CUBO A Mathematical Journal Vol.19, No¯ 03, (01–14). October 2017 The Solvability and Fractional Optimal Control for Semilinear Stochastic Systems Surendra Kumar Department of Mathematics, University of Delhi, Delhi-110007, India. mathdma@gmail.com ABSTRACT This paper deals with fractional optimal control for a class of semilinear stochastic equation in Hilbert space setting. To ensure the existence and uniqueness of mild solution, a set of sufficient conditions is constructed. The existence of fractional optimal control for semilinear stochastic system is also discussed. Finally, an example is included to show the applications of the developed theory. RESUMEN Este art́ıculo estudia el control óptimo fraccional para una clase de ecuaciones es- tocásticas semilineales en un contexto de espacios de Hilbert. Para asegurar la ex- istencia y la unicidad de soluciones blandas, construimos un conjunto de condiciones suficientes. La existencia del control óptimo fraccional para sistemas estocásticos semi- lineales también es discutido. Finalmente, incluimos un ejemplo para mostrar la apli- cabilidad de la teoŕıa desarrollada. Keywords and Phrases: Fractional calculus, Semilinear stochastic system, Mild solution, Opti- mal control, Fixed point theory 2010 AMS Mathematics Subject Classification: 26A33, 34A08, 34A12, 47H10, 49J99, 93C23 2 Surendra Kumar CUBO 19, 3 (2017) 1 Introduction Fractional calculus is about to generalization of the integer order integral and derivative to ar- bitrary order. The potential applications of fractional calculus are in many fields of science and engineering including fluid flow, electrical networks, diffusive transport, rheology, control theory, electromagnetic theory, and probability [13, 10, 9, 5, 7, 8, 15]. It is well known that many real world problems in science and engineering are modeled as stochastic differential equations [4]. Since fractional stochastic differential equations describe a physical dynamical system more accurately, it seems necessary to discuss the qualitative properties for such systems. If a fractional differential equation describes the performance index and system dynamics, then an optimal control problem is known as a fractional optimal control problem. Using the frac- tional variational principle and Lagrange multiplier technique, Agrawal [1] discussed the general formulation and solution scheme for Riemann-Liouville fractional optimal control problems. It is remarkable thathe fixed point technique, which is used to establish the existence result for ab- stract fractional differential equations, could be extended to address the fractional optimal control problems. Wang et. al. [19, 20] discussed the existence of local and global solutions for fractional semilinear systems, and extended the results for fractional optimal control. Using fractional pow- ers of the linear operator, the existence of fractional optimal controls for the Lagrange problem investigated in infinite dimensional reflexive Banach space [21, 22, 24]. Wang et al. [23] studied the solvability and the existence of optimal controls for fractional integro-differential systems with infinite delay via contraction mapping principle. Pan et al. [14] constructed a set of sufficient conditions that guarantees the existence of optimal control to the Riemann-Liouville fractional control systems in the Banach space. Li and Liu [11] presented the optimal control for nonlinear impulsive differential equations in Banach space setting. Yan and Lu [25] investigated the opti- mal control problems for fractional stochastic neutral integro-differential equations with infinite delay in a Hilbert space by using the fractional calculus, stochastic analysis theory, and fixed point theorem. Using the LeraySchauder fixed point theorem, Balasubramaniam and Tamilalagan [2] studied the solvability and optimal controls for impulsive fractional stochastic integro-differential equations. Recently, Tamilalagan and Balasubramaniam [18] investigated the solvability and optimal controls for fractional stochastic differential equations driven by Poisson jumps in Hilbert space by using analytic resolvent operators and classical Banach contraction mapping principle. By motivated from the above work, the main objective of this paper is to obtain sufficient conditions for existence and uniqueness of mild solution of fractional stochastic semilinear system of fractional order via contraction mapping principle. To prove the existence and uniqueness of mild solution it is assumed that nonlinear functions satisfying Lipschitz continuity and linear growth conditions. Next, we introduce a formulation for fractional optimal control governed by the fractional stochastic semilinear systems, where the fractional derivative is defined in the sense of the Caputo. Finally, The existence of fractional optimal controls for the Lagrange problem CUBO 19, 3 (2017) The Solvability and Fractional Optimal Control 3 is investigated. The obtained results are new and considered as a contribution to the theory of stochastic fractional optimal control. The rest of the paper is organized as follows: in Sect. 2, we present some basic definitions, notations and lemmas as preliminaries. In Sect. 3, the existence and uniqueness of mild solution are proved. Existence of fractional optimal control is shown in Sect. 4. In Sect. 5, an example is given to illustrate the theory. 2 Preliminaries This section contains basic definitions, notations and preliminary results, which help us to obtain desired results. Throughout this paper, we use the following notations: Let H and K be separable Hilbert spaces. For convenience, we denote the inner products and norms in all spaces by 〈·, ·〉 and ‖ · ‖, respectively. Let (Ω,F,P) be a complete probability space furnished with complete family of right continuous increasing sub σ-algebras {Ft : 0 ≤ t ≤ τ} satisfying Ft ⊂ F. Let {en} ∞ n=1 be a complete orthonormal system in K, and {βn} ∞ n=1 a sequence of independent Brownian motions such that ω(t) := ∞∑ n=1 √ λnenβn(t), t ∈ [0,τ], where {λn} ∞ n=1 is a bounded sequence of non-negative real numbers such that Qen = λnen, n = 1,2, · · · with tr(Q) = ∑ ∞ n=1 λn < ∞ (tr(Q) denotes the trace of the operator Q). Then the above K-valued stochastic process ω(t) is called a Wiener process. The normal filtration Ft is the sigma algebra generated by {ω(s) : 0 ≤ s ≤ t} and Fτ = F. Denoted by L(K,H) the space of all bounded operators from K into H equipped with the usual operator norm. For ψ ∈ L(K,H), we define ‖ψ‖2Q = tr(ψQψ ∗ ) = ∞∑ n=1 ‖ √ λnψen‖ 2. If ‖ψ‖2Q < ∞, then ψ is called a Q-Hilbert Schmidt operator. Let LQ(K,H) be the space of all Q-Hilbert Schmidt operators ψ : K → H. The completion LQ(K,H) of L(K,H) with respect to the topology induced by the norm ‖·‖Q, where ‖ψ‖ 2 Q = 〈ψ,ψ〉 is a Hilbert space with the above norm topology. The space of strongly measurable, H-valued, square integrable random variables, denoted by L2(Ω,H), is a Banach space equipped with the norm topology ‖x(·)‖ = (E‖x(t)‖ 2)1/2, where E(·) is the expectation with respect to the measure P. Let C([0,τ],L2(Ω,H)) be the Banach space of continuous maps from [0,τ] into L2(Ω,H) satisfying sup0≤t≤τ E‖x(t)‖ 2 < ∞. Let H2 be the closed subspace of C([0,τ],L2(Ω,H)) consisting 4 Surendra Kumar CUBO 19, 3 (2017) of measurable, Ft-adapted, H-valued processes x ∈ C([0,τ],L2(Ω,H)) equipped with the norm ‖x‖H2 := ( sup 0≤t≤τ E‖x(t)‖2 )1/2 . Consider the following integral cost functional J(x,u) := E {∫τ 0 ℓ(t,xu(t),u(t))dt } , (2.1) subject to the state equation CDαt x(t) = Ax(t) + B(t)u(t) + f(t,x(t)) + g(t,x(t)) dω(t) dt , t ∈]0,τ], x(0) = x0 ∈ H, } (2.2) where the integrand ℓ : [0,τ] × H × U → R ∪ {∞} is specified latter; CDαt is the Caputo fractional derivative of order 0 < α < 1, the state x(·) is H-valued stochastic process; the control function u(·) takes its values from a separable reflexive Hilbert space U; A : D(A) ⊆ H → H is the infinitesimal generator of a resolvent Sα(t), t ≥ 0 on H; {B(t) : t ≥ 0} is a family of linear operators from U to H; the functions f : [0,τ] × H → H and g : [0,τ] × H → L(K,H) are nonlinear, L(K,H) denotes the space of all bounded linear operators form E into H, x0 is F0-measurable H-valued random variable independent of ω. Define the admissible set Uad, the set of all v(·) : [0,τ] × Ω → U such that v is Ft adapted stochastic process and E ∫τ 0 ‖v(t)‖pdt < ∞. Clearly Uad 6= ∅ and Uad ⊂ L p([0,τ];U) (1 < p < +∞) is bounded, closed, and convex. Denoted by the set of all admissible state-control pairs (x,u) by Aad, where x is the mild solution of the system (2.2) corresponding to the control u ∈ Uad. The main objective of this paper is to find a pair (x0,u0) ∈ Aad such that J(x0,u0) := inf{J(x,u) : (x,u) ∈ Aad} = ε. We now recall the following well known definitions related to fractional order differentiation and integration. For more details on fractional calculus one can see [13, 10]. Definition 2.1 The Riemann-Liouville fractional integral operator of order α > 0 of a function f : [0,∞) → R with the lower limit 0 is defined as Iαf(t) = 1 Γ(α) ∫t 0 (t − s)α−1f(s)ds, where Γ is the Euler gamma function. Definition 2.2 The Caputo fractional derivative of order α > 0 for the function f ∈ Cm([0,τ],R) is defined by CDαt f(t) = 1 Γ(m − α) ∫t 0 (t − s)m−α−1f(m)(s)ds, m − 1 < α < m ∈ N. CUBO 19, 3 (2017) The Solvability and Fractional Optimal Control 5 If f is an abstract function with values in H, then the integrals appearing in Definition 2.1 and Definition 2.2 are taken in the Bochner sense. Moreover, the Caputo derivative of a constant is always zero. The two-parameter function of the Mittag-Leffler type is defined by the series expansion Eα,β(z) = ∞∑ j=1 zj Γ(αj + β) = 1 2πi ∫ C eλ λα−β λα − z dλ; α,β > 0, z ∈ C, where C is a contour that start and end at −∞ and encircles the disc ‖λ‖ ≤ |z|1/2 counterclockwise (see [6] for more results on Mittag-Leffler function). Definition 2.3 [16, 17] A closed and linear operator A is said to be sectorial of type µ if there exist π/2 ≤ θ ≤ π, M̃ > 0 and µ ∈ R such that the following conditions are satisfied: ρ(A) ⊂ ∑ (θ,µ) := {λ ∈ C : λ 6= µ, |arg(λ − µ)| < θ}, and ‖R(λ,A)‖ := ‖(λ − A) −1‖ ≤ M̃ |λ−µ| , λ ∈ ∑ (θ,µ) . Lemma 2.1 [17] For 0 < α < 2, a linear closed densely defined operator A belongs to Aα(θ0,µ0) if and only if λα ∈ ρ(A) for each λ ∈ Σ(θ0+π/2,µ) and for any µ > µ0, θ < θ0 there is a constant C0 = C0(θ,µ) such that ‖λα−1R(λα,A)‖ ≤ C0 |λ − µ| , for λ ∈ Σ(θ+π/2,µ). Lemma 2.2 [17] If F satisfies the uniform Hölder condition with the exponent 0 < γ ≤ 1 and A is a sectorial operator, then the unique solution of the Cauchy problem CDαt y(t) = Ay(t) + F(t), 0 < α < 1, t ∈ (0,τ], y(0) = y0, } (2.3) is given by y(t) = Sα(t)y0 + ∫t 0 Tα(t − s)F(s)ds, where Sα(t) = Eα,1(At α) = 1 2πi ∫ B̂ρ eλt λα−1 λα − A dλ, Tα(t) = t α−1Eα,α(At α ) = 1 2πi ∫ B̂ρ eλt 1 λα − A dλ, B̂ρ is the Bromwich path, Tα(t) is called the α-resolvent family, and Sα(t) is the solution operator generated by A. An operator A is said to belong to Cα(M̃,µ) if problem (2.3) with F = 0 has a solution operator Sα(t) satisfying ‖Sα(t)‖ ≤ M̃e µt. Denote Cα(µ) := ∪{Cα(M̃,µ) : M̃ ≥ 1}, Cα := {Cα(µ) : µ ≥ 0}, and Aα(θ0,µ0) = {A ∈ C α : A generates analytic solution operators Sα(t) of type (θ0,µ0)}. 6 Surendra Kumar CUBO 19, 3 (2017) If 0 < α < 1 and A ∈ Aα(θ0,µ0), then we have ‖Sα(t)‖ ≤ M̃e µt and ‖Tα(t)‖ ≤ Ce µt(1 + tα−1), t > 0, µ > µ0. If MS := sup 0≤t≤τ ‖Sα(t)‖, MT := sup 0≤t≤τ Ceµt(1 + t1−α), then, we have ‖Sα(t)‖ ≤ MS, ‖Tα(t)‖ ≤ t α−1MT. By Lemma 2.2, a mild solution of the system (2.2) is defined as Definition 2.4 An Ft-adapted stochastic process x(t) ∈ C([0,τ],L 2(Ω,F,H) is called a mild solu- tion of system (2.2) if for each u(·) ∈ Lp([0,τ];U), x(t) is measurable and the following stochastic integral equation is satisfied: x(t) = Sα(t)x0 + ∫t 0 Tα(t − s)[B(s)u(s) + f(s,x(s))]ds + ∫t 0 Tα(t − s)g(s,x(s))dω(s). (2.4) Lemma 2.3[12] A measurable function χ : [0,τ] → V is Bochner integrable, if ‖χ‖ is Lebesgue integrable. 3 Existence and Uniqueness of Mild Solution To prove the existence and uniqueness of mild solution of the system (2.2), we impose the following conditions to the system parameter: [H0] For any x ∈ H, the function t → f(t,x(t)) and t → g(t,x(t)) are Ft-measurable. [H1] The functions f : [0,τ] ×H → H, g : [0,τ] ×H → L(K,H) are continuous, and satisfying linear growth and Lipschitz conditions. Moreover, without loss of generality, we may assume that there are positive constants Lf and Lg such that ‖f(t,x) − f(t,y)‖2 ≤ Lf‖x − y‖ 2, ‖f(t,x)‖2 ≤ Lf ( 1 + ‖x‖2 ) , ‖g(t,x) − g(t,y)‖2Q ≤ Lg‖x − y‖ 2, ‖g(t,x)‖2Q ≤ Lg ( 1 + ‖x‖2 ) . [H3] The operator B ∈ L∞([0,τ]; L(U,H)) and ‖B‖∞ stand for the norm of operator B in the Banach space L∞([0,τ]; L(U,H)). [H4] The multi-valued map U(·) : [0,τ] ⇒ 2U \ {∅} has closed, convex and bounded values; U(·) is graph measurable and U(·) ⊆ Ξ, where Ξ is a bounded subset of U. First we show that the system (2.4) has at least one solution. Theorem 3.1 Under assumptions [H0]–[H3] the system (2.4) admits a unique mild solution on [0,τ] for each control function u(·) ∈ Uad and for some p such that pα > 1. CUBO 19, 3 (2017) The Solvability and Fractional Optimal Control 7 Proof. Define an operator Im : H2 → H2 as (Im x)(t) = Sα(t)x0 + ∫t 0 Tα(t − s)[B(s)u(s) + f(s,x(s))]ds + ∫t 0 Tα(t − s)g(s,x(s))dω(s). To show that (2.4) is the mild solution of the system (2.4) on [0,τ], it is enough to prove that Im has a fixed point in the space H2. For this purpose, we will employ the classical fixed point theorem for contractions. We first show that Im(H2) ⊂ H2. Let x ∈ H2, then we have E‖(Im x)(t)‖2 ≤ 4[I0 + I1 + I2 + I3] (3.1) Clearly I0 ≤ M 2 SE‖x0‖ 2. Next, using the Cauchy-Schwartz inequality, we have I1 = ‖ ∫t 0 Tα(t − s)B(s)u(s)ds‖ 2 ≤ M2T ‖B‖ 2 ∞ [∫t 0 (t − s)α−1‖u(s)‖ds ]2 ≤ M2T ‖B‖ 2 ∞   (∫t 0 (t − s) p(α−1) p−1 ds ) p−1 p (∫t 0 ‖u(s)‖ p Uds ) 1 p   2 ≤ M2T ‖B‖ 2 ∞ ‖u‖2Lp([0,τ];U)τ 2( pα−1p ) ( p − 1 pα − 1 ) 2(p−1) p The Cauchy-Schwartz inequality, and hypothesis (H1) imply that I2 = E ∥ ∥ ∥ ∥ ∫t 0 Tα(t − s)Ef(s,x(s))ds ∥ ∥ ∥ ∥ 2 ≤ (∫t 0 ‖Tα(t − s)‖E‖f(s,x(s))‖ds )2 ≤ (∫t 0 MT (t − s) α−1 2 (t − s) α−1 2 E‖f(s,x(s))‖ds )2 ≤ M2T (∫t 0 (t − s)α−1ds ) (∫t 0 (t − s)α−1‖E‖f(s,x(s))‖2ds ) ≤ M2TLf τα α ∫t 0 (t − s)α−1(1 + E‖x(s)‖2)ds ≤ M2TLf τ2α α2 (1 + ‖x‖2H2), 8 Surendra Kumar CUBO 19, 3 (2017) and I3 = E ∥ ∥ ∥ ∥ ∫t 0 Tα(t − s)Eg(s,x(s))dω(s) ∥ ∥ ∥ ∥ 2 ≤ M2Ttr(Q) (∫t 0 (t − s)α−1ds ) (∫t 0 (t − s)α−1E‖g(s,x(s))‖2Qds ) ≤ M2Ttr(Q)Lg τ2α α2 (1 + ‖x‖2H2). Thus 3.1 becomes E‖(Im x)(t)‖2 ≤ a + b‖x‖2H2, where a and b are suitable positive constants. This implies that Im map H2 into itself. Next, we show that Im is a contraction map. For x, y ∈ H2, the Cauchy-Schwartz inequality, and hypothesis (H1) yield that E‖(Im x)(t) − (Im y)(t)‖2 ≤ 2E‖ ∫t 0 Tα(t − s)[f(s,x(s)) − f(s,y(s))]ds‖ 2 +2E‖ ∫t 0 Tα(t − s)[g(s,x(s)) − g(s,y(s))]dω(s)‖ 2 ≤ 2M2T (Lf + Lgtr(Q)) τ2α α2 ‖x − y‖2H2. Consequently if 2M2T (Lf + Lgtr(Q)) τ2α α2 < 1, (3.2) then the operator Im has a unique fixed point in H2, which is a solution of the system (2.2). The extra condition on τ can be easily removed by considering the equation on intervals [0, τ̃], [τ̃,2τ̃], · · · with τ̃ satisfying (3.2). We now obtain a priori estimate of mild solution for the system (2.2), that helps us to obtain our main result. Lemma 3.1 (A priori estimate). Assuming that system (2.4) is the mild solution of system (2.2) on [0,τ] corresponding to the control u. Then there exists a constant M > 0 such that E‖x(t)‖2 ≤ M, ∀ t ∈ [0,τ]. CUBO 19, 3 (2017) The Solvability and Fractional Optimal Control 9 Proof. Using condition [H1] and Hölder’s inequality, we obtain E‖x(t)‖2 ≤ 4E‖Sα(t)x0‖ 2 + 4E‖ ∫t 0 Tα(t − s)B(s)u(s)ds‖ 2 +4E‖ ∫t 0 Tα(t − s)f(s,x(s))ds‖ 2 + 4E‖ ∫t 0 Tα(t − s)g(s,x(s))dω(s)‖ 2 ≤ 4M2S‖x0‖ 2 + 4M2T ‖B‖ 2 ∞ [∫t 0 (t − s)α−1‖u(s)‖ds ]2 +4M2T (Lf + Lgtr(Q)) (∫t 0 (t − s)α−1ds ) ∫t 0 (t − s)α−1{1 + E‖x(s)‖2}ds ≤ 4M2S‖x0‖ 2 + 4M2T ‖B‖ 2 ∞   (∫t 0 (t − s) p(α−1) p−1 ds ) p−1 p (∫t 0 ‖u(s)‖ p Uds ) 1 p   2 +4M2T (Lf + Lgtr(Q)) τα α ∫t 0 (t − s)α−1{1 + E‖x(s)‖2}ds ≤ 4M2S‖x0‖ 2 + 4M2T ‖B‖ 2 ∞ ‖u‖2Lp([0,τ];U)τ 2( pα−1p ) ( p − 1 pα − 1 ) 2(p−1) p +4M2T (Lf + Lgtr(Q)) τ2α α2 + 4M2T (Lf + Lgtr(Q)) τα α ∫t 0 (t − s)α−1E‖x(s)‖2ds. Now using the Gronwall inequality, one can easily obtain the boundedness of x in H2. 4 Existence of Fractional Optimal Control In this section, we prove the existence of fractional optimal control under the hypothesis: [HL] Following conditions are imposed on the integrand ℓ : [0,τ] × H × U → R ∪ {∞} such that (i) The integrand ℓ : [0,τ] × H × U → R ∪ {∞} is Ft-measurable. (ii) The integrand ℓ(t, ·, ·) is sequentially lower semicontinuous on H × U for almost all t ∈ [0,τ]; (iii) The integrand ℓ(t,x, ·) is convex on U for each x ∈ H and almost all t ∈ [0,τ]. (iv) There exist constants d ≥ 0, e > 0, µ0 is nonnegative and µ0 ∈ L 1([0,τ]; R) such that µ0(t) + dE‖x‖ 2 + eE‖u‖ p U ≤ ℓ(t,x,u). 10 Surendra Kumar CUBO 19, 3 (2017) Theorem 4.1 Suppose hypothesis of Theorem 3.1 and [HL] hold, then Lagrange problem (2.1) admits at least one optimal pair, that is, there exists an admissible state-control pair (x0,u0) ∈ Aad such that J(x0,u0) := E {∫τ 0 ℓ(t,x0(t),u0(t))dt } ≤ J(x,u), ∀ (x,u) ∈ Aad. Proof. If inf{J(x,u)|(x,u) ∈ Aad} = +∞, then there is nothing to prove. Without any loss of generality, we may assume that inf{J(x,u)|(x,u) ∈ Aad} = ε < +∞. Now assumption [HL] implies that ε > −∞. By definition of infimum, there is a minimizing sequence of feasible pairs (xm,um) ∈ Aad, such that J(x m,um) → ε as m → +∞. Since {um} ⊆ Uad, m = 1,2, · · · , {u m} is a bounded subset of the separable reflexive Banach space Lp([0,τ];U), there exists a subsequence, relabeled as {um} and u0 ∈ Lp([0,τ];U) such that um w −→ u0 (um → u0 weakly as m → +∞) in Lp([0,τ];U). Since Uad is closed and convex, the Mazur lemma forces us to conclude that u 0 ∈ Uad. Let {xm} be the sequence of solutions of the system (2.2) corresponding to {um}, that is xm(t) : = Sα(t)x0 + ∫t 0 Tα(t − s)[B(s)u m(s) + f(s,xm(s))]ds + ∫t 0 Tα(t − s)g(s,x m(s))dω(s). By Lemma 3.1, it is easy to see that there exists δ > 0 such that E‖xm‖2 ≤ δ, m = 0,1,2, · · · , where x0 is the mild solution of the system (2.2) corresponding to the control u0 ∈ Uad given by x0(t) : = Sα(t)x0 + ∫t 0 Tα(t − s)[B(s)u 0(s) + f(s,x0(s))]ds + ∫t 0 Tα(t − s)g(s,x 0(s))dω(s). For all t ∈ [0,τ], using condition [H1], the Cauchy-Schwartz inequality and the Hölder inequality, we obtain E‖xm(t) − x0(t)‖2 ≤ 3E‖ ∫t 0 Tα(t − s)[B(s)u m(s) − B(s)u0(s)]ds‖2 +3E‖ ∫t 0 Tα(t − s)[f(s,x m(s)) − f(s,x0(s))]ds‖2 +3E‖ ∫t 0 Tα(t − s)[g(s,x m (s)) − g(s,x0(s))]dω(s)‖2 ≤ 3M2T ( p − 1 pα − 1 ) 2p−2 p τ 2α− 2 p (∫t 0 ‖B(s)um(s) − B(s)u0(s)‖pds ) 2 p +3M2T τα α (Lf + Lgtr(Q)) ∫t 0 (t − s)α−1E‖xm(s) − x0(s)‖2ds. CUBO 19, 3 (2017) The Solvability and Fractional Optimal Control 11 By the well known singular version of Gronwall inequality, there exists a constant K∗(α) indepen- dent of u, m and t such that E‖xm(t) − x0(t)‖2 ≤ K∗(α) (∫τ 0 ‖B(s)um(s) − B(s)u0(s)‖pds ) 2 p ≤ K∗(α)‖Bum − Bu0‖2Lp([0,τ];U). (4.1) Since B is strongly continuous, we get ‖Bum − Bu0‖2Lp([0,τ];U) s −→ 0 as m → ∞. (4.2) From (4.1) and (4.2), we conclude that E‖xm(t) − x0(t)‖2 → 0 as m → ∞. This implies that E‖xm − x0‖2 → 0 in C([0,τ];L2(Ω,H)) as m → ∞. Note that [HL] implies the assumptions of Balder (see Theorem 2.1, [3]). Hence, by Balder’s theorem, we can conclude that (x,u) → E ∫τ 0 L(t,x(t),u(t))dt is sequentially lower semicontinuous in the strong topology of L1([0,τ];H) and weak topology of Lp([0,τ];U) ⊂ L1([0,τ];U). Hence, J is weakly lower semicontinuous on Lp([0,τ];U), and since by [HL] (iv), J > −∞, J attains its infimum at u0 ∈ Uad, that is, ε := lim m→∞ E ∫τ 0 ℓ(t,xm(t),um(t))dt ≥ E ∫τ 0 ℓ(t,x0(t),u0(t))dt = J(x0,u0) ≥ ε. This completes the proof. 5 Applications Let Ω1 ⊂ R 3 be a bounded domain and ∂Ω1 ∈ C 3. Further let H = U := L2(Ω1), ω(t) is a standard cylindrical Wiener process in H defined on a stochastic space (Ω,F,P). Suppose D(A) := H2(Ω1) ∩ H 1 0(Ω1) and for z ∈ D(A), Az := ( ∂ 2 ∂z2 1 + ∂ 2 ∂z2 2 + ∂ 2 ∂z2 3 ) z. The admissible control set Uad := {u ∈ U : ‖u‖Lp([0,1]:U) ≤ 1}. Consider the following fractional stochastic equation    CD 2 3 t x(t,z) = ∆x(t,z) + ∫1 0 K(z,s)u(s,t)ds + ∫1 0 ν(z,s) sin(x,s)ds + (x(t,z))2 1+(x(t,z))2 dω(t), x(0,z) = x0(z), z ∈ Ω1, x(t,z)|z∈∂Ω = 0, t > 0, (5.1) 12 Surendra Kumar CUBO 19, 3 (2017) Define x(t)(z) = x(t,z), (Bu)(t)(z) = ∫1 0 K(z,s)u(s,t)ds, f(t,x(t))(z) = f(t,x(t,z)) = ∫1 0 ν(z,s) sin(x,s)ds, g(t,x(t))(z) = g(t,x(t,z)) = (x(t,z)) 2 1+(x(t,z))2 , and x(0)(z) = x(0,z) = x0(z). Moreover, we assume that K : Ω1 × [0,1] → R is continuous. The function ν is measurable and ∫ Ω1 ∫1 0 ν(z,s)dsdz < ∞. The one-dimensional standard Brownian motion is denoted by ω(t). Thus, for α = 2/3 the problem (5.1) can be written as the abstract form of system (2.2) with the cost function J(x,u) := E {∫1 0 ℓ(t,x(t),u(t))dt } , where ℓ(t,x(t),u(t))(z) = ∫ Ω1 |x(t,z)|2dz + ∫ Ω1 |u(t,z)|2dz. It is easy to see that the assumptions [H0]–[H4] are satisfied. 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Appl. 8, 557–577 (2015) Introduction Preliminaries Existence and Uniqueness of Mild Solution Existence of Fractional Optimal Control Applications Conclusions