞h(t)dt < ∞ is a continuous function. We define the abstract phase space B_h by

B_h= (

φ : (−∞, 0] → H, for any a > 0, (IE|φ(θ)|²)^1/2 is bounded and measurable function on [−a, 0] with φ(0) = 0 and

Z 0

−∞

h(s) sup

s≤θ≤0(IE|φ(θ)|²)^1/2ds < ∞ )

.

If B_h is endowed with the norm kφk_Bh =

Z 0

−∞

h(s) sup

s≤θ≤0

(IE|φ(θ)|²)^1/2ds, φ ∈ B_h, then (B_h, k.kB_h) is a Banach space.

We consider the space B_b =

(

x : (−∞, T ] → H such that x|_Jk ∈ C(J_k, H) and there exist x(t⁺_k) and x(t⁻_k) with x(t_k) = x(t⁻_k), x₀ = φ ∈ B_h, k = 1, 2, . . . , m

) , where x|Jk is the restriction of x to Jk = (tk, tk+1], k = 1, 2, . . . , m. The function k.k_Bh defined by

kxk_Bb = kφkBh+ sup

0≤s≤T(IEkx(s)k²)^1/2, x ∈ Bb

is a seminorm in B_b.

Lemma 2.1 [17]. Assume that x ∈ Bh. Then xt∈ B_h for t ∈ J . Moreover, l(IEkx(t)k²)^1/2≤ l sup

0≤s≤T(IEkx(s)k²)^1/2+ kx0k_Bh, where l =R0

−∞h(s)ds < ∞.

Let us recall the following known definitions. For details see [5].

Definition 2.2. The fractional integral of order α with the lower limit 0 for a function f is defined as

I^αf (t) = 1 Γ(α)

Z t 0

f (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 2.3. Riemann-Liouville derivative of order α with lower limit zero for a function f : [0, ∞) → IR can be written as

(2) ^LD^αf (t) = 1 Γ(n − α)

dⁿ dtⁿ

Z t 0

f (s)

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

Definition 2.4. The Caputo derivative of order α for a function f : [0, ∞) → IR can be written as

(3) ^cD^αf (t) =^LD^α f (t) −

n−1

X

k=0

t^k k!f^k(0)

!

, t > 0, n − 1 < α < n.

If f (t) ∈ Cⁿ[0, ∞), then

cD^αf (t) = 1 Γ(n − α)

Z t 0

(t−s)^n−α−1fⁿ(s)ds = I^n−αfⁿ(s), t > 0, n−1 < α < n.

Obviously, the Caputo derivative of a constant is equal to zero. The Laplace transform of the Caputo derivative of order α > 0 is given as

L{^cD^αf (t); s} = s^αf (s) −ˆ

n−1

X

k=0

s^α−k−1f^(k)(0); n − 1 ≤ α < n.

Definition 2.5. A two parameter function of the Mittag-Leffler type is defined by the series expansion

E_α,β(z) =

∞

X

k=0

z^k

Γ(αk + β) = 1 2πi

Z

C

µ^α−βe^µ

µ^α− z dµ, α, β ∈ C, R(α) > 0, where C is a contour which starts and ends at −∞ end encircles the disc |µ| ≤

|z|^1/2 counter clockwise.

For short, Eα(z) = Eα,1(z). It is an entire function which provides a simple generalization of the exponent function: E₁(z) = e^z and the cosine function:

E2(z²) = cos h(z), E2(−z²) = cos(z), and plays a vital role in the theory of fractional differential equations. The most interesting properties of the Mittag- Leffler functions are associated with their Laplace integral

Z ∞ 0

e^−λtt^β−1E_α,β(ωt^α)dt = λ^α−β

λ^α− ω, Reλ > ω^α1, ω > 0, and for more details see [5].

Definition 2.6 [25]. A closed and linear operator A is said to be sectorial if there are constants ω ∈ IR, θ ∈ [^π₂, π], M > 0, such that the following two conditions are satisfied:

• ρ(A) ⊂ Σ_θ,ω = {λ ∈ C : λ 6= ω, |arg(λ − ω)| < θ},

• kIR(λ, A)k ≤ _|λ−ω|^M , λ ∈ Σθ,ω.

Definition 2.7. Let A be a closed and linear operator with the domain D(A) defined in a Banach space H. Let ρ(A) be the resolvent set of A. We say that A is the generator of an α-resolvent family if there exist ω ≥ 0 and a strongly continuous function S_α : IR+ → L(H), where L(H) is a Banach space of all bounded linear operators from H into H and the corresponding norm is denoted by k.k, such that {λ^α: Reλ > ω} ⊂ ρ(A) and

(4) (λ^αI − A)⁻¹x = Z ∞

0

e^λtS_α(t)xdt, Reλ > ω, x ∈ H, where Sα(t) is called the α-resolvent family generated by A.

Definition 2.8. Let A be a closed and linear operator with the domain D(A) defined in a Banach space H and α > 0. We say that A is the generator of a solution operator if there exist ω ≥ 0 and a strongly continuous function

Sα : IR+→ L(H) such that {λ^α : Reλ > ω} ⊂ ρ(A) and (5) λ^α−1(λ^αI − A)⁻¹x =

Z ∞ 0

e^λtSα(t)xdt, Reλ > ω, x ∈ H, where Sα(t) is called the solution operator generated by A.

The concept of the solution operator is closely related to the concept of a resolvent family. For more details on α-resolvent family and solution operators, we refer the reader to [5].

Lemma 2.9 [3]. If f satisfies the uniform H¨older condition with the exponent β ∈ (0, 1] and A is a sectorial operator, then the unique solution of the Cauchy problem

(6)

cD_t^αx(t) = Ax(t) + f (t, xt, F x(t)), t > t0, t0 ≥ 0, 0 < α < 1, x(t) = φ(t), t ≤ t₀,

is given by

(7) x(t) = T_α(t − t₀)(x(t⁺₀)) + Z t

t0

S_α(t − s)f (s, x_s, F x(s))ds,

where

(8) T_α(t) = E_α,1(At^α) = 1 2πi

Z

Bbr

e^λt λ^α−1 λ^α− Adλ,

(9) S_α(t) = t^α−1E_α,α(At^α) = 1 2πi

Z

Bbr

e^λt 1 λ^α− Adλ,

here bBr denotes the Bromwich path; Sα(t) is called the α-resolvent family and Tα(t) is the solution operator generated by A.

Now, we present the definition of mild solutions for the system (1).

Definition 2.10 [23]. An F_t-adapted stochastic process x : (−∞, T ] → H is called a mild solution of the system (1) if x0 = φ ∈ Bh satisfying x0 ∈ L⁰₂(Ω, H) and the following conditions:

(i) x(t) is B_h-valued and the restriction of x(.) to (t_k, t_k+1], k = 1, 2, . . . , m is continuous.

(ii) For each t ∈ J , x(t) satisfies the following integral equation

(10)

x(t) =



























































φ(t), t ∈ (−∞, 0], Z t

0

S_α(t − s)[Bu(s) + f (s, x_s, F x(s))]ds +

Z t 0

S_α(t − s)σ(s, x_s, Gx(s))dω(s), t ∈ [0, t₁], Tα(t − t1)(x(t⁻₁) + I1(x(t⁻₁))) +

Z t t1

Sα(t − s)[Bu(s) + f (s, xs, F x(s))]ds +

Z t t1

Sα(t − s)σ(s, xs, Gx(s))dω(s), t ∈ (t1, t2], ...

Tα(t−tm)(x(t⁻_m) + Im(x(t⁻_m))) + Z t

tm

Sα(t−s)[Bu(s) + f (s, xs, F x(s))]ds +

Z _t

tm

S_α(t − s)σ(s, x_s, Gx(s))dω(s), t ∈ (t_m, T ].

(iii) ∆x|t=t_k = I_k(x(t⁻_k)), k = 1, 2, . . . , m the restriction of x(.) to the interval [0, T )\{t₁, . . . , t_m} is continuous.

3. Main results

In the present section, we shall formulate and prove sufficient conditions for the approximate controllability of the system (1). To do this, we first prove the ex- istence of solutions for fractional impulsive control system. Then, we show that under certain assumptions, the approximate controllability of semilinear frac- tional impulsive control system (1) is implied by the approximate controllability of the associated linear system.

Definition 3.1 [22]. Let x_T(φ; u) be the state value of (1) at the terminal time T corresponding to the control u and the initial value φ. Introduce the set

R(T, φ) = {x_T(φ; u)(0); u(.) ∈ L²(J, U )},

which is called the reachable set of (1) at the terminal time T and its closure in H is denoted by R(T, φ). The system (1) is said to be approximately controllable on the interval J if R(T, φ) = H.

In order to study the approximate controllability for the impulsive fractional control system (1), we introduce the approximate controllability of its linear part

(11)











cD_t^αx(t) = Ax(t) + (Bu)(t), t ∈ J = [0, T ], T > 0, t 6= t_k,

∆x(t_k) = I_k(x(t⁻_k)), k = 1, 2, . . . , m, x(0) = φ(0).

For this purpose, we need to introduce the relevant operator Γ^T₀ =

Z _T

0

S_α(T − s)BB^?S_α^?(T − s)ds, R(q, Γ^T₀) = (qI + Γ^T₀)⁻¹,

where B^? denotes the adjoint of B and S_α^?(t) is the adjoint of S_α(t). It is straight- forward that the operator Γ^T₀ is a linear bounded operator.

(H0) qR(q, Γ^T₀) → 0 as α → 0⁺ in the strong operator topology.

The hypothesis (H0) is equivalent to the fact that the linear fractional control system (11) is approximately controllable on [0, T ] (see [13], Theorem 2).

In order to establish the result, we impose the following conditions.

(iv) If α ∈ (0, 1) and A ∈ A^α(θ₀, ω₀), then for x ∈ H and t > 0 we have kT_α(t)k ≤ M e^ωt and kS_α(t)k ≤ Ce^ωt(1 + t^α−1), ω > ω₀. Thus we have

kT_α(t)k ≤ fMT and kS_α(t)k ≤ t^α−1MfS,

where fMT = sup_0≤t≤TkT_α(t)k, and fMS = sup_0≤t≤TCe^ωt(1 + t^1−α) (fore more details, see [25]).

(v) There exist µ₁, µ₂ > 0 such that

IEkf (t, γ, x) − f (t, ψ, y)k²H≤ µ₁kγ − ψk²_B

h+ µ₂IEkx − yk²H. (vi) There exist ν₁, ν₂ > 0 such that

IEkσ(t, γ, x) − σ(t, ψ, y)k²_L⁰

2

≤ ν₁kγ − ψk²_B

h+ ν₂IEkx − yk²H.

(vii) f : J × Bh× H → H is continuous and there exist two continuous functions µ1, µ2 : J → (0, ∞) such that

IEkf (t, ψ, x)k²H ≤ µ₁(t)kψk²_Bh+ µ2(t)IEkxk²H, (t, ψ, x) ∈ J × Bh× H, and µ^?₁= sup_s∈[0,t]µ1(s), µ^?₂= sup_s∈[0,t]µ2(s).

(viii) σ : J × Bh× L⁰₂→ H is continuous and there exist two continuous functions ν1, ν2: J → (0, ∞) such that

IEkσ(t, ψ, x)k²_L⁰

2

≤ ν₁(t)kψk²_B

h+ ν₂(t)IEkxk²H, (t, ψ, x) ∈ J × B_h× L⁰₂, and ν₁^? = sup_s∈[0,t]ν₁(s), ν₂^?= sup_s∈[0,t]ν₂(s).

(ix) The functions I_k: H → H is continuous and there exists Λ, ˆΛ > 0 such that

Λ = max

1≤k≤m,x∈ ˆBp

{IEkIk(x)k²_H}, Λ =ˆ max

1≤k≤m;x,˜x∈ ˆBp

{IEkIk(x) − Ik(ˆx)k²_H},

where ˆB_p= {y ∈ B_b⁰, kyk²_B0 b

≤ p, p > 0}.

The set ˆBp is clearly a bounded closed convex set in B⁰_b for each p and for each y ∈ ˆBp. From Lemma 2.1, we have

ky_t+ ¯y_tk²_B

h ≤ 2(ky_tk²_B

h+ k¯y_tk²_B

h)

≤ 4 l² sup

0≤t≤TIEky(t)k²H+ ky0k²_B

h

!

+ 4 l² sup

0≤t≤TIEky(t)k²H+ k¯y0k²_B

h

! (12)

≤ 4(kφk²_B

h+ l²p).

The following lemma is required to define the control function. The reader can refer to [19] for the proof.

Lemma 3.2. For any ˜x_T ∈ L²(F_T, H), there exists ˜g ∈ L²_F(Ω; L²(0, T ; L⁰₂)) such that

˜

xT = IE˜xT + Z T

0

˜

g(s)dω(s).

Now for any q > 0, k = 1, 2, . . . , m and ˜x ∈ L²(F_T, H), we define the control function

u(t) = u^q(t) = B^?S_α^?(T − t)(qI + Γ^T_t

k)⁻¹

× (

IE˜xT + Z T

tk

˜

g(s)dω(s) − Tα(T − tk)[x(t⁻_k) + Ik(x(t⁻_k))]

)

− B^?S_α^?(T − t) Z T

tk

(qI + Γ^T₀)⁻¹Sα(T − s)f (s, xs, F x(s))ds

− B^?S_α^?(T − t) Z T

t_k

(qI + Γ^T₀)⁻¹Sα(T − s)σ(s, xs, Gx(s))dω(s).

Next, we mention the statement of Krasnoselskii’s fixed point theorem [23].

Theorem 3.3. Let ˆB be a nonempty closed convex subset of a Banach space (X, k.k). Suppose that P and Q map ˆB into X and satisfy

(a) P x + Qy ∈ ˆB whenever x, y ∈ ˆB;

(b) P is compact and continuous;

(c) Q is a contraction mapping.

Then there exists z ∈ ˆB such that z = P z + Qz.

Theorem 3.4. Suppose that the assumptions (iv) − (ix) are satisfied with (13) p ≥ 5 fM_T²(p + Λ)M₅+ 5 fM_S²T^2αλ₁

α² + λ₂ T (2α − 1)



M₆+ M₇ and

(14) h

M₁₁+ 3M₁₀Mf_S²T^2α 1

α²(µ₁l + µ₂F^?) + 1

T (2α − 1)(ν₁l + ν₂G^?)i

< 1.

Then the impulsive stochastic fractional differential equation (1) has a mild solu- tion on (−∞, T ].

Proof . Let P₁ : ˆB_p → ˆB_p and P₂ : ˆB_p → ˆB_p be defined as

(15) (P₁z)(t) =











0, t ∈ [0, t₁]

Tα(t − t1)(Z(t⁻₁) + I1(Z(t⁻₁))), t ∈ (t1, t2] ...

Tα(t − tm)(Z(t⁻_m) + Im(Z(t⁻_m))), t ∈ (tm, T ] and

(16)

(P2z)(t) =

















































 Z t

0

Sα(t − s)[Bu^q(s) + f (s, gs+ ¯zs, F (g(s) + ¯z(s)))]ds +

Z t 0

Sα(t − s)σ(s, gs+ ¯zs, G(g(s) + ¯z(s)))dω(s), t ∈ [0, t1], Z t

t1

S_α(t − s)[Bu^q(s) + f (s, g_s+ ¯z_s, F (g(s) + ¯z(s)))]ds +

Z t t1

Sα(t − s)σ(s, gs+ ¯zs, G(g(s) + ¯z(s)))dω(s), t ∈ (t1, t2], ...

Z t tm

Sα(t − s)[Bu^q(s) + f (s, gs+ ¯zs, F (g(s) + ¯z(s)))]ds +

Z t tm

S_α(t − s)σ(s, g_s+ ¯z_s, G(g(s) + ¯z(s)))dω(s), t ∈ (t_m, T ].

It will be shown that the impulsive stochastic fractional differential equation (1) is approximately controllable if for all q > 0 there exists a fixed point of the operator P = P₁+ P₂. To prove this result, we use Krasnoselskii’s fixed point theorem (Theorem 3.3). We shall show that the operator P = P1 + P2 has a fixed point, which is a solution of (1).

In order to use Theorem 3.3, we will verify that P₁ is compact and continuous while P₂ is a contraction operator. For the sake of convenience, we divide the proof into several steps.

Step 1. We show that P₁z + P₂z^? ∈ ˆB_p for z, z^? ∈ ˆB_p. For t ∈ [0, t₁], we have IEk(P1z)(t) + (P2z^?)(t)k²_H

≤ 3IE

Z _t

0

S_α(t − s)f (s, g_s+ 4¯z_s^?, F (g(s) + ¯z^?(s)))ds

2

H

+ 3IE

Z t 0

S_α(t − s)Bu^q(s)ds

2

H

+ 3IE

Z t 0

S_α(t − s)σ(s, g_s+ ¯z_s^?, G(g(s) + ¯z^?(s)))dω(s)

2

H

≤ 3 Z t

0

kS_α(t − s)kds Z t

0

kS_α(t − s)kIEkf (s, gs+ ¯z_s^?, F (g(s) + ¯z^?(s)))k²_Hds

+ 3 Z t

0

kS_α(t − s)k²IEkBu^q(s)k²ds

+ 3 Z t

0

kS_α(t − s)k²IEkσ(s, gs+ ¯z_s^?, G(g(s) + ¯z^?(s)))k²_L0 2ds.

We have for t ∈ [0, t₁] and M_B= kBk,

IEku^q(s)k² ≤ T^2α−2

q² M_B²Mf_S² (

3kIE˜xT + Z t

0

˜

g(s)dω(s)k²

+ 3IEk Z t

0

Sα(T − s)f (s, gs+ ¯z_s^?, F (g(s) + ¯z^?(s)))dsk²

+ 3IEk Z t

0

Sα(T − s)σ(s, gs+ ¯z^?_s, G(g(s) + ¯z^?(s)))dω(s)k² )

≤ 3T^2α−2

q² M_B²Mf_S²

"

2kIE˜x_Tk²+ 2 Z t

0

IEk˜g(s)k²ds

+ Z t

0

kS_α(T − s)kds Z t

0

kS_α(T − s)kIEkf (s, gs+ ¯z_s^?, F (g(s) + ¯z^?(s)))k²ds

+ Z t

0

kS_α(T − s)k²IEkσ(s, gs+ ¯z_s^?, G(g(s) + ¯z^?(s)))k²ds

# .

By using (vii) and (viii), we get IEku^q(s)k²

≤ 3T^2α−2

q² M_B²Mf_S²h

2kIE˜x_Tk²+ 2T M_˜g+ fM_S² Z t

0

(T − s)^α−1ds

× Z t

0

(T − s)^α−1[µ1(s)kgs+ ¯z^?_sk²_B

h+ µ2(s)IEkF (g(s) + ¯z^?(s))k²_H]ds + fM_S²

Z t 0

(T − s)^2α−2[ν₁(s)kg_s+ ¯z_s^?k²_B

h+ ν₂(s)IEkG(g(s) + ¯z^?(s))k²_H]dsi

≤ 3T^2α−2

q² M_B²Mf_S² h

2kIE˜xTk²+ 2T M˜g+ fM_S²T^α α

Z t 0

(T − s)^α−1

× [4µ^?₁(kφk²_B

h+ l²p) + µ^?₂F^? sup

s∈[0,T ]

IEkz^?k²_H]ds

+ fM_S² Z t

0

(T − s)^2α−2[4ν₁^?(kφk²_B

h+ l²p) + ν₂^?G^? sup

s∈[0,T ]

IEkz^?k²_H]dsi

≤ 3T^2α−2

q² M_B²Mf_S² h

2kIE˜xTk²+ 2T M˜g+ fM_S²T^2α

α² [4µ^?₁(kφk²_Bh+ l²p) + µ^?₂F^?p]

× fM_S²T^2α−1

2α − 1[4ν₁^?(kφk²_Bh+ l²p) + ν₂^?G^?p]

i

= M₁+ 3T^2(2α−1)

q² M_B²Mf_S⁴ λ₁

α² + λ₂ T (2α − 1)

! ,

where Mg˜= max{k˜g(s)k²; s ∈ [0, t]} and M1 = 3^T2α−2_q2 M_B²Mf_S²[2kIE˜xTk²+ 2T M˜g].

Now, we have

IEk(P1z)(t) + (P2z^?)(t)k²_H

≤ 3 fM_S²T^2α λ1

α² + λ2

T (2α − 1)



+ 3 fM_S²M_B^2T_2α−1^2α−1



M1+ 3T^2(2α−1)

q² M_B²Mf_S⁴

λ1

α² + λ2

T (2α − 1)



= 3 fM_S²M2T^2α λ1

α² + λ2

T (2α − 1)

 + M3,

with M2 =



1 + 3 fM_S⁴M_B²_q2^T(2α−1)^4α−3



and M3 = 3 fM_S²M_B²M1T^2α−1

2α−1. Thus, by the condition (13), we obtain kP₁z + P₂z^?k_B0

b ≤ p.

Similarly, for t ∈ (ti, ti+1], i = 1, . . . , m, we get the estimate IEk(P1z)(t) + (P2z^?)(t)k²_H

≤ 5kT_α(t − t_i)k²IEkz(t⁻_i )k²_H+ 5kT_α(t − t_i)k²IEkIi(z(t⁻_i ))k²_H + 5IE

Z t ti

Sα(t − s)Bu^q(s)ds

2 H

+ 5IE

Z t ti

S_α(t − s)f (s, g_s+ ¯z_s^?, F (g(s) + ¯z^?(s)))ds

2

H

+ 5IE

Z t ti

S_α(t − s)σ(s, g_s+ ¯z_s^?, G(g(s) + ¯z^?(s)))dω(s)

2 H

.

Moreover,

IEku^q(s)k²

≤ T^2α−2

q² M_B²Mf_S² (

5kIE˜x_T + Z t

ti

˜

g(s)dω(s)k²+ 5kTα(T − ti)k²IEkz(t⁻_i )k²_H + 5kTα(T − ti)k²IEkIi(z(t⁻_i ))k²_H

+ 3IE

Z t ti

S_α(T − s)f (s, g_s+ ¯z_s^?, F (g(s) + ¯z^?(s)))ds

2

+ 3IE

Z t ti

S_α(T − s)σ(s, g_s+ ¯z^?_s, G(g(s) + ¯z^?(s)))dω(s)

2)

≤ 5T^2α−2

q² M_B²Mf_S²Mf_T²h kzk²_B0

b + IEkIi(z(t⁻_i ))k²_Hi + M4+ 5T^2(2α−1)

q² M_B²Mf_S⁴ λ₁

α² + λ₂ T (2α − 1)

! ,

M4 = 5^T2α−2_q2 M_B²Mf_S² h

2kIE˜x_Tk²+ 2T M˜g

i . Now, we have

IEk(P1z)(t) + (P₂z^?)(t)k²_H

≤ 5M_B²Mf_S²T^2α−1 2α − 1



M4+ 5T^2α−2

q² M_B²Mf_S²Mf_T²(p + Λ) + 5T^2(2α−1)

q² M_B²Mf_S⁴λ₁

α² + λ₂ T (2α − 1)



+ 5 fM_T²(p + Λ) + 5 fM_S²T^2α

λ1

α² + λ2

T (2α − 1)



= 5 fM_T²(p + Λ)M5+ 5 fM_S²T^2α

λ1

α² + λ2

T (2α − 1)



M6+ M7 ≤ p,

where M₅ = 1 + 5M_B⁴Mf_S⁴_q2^T(2α−1)^4α−3
, M₆ = 1 + 5M_B⁴Mf_S⁴_q^T2(2α−1)^2(α−2)

and M₇ = 5M_B²Mf_S^2T_2α−1^2α−3M₄.

This implies that kP1z + P2z^?k_B0

b ≤ p with λ₁ = 4µ^?₁(kφk²_B

h+ l²p) + µ^?₂F^?p and λ₂ = 4ν₁^?(kφk²_B

h+ l²p) + ν₂^?G^?p. Hence, we get P₁z + P₂z^?∈ B_p. Step 2. The map P1 is continuous on Bp.

Let {zⁿ}^∞_n=1 be a sequence in B_p with lim zⁿ → z ∈ B_p. Then for t ∈ (t_i, t_i+1], i = 0, 1, . . . , m, we have

IEk(P1zⁿ)(t) − (P1z)(t)k²_H

≤ 2kT_α(t − t_i)k²h

IEkzⁿ(t⁻_i ) − z(t⁻_i )k²_H+ IEkIi(zⁿ(t⁻_i )) − I_i(z(t⁻_i ))k²_Hi . Since the functions Ii, i = 0, 1, . . . , m are continuous, then

lim_n→∞IEkP1zⁿ− P₁zk² = 0 which implies that the mapping P₁ is continuous on Bp.

Step 3. P1 maps bounded sets into bounded sets in Bp.

Let us prove that for p > 0 there exists a ϑ > 0 such that for each z ∈ Bp, we have IEk(P1z)(t)k²_H ≤ ϑ for t ∈ (t_i, ti+1], i = 0, 1, . . . , m. We obtain

IEk(P1z)(t)k²_H ≤ 2kT_α(t − ti)k² h

IEkz(t⁻_i )k²_H+ IEkIi(z(t⁻_i ))k²_H i

≤ 2 fM_T²(p + Λ) = ϑ, which proves the result.

Step 4. The map P₁ is equicontinuous.

Let τ₁, τ₂ ∈ (t_i, t_i+1], t_i ≤ τ₁ < τ₂≤ t_i+1, i = 0, 1, . . . , m, z ∈ B_p. We have IEk(P1z)(τ2) − (P1z)(τ1)k²_H

≤ 2kT_α(τ₂− t_i) − T_α(τ₁− t_i)k²h

IEkz(t⁻_i )k²_H+ IEkIi(z(t⁻_i ))k²_Hi

≤ 2(p + Λ)kT_α(τ2− t_i) − Tα(τ1− t_i)k².

Since T_α is strongly continuous it allows us to conclude that

lim_τ2→τ₁kT_α(τ₂− t_i) − T_α(τ₁− t_i)k² = 0, which implies that P₁(B_p) is equicon- tinuous. Finally, combining Step 1 to Step 4 together with Ascoli’s theorem, we conclude that the operator P₁ is compact.

Next, we show that the map P2 is a contraction mapping. Let z, z^? ∈ B_p and t ∈ (ti, ti+1], i = 0, 1, . . . , m. We have

IEk(P2z)(t) − (P₂z^?)(t)k²_H

≤ 3IE

Z t ti

S_α(t − s)h

f (s, g_s+ ¯z_s, F (g(s) + ¯z(s)))

−f (s, g_s+ ¯z^?_s, F (g(s) + ¯z^?(s))) i

ds

2

H

+ 3IE

Z t ti

S_α(t − s)h

σ(s, g_s+ ¯z_s, G(g(s) + ¯z(s)))

−σ(s, g_s+ ¯z^?_s, G(g(s) + ¯z^?(s)))i dω(s)

2

H

+ 3IE

Z t ti

S_α(t − s)Bu^q(s)ds

2

H

≤ 3 Z t

ti

kS_α(t − s)kds Z t

ti

kS_α(t − s)k

× IEkf(s, gs+ ¯z_s, F (g(s) + ¯z(s))) − f (s, g_s+ ¯z_s^?, F (g(s) + ¯z^?(s)))k²_Hds

+ 3 Z t

ti

kS_α(t − s)k²IEkσ(s, gs+ ¯z_s, G(g(s) + ¯z(s)))

− σ(s, g_s+ ¯z_s^?, G(g(s) + ¯z^?(s)))k²_L0 2

ds + 3

Z t ti

kS_α(t − s)k²IEkBu^q(s)k²ds.

Moreover, IEku^q(s)k²

≤ T^2α−2

q² M_B²Mf_S² (

5kIE˜x_T + Z t

ti

˜

g(s)dω(s)k²+ 5kT_α(T − t_i)k²IEkz(t⁻_i ) − z^?(t⁻_i )k²_H + 5kTα(T − ti)k²IEkIi(z(t⁻_i )) − Ii(z^?(t⁻_i ))k²_H

+ 5IE

Z t ti

S_α(T − s)h

f (s, g_s+ ¯z_s, F (g(s) + ¯z(s)))

−f (s, g_s+ ¯z_s^?, F (g(s) + ¯z^?(s)))i ds

2

H

+ 5IE

Z t ti

S_α(T − s)h

σ(s, g_s+ ¯z_s, G(g(s) + ¯z(s)))

−σ(s, g_s+ ¯z^?_s, G(g(s) + ¯z^?(s)))i dω(s)

2

H

)

≤ M₄+ 5T^2α−2

q² M_B²Mf_S²Mf_T² h

kz − z^?k²_B0 b + ˆΛ

i

+ 5T^2α−2

q² M_B²Mf_S⁴ Z t

ti

(T − s)^α−1ds

× Z t

ti

(T − s)^α−1h

µ₁k¯z_s− ¯z_s^?k²_B

h+ µ₂IEkF (g(s) + ¯z(s)) − F (g(s) + ¯z^?(s))k²_Hi ds

+ 5T^2α−2

q² M_B²Mf_S⁴ Z t

ti

(T − s)^2(α−1) h

ν1k¯zs− ¯z_s^?k²_B

h

+ ν₂IEkG(g(s) + ¯z(s)) − G(g(s) + ¯z^?(s))k²_Hi ds

≤ M₄+ 5T^2α−2

q² M_B²Mf_S²Mf_T² h

kz − z^?k²_B0 b + ˆΛ

i

+ 5T^2α−2

q² M_B²Mf_S⁴T^α α

Z t ti

(T − s)^α−1

×h

µ₁l sup IEkz(s) − z^?(s)k²_H+ µ₂F^?sup IEkz(s) − z^?(s)k²_Hi ds

+ 5T^2α−2

q² M_B²Mf_S⁴T^α α

Z t ti

(T − s)^2(α−1)

×h

ν₁l sup IEkz(s) − z^?(s)k²_H+ ν₂G^?sup IEkz(s) − z^?(s)k²_Hi ds

≤ M₄+ 5T^2α−2

q² M_B²Mf_S²Mf_T² h

kz − z^?k²_B0 b + ˆΛ

i

+ 5T^2α−2

q² M_B²Mf_S⁴T^2αh 1

α²(µ₁l + µ₂F^?) + 1

T (2α − 1)(ν₁l + ν₂G^?)i

kz − z^?k²_B0 b

. Thus

IEk(P2z)(t) − (P₂z^?)(t)k²_H

≤ 3M_B²Mf_S²T^2α−1

2α − 1M4+ 15M_B⁴Mf_S⁴ T^4α−3 (2α − 1)q²



kz − z^?k²_B0 b + ˆΛ



+ 3 fM_S²T^2α

1 + 5M_B⁴Mf_S² T^4α−3 (2α − 1)q²

 1

α²(µ₁l + µ₂F^?)

+ 1

T (2α − 1)(ν1l + ν2G^?)



kz − z^?k²_B0 b

= M8+ M9



kz − z^?k²_B0 b + ˆΛ

 + 3M₁₀Mf_S²T^2α 1

α²(µ₁l + µ₂F^?) + 1

T (2α − 1)(ν₁l + ν₂G^?)

kz − z^?k²_B0 b

≤ h

M11+ 3M10Mf_S²T^2α

 1

α²(µ1l + µ2F^?) + 1

T (2α − 1)(ν1l + ν2G^?)

i

kz − z^?k²_B0 b, M₁₁= M₈+ (1 + ˆΛ)M₉.

By the condition (14), we deduce that P₂ is a contraction mapping. Hence, by Krasnoselskii’s fixed point theorem we conclude that problem (1) has at least one solution on (−∞, T ]. This completes the proof of the theorem.

Theorem 3.5. Assume that the assumptions of Theorem 3.4, hold and, in addi- tion, the functions f and σ are uniformly bounded on their respective domains.

Further, if S_α(t) is compact, then the fractional stochastic impulsive control sys- tem (1) is approximately controllable on (−∞, T ].

Proof. Let x^q ∈ ˆBp be a fixed point of the operator P = P1 + P2. By the stochastic Fubini theorem, it is easy to see that for all i = 1, 2, . . . , m

x^q(T )

= ˜xT − q(qI + Γ^T_t

i)⁻¹

 IEˆxT +

Z T ti

˜

g(s)dω(s) − Tα(T − ti)[x^q(t⁻_i ) + Ii(x^q(t⁻_i ))]