Existence of Fractional Stochastic Schr¨odinger
Evolution Equations with Potential and Optimal
Controls
Zuomao Yan and Xiumei Jia
Abstract—In this paper, we study the fractional stochastic nonlinear Schr¨odinger evolution equations with potential and optimal controls in Hilbert spaces. The existence of mild solutions is proved by means of fractional calculus, stochastic analysis, and fixed point theorems with the semigroup theory. Using these results, the existence of optimal pairs of system governed by fractional stochastic nonlinear Schr¨odinger equa-tions is also presented. An example is given for demonstration. Index Terms—fractional stochastic schr¨odinger evolution equations, existence, optimal controls, potential, fixed point theorem.
I. INTRODUCTION
T
HE theory of fractional differential equations has re-ceived increasing attention during recent years since they can be used to describe many phenomena arising in viscoelasticity, electrochemistry, control, porous media, elec-tromagnetic, etc. Actually, for various real world problems in science and engineering, fractional derivatives describe certain physical phenomena more accurately than integer order derivatives. Some works have done on the qualitative properties of solutions for these equations; see [1], [2], [3] and the references therein. Recently, the existence of solu-tions for fractional semilinear differential equasolu-tions including delay systems is one of the theoretical fields that investigated by many authors [4], [5], [6], [7], [8], [9]. As a result of its widespread use, the existence of fractional optimal control systems have been discussed in publications( see [10], [11], [12]). On the other hand, the nonlinear Schr¨odinger equation is a model of the evolution of a one-dimensional packet of surface waves on sufficiently deep water. It arises from the study of nonlinear wave propagation in dispersive and inhomogeneous media, such as plasma phenomena and non-uniform dielectric media. Therefore, it is a generic equation describing the evolution of the slowly varying amplitude of a nonlinear wave train in weakly nonlinear, strongly dispersive, and hyperbolic systems [13]. In recent years, the fractional Schr¨odinger equations have become a field of increasing interest(see [14] and references therein). Particularly, Guo and Xu [15] considered the fractional Schr¨odinger equation with a free particle and an infinite square potential. de Oliveira [16] derived the existence of solutions to fractionalManuscript received October 13, 2015; revised January 2, 2016. This work was supported the National Natural Science Foundation of China (11461019), the President Fund of Scientific Research Innovation and Application of Hexi University (xz2013-10, XZ2014-22), the Scientific Research Project of Universities of Gansu Province (2014A-110).
Z. Yan and X. Jia are with the Department of Mathematics, Hexi Univer-sity, Zhangye, Gansu 734000, P.R. China, e-mail: yanzuomao@163.com.
Schr¨odinger equation for delta potentials. Wang et al. [17] discussed the existence, uniqueness, local stability and at-tractivity, and data continuous dependence of mild solution for fractional Schr¨odinger equations with potential. Also, the existence and uniqueness of optimal pairs for the fractional controlled systems are obtained.
Stochastic Schr¨odinger equations are frequently used to describe many quantum measurement processes and in gen-eral, quantum systems that are sensitive to the environ-ment influence (see [18]). Moreover, nonlinear stochastic Schr¨odinger equations are becoming an established tool for numerical simulation of the evolution of open quantum systems (see [19]). Many authors investigated the existence of a mild solution of stochastic equations of Schr¨odinger in Hilbert spaces. Grecksch and Lisei [20] considered a class of stochastic evolution equation of Schr¨odinger type over a triplet of rigged Hilbert spaces, which includes as special cases stochastic Schr¨odinger equations. Further, the authors [21] also studied the approximation of stochastic nonlinear equations of Schr¨odinger type by the splitting method. Keller [22] discussed the existence of optimal controls for a linear stochastic Schr¨odinger equation. Pinaud [23] studied the existence of stochastic nonlinear Schr¨oinger equations driven by a fractional noise.
we will consider the Lagrange problem of systems governed by fractional stochastic Schr¨odinger evolution equations with potential and the existence result of optimal controls will be presented. Since many control systems arising from realistic models can be described as fractional stochastic Schr¨odinger systems. So it is natural to extend the concept of existence of optimal controls to dynamical systems represented by these systems. The known results appeared in [15], [16], [17] are generalized to the stochastic systems settings, and the results appear are also new for deterministic Schr¨odinger systems with potential.
The rest of this paper is organized as follows. In Section 2, we introduce some notations and necessary preliminaries. Section 3 verifies the existence of solutions for fractional sto-chastic control system. Section 4 we establish the existence results for optimal pairs of system governed by fractional stochastic control system. In Section 5, an example is given to illustrate our results.
II. PRELIMINARIES
The purpose of this paper is to study the existence of solutions for the following nonlinear fractional stochastic Schr¨odinger evolution equations with potential of the form
i(J1−α
t x)(t, y) +△(t, y)−kV(y)x(t, y)
=g(t, x(t, y)) +f(t, x(t, y))w(t)
dt , (1) y∈Π, t∈J = [0, b],
x(0, y) =x0(y), y∈Π, (2)
where 12 < α < 1, J1−α is the (1−α)-order
Riemann-Liouville fractional integral operator, Π⊆R2 is a bounded
domain with a smooth boundary∂Π,∆denotes the Laplace operator in R2, x is a complex valued function inJ×R2,
k := maxt∈[0,b]|χ(t)| with χ ∈ C(J, R), is a positive
constant, and the function V is called potential. x0 is the
initial condition, and g, f : J ×R → R are continuous nonlinear functions.
The state x(t,·) takes values in a separable real Hilbert space H = L2(Π) with inner product h·,·i
H and norm k
· kH . Define an operatorA onL2(Π) with domain D(A)
given by
D(A) =H2(Π)∩H01(Π),
such that Ax = i∆x. By virtue of the well known Hille-Yosida theorem (see Pazy [34], it is obvious that A is the infinitesimal generator of a strongly continuous group
{T(t),−∞< t < ∞} in H. Moreover {T(t),−∞< t <
∞} can be given by
(T(t)x)(y) = 1 4πit
Z
Π
ei|y4−tz|x(z)dz. (3)
Furthermore, we have the following properties.
Lemma 1([34]). Let{T(t), t≥0}be the strongly continu-ous semigroup given by (3). ThenT(·)can be extended in a unique way to a bounded operator from L2(Π)into L2(Π)
and
kT(t)xkL2(Π)≤kxkL2(Π).
Lemma 2 ([17]). LetΠbe a measurable subset of R2, k= max[0,b]|χ(t)| andV ∈H2(Π).Then, we have
kkV xkL2(Π)≤kkV kL∞(Π)kxkL2(Π).
Throughout this paper, we use the following notations. Let
(Ω,F, P)be a complete probability space with probability measure P on Ω and a filtration {Ft}t≥0 satisfying the
usual conditions, that is the filtration is right continuous and F0 contains all P-null sets. Let H, K be two real
separable Hilbert spaces and we denote by h·,·iH,h·,·iK
their inner products and by k · kH,k · kK their vector
norms, respectively.L(K, H)be the space of linear operators mappingK intoH,andLb(K, H)be the space of bounded linear operators mappingK intoH equipped with the usual norm k · kH and Lb(H) denotes the Hilbert space of
bounded linear operators fromH to H.Let {w(t) :t≥0}
denote an K-valued Wiener process defined on the prob-ability space (Ω,F, P) with covariance operator Q, that is
Ehw(t), xiKhw(s), yiK = (t∧s)hQx, yiK,for allx, y∈K,
whereQis a positive, self-adjoint, trace class operator onK.
In particular, we denotew(t)anK-valuedQ-Wiener process with respect to{Ft}t≥0,andx0is anF0-adapted,H-valued
random variables independent ofw.
In order to define stochastic integrals with respect to the Q -Wiener process w(t), we introduce the subspace
K0 = Q1/2(K) of K which is endowed with the inner
producth˜u,v˜iK0 =hQ
−1/2u, Q˜ −1/2v˜i
K is a Hilbert space.
We assume that there exists a complete orthonormal system
{en}∞
n=1 in K, a bounded sequence of nonnegative real
numbers {λn}∞
n=1 such that Qen =λnen. and a sequence
βn of independent Brownian motions such that
hw(t), ei= ∞
X
n=1
p
λnhen, eiβn(t), e∈K, t∈J,
and Ft = Ftw, where Ftw is the σ-algebra generated by
{w(s) : 0≤s≤t}. Let L0
2 =L2(K0, H) be the space of
all Hilbert-Schmidt operators from K0 toH with the norm k ψk2
L0
2=Tr((ψQ
1/2)(ψQ1/2)∗)
for anyψ ∈L0
2. Clearly
for any bounded operatorsψ∈Lb(K, H)this norm reduces tokψk2
L0
2=Tr(ψQψ
∗).
LetLp(Fb, H)be the Banach space of all F
b-measurable pth power integrable random variables with values in the Hilbert space H. Let C([0, b];Lp(F, H)) be the Banach
space of continuous maps from[0, b]intoLp(F, H) satisfy-ing the conditionsupt∈JEkx(t)kp<∞. In particular, we
introduce the space C(J, H) denote the closed subspace of
C([0, b];Lp(F, H))consisting of measurable andF
t-adapted H-valued stochastic processes x ∈C([0, b];Lp(F, H))
en-dowed with the norm k x kC= (sup0≤t≤bE k x(t) kpH)
1
p.
Then(C,k · kC)is a Banach space.
LetXbe a Banach space, we recall some basic definitions in fractional calculus. For more details, see [1], [2]. Definition 1. The Riemann-Liouville fractional integral of the orderα >0of h:J →X is defined by
Jtαh(t) =
1 Γ(µ)
Z t
0
(t−s)α−1h(s)ds.
Definition 2.The Riemann-Liouville fractional derivative of the orderα∈(0,1)of h:J →X is defined by
Dtαh(t) = d dtJ
1−α t h(t).
Definition 3. The Caputo fractional derivative of the order
α∈(0,1)of h:J →X is defined by
CDα
Definition 4 ([35]). The Mittag-Leffler function is defined by
Eα,β(z) = ∞
X
n=0
zn
Γ(αn+β)), α, β >0, z∈C,˜ (4)
where C˜ denotes the complex plane. When β = 1, set
Eα(z) =Eα,1(z).
Definition 5 ([1]). The Mainardi’s function is defined by
Mα(z) = ∞
X
n=0
(−z)n
n!Γ(−αn−α+ 1), 0< α <1, z∈C.˜ (5)
The Laplace transform of the Mainardi’s functionMα(ξ)
is (see [36]):
Z ∞
0
e−ξλMα(ξ)dξ=Eα(−λ). (6)
By (4) and (6), it is clear that
Z ∞
0
Mα(ξ)dξ= 1, 0< α <1. (7)
On the other hand, Mα(z) satisfies the following equality (see [32])
Z ∞
0
α
ξα+1Mα(1/ξ
α)e−λξdξ=e−λα
. (8)
and the equality (see [36])
Z ∞
0
ξθMα(ξ)dξ= Γ(θ+ 1)
Γ(αθ+ 1), θ >−1, 0< α <1. (9)
For0< α <1, set
Tα(t)x=
Z ∞
0
Mα(ξ)T(tαξ)xdξ, t≥0, x∈H, (10)
and
Sα(t)x=
Z ∞
0
αξMα(ξ)T(tαξ)xdξ, t≥0, x∈H. (11)
It is easy to see thatTα(t)andSα(t)are strongly continuous onR+.Motivated by the Lemma 3.1 in [4], we present the
following definition of mild solutions to (1)-(2).
Definition 6. An Ft-adapted stochastic process x(·, y) :
[0, b] → H is called a mild solution of the system (1)-(2) if x(0, y) =x0(y)∈H, x(·, y)∈ C(J, H)and
(i) x(t, y) is measurable and adapted toFt, t≥0.
(ii) x(t, y)∈H has c`adl`ag paths ont∈J a.s and for each
t∈J,x(t, y)satisfies
x(t, y) =Tα(t)x0(y)
+
Z t
0
(t−s)α−1Sα(t−s)kV(y)x(s, y)ds
+
Z t
0
(t−s)α−1Sα(t−s)g(s, x(s, y))ds
+
Z t
0
(t−s)α−1Sα(t−s)f(s, x(s, y))dw(s).
Lemma 3(Bochner’s Theorem [37]). A measurable function
Λ : J → H is Bochner integrable, if k Λ kH is Lebesgue
integrable.
Lemma 4([38]). For anyp≥1and for arbitraryL0 2-valued
predictable processφ(·)such that
sup
s∈[0,t]
E
w w w w
Z s
0
φ(v)dw(v)
w w w w
2p
H
≤(p(2p−1))p
×
µ Z t
0
(Ekφ(s)k2Lp0 2)
1/pds
¶p
, t∈[0,∞).
In the rest of this paper, we denote by Cp = (p(p− 1)/2)p/2.
Lemma 5(Schaefer’s fixed point theorem [39]). Let X be a normed linear space. Let G : X → X be a completely continuous operator, that is, it is continuous and the image of any bounded set is contained in a compact set and let
ζ(G) ={x∈X:x=λGxfor some 0< λ <1}.
Then eitherζ(G)is unbounded or Ghas a fixed point.
III. EXISTENCE OF SOLUTIONS FOR FRACTIONAL STOCHASTIC CONTROL SYSTEMS
In this section, we prove the existence of solutions for fractional stochastic control system (1)-(2). Consider the space Z ={x(t, y) : [0, b]→H;x(0, y) =x0(y), x(t, y)∈ C(J, H)} endowed with the uniform convergence topology. In order to obtain the result, we introduce the following assumptions:
(H1) The functiong :J ×H →H is continuous and there exists a constantLg>0such that
Ekg(t, x1)−g(t, x2)kpH≤LgEkx1−x2kpH
for allt∈J, x1, x2∈H.
(H2) The functionf :J×H →L0
2 is continuous and there
exists a constantsLf >0 such that
Ekf(t, x1)−f(t, x2)kpL0
2≤LfEkx1−x2k
p H
for allt∈J, x1, x2∈H.
(H3) LetL0= 3p−1(Γ(α1))p
bpα
p(α−1)+1(kpkV k
p
L∞(Π) +Lpg+ CpLpfb−p/2)be such that 0< L
0<1.
Theorem 1. Let V ∈H2(Π).Assume that (H1)-(H3) are satisfied, then system (1)-(2) has a unique mild solution
x(·, y)∈Z.
Proof.Define the operatorΦ :Z→Z by
(Φx)(t, y)
=Tα(t)x0(y)
+
Z t
0
(t−s)α−1Sα(t−s)kV(y)x(s, y)ds
+
Z t
0
(t−s)α−1Sα(t−s)g(s, x(s, y))ds
+
Z t
0
(t−s)α−1Sα(t−s)
×f(s, x(s, y))dw(s), t∈J. (12)
It is clear that Φis a well-defined operator from Z intoZ. We show that Φhas a fixed point, which in turn is a mild solution of the problem (1)-(2).
By (9)-(11), and Lemma 1, we have
kTα(t)kLb(H)≤1, kSα(t)kLb(H)≤
1
Let t∈[0, b]andx∗(·, y), x∗∗(·, y)∈Z.From (H1),(H2)
and Lemmas 2, 4, we have
E k(Φx∗)(t, y)−(Φx∗∗)(t, y)kp H
≤3p−1E
w w w w
Z t
0
(t−s)α−1Sα(t−s)kV(y)
×[x∗(s, y)−x∗∗(s, y)]ds
w w w w
p
H
+3p−1E
w w w w
Z t
0
(t−s)α−1Sα(t−s)
×[g(s, x∗(s, y))−g(s, x∗(s, y))]ds
w w w w
p
H
+3p−1E
w w w w
Z t
0
(t−s)α−1Sα(t−s)
×[f(s, x∗(s, y))−f(s, x∗(s, y))]dw(s)
w w w w
p
H
≤3p−1 1 (Γ(α))pk
pbp−1kV kp L∞(Π)
×
Z t
0
(t−s)p(α−1)
×Ekx∗(s, y)−x∗∗(s, y)kp Hds
+3p−1 1 (Γ(α))pb
p−1
Z t
0
(t−s)p(α−1)
×Ekg(s, x∗(s, y))−g(s, x∗(s, y))kpH ds
+3p−1Cp 1
(Γ(α))p
· Z t
0
[(t−s)p(α−1)
×Ekf(s, x∗(s, y))
−f(s, x∗(s, y))kp L0
2]
2/pds
¸p/2
≤3p−1 1 (Γ(α))pk
pbp−1kV kp L∞(Π)
×
Z t
0
(t−s)p(α−1)ds
× sup
s∈[0,b]
Ekx∗(s, y)−x∗∗(s, y)kp H
+3p−1 1 (Γ(α))pb
p−1Lp g
Z t
0
(t−s)p(α−1)
×Ekx∗(s, y)−x∗∗(s, y)kp Hds
+3p−1Cp 1 (Γ(α))pL
p fb
p/2−1Z t
0
(t−s)p(α−1)
×Ekx∗(s, y)−x∗∗(s, y)kp Hds
≤3p−1 1 (Γ(α))p
bp(α−1)+1
p(α−1) + 1(k
pbp−1kV kp L∞(Π)
+Lpgbp−1+CpL p fb
p/2−1)
× sup
s∈[0,b]
Ekx∗(s, y)−x∗∗(s, y)kp H
= 3p−1 1 (Γ(α))p
bpα p(α−1) + 1(k
pkV kp L∞(Π)
+Lpg+CpL p fb
−p/2)kx∗
−x∗∗ kpC .
Taking supremum overt,
kΦ1x∗−Φ1x∗∗kpC≤L0kx∗−x∗∗kpC,
where L0 = 3p−1(Γ(α1))p b pα
p(α−1)+1(k
p k V kp
L∞(Π) +Lpg+ CpLpfb−p/2) < 1, which implies that Φ is a contraction
on Z. Hence by the Banach fixed point theorem, Φ has a unique fixed point x(·, y) in Z, and x(·, y) is the unique mild solution of system (1)-(2). The proof is complete.
We use the below condition instead of (H1) and (H2) to avoid the Lipschitz continuity ofg, f used in Theorem 1. (B1) The operator families Tα(t)andSα(t)are compact for
allt >0.
(B2) The function g(t,·) : H → H is continuous for each
t∈J, and for everyx∈H,the functiont→g(t, x)is strongly measurable.
(B3) There exists a positive function mg ∈Lp(J, R+)such
that
Ekg(t, x)kpH≤mg(t)
for allt∈J, x∈H.
(B4) The function f(t,·) : H →L0
2 is continuous for each
t∈J,and for every x∈H,the functiont→f(t, x)is strongly measurable.
(B5) There exists a positive function mf ∈Lp(J, R+)such
that
Ekf(t, x)kpL0
2≤mf(t)
for allt∈J, x∈H.
Theorem 2.Let V ∈H2(Π). If the assumptions (B1)-(B5) are satisfied. Then the system(1)-(2) has at least one mild solution onJ provided thatp2(α−1) +p >1.
Proof.We define the mapΦon the spaceZ as in Eq. (12). We shall show that Φ satisfies all conditions of Lemma 5. The proof will be given in several steps.
Step 1. The set ζ = {x(·, y) ∈Z : λ ∈(0,1), x(·, y) =
λΦ(x(·, y))}is bounded.
Indeed, let λ ∈ (0,1) and let x(·, y) ∈ Z be a possible solution ofx(·, y) =λΦ(x(·, y))for some 0< λ <1. This implies by (12) that for eacht∈[0, b]we have
x(t, y)
=λTα(t)x0(y)
+λ
Z t
0
(t−s)α−1Sα(t−s)kV(y)x(s, y)ds
+λ
Z t
0
(t−s)α−1Sα(t−s)g(s, x(s, y))ds
+λ
Z t
0
(t−s)α−1Sα(t−s)
×f(s, x(s, y))dw(s), t∈J.
By (B2), (B5), and Lemmas 2 and 4, we have fort∈J
Ekx(t, y)kpH
≤4p−1EkTα(t)x 0(y)kpH
+4p−1E
w w w w
Z t
0
(t−s)α−1Sα(t−s)
×kV(y)x(s, y)ds
w w w w
p
H
+4p−1E
w w w w
Z t
0
(t−s)α−1Sα(t−s)
×g(s, x(s, y))ds
w w w w
p
H
+4p−1E
w w w w
Z t
0
×f(s, x(s, y))dw(s)
w w w w
p
H
≤4p−1E kx 0kpH
+4p−1 1 (Γ(α))pk
pbp−1Z t
0
(t−s)p(α−1)
× kV(y)kpHEkx(s, y)kpHds
+4p−1 1 (Γ(α))pb
p−1
Z t
0
(t−s)p(α−1)
×Ekg(s, x(s, y))kpH ds
+4p−1Cp 1 (Γ(α))p
· Z t
0
[(t−s)p(α−1)
×Ekf(s, x(s, y))kpL0 2]
2/pds
¸p/2
≤4p−1E kx0kpH
+4p−1 1 (Γ(α))pk
pbp−1kV kp L∞(Π)
×
Z t
0
(t−s)p(α−1)Ekx(s, y)kpHds
+4p−1 1 (Γ(α))pb
p−1Z t
0
(t−s)p(α−1)mg(s, y)ds
+4p−1Cp 1 (Γ(α))pb
p/2−1
×
Z t
0
(t−s)p(α−1)mf(s, y)ds
≤4p−1E kx0kpH
+4p−1 1 (Γ(α))pk
pbp−1kV kp L∞(Π)
×
Z t
0
(t−s)p(α−1)Ekx(s, y)kp Hds
+4p−1 1
(Γ(α))pb
p−1µ Z t
0
(t−s)p2(pα−1−1)ds ¶p−1
p
×
µ Z t
0
(mg(s, y))pds
¶1
p
+4p−1Cp 1
(Γ(α))pb p/2−1
×
µ Z t
0
(t−s)p2(pα−1−1)ds ¶p−1
p
×
µ Z t
0
(mf(s, y))pds
¶1
p
≤Mf(y) + 4p−1 1 (Γ(α))pk
pbp−1kV kp L∞(Π)
×
Z t
0
(t−s)p(α−1)Ekx(s, y)kpHds,
where
f
M(y) = 4p−1Ekx
0kpH +4p−1
1 (Γ(α))pb
p−1
×
µ
p−1
p2(α−1) +p−1
¶p−1
p
×bp2(α−1)+p p−1
µ Z b
0
(mg(s, y))pds
¶1
p
+4p−1Cp 1
(Γ(α))pb p/2−1
×
µ
p−1
p2(α−1) +p−1
¶p−1
p
bp2(α−1)+p p−1
×
µ Z b
0
(mf(s, y))pds
¶1
p
.
Consider the function defined by
v(t, y) = sup{Ekx(s, y)kHp: 0≤s≤t}, 0≤t≤b.
For eacht∈[0, b],we have
v(t, y)≤Mf(y) + 4p−1 1 (Γ(α))pk
pbp−1
× kV kpL∞(Π) Z t
0
(t−s)p(α−1)v(s, y)ds.
Applying Gronwall’s inequality in the above expression, we obtain
v(t, y)
≤Mf(y)exp
½
4p−1 1 (Γ(α))pk
pbp−1
× kV kpL∞(Π) Z b
0
(t−s)p(α−1)ds
¾
≤Mf(y)exp
½
4p−1 1 (Γ(α))p
bpα p(α−1) + 1k
p
× kV kpL∞(Π) ¾
,
and therefore
kx(·, y)kpC≤Mf(y)exp
½
4p−1 1 (Γ(α))p
× b
pα
p(α−1) + 1k
pkV kp L∞(Π)
¾
<∞.
Thus the proof of boundedness of the setζ is complete.
Step2. The set{(Φx)(t, y) :x∈Br}is relatively compact inH.
We note that(Φ(Br))(t, y)is relatively compact inZ for
t = 0. Let 0 < t ≤ s ≤ b be fixed and ε a real number satisfying0< ε < tfor x∈Br. We define
(Φεx)(t, y)
=Tα(t)x0(y) +
Z t−ε
0
(t−s)α−1Sα(t−s)
×kV(y)x(s, y)ds
+
Z t−ε
0
(t−s)α−1Sα(t−s)g(s, x(s, y))ds
+
Z t−ε
0
(t−s)α−1Sα(t−s)f(s, x(s, y))dw(s).
Using the compactness ofTα(t), Sα(t)fort >0,we deduce that the set Uε(t) = {(Φεx)(t, y) : x ∈ Br} is relatively
x∈Br we have
Ek(Φx)(t, y)−(Φεx)(t, y)kpH
≤3p−1E
w w w w
Z t
t−ε
(t−s)α−1Sα(t−s)
×kV(y)x(s, y)ds
w w w w p H
+3p−1E
w w w w
Z t
t−ε
(t−s)α−1Sα(t−s)
×g(s, x(s, y))ds
w w w w p H
+3p−1E
w w w w
Z t
t−ε
(t−s)α−1Sα(t−s)
×f(s, x(s, y))dw(s)
w w w w p H
≤3p−1 1 (Γ(α))pk
pεp−1
Z t
t−ε
(t−s)p(α−1)
× kV(y)kpH Ekx(s, y)k p Hds
+3p−1 1 (Γ(α))pε
p−1
Z t
t−ε
(t−s)p(α−1)
×Ekg(s, x(s, y))kpH ds
+3p−1Cp 1
(Γ(α))p
· Z t
t−ε
[(t−s)p(α−1)
×Ekf(s, x(s, y))kpL0 2]
2/pds
¸p/2
≤3p−1 1 (Γ(α))pk
pεp−1kV kp L∞(Π)
×
Z t
t−ε
(t−s)p(α−1)Ekx(s, y)kpHds
+3p−1 1 (Γ(α))pε
p−1Z t
t−ε
(t−s)p(α−1)mg(s, y)ds
+3p−1Cp 1 (Γ(α))pε
p/2−1
×
Z t
t−ε
(t−s)p(α−1)mf(s, y)ds
≤3p−1 1 (Γ(α))pk
pεp−1kV kp L∞(Π)
×
Z t
t−ε
(t−s)p(α−1)rds
+3p−1 1
(Γ(α))pε p−1µ Z
t
t−ε
(t−s)p2(pα−1−1)ds ¶p−1
p
×
µ Z t
t−ε
(mg(s, y))pds
¶1
p
+3p−1Cp 1
(Γ(α))pε p/2−1
×
µ Z t
t−ε
(t−s)p2(pα−1−1)ds ¶p−1
p
×
µ Z t
t−ε
(mf(s, y))pds
¶1
p
,
and there are relatively compact sets arbitrarily close to the set {(Φx)(t, y) : x ∈ Br}, and (Φ(Br)(t) is a relatively compact in H. By the Arzel´a-Ascoli theorem, we can con-clude that the set{(Φx)(t, y) :x∈Br}is relatively compact
inH for every t∈J.
Step3. Φmaps bounded sets into equicontinuous sets of
Z.
Let 0 < ε < t < b. From step 2, (ΦBr)(t, y) is relatively compact for each t and by the strong continuity ofTα(t), Sα(t),we can choose 0< δ < b−t with
kTα(t+h)x−Tα(t)xkH≤ε,
kSα(t+h)x−Sα(t)xkH≤ε
forx∈(ΦBr)(t, y)when 0< h < δ.
For anyx∈Br,we have
Ek(Φ2x)(t+h, y)−(Φ2x)(t, y)kpH
≤4p−1Ek[Tα(t+h−s)−Tα(t−s)]x 0(y)kpH
+4p−1E
w w w w
Z t+h
0
(t+h−s)α−1Sα(t+h−s)
×kV(y)x(s, y)ds
−
Z t
0
(t−s)α−1Sα(t−s)kV(y)x(s, y)ds
w w w w p H
+4p−1E
w w w w
Z t+h
0
(t+h−s)α−1Sα(t+h−s)
×g(s, x(s, y))ds
−
Z t
0
(t−s)α−1Sα(t−s)g(s, x(s, y))ds
w w w w p H
+4p−1
w w w w
Z t+h
0
(t+h−s)α−1Sα(t+h−s)
×f(s, x(s, y))dw(s)
−
Z t
0
(t−s)α−1Sα(t−s)f(s, x(s, y))dw(s)
w w w w p H = 4 X i=1 Ii.
In view of (B3), (B5) and H¨older’s inequality, it follows that
I1= 4p−1kTα(t+h−s)x0(y) −Tα(t−s)x0(y)kpH≤4
p−1ε,
I2
≤3p−1E
w w w w Z t 0
[(t+h−s)α−1−(t−s)α−1]
×Sα(t+h−s)kV(y)ds
w w w w p H
+3p−1E
w w w w Z t 0
[Sα(t+h−s)−Sα(t−s)]
×(t−s)α−1kV(y)ds
w w w w p H
+3p−1E
w w w w
Z t+h
t
(t+h−s)α−1Sα(t+h−s)
×kV(y)ds
w w w w p H
≤3p−1bp−1
Z t
0
|(t+h−s)α−1−(t−s)α−1|p
× kSα(t−s)kpH EkkV(y)kpHds
+3p−1bp−1Z t 0
−Sα(t−s)kkpH (t−s)p(α−1)EkV(y)kp Hds
+3p−1bp−1Z t+h
t
(t+h−s)p(α−1)
× kSα(t+h−s)kHp EkkV(y)kpH ds
≤3p−1bp−1 1
(Γ(α))pk
pkV kp L∞(Π)
×
Z t
0
|(t+h−s)α−1−(t−s)α−1|pds
+3p−1bp−1εp kV kp L∞(Π)
Z t
0
(t−s)p(α−1)ds
+3p−1bp−1 1 (Γ(α))pk
p kV kp L∞(Π)
×
Z t+h
t
(t+h−s)p(α−1)ds,
I3
≤3p−1E
w w w w
Z t
0
[(t+h−s)α−1−(t−s)α−1]
×Sα(t+h−s)g(s, x(s, y))ds
w w w w
p
H
+3p−1E
w w w w
Z t
0
[Sα(t+h−s)−Sα(t−s)]
×(t−s)α−1g(s, x(s, y))ds
w w w w
p
H
+3p−1E
w w w w
Z t+h
t
(t+h−s)α−1Sα(t+h−s)
×g(s, x(s, y))ds
w w w w
p
H
≤3p−1bp−1
Z t
0
|(t+h−s)α−1−(t−s)α−1|p
× kSα(t−s)kpHEkg(s, x(s, y))kpH ds
+3p−1Z t 0
EkSα(t+h−s)g(s, x(s, y))
−Sα(t−s)g(s, x(s, y))kpH (t−s)p(α−1)ds
+3p−1Z t+h
t
(t+h−s)p(α−1)
× kSα(t+h−s)kpHEkg(s, x(s, y))kpHds
≤3p−1bp−1 1
(Γ(α))p
×
Z t
0
|(t+h−s)α−1−(t−s)α−1|pmg(s)ds
+3p−1bp−1εpZ t
0
(t−s)p(α−1)ds
+3p−1bp−1 1 (Γ(α))p
×
Z t+h
t
(t+h−s)p(α−1)mg(s, y)ds,
I4
≤3p−1E
w w w w
Z t
0
[(t+h−s)α−1−(t−s)α−1]
×Sα(t+h−s)f(s, x(s, y))dw(s)
w w w w
p
H
+3p−1E
w w w w
Z t
0
[Sα(t+h−s)−Sα(t−s)]
×(t−s)α−1f(s, x(s, y))dw(s)
w w w w
p
H
+3p−1E
w w w w
Z t+h
t
(t+h−s)α−1Sα(t+h−s)
×f(s, x(s, y))dw(s)
w w w w
p
H
≤3p−1Cp
· Z t
0
[|(t+h−s)α−1−(t−s)α−1|p
× kSα(t−s)kpH Ekf(s, x(s, y))kpL0 2]
2/pds
¸p/2
+3p−1Cp· Z t 0
[kSα(t+h−s)f(s, x(s, y))
−Sα(t−s)f(s, x(s, y))kpH(t−s)p(α
−1)]2/pds
¸p/2
+3p−1Cp· Z t+h
t
[(t+h−s)p(α−1)
× kSα(t+h−s)kpH
×Ekf(s, x(s, y))kpL0 2]
2/pds
¸p/2
≤3p−1Cp 1
(Γ(α))pb p/2−1
×
Z t
0
|(t+h−s)α−1−(t−s)α−1|pmf(s, y)ds
+3p−1Cpεpbp/2−1Z t 0
(t−s)p(α−1)ds
+3p−1Cp 1 (Γ(α))pb
p/2−1
×
Z t+h
t
(t+h−s)p(α−1)mf(s, y)ds.
Since
Z t
0
|(t+h−s)α−1−(t−s)α−1|pmg(s, y)ds
≤2p−1
Z t
0
(t+h−s)p(α−1)mg(s, y)ds
+2p−1
Z t
0
(t−s)p(α−1)mg(s, y)ds,
and by H¨older’s inequality, we have
Z t
0
(t−s)p(α−1)mg(s)ds
≤
µ Z t
0
(t−s)p2(pα−1−1)ds ¶p−1
p µ Z t
0
(mg(s, y))pds
¶1
p
≤
µ
p−1
p2(α−1) +p−1
¶p−1
p
bp2(α−1)+p p−1
×
µ Z b
0
(mg(s, y))pds
¶1
p
<+∞.
Similarly, we have
Z t
0
Then by the dominated convergence theorem,
Z t
0
|(t+h−s)α−1−(t−s)α−1|p
×mg(s, y)ds→0 as h→0.
In the same way, we can get
Z t
0
|(t+h−s)α−1−(t−s)α−1|pds→0as h→0,
Z t
0
|(t+h−s)α−1−(t−s)α−1|p
×mf(s, y)ds→0 ash→0.
We see that E k Φx(t+h, y)−Φx(t, y)kpH tends to zero independently of x ∈ Br as h → 0 and sufficiently small positive number ε. Thus, the set {Φx(·, y) : x ∈ Br} is equicontinuous.
Step4. Φ :Z→Z is continuous.
Let {x(n)} ⊆ Br with x(n) →x(n→ ∞) in Z.By the
assumptions (B1) and (B4), we have
g(s, x(n)(s, y))→g(s, x(s, y))as n→ ∞,
f(s, x(n)(s, y))→f(s, x(s, y))as n→ ∞
for eachs∈[0, t],and since
Ekg(s, x(n)(s, y))−g(s, x(s, y))kpH ≤2p−1mg(s, y), s∈[0, b],
Ekf(s, x(n)(s, y))−f(s, x(s, y))kp L0
2
≤2p−1mf(s, y), s∈[0, b],
andR0t(t−s)p(α−1)ds <+∞,Rt 0(t−s)
p(α−1)mg(s, y)ds <
+∞,R0t(t−s)p(α−1)mf(s, y)ds <+∞.Then the dominated
convergence theorem ensures that
Ek(Φx(n))(t, y)−(Φx)(t, y)kpH
≤3p−1E
w w w w
Z t
0
(t−s)α−1Sα(t−s)kV(y)
×[x(n)(s, y)−x(s, y)]ds
w w w w
p
H
+3p−1E
w w w w
Z t
0
(t−s)α−1Sα(t−s)
×[g(s, x(n)(s, y))−g(s, x(s, y))]ds
w w w w
p
H
+3p−1E
w w w w
Z t
0
(t−s)α−1Sα(t−s)
×[f(s, x(n)(s, y))−f(s, x(s, y))]dw(s)
w w w w
p
H
≤3p−1bp−1 1
(Γ(α))p
Z t
0
(t−s)p(α−1)kp
× kV(y)kpHEkx(n)(s, y)−x(s, y)kpHds
+3p−1bp−1 1 (Γ(α))p
Z t
0
(t−s)p(α−1)
×Ekg(s, x(n)(s, y))−g(s, x(s, y))kpH ds
+3p−1Cp 1 (Γ(α))p
· Z t
0
[(t−s)p(α−1)
×Ekf(s, x(n)(s, y))
−f(s, x(s, y))kpL0 2]
2/pds
¸p/2
≤3p−1bp−1 1 (Γ(α))pk
pkV(y)kp
∞
Z t
0
(t−s)p(α−1)
×Ekx(n)(s, y)−x(s, y)kpHds
+3p−1bp−1 1
(Γ(α))p
Z t
0
(t−s)p(α−1)
×Ekg(s, x(n)(s, y))−g(s, x(s, y))kpHds
+3p−1Cp 1 (Γ(α))pb
p/2−1Z t
0
(t−s)p(α−1)
×Ekf(s, x(n)(s, y))−f(s, x(s, y))kpL0 2 ds.
Then
kΦ2x(n)(·, y)−Φ2x(·, y)kpC = sup
t∈J
Ek(Φ2x(n))(t, y)−(Φ2x)(t, y)kpH
→0as n→ ∞.
Therefore,Φis continuous.
These arguments enable us to conclude that Φ is com-pletely continuous. We can now apply Lemma 5 to conclude that Φhas at least fixed point x(·, y)∈Z,which is a mild solution of problem (1)-(2). The proof is complete.
IV. EXISTENCE OF FRACTIONAL STOCHASTIC OPTIMAL CONTROLS
In this section we consider a control problem and present a result on the existence of fractional stochastic optimal controls. let Y is a separable reflexive Hilbert space from which the controlsutake the values.L∞(J, L(Y, H))denote the space of operator valued functions which are measurable in the strong operator topology and uniformly bounded on the interval J. Let LpF(J, Y) is the closed subspace
of LpF(J ×Ω, Y), consisting of all measurable and Ft
-adapted, Y-valued stochastic processes satisfying the con-ditionER0bku(t)kpY dt <∞,and endowed with the norm
kukLpF(J,Y)= (E
Rb
0 ku(t)k
p Y dt)
1
p.LetU be a nonempty
closed bounded convex subset ofY.We define the admissible control set
Uad={v(·, y)∈LpF(J, Y);v(t, y)∈U a.e.t∈J}.
Consider the following controlled nonlinear fractional sto-chastic Schr¨odinger evolution equations with potential of the form
i(Jt1−αx)(t, y) +△(t, y)−kV(y)x(t, y)
=B(t, y)u(t, y) +g(t, x(t, y))
+f(t, x(t, y))w(t)
dt , (13)
y∈Π, t∈(0, b], u∈Uad,
x(0, y) =x0(y), y∈Π. (14)
We will assume that (S) B∈L∞(J, L(Y, H)).
Then, Bu∈Lp(J, H) for all u∈Uad. By Theorem 2, we
Theorem 3. Assume that assumptions of Theorem 2 hold and, in addition, the assumption (S) is satisfied. For every
u∈Uad,system(13)-(14) has a mild solution corresponding tougiven by the solution of the following integral equation
xu(t, y)
=Tα(t)x0(y)
+
Z t
0
(t−s)α−1Sα(t−s)kV(y)x(s, y)ds
+
Z t
0
(t−s)α−1Sα(t−s)B(s, y)u(s, y)ds
+
Z t
0
(t−s)α−1Sα(t−s)g(s, x(s, y))ds
+
Z t
0
(t−s)α−1Sα(t−s)
×f(s, x(s, y))dw(s), t∈J.
Proof. Consider the space Z endowed with the uniform convergence topology and define the operator Θ : Z → Z
by
(Θx)(t, y)
=Tα(t)x0(y)
+
Z t
0
(t−s)α−1Sα(t−s)kV(y)x(s, y)ds
+
Z t
0
(t−s)α−1Sα(t−s)B(s, y)u(s, y)ds
+
Z t
0
(t−s)α−1Sα(t−s)g(s, x(s, y))ds
+
Z t
0
(t−s)α−1Sα(t−s)
×f(s, x(s, y))dw(s), t∈J.
Using (S) and the H¨older inequality, we have
E
w w w w
Z t
0
(t−s)α−1Sα(t−s)B(s, y)u(s, y)ds
w w w w
p
H
≤E
· Z t
0
(t−s)α−1kSα(t−s)k
H
× kB(s, y)kHku(s, y)kH ds
¸p
≤ 1
(Γ(α))p kBk p
∞
×E
· Z t
0
(t−s)α−1ku(s, y)k
Y ds
¸p
≤ 1
(Γ(α))p kBk p
∞
µ Z t
0
(t−s)p(pα−1−1)ds ¶p−1
×E
Z t
0
ku(s, y)kpY ds
≤ 1
(Γ(α))p kBk p
∞
µ
p−1
pα−1
¶p−1
×bpα−1kukp LpF(J,Y),
wherek B k∞ is the norm of operator B in Banach space
L∞(J, L(Y, H))andpα >1.From Lemma 3, it follows that
(t−s)α−1Sα(t−s)B(s, y)u(s, y)is Lebesgue integrable with
respect tos∈[0, t]for allt∈J.Hence we conclude thatΘ
is a well-defined operator fromZ into Z.The proofs of the
other steps are similar to those in Theorem 2. Therefore, we omit the details. The proof is complete.
Letxu(·, y)denote the mild solution of system (13)-(14)
corresponding to the controlu(·, y)∈Uad.We consider the Lagrange problem (P):
Find an optimal pair(x0(·, y), u0(·, y))∈ BC×Uadsuch
that
J(x0(·, y), u0(·, y))
≤ J(xu(·, y), u(·, y))for all u(·, y)∈Uad,
where the cost function
J(xu(·, y), u(·, y)) =E
Z b
0
l(t, xu(t, y), u(t, y))dt,
andxu(·, y)denotes the mild solution of system (13)- (14)
corresponding to the controlu(·, y)∈Uad.
We introduce the following assumption onl.
(P1) The functional l : J ×H ×Y → R∪ {∞} is Borel measurable.
(P2) l(t,·,·)is sequentially lower semicontinuous onH×Y
for almost allt∈J.
(P3) l(t, x,·)is convex onY for eachx∈H and almost all
t∈J.
(P4) There exist constants d1, d2>0, µis nonnegative and
µ∈ L1(J, R) such that l(t, x, u)≥µ(t) +d
1 k xkH
+d2kukpY .
To prove the existence of solution for problem (P), we need the following important lemma.
Lemma 6. Operator Ψ : Lp(J, Y)→ Z for some pα(1− ϑ)>1 given by
(Ψu)(·, y) =
Z ·
0
Sα(· −s)B(s)u(s)ds
is completely continuous. Proof.Suppose that un ⊆Lp
F(J, Y)is bounded, we define Λn(t, y) = (Ψun)(t, y), t ∈ J. Similar to the proof of
Theorem 2, one can know that for any fixed t ∈ J and,
EkΛn(t, y)kpH is bounded. By using (B1)-(B5), it is ease
to verify that Λn(t, y) is relatively compact in H and is
also equicontinuous. Due to Ascoli-Arzela Theorem again,
{Λn(t, y)} is compact in H. Obviously, Ψ is linear and
continuous. Hence, Ψ is a completely continuous operator. The proof is complete.
Next we can give the following result on existence of optimal controls for problem (P).
Theorem 4. If the assumptions(P1)-(P4) and the assump-tions of Theorem 3 hold. Then the Lagrange problem (P)
admits at least one optimal pair onZ×Uad.
Proof. Without loss of generality, we assume that inf{J(xu(·, y), u(·, y))|u(·, y) ∈ Uad} = ε < +∞.
Oth-erwise, there is nothing to prove. By assumptions (P1)-(P4), we have
J(xu(·, y), u(·, y))
≥
Z b
0
ϕ(t)dt+d1
Z b
0
Ekxu(t, y)kH dt
+d2
Z b
0
where η > 0 is a constant. Hence, ε ≥ −η >
−∞. On the other hand, by using definition of infi-mum, there exists a minimizing sequence of feasible pair
{(xm(·, y), um(·, y))} ⊂ Aad,such that
J(xm(·, y), um(·, y))→εas m→+∞,
whereAad={(x(·, y), u(·, y))|x(·, y) is a mild solution of
system (13)-(14) corresponding to u(·, y)∈Uad}.
For {um(·, y)} ⊆ Uad. {um(·, y)} is bounded in Lp
F, so
there exists a subsequence, relabeled as{um(·, y)},andu0∈
LpF(J,Y) such that
um(·, y)(w)−→u0(·, y)inLpF asm→ ∞.
Since Uad is closed and convex, by Marzur Lemma, we conclude thatu0(·, y)∈Uad.
Now we suppose that xm(·, y) are the mild solutions of
system (13)-(14) corresponding toum(·, y)(m= 0,1,2, . . .),
andxm(·, y)satisfied the following integral equation
xm(t, y)
=Tα(t)x0(y)
+
Z t
0
(t−s)α−1Sα(t−s)kV(y)xm((s, y)ds
+
Z t
0
(t−s)α−1Sα(t−s)B(s, y)um(s, y)ds
+
Z t
0
(t−s)α−1Sα(t−s)g(s, xm(s, y))ds
+
Z t
0
(t−s)α−1Sα(t−s)
×f(s, xm(s, y))dw(s), t∈J.
Letgm(s, y)≡g(s, xm(s, y)), fm(s, y)≡f(s, xm(s, y)).
Then by (B4) and (B5), we obtain that
kgm(·, y)kpLp(J,H)
=E
µ Z b
0
kgm(s, y)kpH ds
¶
=
Z b
0
E kgm(s, y)kpH ds
≤
Z b
0
mh(s, y)ds≤bp−1p
µ Z b
0
(mg(s, y))pds
¶1
p
,
kfm(·, y)kpLp(J,L b(K,H)
=E
µ Z b
0
kfm(s, y)kpL
b(K,H)ds
¶
=
Z b
0
Ekfm(s, y)kpLb(K,H)ds
≤
Z b
0
mf(s, y)ds≤bp−1p
µ Z b
0
(mf(s, y))pds
¶1
p
.
That is to say, hm(·, y) : J → H and fm(·, y) :
J → Lb(K, H) are bounded continuous operators. Hence,
hm(·, y)∈Lp(J, H), fm(·, y)∈Lp(J, Lb(K, H)). Further-more, {hm(·, y)}, {fm(·, y)} is bounded in Lp(J, H) and
in Lp(J, Lb(K, H)), and there are subsequences, relabeled
as {hm(·, y)},{fm(·, y)}, and bh(·, y)∈ Lp(J, H),fb(·, y)∈ Lp(J, Lb(K, H))such that
hm(·, y)(w)−→bh(·, y) inLp(J, H)as m→ ∞,
fm(·)(w)−→fb(·, y)inLp(J, Lb(K, H))as m→ ∞.
Next we turn to consider the following controlled system
i(J1−α
t x)(t, y) +△(t, y)−kV(y)x(t, y)
=B(y, t)u0(y, t) +bg(t, y) +fb(t, y)w(t)
dt (15) y∈Π, t∈[0, b], u∈Uad,
x(0, y) =x0(y), y∈Π. (16)
By Theorem 3, it is easy to see that system (15)-(16) has a mild solution
b
x(t, y)
=Tα(t)x0(y)
+
Z t
0
(t−s)α−1Sα(t−s)kV(y)xb(s, y)ds
+
Z t
0
(t−s)α−1Sα(t−s)B(s, y)u0(s, y)ds
+
Z t
0
(t−s)α−1Sα(t−s)
b
g(s, y)ds
+
Z t
0
(t−s)α−1Sα(t−s)fb(s, y)dw(s), t∈J.
For eacht∈J, xm(·, y),xb(·, y)∈Z,we have
Ekxm(t, y)−xb(t, y)kpH
≤µ(1)
m(t, y) +µ(2)m(t, y) +µ(3)m (t, y) +µ(4)m(t, y),
where
µ(1)m(t, y)
= 4p−1E
w w w w
Z t
0
(t−s)α−1Sα(t−s)
×[kV(y)xm(s, y)−kV(y)xb(s, y)]ds
w w w w
p
H ,
µ(2)m(t, y)
= 4p−1E
w w w w
Z t
0
(t−s)α−1Sα(t−s)
×B(s, y)[um(s, y)−u0(s, y)]ds
w w w w
p
H ,
µ(3)
m(t, y)
= 4p−1E
w w w w
Z t
0
(t−s)α−1Sα(t−s)
×[gm(s, y)−bg(s, y)]ds
w w w w
p
H ,
µ(4)m(t, y)
= 4p−1E
w w w w
Z t
0
(t−s)α−1Sα(t−s)
×[fm(s, y)−fb(s, y)]dw(s)
w w w w
p
Using the H¨older inequality, we can obtain
µ(1)m(t, y)
≤4p−1bp−1
Z t
0
(t−s)p(α−1)kSα(t−s)kp H
× kkV(y)kpHEkxm(s, y)−xb(s, y)kpHds
≤4p−1bp−1 1 (Γ(α))pk
pkV kp L∞(Π)
×
Z t
0
(t−s)p(α−1)Ekxm(s, y)−xb(s, y)kp H ds.
By (S) and using the H¨older inequality again, we have
µ(2)m(t, y)
≤4p−1E· Z t 0
(t−s)α−1kB(s, y)k
H
× kSα(t−s)[um(s, y)−u0(s, y)]kHds
¸p
≤4p−1kBkp∞
µ Z t
0
(t−s)p(pα−1−1)ds ¶p−1
×
Z t
0
EkSα(t−s)[um(s, y)−u0(s, y)]kp Hds
≤4p−1kBkp
∞
µ
p−1
pα−1
¶p−1
bpα−1
×
Z b
0
EkSα(t−s)[um(s, y)−u0(s, y)]kpHds,
and
µ(3)m (t, y)
≤4p−1E
· Z t
0
(t−s)α−1kSα(t−s)
×[gm(s, y)−bg(s, y)]kHds
¸p
≤4p−1µ Z t 0
(t−s)p(pα−1−1)ds ¶p−1
×
Z t
0
EkSα(t−s)[gm(s, y)−bg(s, y)]kpH ds
≤4p−1
µ
p−1
pα−1
¶p−1
bpα−1
×
Z b
0
EkSα(t−s)[gm(s, y)−bg(s, y)]kpH ds,
µ(4)
m(t, y)
≤4p−1Cp
· Z t
0
[(t−s)p(α−1)EkSα(t−s)
×[fm(s, y)−fb(s, y)]kpL0 2 ds]
2/p
¸p/2
≤4p−1
µ Z t
0
(t−s)p/p/2(2−1α−1)ds ¶p/2−1
×
Z t
0
EkSα(t−s)[fm(s, y)−fb(s, y)]kpH ds
≤4p−1
µ
p/2−1 (p/2)α−1
¶p/2−1
b(p/2)α−1
×
Z b
0
EkSα(t−s)[fm(s, y)−fb(s, y)]kpHds,
By Lemma 6 and Lebesgue’s dominated convergence theo-rem,
Z t
0
EkSα(t−s)[um(s, y)−u0(s, y)]kpHds
→0 as m→ ∞,
Z t
0
EkSα(t−s)[gm(s, y)−bg(s, y)kpH]ds
→0 asm→ ∞,
Z t
0
EkSα(t−s)[fm(s, y)−fb(s, y)kpH]ds
→0 as m→ ∞.
Therefore, we obtain
µ(2)m(t, y), µ(3)m(t, y), µ(4)m(t, y)→0as m→ ∞.
Then we have
Ekxm(t, y)−xb(t, y)kpH
≤4p−1bp−1 1
(Γ(α))pk
pkV kp L∞(Π)
×
Z t
0
(t−s)p(α−1)Ekxm(s, y)−
b
x(s, y)kpHds
+µ(2)
m(t, y) +µ(3)m(t, y) +µ(4)m(t, y).
Using Gronwall’s inequality again,
E kxm(t, y)−bx(t, y)kp H
≤[µ(2)
m(t, y) +µ(3)m(t, y) +µ(4)m(t, y)]
×exp
½
4p−1 1 (Γ(α))p
bpα p(α−1) + 1
×kpkV kpL∞(Π) ¾
,
which implies that
xm(·, y)→xb(·, y)inZ as m→ ∞.
Further, by (B3) and (B5), we can obtain
gm(y,·)→g(·, x(·, y)inZ as m→ ∞,
fm(·, y)→f(·, x(y,·)inZ asm→ ∞.
Using the uniqueness of limit, we have
b
g(t, y) =g(t,bx(t, y)), fb(t, y) =f(t,xb(t, y)).
Thus,bx(·, y)can be given by
b
x(t, y)
=Tα(t)x0(y)
+
Z t
0
(t−s)α−1Sα(t−s)kV(y)
b
x(s, y)ds
+
Z t
0
(t−s)α−1Sα(t−s)B(y, s)u0(s, y)ds
+
Z t
0
(t−s)α−1Sα(t−s)g(s,
b
x(s, y))ds
+
Z t
0
(t−s)α−1Sα(t−s)
which is just a mild solution of system (13)-(14) correspond-ing to u0(·, y). Since Z ֒→ L1(J, H), using (P1)-(P4) and
Balder’s theorem ([40]), we can obtain
ε= lim
m→∞E
Z b
0
l(t, xm(t, y), um(t, y))dt
≥E
Z b
0
l(t,bx(t, y), u0(t, y))dt
≥ J(xb(·, y), u0(·, y))≥ε.
This shows thatJ attains its minimum at(bx(·, y), u0(·, y))∈
Z×Uad and the proof is complete.
V. APPLICATION
Consider the following controlled fractional stochastic Schr¨odinger equation of the form
i(J1−α
t z)(t, y) +△(t, y)−kV(y)z(t, y)
=
Z
Π ˜
k(y, τ)u(t, τ)dτ+ e −γ1t+t
et+e−tsinx(t, y)
+e −γ2t+t
et+e−tcosx(t, y) w(t)
dt , (17)
y∈Π, t∈(0, b], u∈Uab,
x(0, y) =x0(y), y∈Π, (18)
where 12 < α < 1, J1−α is the (1−α)-order
Riemann-Liouville fractional integral operator, Π⊆R2 is a bounded
domain with a smooth boundary∂Π,∆denotes the Laplace operator in R2,and ˜k∈C(Π×Π, R), γ
1 andγ2 are
posi-tive real constant. w(t)denotes a one-dimensional standard Wiener process inHdefined on a stochastic space(Ω,F, P).
Let X =Y = (L2(Π),k · k). The operatorA:D(A)→
H defined by D(A) = H2(Π)∩H1
0(Π) such that Ax =
i∆x. It is obvious that A is the infinitesimal generator of a compact, analytic semigroup T(·) of uniformly bounded linear operator. Ford >0,we define the admissible control setUad={u(·, y)|J →Y measurable,Ft-adapted stochastic
processes, andkukLpF(J,Y)≤d}.
Definex(t)y=x(t, y),and
g(t, x(t, y)) = e −γ1t+t
et+e−tsinx(t, y),
f(t, x(t, y)) = e −γ2t+t
et+e−tcosx(t, y),
and
(Bu)(t, y) =
Z
Π ˜
k(y, τ)u(t, τ)dτ.
Thus system (17)-(18) can be transformed into (13)-(14) with the cost function
J(x, u)
=E
Z b
0
Z
Π
|x(t, y)|2dy+E
Z b
0
Z
Π
|u(t, y)|2dy.
Obviously,E kg(t, x)kp=e−pγ1t, Ekf(t, x)kp=e−pγ2t.
Foru∈L2([0, b]×Π),we have
Z
Π
Z b
0
|Bu(t, y)|2dtdy
≤
Z
Π
Z b
0
· Z
Π
Z b
0
|k˜(y, τ)u(s, τ)|2dsdτ
¸
dtdy
≤M˜k2b
2·(
mes(Π))2
Z
Π
Z b
0
|u(s, τ)|2dsdτ <∞,
whereM˜k = max(y,τ)∈Π×Π|k˜(y, τ)|. This implies that the
operatorB:L2([0, b]×Π)→L2([0, b]×Π), and
kBukL2(Π×[0,b])
≤M˜kb·mes(Π)kukL2([0,b]×Π).
Then we can conclude that B is a bounded operator in
L2([0, b]×Π). Further, we can impose some suitable
con-ditions on the above-defined functions to verify the assump-tions on Theorem 4. Therefore, the problem (17)-(18) has at least one optimal pair.
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