# Optimal control of quantum system in fermion fields: Pontryagin-type maximum
principle (I)

Penghui Wang, Shan Wang
School of Mathematics, Shandong University, Jinan, 250100, China E-mail
addresses: phwang@sdu.edu.cn(P.Wang), 202020244@mail.sdu.edu.cn(S.Wang).

Abstract  In this paper, the Pontryagin-type maximum principle for optimal
control of quantum stochastic systems in fermion fields is obtained. These
systems have gained significant prominence in numerous quantum applications
ranging from physical chemistry to multi-dimensional nuclear magnetic
resonance experiments. Furthermore, we establish the existence and uniqueness
of solutions to backward quantum stochastic differential equations driven by
fermion Brownian motion. The application of noncommutative martingale
inequalities and the martingale representation theorem enables this
achievement.

2020 AMS Subject Classification: 47C15, 49K27, 81S25, 81Q93, 81V74.

Keywords. Infinite-dimensional quantum control system; Pontryagin-type maximum
principle; Backward quantum stochastic differential equations; Noncommutative
martingale representation theorem.

##  1 Introduction

Let $\mathscr{H}$ be a separable complex Hilbert space. The anti-symmetric
Fock space over $\mathscr{H}$ [8, 36] is defined by

$\Lambda(\mathscr{H}):=\bigoplus_{n=0}^{\infty}\Lambda_{n}(\mathscr{H}),$

where $\Lambda_{n}(\mathscr{H})$ is the Hilbert space anti-symmetric $n$-fold
tensor product of $\mathscr{H}$ with itself, and
$\Lambda_{0}(\mathscr{H}):=\mathbb{C}$. For any $z\in\mathscr{H}$, the
creation operator
$C(z):\Lambda_{n}(\mathscr{H})\rightarrow\Lambda_{n+1}(\mathscr{H})$ defined
by $v\mapsto\sqrt{n+1}\ z\wedge v$, is a bounded operator on
$\Lambda(\mathscr{H})$ with $\|C(z)\|=\|z\|$. The annihilation operator $A(z)$
is the adjoint of $C(z)$, i.e. $A(z)=C(z)^{*}$. The fermion field $\Psi(z)$ is
defined on $\Lambda(\mathscr{H})$ by

$\Psi(z):=C(z)+A(Jz),$ (1.1)

where the map $J:\mathscr{H}\rightarrow\mathscr{H}$ is a conjugation operator
(i.e., $J$ is antilinear, antiunitary, and $J^{2}=1$). The canonical anti-
commutation relation holds:

$\\{\Psi(z),\Psi(z^{\prime})\\}\equiv\Psi(z)\Psi(z^{\prime})+\Psi(z^{\prime})\Psi(z)=2\langle
Jz^{\prime},z\rangle I,\quad z,z^{\prime}\in\mathscr{H}.$ (1.2)

Denote by $\mathscr{C}$ the von Neumann algebra generated by the bounded
operators $\\{\Psi(z):z\in\mathscr{H}\\}$. For the Fock vacuum
$\Omega\in\Lambda(\mathscr{H})$, define

$m(\cdot):=\langle\Omega,\cdot\Omega\rangle_{\Lambda(\mathscr{H})}$

on $\mathscr{C}$. By [8, 14, 27, 36], $m(\cdot)$ is a faithful, normal,
central state on $\mathscr{C}$, which is also called the quantum expectation
with respect to the Fock vacuum and is denoted by $\mathbb{E}(\cdot)$. The
space $(\mathscr{C},m)$ is a quantum (noncommutative) probability space. For
$p\in[1,\infty)$, we define the noncommutative $L^{p}$-norm on $\mathscr{C}$
by

$\|f\|_{p}:=m\left(|f|^{p}\right)^{\frac{1}{p}}=\left\langle\Omega,|f|^{p}\Omega\right\rangle_{\Lambda(\mathscr{H})}^{\frac{1}{p}},$

where $|f|=(f^{*}f)^{\frac{1}{2}}$. The space $L^{p}(\mathscr{C},m)$ is the
completion of $(\mathscr{C},\|\cdot\|_{p})$, which is the noncommutative
$L^{p}$-space, abbreviated as $L^{p}(\mathscr{C})$.

Quantum (Noncommutative) probability theory has attracted significant
attention since it was recognized as a new branch of mathematics. In
particular, quantum stochastic calculus has also received attention with
various degrees of completeness in the setting of the bosonic and fermionic
Fock space and for Clifford algebras in the works of Segal[39], Barnett,
Streater and Wilde [8, 9, 10, 40], Applebaum and Hudson [1, 2, 3, 23],
Parthasarathy [36, 37, 24], Belavkin [4], Gordina [18], Sinha and Goswami
[38]. Among them, Haudson, Lindsay, Barnett, Streater and Wilde investigated
the solutions to quantum stochastic differential equations (QSDEs for short).
Belavkin [4] employed quantum stochastic methods that were developed in the
1980s to describe quantum noise and a quantum generalization of the Itô
calculus. Later, Gough, Guta, James and Nurdin [22] developed fermion
filtering theory using the fermion quantum stochastic calculus.

In what follows, let $\mathscr{H}=L^{2}(\mathbb{R}^{+})$, and $Jf=\bar{f}$ for
$f\in L^{2}(\mathbb{R}^{+})$. Let $\mathscr{C}$ be von Neumann algebra
generated by $\\{\Psi(v):v\in L^{2}(\mathbb{R}^{+})\\}$, and
$\\{\mathscr{C}_{t}\\}_{t\geq 0}$ be an increasing family of von Neumann
subalgebras of $\mathscr{C}$ generated by $\\{\Psi(v):v\in
L^{2}(\mathbb{R}^{+})\ \textrm{and}\ \textrm{ess supp}\ v\subset[0,t]\\}$. The
family of von Neumann subalgebras $\\{\mathscr{C}_{t}\\}_{t\geq 0}$ is called
the filtration of $\mathscr{C}$ [35]. The fermion Brownian motion $W(t)$ is
defined by

$W(t):=\Psi(\chi_{[0,t]})=C(\chi_{[0,t]})+A(J\chi_{[0,t]}),\quad t\geq 0.$

Obviously, $W(t)$ is self-adjoint. In this paper, we consider the controlled
QSDE driven by the fermion Brownian motion $W(t)$ in $L^{p}(\mathscr{C})$:

$\left\\{\begin{aligned}
dx(t)=&D(t,x(t),u(t))dt+F(t,x(t),u(t))dW(t)+dW(t)G(t,x(t),u(t)),\ \textrm{in}\
(0,T],\\\ x(0)=&x_{0},\end{aligned}\right.$ (1.3)

where $x_{0}\in L^{p}(\mathscr{C}_{t_{0}})$ is the initial condition,
$u(\cdot)\in\mathcal{U}(0,T)$ is the control variable. Here, the control
domain $\mathcal{U}(0,T)$ is defined by

$\mathcal{U}(0,T):=\left\\{u:[0,T]\to U;\ u(\cdot)\ \textrm{is}\
\\{\mathscr{C}_{t}\\}_{t\geq 0}\textrm{-adapted, continuous}\right\\},$ (1.4)

where $U$ is a linear subspace of $L^{p}(\mathscr{C})$. The control domain $U$
is also a metric space with the metric $d(u_{1},u_{2})=\|u_{1}-u_{2}\|_{p}$.
As in [32, (S1), Page 388], the maps
$D(\cdot,\cdot,\cdot),F(\cdot,\cdot,\cdot),G(\cdot,\cdot,\cdot):[0,T]\times
L^{p}(\mathscr{C})\times U\to L^{p}(\mathscr{C})$ are adapted, where the
definition of the adapted maps is provided in Definition 2.1. In (1.3),
$u(\cdot)$ is called the control, while $x(\cdot)=x(\cdot;x_{0},u(\cdot))$ is
the corresponding state process. It should be noted that in the framework of
quantum stochastic calculus, based on the noncommutativity of operators, both
the right integral $\int_{0}^{t}f(s)dW(s)$ and the left integral
$\int_{0}^{t}dW(s)f(s)$ exist.

Let $x(\cdot)$ be the solution to (1.3) corresponding to the control
$u(\cdot)$. Define the cost functional $\mathcal{J}(\cdot)$ as follows:

$\mathcal{J}(u(\cdot)):=\int_{0}^{T}L(t,x(t),u(t))dt+h(x(T)),\quad
u(\cdot)\in\mathcal{U}(0,T),$ (1.5)

where the maps $L(\cdot,\cdot,\cdot):[0,T]\times L^{p}(\mathscr{C})\times
U\to\mathbb{R}$ and $h(\cdot):L^{p}(\mathscr{C}_{T})\to\mathbb{R}$. In
particular, the maps $\widehat{L}(\cdot,\cdot,\cdot):[0,T]\times
L^{p}(\mathscr{C})\times U\to L^{p}(\mathscr{C})_{sa}$ and
$\widehat{h}(\cdot):L^{p}(\mathscr{C}_{T})\to L^{p}(\mathscr{C}_{T})_{sa}$,
and elements of $L^{p}(\mathscr{C})_{sa}$ are observables. And for any
$u(\cdot)\in\mathcal{U}(0,T)$, the cost functional

$\mathcal{J}(u(\cdot))=\mathbb{E}\left(\int_{0}^{T}\widehat{L}(t,x(t),u(t))dt+\widehat{h}(x(T))\right)$

was considered by[26, 27, 41].

Similar to [21, 27, 33], we consider quantum optimal control problem for
(1.3):
Problem(QOC). Find a control $\bar{u}(\cdot)\in\mathcal{U}(0,T)$ such that

$\mathcal{J}(\bar{u}(\cdot))=\inf_{u(\cdot)\in\mathcal{U}(0,T)}\mathcal{J}(u(\cdot)).$
(1.6)

Any $\bar{u}(\cdot)\in\mathcal{U}(0,T)$ satisfying (1.6) is called an optimal
control. The corresponding $\bar{x}(\cdot)$ and
$(\bar{x}(\cdot),\bar{u}(\cdot))$ are called an optimal state process and
optimal pair of quantum control systems, respectively.

Optimal control theory is a powerful mathematical tool, which has been rapidly
developed since the 1950s, mainly for engineering applications. Recently, this
method has become widely used to improve process performance in quantum
technologies by means of the highly efficient control of quantum dynamics [44,
15]. Pontryagin’s maximum principle and Bellman’s dynamic programming
principle are two of the most important tools for solving optimal control
problems. Belavkin, Smolyanov, James et al. [14, 7, 5, 11, 15, 21, 19, 20, 26,
33, 22, 41] tackled the optimal control problem by quantum stochastic calculus
and dynamic programming methods in the case that the control domain $U$ is of
finite dimension. More precisely, Belavkin, Gough and Smolyanov [19, 20]
investigated the quantum optimal control problem, which is the evolution of a
quantum system subject to continuous measurements governed by the QSDE

$dX_{t}=\omega(t,u_{t},X_{t})dt+\sum_{\alpha=1}^{m}\sigma_{\alpha}(t,u_{t},X_{t})dM^{\alpha}_{t},$
(1.7)

where $\\{X_{t}\\}$ denotes a stochastic process,
$\\{M^{\alpha};\alpha=1,\cdots,m\\}$ denotes a sequence of martingales. In
this case, the optimal cost is defined by

$\mathcal{J}[\\{u\\};t_{0},X_{0}]=\int_{t_{0}}^{T}\textit{l}(s,u_{s},X_{s})ds+g(X_{T}),$

where $\\{X_{s}:s\in[t_{0},T]\\}$ is the solution to (1.7) with the initial
condition $X_{0}$, $u$ denotes a continuous control function, taking values in
$\mathbb{R}^{n}$. James [26], Sharifi and Momeni [41] investigated the optimal
control problems of the following quantum systems

$dX(t)=\omega(t,X(t),u(t))dt+\sigma(t,X(t))dw(t),$

where the maps $\omega$ and $\sigma$ are Lipschitz continuous with respect to
$X(t)$, $w(t)$ is a martingale. They derived Bellman equation and identified
the optimal control $u$ with the cost functional
$\mathcal{J}[\\{u\\};t_{0},X_{0}]$, such that

$S(t_{0},X_{0})=\inf_{u}{\mathcal{J}}[\\{u\\};t_{0},X_{0}]=\inf_{u\in\mathcal{U}(t)}\mathbb{E}\left\\{\int_{t_{0}}^{T}C(u(t),X(t))dt+G(u(T),X(T))\right\\},$

where $S(t_{0},X_{0})$ is the value function, the cost density $C(u(t),X(t))$
and the terminal cost $G(u(T),X(T))$ are observables, and the control domain
$\mathcal{U}(t)$ is a subspace of $\mathbb{R}^{n}$. The adaptedness of the
control variable $u$ is not considered. And then, Mulero-Martínez and Molina-
Vilaplana [33] derived the quantum Pontryagin maximum principle in a global
form from the Hamilton-Jacobi-Bellman equation for quantum optimal control
with adapted control variable $u$. Boscain, Sigalotti and Sugny [6] described
modern aspects of optimal control theory, with a particular focus on the
Pontryagin maximum principle, where the finite-dimensional quantum control
system is considered. The aforementioned results are about quantum optimal
control problems with finite-dimensional control domains. However, quantum
optimal control problems with infinite-dimensional control domain have rarely
been investigated.

In classical probability theory, there is a great deal of research on optimal
control theory. Since the 1970s, the maximum principle has been extensively
studied for stochastic control systems. Peng, Yong, Zhang et al. obtained
corresponding results on the Pontryagin-type maximum principle in [16, 25, 29,
30, 31, 32, 13, 17, 47]. Peng [34] investigated optimal control problems for
control systems when the control enters the diffusion term and the control
domain $U$ is nonconvex. Lü and Zhang [29, 30, 31] derived the necessary
conditions for optimal controls under the case of convex and nonconvex control
domains, respectively. Furthermore, Du and Meng [13], Fuhrman, Hu and
Tessitore [17], Lü and Zhang [30] were concerned with the general stochastic
maximum principle of infinite dimensions.

Inspired by the classical infinite-dimensional optimal control theory, this
paper investigates the optimal control problem of infinite-dimensional quantum
systems with the infinite-dimensional control domains. To obtain the
Pontryagin-type maximum principle, the following main ideas play key roles.

  * •

For $p\in[1,\infty)$, the noncommutative space $L^{p}(\mathscr{C})$ can be
decomposed into

$L^{p}(\mathscr{C})=L^{p}(\mathscr{C}_{e})\oplus L^{p}(\mathscr{C}_{o}),$

and the details are listed in [35, Proposition 3.3]. The fermion Brownian
motion $\\{W(t);t\geq 0\\}$ commutes with the elements of
$L^{p}(\mathscr{C}_{e})$, and anti-commutes with the elements of
$L^{p}(\mathscr{C}_{o})$. This allows us to overcome the difficulty of the
noncommutativity of QSDEs in fermion fields.

  * •

To address quantum stochastic calculus with respect to the fermion Brownian
motion $\\{W(t);t\geq 0\\}$, the canonical anti-commutation relation (1.2) is
critical, which implies that

$W(t)^{*}W(t)=W(t)^{2}=t\textrm{I},\quad t\in[0,\infty).$

  * •

The Burkholder-Gundy inequality with respect to noncommutative martingales,
given by Pisier and Xu [35], plays a significant role in solving the QSDEs in
fermion fields and obtaining the corresponding estimates. Based on this, we
prove that the solution to the corresponding backward quantum stochastic
differential equation (BQSDE for short) is a necessary condition of Problem
(QOC) with an infinite-dimensional control domain $U$. This is the Pontryagin-
type maximum principle of quantum control systems.

  * •

Finally, we investigate the solution to BQSDEs by using the noncommutative
martingales representation theorem and noncommutative martingale inequalities.

The paper is organized as follows. In section 2, we review some basic
notations on fermion fields and main lemmas. Section 3 formulates the
Pontryagin-type maximum principle for quantum optimal control problems in
$L^{p}(\mathscr{C})$. In section 4, we prove the existence and uniqueness of
the solution to BQSDEs in noncommutative space.

##  2 Preliminaries

In this section, we introduce notations based on references [8, 9, 10, 35, 12,
42, 43] and some lemmata, which will be used later.

######  Definition 2.1.

[8, 9, 35] A map $x:\mathbb{R}^{+}\rightarrow L^{p}(\mathscr{C})$ is said to
be adapted if $x(t)\in L^{p}(\mathscr{C}_{t})$ for each $t\in\mathbb{R}^{+}$.
A map $F:\mathbb{R}^{+}\times L^{p}(\mathscr{C})\rightarrow
L^{p}(\mathscr{C})$ is said to be adapted if $F(t,u)\in
L^{p}(\mathscr{C}_{t})$ for any $t\in\mathbb{R}^{+}$ and $u\in
L^{p}(\mathscr{C}_{t})$.

Throughout this paper, we let $T>0$ be a fixed time horizon and denote

$\displaystyle C_{\mathbb{A}}(0,T;L^{p}(\mathscr{C})):=\left\\{f:[0,T]\to
L^{p}(\mathscr{C})\ |\ f(\cdot)\ {\rm is}\ \\{\mathscr{C}_{t}\\}_{t\geq
0}{\rm-adapted\ and\ continuous}\right\\},$

with the following norm

$\|f(\cdot)\|_{C_{\mathbb{A}}(0,T;L^{p}(\mathscr{C}))}:=\sup_{t\in[0,T]}\|f(t)\|_{p}.$

It is clear that $C_{\mathbb{A}}(0,T;L^{p}(\mathscr{C}))$ is a Banach space.
As usual, for any Banach space $X_{1}$ and $X_{2}$, $\mathcal{L}(X_{1};X_{2})$
is the Banach space of all bounded linear operators from $X_{1}$ to $X_{2}$.
For $p_{1},p_{2}\in(1,\infty)$, put

$\displaystyle
L^{\infty}_{\mathbb{A}}(0,T;\mathcal{L}(L^{p_{1}}(\mathscr{C});L^{p_{2}}(\mathscr{C})))$
$\displaystyle:=\Big{\\{}T:[0,T]\to\mathcal{L}(L^{p_{1}}(\mathscr{C});L^{p_{2}}(\mathscr{C}));T(\cdot)\
\textrm{is measurable and essentially bounded},$
$\displaystyle\indent\indent\indent\indent\indent\indent\indent\textrm{and}\
T(t)\in\mathcal{L}(L^{p_{1}}(\mathscr{C}_{t});L^{p_{2}}(\mathscr{C}_{t}))\
\textrm{a.e. on }[0,T]\Big{\\}}.$

Recall that in [35] the grading automorphism $\Upsilon$ on
$L^{p}(\mathscr{C})$ is uniquely determined by

$\Upsilon(\Psi(v_{1})\Psi(v_{2})\cdots\Psi(v_{n}))=(-1)^{n}\Psi(v_{1})\Psi(v_{2})\cdots\Psi(v_{n}),\quad
v_{i}\in\mathscr{H},\ 1\leq i\leq n.$

######  Definition 2.2.

[35, 8, 9] An element $f\in L^{p}(\mathscr{C})$ is said to be even (resp. odd)
if $\Upsilon(f)=f$ (resp. $\Upsilon(f)=-f$).

######  Lemma 2.1.

[46, 35] Let $p\in[1,\infty)$. For $f\in\mathcal{H}^{p}(0,T)$, its Itô-
Clifford integral $\int_{0}^{t}f(s)dW(s)$ and $\int_{0}^{t}dW(s)f(s)$ are
$L^{p}$-martingales for $t\in[0,T]$. Moreover, it holds that

$\left\|\int_{0}^{t}dW(s)f(s)\right\|_{p}\simeq_{p}\left\|\int_{0}^{t}f(s)dW(s)\right\|_{p}\simeq_{p}\|f\|_{\mathcal{H}^{p}(0,t)},\
t\in[0,T].$ (2.1)

If $p\in(1,2]$, then

$\left(\int_{0}^{t}\|f(s)\|_{p}^{2}ds\right)^{\frac{1}{2}}\leq\mathcal{C}_{p}\left\|\int_{0}^{t}f(s)dW(s)\right\|_{p}.$
(2.2)

If $p\in[2,\infty)$, then

$\left\|\int_{0}^{t}f(s)dW(s)\right\|_{p}\leq\mathcal{C}_{p}\left(\int_{0}^{t}\|f(s)\|_{p}^{2}ds\right)^{\frac{1}{2}}.$
(2.3)

##  3 The Pontryagin-type Maximum Principle

This section is devoted to obtaining the Pontryagin-type maximum principle for
quantum optimal control of (1.3) in $L^{p}(\mathscr{C})$ for $p\in[2,\infty)$.
For the state equation (1.3) and the cost functional (1.5), we impose the
following assumptions.

######  Assumption 3.1.

(A1)

The maps
$D(\cdot,\cdot,\cdot),F(\cdot,\cdot,\cdot),G(\cdot,\cdot,\cdot):[0,T]\times
L^{p}(\mathscr{C})\times U\to L^{p}(\mathscr{C})$ are adapted, and there
exists a constant $\mathcal{C}>0$ such that for any $(t,u)\in[0,T]\times U$,
$x,\hat{x}\in L^{p}(\mathscr{C})$,

$\left\\{\begin{aligned}
&\|D(t,x,u)-D(t,\hat{x},u)\|_{p}\leq\mathcal{C}\|x-\hat{x}\|_{p},\\\
&\|F(t,x,u)-F(t,\hat{x},u)\|_{p}\leq\mathcal{C}\|x-\hat{x}\|_{p},\\\
&\|G(t,x,u)-G(t,\hat{x},u)\|_{p}\leq\mathcal{C}\|x-\hat{x}\|_{p},\\\
&\|D(t,0,u)\|_{p}+\|F(t,0,u)\|_{p}+\|G(t,0,u)\|_{p}\leq\mathcal{C}.\end{aligned}\right.$

(A2)

The maps $L:[0,T]\times L^{p}(\mathscr{C})\times U\to\mathbb{R}$ and
$h:L^{p}(\mathscr{C}_{T})\to\mathbb{R}$ are measurable, and there exists a
constant $\mathcal{C}>0$ such that for any $(t,u)\in[0,T]\times U$,
$x,\hat{x}\in L^{p}(\mathscr{C})$,

$\left\\{\begin{aligned}
&|L(t,x,u)-L(t,\hat{x},u)|\leq\mathcal{C}\|x-\hat{x}\|_{p},\\\
&|h(x)-h(\hat{x})|\leq\mathcal{C}\|x-\hat{x}\|_{p},\\\
&|L(t,0,u)|+|h(0)|\leq\mathcal{C}.\end{aligned}\right.$

(A3)

The maps $D,\ F,\ G,\ L$ and $h$ are second order Fréchet differentiable on
$L^{p}(\mathscr{C})$. For any $(t,u)\in[0,T]\times U$, the maps
$D_{x}(t,\cdot,u),\ F_{x}(t,\cdot,u),\
G_{x}(t,\cdot,u):L^{p}(\mathscr{C})\to\mathcal{L}(L^{p}(\mathscr{C}))$ and
$L_{x}(t,\cdot,u),\ h_{x}(\cdot):L^{p}(\mathscr{C})\to
L^{p^{\prime}}(\mathscr{C})$ are continuous, the maps $D_{xx}(t,\cdot,u),\
F_{xx}(t,\cdot,u)$,
$G_{xx}(t,\cdot,u):L^{p}(\mathscr{C})\to\mathcal{L}(L^{p}(\mathscr{C}),L^{p}(\mathscr{C});L^{p}(\mathscr{C}))$
and $L_{xx}(t,\cdot,u),\
h_{xx}(\cdot):L^{p}(\mathscr{C})\to\mathcal{L}(L^{p}(\mathscr{C});L^{p^{\prime}}(\mathscr{C}))$
are continuous. Moreover, for any $(t,x,u)\in[0,T]\times
L^{p}(\mathscr{C})\times U$,

$\left\\{\begin{aligned}
&\|D_{x}(t,x,u)\|_{\mathcal{L}(L^{p}(\mathscr{C}))}+\|F_{x}(t,x,u)\|_{\mathcal{L}(L^{p}(\mathscr{C}))}+\|G_{x}(t,x,u)\|_{\mathcal{L}(L^{p}(\mathscr{C}))}\leq\mathcal{C},\\\
&\|D_{xx}(t,x,u)\|_{\mathcal{L}(L^{p}(\mathscr{C}),L^{p}(\mathscr{C});L^{p^{\prime}}(\mathscr{C}))}+\|F_{xx}(t,x,u)\|_{\mathcal{L}(L^{p}(\mathscr{C}),L^{p}(\mathscr{C});L^{p^{\prime}}(\mathscr{C}))}\\\
&\indent\indent+\|G_{xx}(t,x,u)\|_{\mathcal{L}(L^{p}(\mathscr{C}),L^{p}(\mathscr{C});L^{p^{\prime}}(\mathscr{C}))}\leq\mathcal{C},\\\
&\|L_{x}(t,x,u)\|_{p^{\prime}}+\|h_{x}(x)\|_{p^{\prime}}\leq\mathcal{C},\\\
&\|L_{xx}(t,x,u)\|_{\mathcal{L}(L^{p}(\mathscr{C});L^{p^{\prime}}(\mathscr{C}))}+\|h_{xx}(x)\|_{\mathcal{L}(L^{p}(\mathscr{C});L^{p^{\prime}}(\mathscr{C}))}\leq\mathcal{C}.\end{aligned}\right.$

In what follows, when there is no confusion, denote $p^{\prime}$ the conjugate
number of $p$, i.e. $\frac{1}{p}+\frac{1}{p^{\prime}}=1$.

Before stating the main result, we introduce the following auxiliary result.

######  Lemma 3.1.

Under the above conditions, there exists a unique solution $x(\cdot)\in
C_{\mathbb{A}}(0,T;L^{p}(\mathscr{C}))$ to (1.3) for $p\in[2,\infty)$.
Moreover,

$\sup_{t\in[0,T]}\|x(t)\|_{p}^{2}\leq\mathcal{C}\left(1+\|x_{0}\|_{p}^{2}\right).$
(3.1)

###### Proof.

By [28, Theorem 3.1], there exists a unique solution $x(\cdot)\in
C_{\mathbb{A}}(0,T;L^{p}(\mathscr{C}))$ to (1.3), and $\\{x(t)\\}_{t\geq 0}$
satisfies the integral equation

$\displaystyle
x(t)=x_{0}+\int_{0}^{t}D(\tau,x(\tau),u(\tau))d\tau+\int_{0}^{t}F(\tau,x(\tau),u(\tau))dW(\tau)+\int_{0}^{t}dW(\tau)G(\tau,x(\tau),u(\tau)).$

By Assumption 3.1 (A1), we have

$\displaystyle\|D(t,x(t),u(t))\|_{p}\leq$
$\displaystyle\|D(t,x(t),u(t))-D(t,0,u(t))\|_{p}+\|D(t,0,u(t))\|_{p}$ (3.2)
$\displaystyle\leq$ $\displaystyle\mathcal{C}(\|x(t)\|_{p}+1),\ {\rm a.e.}\
t\in[0,T].$

Similarly, for $t\in[0,T]$,

$\|F(t,x(t),u(t))\|_{p}\leq\mathcal{C}(\|x(t)\|_{p}+1)\quad{\rm
and}\quad\|G(t,x(t),u(t))\|_{p}\leq\mathcal{C}(\|x(t)\|_{p}+1).$ (3.3)

Therefore, by the Minkowski inequality, the Hölder inequality, (2.3), (3.2)
and (3.3),

$\displaystyle\|x(t)\|_{p}^{2}\leq$ $\displaystyle
4\|x_{0}\|_{p}^{2}+4T\int_{0}^{t}\|D(\tau,x(\tau),u(\tau))\|_{p}^{2}d\tau$
$\displaystyle+4\mathcal{C}_{p}\int_{0}^{t}\|F(\tau,x(\tau),u(\tau))\|_{p}^{2}d\tau+4\mathcal{C}_{p}\int_{0}^{t}\|G(\tau,x(\tau),u(\tau))\|_{p}^{2}d\tau$
$\displaystyle\leq$ $\displaystyle
4\|x_{0}\|_{p}^{2}+4\mathcal{C}_{p,D,F,G,T}\int_{0}^{t}(1+\|x(t)\|_{p}^{2})d\tau.$

By the Gronwall inequality and the Lebesgue Dominated Convergence Theorem, we
conclude that

$\sup_{t\in[0,T]}\|x(t)\|_{p}^{2}\leq\mathcal{C}(\|x_{0}\|_{p}^{2}+1).$

∎

Let $(\bar{x}(\cdot),\bar{u}(\cdot))$ be the given optimal pair of Problem
(QOC) of (1.3) in $L^{p}(\mathscr{C})$ with the cost functional

$\mathcal{J}(\bar{u}(\cdot))=\int_{0}^{T}L(t,\bar{x}(t),\bar{u}(t))dt+h(\bar{x}(T)).$
(3.4)

Then it satisfies the following quantum control system

$\left\\{\begin{aligned}
d\bar{x}(t)=&D(t,\bar{x}(t),\bar{u}(t))dt+F(t,\bar{x}(t),\bar{u}(t))dW(t)+dW(t)G(t,\bar{x}(t),\bar{u}(t)),\
\rm{in}\ (0,T],\\\ \bar{x}(0)=&x_{0}.\end{aligned}\right.$ (3.5)

Fix any $u(\cdot)\in\mathcal{U}(0,T)$, $\varepsilon>0$, we define

$u^{\varepsilon}(t):=\left\\{\begin{aligned}
\bar{u}(t),\quad&t\in[0,T]\backslash E_{\varepsilon},\\\ u(t),\quad&t\in
E_{\varepsilon},\end{aligned}\right.$ (3.6)

where $E_{\varepsilon}\subseteq[0,T]$ is a measurable set with
$|E_{\varepsilon}|=\varepsilon$. Let $x^{\varepsilon}(\cdot)$ be the state
process of (1.3) corresponding to the control variable
$u^{\varepsilon}(\cdot)$, that is,

$\left\\{\begin{aligned}
dx^{\varepsilon}(t)=&D(t,x^{\varepsilon}(t),u^{\varepsilon}(t))dt+F(t,x^{\varepsilon}(t),u^{\varepsilon}(t))dW(t)+dW(t)G(t,x^{\varepsilon}(t),u^{\varepsilon}(t)),\quad{\rm
in}\ (0,T],\\\ x^{\varepsilon}(0)=&x_{0}.\end{aligned}\right.$ (3.7)

For the sake of convenience, we denote for $\varphi=D,F,G,L$,

$\left\\{\begin{aligned}
&\varphi_{x}(t):=\varphi_{x}(t,\bar{x}(t),\bar{u}(t)),\\\
&\varphi_{xx}(t):=\varphi_{xx}(t,\bar{x}(t),\bar{u}(t)),\\\
&\delta\varphi(t):=\varphi(t,\bar{x}(t),u(t))-\varphi(t,\bar{x}(t),\bar{u}(t)),\\\
&\delta\varphi_{x}(t):=\varphi_{x}(t,\bar{x}(t),u(t))-\varphi_{x}(t,\bar{x}(t),\bar{u}(t)),\\\
&\delta\varphi_{xx}(t):=\varphi_{xx}(t,\bar{x}(t),u(t))-\varphi_{xx}(t,\bar{x}(t),\bar{u}(t)),\\\
&\widetilde{\varphi}_{x}(t):=\int_{0}^{1}\varphi_{x}(t,\bar{x}(t)+\theta(x^{\varepsilon}(t)-\bar{x}(t)),u^{\varepsilon}(t))d\theta,\\\
&\widetilde{\varphi}_{xx}(t):=2\int_{0}^{1}(1-\theta)\varphi_{xx}(t,\bar{x}(t)+\theta(x^{\varepsilon}(t)-\bar{x}(t)),u^{\varepsilon}(t))d\theta.\end{aligned}\right.$
(3.8)

Let $y^{\varepsilon}(\cdot)$ and $z^{\varepsilon}(\cdot)$ be respectively the
solution to the following QSDEs:

$\left\\{\begin{aligned}
dy^{\varepsilon}(t)=&D_{x}(t)y^{\varepsilon}(t)dt+\left\\{F_{x}(t)y^{\varepsilon}(t)+\delta
F(t)\chi_{E_{\varepsilon}}(t)\right\\}dW(t)\\\
&\quad+dW(t)\left\\{G_{x}(t)y^{\varepsilon}(t)+\delta
G(t)\chi_{E_{\varepsilon}}(t)\right\\},\quad\rm{in}\ (0,T],\\\
y^{\varepsilon}(0)=&0,\end{aligned}\right.$ (3.9)

and

$\left\\{\begin{aligned}
dz^{\varepsilon}(t)=&\left\\{F_{x}(t)z^{\varepsilon}(t)+\delta
F_{x}(t)y^{\varepsilon}(t)\chi_{E_{\varepsilon}}(t)+\frac{1}{2}F_{xx}(t)\left(y^{\varepsilon}(t),y^{\varepsilon}(t)\right)\right\\}dW(t)\\\
&+dW(t)\left\\{G_{x}(t)z^{\varepsilon}(t)+\delta
G_{x}(t)y^{\varepsilon}(t)\chi_{E_{\varepsilon}}(t)+\frac{1}{2}G_{xx}(t)(y^{\varepsilon}(t),y^{\varepsilon}(t))\right\\}\\\
&+\left\\{D_{x}(t)z^{\varepsilon}(t)+\delta
D(t)\chi_{E_{\varepsilon}}(t)+\frac{1}{2}D_{xx}(t)(y^{\varepsilon}(t),y^{\varepsilon}(t))\right\\}dt\quad\rm{in}\
(0,T],\\\ z^{\varepsilon}(0)=&0,\end{aligned}\right.$ (3.10)

where the maps $D_{xx}(\cdot),\ F_{xx}(\cdot),\ G_{xx}(\cdot)\in
L_{\mathbb{A}}^{\infty}(0,T;\mathcal{L}(L^{p}(\mathscr{C}),L^{p}(\mathscr{C});L^{p}(\mathscr{C})))$
are introduced in Assumption 3.1. This means that, for any
$(t,x,u)\in[0,T]\times L^{p}(\mathscr{C})\times U$ and $x_{1},x_{2}\in
L^{p}(\mathscr{C})$,
$D_{xx}(t,x,u)(x_{1},x_{2}),F_{xx}(t,x,u)(x_{1},x_{2}),G_{xx}(t,x,u)(x_{1},x_{2})\in
L^{p}(\mathscr{C})$.

######  Theorem 3.2.

Suppose that Assumption 3.1 holds. Then, for $p\in[2,\infty)$,

$\displaystyle\sup_{t\in[0,T]}\|x^{\varepsilon}(t)-\bar{x}(t)\|_{p}^{2}=\textbf{o}(\varepsilon);$
(3.11)
$\displaystyle\sup_{t\in[0,T]}\|y^{\varepsilon}(t)\|_{p}^{2}=\textbf{o}(\varepsilon);$
(3.12)
$\displaystyle\sup_{t\in[0,T]}\|z^{\varepsilon}(t)\|_{p}^{2}=\textbf{o}(\varepsilon^{2});$
(3.13)
$\displaystyle\sup_{t\in[0,T]}\|x^{\varepsilon}(t)-\bar{x}(t)-y^{\varepsilon}(t)\|_{p}^{2}=\textbf{o}(\varepsilon^{2});$
(3.14)
$\displaystyle\sup_{t\in[0,T]}\|x^{\varepsilon}(t)-\bar{x}(t)-y^{\varepsilon}(t)-z^{\varepsilon}(t)\|_{p}^{2}=\textbf{o}(\varepsilon^{2}).$
(3.15)

Moreover, the following expansion holds for the cost functional:

$\displaystyle\mathcal{J}(u^{\varepsilon}(\cdot))=$
$\displaystyle\mathcal{J}(\bar{u}(\cdot))+{\rm{Re}}\left\langle
h_{x}(\bar{x}(T),\
y^{\varepsilon}(T)+z^{\varepsilon}(T)\right\rangle+\frac{1}{2}{\rm{Re}}\left\langle
h_{xx}(\bar{x}(T)y^{\varepsilon}(T),\ y^{\varepsilon}(T)\right\rangle$ (3.16)
$\displaystyle+{\rm{Re}}\int_{0}^{T}\Big{\\{}\left\langle
L_{x}(t),y^{\varepsilon}(t)+z^{\varepsilon}(t)\right\rangle+\frac{1}{2}\left\langle
L_{xx}(t)y^{\varepsilon}(t),y^{\varepsilon}(t)\right\rangle+\delta
L(t)\chi_{E_{\varepsilon}}(t)\Big{\\}}dt+\textbf{o}(\varepsilon).$

###### Proof.

Let $\xi^{\varepsilon}(t):=x^{\varepsilon}(t)-\bar{x}(t)$ for $t\in[0,T]$.
Thus, $\xi^{\varepsilon}(\cdot)$ satisfies the following QSDE:

$\left\\{\begin{aligned}
d\xi^{\varepsilon}(t)=&\left\\{\widetilde{D}_{x}(t)\xi^{\varepsilon}(t)+\delta
D(t)\chi_{E_{\varepsilon}}(t)\right\\}dt+\left\\{\widetilde{F}_{x}(t)\xi^{\varepsilon}(t)+\delta
F(t)\chi_{E_{\varepsilon}}(t)\right\\}dW(t)\\\
&+dW(t)\left\\{\widetilde{G}_{x}(t)\xi^{\varepsilon}(t)+\delta
G(t)\chi_{E_{\varepsilon}}(t)\right\\},\quad\textrm{in}\ (0,T],\\\
\xi^{\varepsilon}(0)=&0.\end{aligned}\right.$ (3.17)

From Assumption 3.1 (A3), Lemma 2.1 and the Hölder inequality, we have

$\displaystyle\|\xi^{\varepsilon}(t)\|_{p}^{2}\leq$
$\displaystyle\mathcal{C}\int_{0}^{t}\|\widetilde{F}_{x}(\tau)\xi^{\varepsilon}(\tau)+\delta
F(\tau)\chi_{E_{\varepsilon}}(\tau)\|_{p}^{2}+\|\widetilde{G}_{x}(\tau)\xi^{\varepsilon}(\tau)+\delta
G(\tau)\chi_{E_{\varepsilon}}(\tau)\|_{p}^{2}d\tau$ (3.18)
$\displaystyle+\mathcal{C}\left(\int_{0}^{t}\|\widetilde{D}_{x}(\tau)\xi^{\varepsilon}(\tau)\|_{p}+\|\delta
D(\tau)\chi_{E_{\varepsilon}}(\tau)\|_{p}(\tau)d\tau\right)^{2}$
$\displaystyle\leq$
$\displaystyle\mathcal{C}\int_{0}^{t}\|\xi^{\varepsilon}(\tau)\|_{p}^{2}d\tau+\mathcal{C}\left(\int_{0}^{t}\|\delta
D(\tau)\chi_{E_{\varepsilon}}(\tau)\|_{p}d\tau\right)^{2}$
$\displaystyle+\mathcal{C}\int_{0}^{t}\|\delta
F(\tau)\chi_{E_{\varepsilon}}(\tau)\|_{p}^{2}+\|\delta
G(\tau)\chi_{E_{\varepsilon}}(\tau)\|_{p}^{2}d\tau.$

Under Assumption 3.1 (A1), it follows from (3.1) that, for $t\in[0,T]$

$\displaystyle\|\delta D(t)\|_{p}\leq$
$\displaystyle\|D(t,\bar{x}(t),u(t))-D(t,0,u(t))\|_{p}+\|D(t,0,u(t))-D(t,0,\bar{u}(t))\|_{p}$
(3.19) $\displaystyle+\|D(t,0,\bar{u}(t))-D(t,\bar{x}(t),\bar{u}(t))\|_{p}$
$\displaystyle\leq$ $\displaystyle\mathcal{C}(\|\bar{x}(t)\|_{p}+1)$
$\displaystyle\leq$ $\displaystyle\mathcal{C}(\|x_{0}\|_{p}+1).$

Similarly, we have

$\|\delta F(t)\|_{p}\leq\mathcal{C}(\|x_{0}\|_{p}+1),\quad\|\delta
G(t)\|_{p}\leq\mathcal{C}(\|x_{0}\|_{p}+1).$ (3.20)

From (3.1) and (3.19), we obtain

$\displaystyle\int_{0}^{t}\|\delta
D(\tau)\chi_{E_{\varepsilon}}(\tau)\|_{p}d\tau\leq$
$\displaystyle\int_{0}^{t}\|\delta
D(\tau)\|_{p}\chi_{E_{\varepsilon}}(\tau)d\tau$ (3.21) $\displaystyle\leq$
$\displaystyle\mathcal{C}\int_{0}^{t}(\|x_{0}\|_{p}+1)\chi_{E_{\varepsilon}}(\tau)d\tau.$

Similar to the above reasoning, we have

$\displaystyle\int_{0}^{t}\|\delta
F(\tau)\chi_{E_{\varepsilon}}(\tau)\|_{p}^{2}+\|\delta
G(\tau)\chi_{E_{\varepsilon}}(\tau)\|_{p}^{2}d\tau$
$\displaystyle\leq\int_{0}^{t}\left\\{\|\delta F(\tau)\|_{p}^{2}+\|\delta
G(\tau)\|_{p}^{2}\right\\}\chi_{E_{\varepsilon}}(\tau)d\tau$ (3.22)
$\displaystyle\leq\mathcal{C}\int_{0}^{t}(\|x_{0}\|_{p}^{2}+1)\chi_{E_{\varepsilon}}(\tau)d\tau.$

By the Gronwall inequality, together with (3.18), (3.21), (3.22), we can infer
that

$\sup_{t\in[0,T]}\|\xi^{\varepsilon}(t)\|_{p}^{2}\leq\mathcal{C}(\|x_{0}\|_{p}^{2}+1)\varepsilon.$
(3.23)

This proves (3.11).

Now, we provide estimate on $y^{\varepsilon}$. From (3.9), we obtain that

$\displaystyle\|y^{\varepsilon}(t)\|_{p}\leq$
$\displaystyle\left\|\int_{0}^{t}D_{x}(\tau)y^{\varepsilon}(\tau)d\tau\right\|_{p}+\left\|\int_{0}^{t}F_{x}(\tau)y^{\varepsilon}(\tau)dW(\tau)\right\|_{p}+\left\|\int_{0}^{t}dW(\tau)G_{x}(\tau)y^{\varepsilon}(\tau)\right\|_{p}$
$\displaystyle+\left\|\int_{0}^{t}dW(\tau)G_{x}(\tau)y^{\varepsilon}(\tau)\right\|_{p}+\left\|\int_{0}^{t}dW(\tau)\delta
G(\tau)\chi_{E_{\varepsilon}}(\tau)\right\|_{p}$ $\displaystyle\leq$
$\displaystyle\mathcal{C}\left(\int_{0}^{t}\|y^{\varepsilon}(\tau)\|_{p}^{2}d\tau\right)^{\frac{1}{2}}+\mathcal{C}\left(\int_{0}^{t}\left(\|x_{0}\|^{2}_{p}+1\right)\chi_{E_{\varepsilon}}(\tau)d\tau\right)^{\frac{1}{2}},\quad
t\in[0,T].$

By means of the Gronwall inequality, we yield that

$\sup\limits_{t\in[0,T]}\|y^{\varepsilon}(t)\|_{p}^{2}\leq\mathcal{C}(\|x_{0}\|_{p}^{2}+1)\varepsilon.$
(3.24)

Hence, (3.12) holds.

Next, we prove (3.13). From (3.10), for $t\in[0,T]$,

$\displaystyle z^{\varepsilon}(t)=$
$\displaystyle\int_{0}^{t}\left\\{D_{x}(\tau)z^{\varepsilon}(\tau)+\delta
D(\tau)\chi_{E_{\varepsilon}}(\tau)+\frac{1}{2}D_{xx}(\tau)(y^{\varepsilon}(\tau),y^{\varepsilon}(\tau))\right\\}d\tau$
$\displaystyle+\int_{0}^{t}\left\\{F_{x}(\tau)z^{\varepsilon}(\tau)+\delta
F_{x}(\tau)y^{\varepsilon}(\tau)\chi_{E_{\varepsilon}}(\tau)+\frac{1}{2}F_{xx}(\tau)(y^{\varepsilon}(\tau),y^{\varepsilon}(\tau))\right\\}dW(\tau)$
$\displaystyle+\int_{0}^{t}dW(\tau)\left\\{G_{x}(\tau)z^{\varepsilon}(\tau)+\delta
G_{x}(\tau)y^{\varepsilon}(\tau)\chi_{E_{\varepsilon}}(\tau)+\frac{1}{2}G_{xx}(\tau)(y^{\varepsilon}(\tau),y^{\varepsilon}(\tau))\right\\}.$

Based on Assumption 3.1 (A3), we have

$\displaystyle\left\|\int_{0}^{t}D_{xx}(\tau)(y^{\varepsilon}(\tau),y^{\varepsilon}(\tau))d\tau\right\|_{p}$
$\displaystyle\leq\int_{0}^{t}\|D_{xx}(\tau)\|_{\mathcal{L}(L^{p}(\mathscr{C}),L^{p}(\mathscr{C});L^{p}(\mathscr{C}))}\|y^{\varepsilon}(\tau)\|_{p}^{2}d\tau$
(3.25)
$\displaystyle\leq\mathcal{C}\int_{0}^{t}\|y^{\varepsilon}(\tau)\|_{p}^{2}d\tau.$

By Lemma 2.1, similar to (3.25), we find that

$\displaystyle\left\|\int_{0}^{t}F_{xx}(\tau)(y^{\varepsilon}(\tau),y^{\varepsilon}(\tau))dW(\tau)\right\|_{p}\leq$
$\displaystyle\mathcal{C}_{p}\left(\int_{0}^{t}\|F_{xx}(\tau)(y^{\varepsilon}(\tau),y^{\varepsilon}(\tau))\|_{p}^{2}d\tau\right)^{\frac{1}{2}}$
(3.26) $\displaystyle\leq$
$\displaystyle\mathcal{C}_{p,F}\left(\int_{0}^{t}\|y^{\varepsilon}(\tau)\|_{p}^{4}d\tau\right)^{\frac{1}{2}},$

and

$\left\|\int_{0}^{t}G_{xx}(\tau)(y^{\varepsilon}(\tau),y^{\varepsilon}(\tau))dW(\tau)\right\|_{p}\leq\mathcal{C}_{p,G}\left(\int_{0}^{t}\|y^{\varepsilon}(\tau)\|_{p}^{4}d\tau\right)^{\frac{1}{2}}.$
(3.27)

Combined with Lemma 2.1 and (3.19), one has

$\displaystyle\left\|\int_{0}^{t}\delta
F_{x}(\tau)y^{\varepsilon}(\tau)\chi_{E_{\varepsilon}}(\tau)dW(\tau)\right\|_{p}^{2}\leq$
$\displaystyle\mathcal{C}_{p}\int_{0}^{t}\left\|\delta
F_{x}(\tau)y^{\varepsilon}(\tau)\right\|_{p}^{2}\chi_{E_{\varepsilon}}(\tau)d\tau$
(3.28) $\displaystyle\leq$ $\displaystyle\mathcal{C}_{p}\int_{0}^{t}\|\delta
F_{x}(\tau)\|^{2}_{\mathcal{L}(L^{p}(\mathscr{C}))}\|y^{\varepsilon}(\tau)\|_{p}^{2}\chi_{E_{\varepsilon}}(\tau)d\tau$
$\displaystyle\leq$
$\displaystyle\mathcal{C}\int_{0}^{t}(\|x_{0}\|_{p}^{2}+1)\|y^{\varepsilon}(\tau)\|_{p}^{2}\chi_{E_{\varepsilon}}(\tau)d\tau,$

and

$\displaystyle\left\|\int_{0}^{t}dW(\tau)\delta
G_{x}(\tau)y^{\varepsilon}(\tau)\chi_{E_{\varepsilon}}(\tau)\right\|_{p}^{2}$
$\displaystyle\leq\mathcal{C}_{p}\int_{0}^{t}\left\|\delta
G_{x}(\tau)y^{\varepsilon}(\tau)\right\|_{p}^{2}\chi_{E_{\varepsilon}}(\tau)d\tau$
(3.29)
$\displaystyle\leq\mathcal{C}\int_{0}^{t}(\|x_{0}\|_{p}^{2}+1)\|y^{\varepsilon}(\tau)\|_{p}^{2}\chi_{E_{\varepsilon}}(\tau)d\tau.$

Hence, it follows from Lemma 2.1, (3.19), (3.22), (3.24) and (3.25)-(3.29)
that

$\displaystyle\|z^{\varepsilon}(t)\|_{p}^{2}\leq$
$\displaystyle\mathcal{C}\left(\int_{0}^{t}\|z^{\varepsilon}(\tau)\|_{p}^{2}d\tau+\int_{0}^{t}\|y^{\varepsilon}(\tau)\|_{p}^{4}d\tau+\left(\int_{0}^{t}(\|x_{0}\|_{p}+1)\chi_{E_{\varepsilon}}(\tau)d\tau\right)^{2}\right.$
$\displaystyle\left.+\int_{0}^{t}(\|x_{0}\|_{p}^{2}+1)\|y^{\varepsilon}(\tau)\|_{p}^{2}\chi_{E_{\varepsilon}}(\tau)d\tau\right)$
$\displaystyle\leq$
$\displaystyle\mathcal{C}\left(\int_{0}^{t}\|z^{\varepsilon}(\tau)\|_{p}^{2}d\tau+(\|x_{0}\|_{p}^{2}+1)\varepsilon^{2}\right),$

this, together with the Gronwall inequality, implies that

$\sup_{t\in[0,T]}\|z^{\varepsilon}(t)\|_{p}^{2}\leq\mathcal{C}\left(\|x_{0}\|^{2}_{p}+1\right)\varepsilon^{2}.$

In order to prove (3.14). we set
$\eta^{\varepsilon}(t):=\xi^{\varepsilon}(t)-y^{\varepsilon}(t)=x^{\varepsilon}(t)-\bar{x}(t)-y^{\varepsilon}(t)$,
for $t\in[0,T].$ By calculation, $\eta^{\varepsilon}$ solves the following
QSDE:

$\left\\{\begin{aligned}
d\eta^{\varepsilon}(t)=&\left\\{\widetilde{D}_{x}(t)\eta^{\varepsilon}(t)+\left(\widetilde{D}_{x}(t)-D_{x}(t)\right)y^{\varepsilon}(t)+\delta
D(t)\chi_{E_{\varepsilon}}(t)\right\\}dt\\\
&+\left\\{\widetilde{F}_{x}(t)\eta^{\varepsilon}(t)+\left(\widetilde{F}_{x}(t)-F_{x}(t)\right)y^{\varepsilon}(t)\right\\}dW(t)\\\
&+dW(t)\left\\{\widetilde{G}_{x}(t)\eta^{\varepsilon}(t)+\left(\widetilde{G}_{x}(t)-G_{x}(t)\right)y^{\varepsilon}(t)\right\\},\quad\textrm{in}\
(0,T],\\\ \eta^{\varepsilon}(0)=&0.\end{aligned}\right.$ (3.30)

Hence,

$\displaystyle\|\eta^{\varepsilon}(t)\|_{p}^{2}$
$\displaystyle\leq\mathcal{C}\left\\{\int_{0}^{t}\|\eta^{\varepsilon}(\tau)\|_{p}^{2}+\left\|\left(\widetilde{F}_{x}(\tau)-F_{x}(\tau)\right)y^{\varepsilon}(\tau)\right\|_{p}^{2}+\left\|\left(\widetilde{G}_{x}(\tau)-G_{x}(\tau)\right)y^{\varepsilon}(\tau)\right\|_{p}^{2}d\tau\right.$
(3.31)
$\displaystyle\indent\left.+\left(\int_{0}^{t}\left\|\left(\widetilde{D}_{x}(\tau)-D_{x}(\tau)\right)y^{\varepsilon}(\tau)\right\|_{p}+\|\delta
D(\tau)\chi_{E_{\varepsilon}}(\tau)\|_{p}d\tau\right)^{2}\right\\}.$

The terms on the right side of (3.31) will now be analyzed individually. Under
Assumption 3.1 (A3), we have

$\displaystyle\left\|\widetilde{D}_{x}(\tau)-D_{x}(\tau))\right\|_{\mathcal{L}(L^{p}(\mathscr{C}))}$
(3.32) $\displaystyle=$
$\displaystyle\left\|\int_{0}^{1}\left\\{D_{x}(\tau,\bar{x}(\tau)+\theta\xi^{\varepsilon}(\tau),u^{\varepsilon}(\tau))-D_{x}(\tau)\right\\}d\theta\right\|_{\mathcal{L}(L^{p}(\mathscr{C}))}$
$\displaystyle=$
$\displaystyle\bigg{\|}\int_{0}^{1}\left\\{D_{x}(\tau,\bar{x}(\tau)+\theta\xi^{\varepsilon}(\tau),u^{\varepsilon}(\tau))-D_{x}(\tau,\bar{x}(\tau),u^{\varepsilon}(\tau))\right\\}$
$\displaystyle\indent\indent+\left\\{D_{x}(\tau,\bar{x}(\tau),u^{\varepsilon}(\tau))-D_{x}(\tau)\right\\}d\theta\bigg{\|}_{\mathcal{L}(L^{p}(\mathscr{C}))}$
$\displaystyle=$
$\displaystyle\left\|\int_{0}^{1}\left\\{\int_{0}^{1}D_{xx}(\tau,\bar{x}(\tau)+\sigma\theta\xi^{\varepsilon}(\tau),u^{\varepsilon}(\tau))\sigma\xi^{\varepsilon}(\tau)d\sigma+\chi_{E_{\varepsilon}}(\tau)\delta
D_{x}(\tau)\right\\}d\theta\right\|_{\mathcal{L}(L^{p}(\mathscr{C}))}$
$\displaystyle\leq$
$\displaystyle\mathcal{C}(\|\xi^{\varepsilon}(\tau)\|_{p}+\chi_{E_{\varepsilon}}(\tau)),\
\textrm{a.e.}\ \tau\in[0,T].$

It follows from (3.11), (3.12) and (3.32) that

$\displaystyle\int_{0}^{t}\left\|\left(\widetilde{D}_{x}(\tau)-D_{x}(\tau)\right)y^{\varepsilon}(\tau)\right\|_{p}d\tau\leq$
$\displaystyle\int_{0}^{t}\left\|\widetilde{D}_{x}(\tau)-D_{x}(\tau))\right\|_{\mathcal{L}(L^{p}(\mathscr{C}))}\left\|y^{\varepsilon}(\tau)\right\|_{p}d\tau$
(3.33) $\displaystyle\leq$
$\displaystyle\mathcal{C}\int_{0}^{t}(\|\xi^{\varepsilon}(t)\|_{p}+\chi_{E_{\varepsilon}}(\tau))\|y^{\varepsilon}(t)\|_{p}d\tau$
$\displaystyle\leq$ $\displaystyle\mathcal{C}\varepsilon.$

Similar to (3.32), we have

$\displaystyle\left\|\widetilde{F}_{x}(\tau)-F_{x}(\tau)\right\|_{\mathcal{L}(L^{p}(\mathscr{C}))}\leq$
$\displaystyle\mathcal{C}(\|\xi^{\varepsilon}(\tau)\|_{p}+\chi_{E_{\varepsilon}}(\tau)),\
\textrm{a.e.}\ \tau\in[0,T],$
$\displaystyle\left\|\widetilde{G}_{x}(\tau)-G_{x}(\tau)\right\|_{\mathcal{L}(L^{p}(\mathscr{C}))}\leq$
$\displaystyle\mathcal{C}(\|\xi^{\varepsilon}(\tau)\|_{p}+\chi_{E_{\varepsilon}}(\tau)),\
\textrm{a.e.}\ \tau\in[0,T].$

Hence, similar to (3.33), we obtain that

$\displaystyle\int_{0}^{T}\left\|\left(\widetilde{F}_{x}(\tau)-F_{x}(\tau)\right)y^{\varepsilon}(\tau)\right\|_{p}^{2}d\tau\leq$
$\displaystyle\mathcal{C}\int_{0}^{T}\left\|\widetilde{F}_{x}(\tau)-F_{x}(\tau)\right\|^{2}_{\mathcal{L}(L^{p}(\mathscr{C}))}\|y^{\varepsilon}(\tau)\|_{p}^{2}d\tau$
(3.34) $\displaystyle\leq$ $\displaystyle\mathcal{C}\varepsilon^{2},$
$\int_{0}^{T}\left\|\left(\widetilde{G}_{x}(\tau)-G_{x}(\tau)\right)y^{\varepsilon}(\tau)\right\|_{p}^{2}d\tau\leq\mathcal{C}\varepsilon^{2}.$
(3.35)

From (3.21) (3.31), (3.33)-(3.35), we conclude that

$\sup_{t\in[0,T]}\|\eta^{\varepsilon}(t)\|_{p}^{2}\leq\mathcal{C}\varepsilon^{2}.$
(3.36)

Finally, we prove (3.15). Let

$\zeta^{\varepsilon}(t):=x^{\varepsilon}(t)-\bar{x}(t)-y^{\varepsilon}(t)-z^{\varepsilon}(t)=\eta^{\varepsilon}(t)-z^{\varepsilon}(t),\quad
t\in[0,T].$

A direct calculation gives

$\left\\{\begin{aligned}
d\zeta^{\varepsilon}(t)=&\left\\{D_{x}(t)\zeta^{\varepsilon}(t)+\Theta_{1}(t\right)\\}dt+\left\\{F_{x}(t)\zeta^{\varepsilon}(t)+\Theta_{2}(t)\right\\}dW(t)\\\
&+dW(t)\left\\{G_{x}(t)\zeta^{\varepsilon}(t)+\Theta_{3}(t)\right\\},\quad\textrm{in}\
(0,T],\\\ \zeta^{\varepsilon}(0)=&0,\end{aligned}\right.$ (3.37)

where

$\displaystyle\Theta_{1}(t):=$ $\displaystyle\delta
D_{x}(t)\xi^{\varepsilon}(t)\chi_{E_{\varepsilon}}(t)+\frac{1}{2}\left\\{\widetilde{D}_{xx}(t)-D_{xx}(t,\bar{x}(t),u^{\varepsilon}(t))\right\\}(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))$
$\displaystyle+\frac{1}{2}D_{xx}(t)\left\\{(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))-(y^{\varepsilon}(t),y^{\varepsilon}(t))\right\\}+\frac{1}{2}\delta
D_{xx}(t)(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))\chi_{E_{\varepsilon}}(t),$
$\displaystyle\Theta_{2}(t):=$ $\displaystyle\delta
F_{x}(t)\eta^{\varepsilon}(t)\chi_{E_{\varepsilon}}(t)+\frac{1}{2}\left\\{\widetilde{F}_{xx}(t)-F_{xx}(t,\bar{x}(t),u^{\varepsilon}(t))\right\\}(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))$
$\displaystyle+\frac{1}{2}F_{xx}(t)\\{(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))-(y^{\varepsilon}(t),y^{\varepsilon}(t))\\}+\frac{1}{2}\delta
F_{xx}(t)(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))\chi_{E_{\varepsilon}}(t),$
$\displaystyle\Theta_{3}(t):=$ $\displaystyle\delta
G_{x}(t)\eta^{\varepsilon}(t)\chi_{E_{\varepsilon}}(t)+\frac{1}{2}\left\\{\widetilde{G}_{xx}(t)-G_{xx}(t,\bar{x}(t),u^{\varepsilon}(t))\right\\}(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))$
$\displaystyle+\frac{1}{2}G_{xx}(t)\\{(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))-(y^{\varepsilon}(t),y^{\varepsilon}(t))\\}+\frac{1}{2}\delta
G_{xx}(t)(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))\chi_{E_{\varepsilon}}(t).$

Indeed, from (3.5), (3.7)-(3.10), it is easy to verify that the drift term for
the equation solved by $\zeta^{\varepsilon}(\cdot)$ is

$\displaystyle
D(t,x^{\varepsilon}(t),u^{\varepsilon}(t))-D(t,\bar{x}(t),\bar{u}(t))-D_{x}(t)y^{\varepsilon}(t)-D_{x}(t)z^{\varepsilon}(t)$
(3.38) $\displaystyle-\delta
D(t)\chi_{E_{\varepsilon}}(t)-\frac{1}{2}D_{xx}(t)(y^{\varepsilon}(t),y^{\varepsilon}(t))$
$\displaystyle=$ $\displaystyle
D(t,x^{\varepsilon}(t),u^{\varepsilon}(t))-D(t,\bar{x}(t),u^{\varepsilon}(t))-D_{x}(t)\\{y^{\varepsilon}(t)+z^{\varepsilon}(t)\\}-\frac{1}{2}D_{xx}(t)(y^{\varepsilon}(t),y^{\varepsilon}(t)).$

For $\theta\in[0,1]$, by Taylor’s formula with integral type, we can deduce
that

$\displaystyle
D(t,x^{\varepsilon}(t),u^{\varepsilon}(t))-D(t,\bar{x}(t),u^{\varepsilon}(t))$
(3.39) $\displaystyle=$ $\displaystyle
D_{x}(t,\bar{x}(t),u^{\varepsilon}(t))\xi^{\varepsilon}(t)+\int_{0}^{1}(1-\theta)D_{xx}(t,\bar{x}(t)+\theta\xi^{\varepsilon}(t),u^{\varepsilon}(t))(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))d\theta$
$\displaystyle=$ $\displaystyle
D_{x}(t,\bar{x}(t),u^{\varepsilon}(t))\xi^{\varepsilon}(t)+\frac{1}{2}\widetilde{D}_{xx}(t)(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t)).$

Next,

$\displaystyle
D_{x}(t,\bar{x}(t),u^{\varepsilon}(t))\xi^{\varepsilon}(t)-D_{x}(t)(y^{\varepsilon}(t)+z^{\varepsilon}(t))$
(3.40) $\displaystyle=$
$\displaystyle\\{D_{x}(t,\bar{x}(t),u^{\varepsilon}(t))-D_{x}(t)\\}\xi^{\varepsilon}(t)+D_{x}(t)\\{\xi^{\varepsilon}(t)-y^{\varepsilon}(t)-z^{\varepsilon}(t)\\}$
$\displaystyle=$ $\displaystyle\delta
D_{x}(t)\xi^{\varepsilon}(t)\chi_{E_{\varepsilon}}(t)+D_{x}(t)\zeta^{\varepsilon}(t).$

Moreover,

$\displaystyle\frac{1}{2}\widetilde{D}_{xx}(t)(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))-\frac{1}{2}D_{xx}(t)(y^{\varepsilon}(t),y^{\varepsilon}(t))$
(3.41) $\displaystyle=$
$\displaystyle\frac{1}{2}\widetilde{D}_{xx}(t)(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))-\frac{1}{2}D_{xx}(t,\bar{x}(t),u^{\varepsilon}(t))(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))$
$\displaystyle+\frac{1}{2}D_{xx}(t,\bar{x}(t),u^{\varepsilon}(t))(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))-\frac{1}{2}D_{xx}(t,\bar{x}(t),\bar{u}(t))(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))$
$\displaystyle+\frac{1}{2}D_{xx}(t)(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))-\frac{1}{2}D_{xx}(t)(y^{\varepsilon}(t),y^{\varepsilon}(t))$
$\displaystyle=$ $\displaystyle\frac{1}{2}\delta
D_{xx}(t)(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))\chi_{E_{\varepsilon}}(t)+\frac{1}{2}D_{xx}(t)\left\\{(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))-(y^{\varepsilon}(t),y^{\varepsilon}(t))\right\\}$
$\displaystyle+\frac{1}{2}\left\\{\widetilde{D}_{xx}(t)-D_{xx}(t,\bar{x}(t),u^{\varepsilon}(t))\right\\}(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t)).$

From (3.38)-(3.41), we conclude that

$\displaystyle
D(t,x^{\varepsilon}(t),u^{\varepsilon}(t))-D(t,\bar{x}(t),\bar{u}(t))-D_{x}(t)y^{\varepsilon}(t)-D_{x}(t)z^{\varepsilon}(t)$
$\displaystyle-\delta
D(t)\chi_{E_{\varepsilon}}(t)-\frac{1}{2}D_{xx}(t)(y^{\varepsilon}(t),y^{\varepsilon}(t))$
$\displaystyle=$ $\displaystyle D_{x}(t)\zeta^{\varepsilon}(t)+\delta
D_{x}(t)\xi^{\varepsilon}(t)\chi_{E_{\varepsilon}}(t)+\frac{1}{2}\left\\{\widetilde{D}_{xx}(t)-D_{xx}(t,\bar{x}(t),u^{\varepsilon}(t))\right\\}(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))$
$\displaystyle+\frac{1}{2}\delta
D_{xx}(t)(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))\chi_{E_{\varepsilon}}(t)+\frac{1}{2}D_{xx}(t)\left\\{(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))-(y^{\varepsilon}(t),y^{\varepsilon}(t))\right\\}.$

Similarly, the diffusion terms are

$\displaystyle
F(t,x^{\varepsilon}(t),u^{\varepsilon}(t))-F(t,\bar{x}(t),\bar{u}(t))-F_{x}(t)y^{\varepsilon}(t)-F_{x}(t)z^{\varepsilon}(t)$
$\displaystyle-\delta F(t)\chi_{E_{\varepsilon}}(t)-\delta
F_{x}(t)y^{\varepsilon}(t)\chi_{E_{\varepsilon}}(t)-\frac{1}{2}F_{xx}(t)(y^{\varepsilon}(t),y^{\varepsilon}(t))$
$\displaystyle=$ $\displaystyle F_{x}(t)\zeta^{\varepsilon}(t)+\delta
F_{x}(t)\eta^{\varepsilon}(t)\chi_{E_{\varepsilon}}(t)+\frac{1}{2}\left\\{\widetilde{F}_{xx}(t)-F_{xx}(t,\bar{x}(t),u^{\varepsilon}(t))\right\\}(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))$
$\displaystyle+\frac{1}{2}\delta
F_{xx}(t)(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))\chi_{E_{\varepsilon}}(t)+\frac{1}{2}F_{xx}(t)\left\\{(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))-(y^{\varepsilon}(t),y^{\varepsilon}(t))\right\\},$

and

$\displaystyle
G(t,x^{\varepsilon}(t),u^{\varepsilon}(t))-G(t,\bar{x}(t),\bar{u}(t))-G_{x}(t)y^{\varepsilon}(t)-G_{x}(t)z^{\varepsilon}(t)$
$\displaystyle-\delta G(t)\chi_{E_{\varepsilon}}(t)-\delta
G_{x}(t)y^{\varepsilon}(t)\chi_{E_{\varepsilon}}(t)-\frac{1}{2}G_{xx}(t)(y^{\varepsilon}(t),y^{\varepsilon}(t))$
$\displaystyle=$ $\displaystyle G_{x}(t)\zeta^{\varepsilon}(t)+\delta
G_{x}(t)\eta^{\varepsilon}(t)\chi_{E_{\varepsilon}}(t)+\frac{1}{2}\left\\{\widetilde{G}_{xx}(t)-G_{xx}(t,\bar{x}(t),u^{\varepsilon}(t))\right\\}(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))$
$\displaystyle+\frac{1}{2}\delta
G_{xx}(t)(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))\chi_{E_{\varepsilon}}(t)+\frac{1}{2}G_{xx}(t)\left\\{(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))-(y^{\varepsilon}(t),y^{\varepsilon}(t))\right\\}.$

From (3.37), we have

$\sup_{t\in[0,T]}\|\zeta^{\varepsilon}(t)\|_{p}^{2}\leq\mathcal{C}\left(\left(\int_{0}^{T}\|\Theta_{1}(t)\|_{p}dt\right)^{2}+\int_{0}^{T}\left\\{\|\Theta_{2}(t)\|_{p}^{2}+\|\Theta_{3}(t)\|_{p}^{2}\right\\}dt\right).$
(3.42)

Next, we consider $\Theta_{1}(t),\Theta_{2}(t)$ and $\Theta_{3}(t)$,
respectively. From (3.11)-(3.14), we first estimate $\Theta_{1}(\cdot)$. By
the Minkowski inequality and the Hölder inequality again,

$\displaystyle\int_{0}^{T}\|\Theta_{1}(t)\|_{p}dt$
$\displaystyle\leq\mathcal{C}\int_{0}^{T}\left(\frac{1}{2}\left\|D_{xx}(t)\\{(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))-(y^{\varepsilon}(t),y^{\varepsilon}(t))\\}\right\|_{p}\right.$
(3.43)
$\displaystyle\indent+\frac{1}{2}\left\|\left\\{\widetilde{D}_{xx}(t)-D_{xx}(t,\bar{x}(t),u^{\varepsilon}(t))\right\\}(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))\right\|_{p}$
$\displaystyle\indent\left.+\chi_{E_{\varepsilon}}(t)\left\\{\|\delta
D_{x}(t)\xi^{\varepsilon}(t)\|_{p}+\frac{1}{2}\|\delta
D_{xx}(t)(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))\|_{p}\right\\}\right)dt.$

Now, we now estimate each term in the right of (3.43) separately. Based on
(3.23), we have the following estimate:

$\displaystyle\int_{0}^{T}\chi_{E_{\varepsilon}}(t)\left\\{\left\|\delta
D_{x}(t)\xi^{\varepsilon}(t)\right\|_{p}+\frac{1}{2}\left\|\delta
D_{xx}(t)(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))\right\|_{p}\right\\}dt$
(3.44) $\displaystyle\leq$
$\displaystyle\int_{0}^{T}\chi_{E_{\varepsilon}}(t)\left\\{\|\delta
D_{x}(t)\|_{\mathcal{L}(L^{p}(\mathscr{C}))}\|\xi^{\varepsilon}(t)\|_{p}+\|\delta
D_{xx}(t)\|_{\mathcal{L}(L^{p}(\mathscr{C}),L^{p}(\mathscr{C});L^{p}(\mathscr{C}))}\|\xi^{\varepsilon}(t)\|_{p}^{2}\right\\}dt$
$\displaystyle\leq$
$\displaystyle\mathcal{C}\int_{0}^{t}\chi_{E_{\varepsilon}}(t)\left\\{\sup_{t\in[0,T]}\|\xi^{\varepsilon}(t)\|_{p}+\sup_{t\in[0,T]}\|\xi^{\varepsilon}(t)\|_{p}^{2}\right\\}dt$
$\displaystyle\leq$ $\displaystyle\mathcal{C}\varepsilon^{\frac{3}{2}}.$

From (3.8), we can obtain that, for $t\in[0,T]$

$\displaystyle\left\|\widetilde{D}_{xx}(t)-D_{xx}(t,\bar{x}(t),u^{\varepsilon}(t))\right\|_{\mathcal{L}(L^{p}(\mathscr{C}),L^{p}(\mathscr{C});L^{p}(\mathscr{C}))}$
(3.45) $\displaystyle=$
$\displaystyle\left\|2\int_{0}^{1}(1-\theta)D_{xx}(t,\bar{x}(t)+\theta\xi^{\varepsilon}(t),u^{\varepsilon}(t))d\theta-
D_{xx}(t,\bar{x}(t),u^{\varepsilon}(t))\right\|_{\mathcal{L}(L^{p}(\mathscr{C}),L^{p}(\mathscr{C});L^{p}(\mathscr{C}))}$
$\displaystyle=$
$\displaystyle\left\|2\int_{0}^{1}(1-\theta)\chi_{E_{\varepsilon}}(t)\\{D_{xx}(t,\bar{x}(t)+\theta\xi^{\varepsilon}(t),u(t))-D_{xx}(t,\bar{x}(t),u(t))\\}d\theta\right.$
$\displaystyle\indent\left.+2\int_{0}^{1}(1-\theta)\left\\{D_{xx}(t,\bar{x}(t)+\theta\xi^{\varepsilon}(t),\bar{u}(t))-D_{xx}(t)\right\\}d\theta\right\|_{\mathcal{L}(L^{p}(\mathscr{C}),L^{p}(\mathscr{C});L^{p}(\mathscr{C}))}$
$\displaystyle\leq$
$\displaystyle\mathcal{C}\left(\int_{0}^{1}\left\|D_{xx}(t,\bar{x}(t)+\theta\xi^{\varepsilon}(t),\bar{u}(t))-D_{xx}(t)\right\|_{\mathcal{L}(L^{p}(\mathscr{C}),L^{p}(\mathscr{C});L^{p}(\mathscr{C}))}d\theta+\chi_{E_{\varepsilon}}(t)\right).$

Hence, by (3.11) and the continuity of $D_{xx}(t,x,u)$ with respect to $x$, we
have

$\displaystyle\left(\int_{0}^{T}\left\|\left\\{\widetilde{D}_{xx}(t)-D_{xx}(t,\bar{x}(t),u^{\varepsilon}(t))\right\\}(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))\right\|_{p}dt\right)^{2}$
(3.46) $\displaystyle\leq$
$\displaystyle\mathcal{C}\int_{0}^{T}\left\|\widetilde{D}_{xx}(t)-D_{xx}(t,\bar{x}(t),u^{\varepsilon}(t))\right\|^{2}_{\mathcal{L}(L^{p}(\mathscr{C}),L^{p}(\mathscr{C});L^{p}(\mathscr{C}))}\|\xi^{\varepsilon}(t)\|_{p}^{4}dt$
$\displaystyle\leq$
$\displaystyle\mathcal{C}\sup_{t\in[0,T]}\|\xi^{\varepsilon}(t)\|_{p}^{4}\int_{0}^{T}\bigg{\\{}\chi_{E_{\varepsilon}}(t)$
$\displaystyle\left.+\int_{0}^{1}\|D_{xx}(t,\bar{x}(t)+\theta\xi^{\varepsilon}(t),\bar{u}(t))-D_{xx}(t)\|^{2}_{\mathcal{L}(L^{p}(\mathscr{C}),L^{p}(\mathscr{C});L^{p}(\mathscr{C}))}d\theta\right\\}dt$
$\displaystyle\leq$ $\displaystyle\mathcal{C}\varepsilon^{2}.$

By means of (3.23),(3.24) and (3.36), and using the Hölder inequality again,
we obtain

$\displaystyle\left(\int_{0}^{T}\|D_{xx}(t)\\{(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))-(y^{\varepsilon}(t),y^{\varepsilon}(t))\\}\|_{p}dt\right)^{2}$
(3.47) $\displaystyle=$
$\displaystyle\left(\int_{0}^{T}\|D_{xx}(t)(\xi^{\varepsilon}(t),\eta^{\varepsilon}(t))+D_{xx}(t)(\eta^{\varepsilon}(t),y^{\varepsilon}(t))\|_{p}dt\right)^{2}$
$\displaystyle\leq$
$\displaystyle\mathcal{C}\int_{0}^{T}\|D_{xx}(t)\|^{2}_{\mathcal{L}(L^{p}(\mathscr{C}),L^{p}(\mathscr{C});L^{p}(\mathscr{C}))}\\{\|\xi^{\varepsilon}(t)\|_{p}^{2}+\|y^{\varepsilon}(t)\|_{p}^{2}\\}\|\eta^{\varepsilon}(t)\|_{p}^{2}dt$
$\displaystyle\leq$
$\displaystyle\mathcal{C}\left(\sup_{t\in[0,T]}\|\xi^{\varepsilon}(t)\|_{p}^{2}+\sup_{t\in[0,T]}\|y^{\varepsilon}(t)\|_{p}^{2}\right)\sup_{t\in[0,T]}\|\eta^{\varepsilon}(t)\|_{p}^{2}$
$\displaystyle\leq$ $\displaystyle\mathcal{C}\varepsilon^{3}.$

From (3.43)-(3.47), we infer that

$\left(\int_{0}^{T}\|\Theta_{1}(t)\|_{p}dt\right)^{2}\leq\mathcal{C}\varepsilon^{2}.$
(3.48)

By virtue of (3.23) and (3.36) again, similar to (3.44), we obtain that

$\displaystyle\int_{0}^{T}\chi_{E_{\varepsilon}}(t)\left\\{\|\delta
F_{x}(t)\eta^{\varepsilon}(t)\|_{p}^{2}+\|\delta
F_{xx}(t)(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))\|_{p}^{2}\right\\}dt$
(3.49) $\displaystyle\leq$
$\displaystyle\mathcal{C}\int_{0}^{T}\chi_{E_{\varepsilon}}(t)\left\\{\sup_{t\in[0,T]}\|\eta^{\varepsilon}(t)\|_{p}^{2}+\sup_{t\in[0,T]}\|\xi^{\varepsilon}(t)\|_{p}^{4}\right\\}dt$
$\displaystyle\leq$ $\displaystyle\mathcal{C}\varepsilon^{3}.$

Similar to (3.46), we find that

$\displaystyle\int_{0}^{T}\left\|\left(\widetilde{F}_{xx}(t)-F_{xx}(t,\bar{x}(t),u^{\varepsilon}(t))\right)(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))\right\|_{p}^{2}dt$
(3.50) $\displaystyle\leq$
$\displaystyle\int_{0}^{T}\left\|\widetilde{F}_{xx}(t)-F_{xx}(t,\bar{x}(t),u^{\varepsilon}(t))\right\|_{\mathcal{L}(L^{p}(\mathscr{C}),L^{p}(\mathscr{C});L^{p}(\mathscr{C}))}^{2}\|\xi^{\varepsilon}(t)\|_{p}^{4}dt$
$\displaystyle\leq$ $\displaystyle\mathcal{C}\varepsilon^{2}.$

Similar to (3.47), it holds that

$\displaystyle\int_{0}^{T}\|F_{xx}(t)\\{(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))-(y^{\varepsilon}(t),y^{\varepsilon}(t))\\}\|_{p}^{2}dt$
(3.51) $\displaystyle\leq$
$\displaystyle\int_{0}^{T}\|F_{xx}(t)\|^{2}_{\mathcal{L}(L^{p}(\mathscr{C}),L^{p}(\mathscr{C});L^{p}(\mathscr{C}))}\left\\{\|\xi^{\varepsilon}(t)\|_{p}^{2}+\|y^{\varepsilon}(t)\|_{p}^{2}\right\\}\|\eta^{\varepsilon}(t)\|_{p}^{2}dt$
$\displaystyle\leq$ $\displaystyle\mathcal{C}\varepsilon^{3}.$

Thus, from (3.49)-(3.51), we conclude that

$\displaystyle\int_{0}^{T}\|\Theta_{2}(t)\|_{p}^{2}dt$
$\displaystyle\leq\mathcal{C}\int_{0}^{T}\Big{(}\|F_{xx}(t)\\{(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))-(y^{\varepsilon}(t),y^{\varepsilon}(t))\\}\|_{p}^{2}$
(3.52)
$\displaystyle\indent\indent+\left\|\left\\{\widetilde{F}_{xx}(t)-F_{xx}(t,\bar{x}(t),u^{\varepsilon}(t))\right\\}(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))\right\|_{p}^{2}$
$\displaystyle\indent+\chi_{E_{\varepsilon}}(t)\left\\{\|\delta
F_{x}(t)\eta^{\varepsilon}(t)\|_{p}^{2}+\|\delta
F_{xx}(t)(\xi^{\varepsilon}(t),\xi^{\varepsilon}(t))\|_{p}^{2}\right\\}\Big{)}dt$
$\displaystyle\leq\mathcal{C}\varepsilon^{2}.$

By a similar arguments, we have

$\int_{0}^{T}\|\Theta_{3}(t)\|_{p}^{2}dt\leq\mathcal{C}\varepsilon^{2}.$
(3.53)

Substituting (3.48), (3.52), (3.53) into (3.42), we obtain that

$\sup_{t\in[0,T]}\|\zeta^{\varepsilon}(t)\|_{p}^{2}\leq\mathcal{C}\varepsilon^{2}.$

Then (3.15) holds.

Finally, we prove (3.16). By Taylor expansion, we have

$\displaystyle\mathcal{J}(u^{\varepsilon}(\cdot))-\mathcal{J}(\bar{u}(\cdot))$
$\displaystyle=h(x^{\varepsilon}(T))-h(\bar{x}(T))+\int_{0}^{T}\left\\{L(t,x^{\varepsilon}(t),u^{\varepsilon}(t))-L(t,\bar{x}(t),\bar{u}(t))\right\\}dt$
$\displaystyle=\textrm{Re}\int_{0}^{T}\left\\{\delta
L(t)\chi_{E_{\varepsilon}}(t)+\left\langle
L_{x}(t,\bar{x}(t),u^{\varepsilon}(t)),\xi^{\varepsilon}(t)\right\rangle\right\\}dt$
$\displaystyle\indent+\textrm{Re}\int_{0}^{T}\left\\{\int_{0}^{1}\left\langle(1-\theta)L_{xx}(t,\bar{x}(t)+\theta\xi^{\varepsilon}(t),u^{\varepsilon}(t))\xi^{\varepsilon}(t),\xi^{\varepsilon}(t)\right\rangle
d\theta\right\\}dt$ $\displaystyle\indent+\textrm{Re}\left\langle
h_{x}(\bar{x}(T)),\xi^{\varepsilon}(T)\right\rangle+\textrm{Re}\int_{0}^{1}\left\langle(1-\theta)h_{xx}(\bar{x}(T)+\theta\xi^{\varepsilon}(T))\xi^{\varepsilon}(T),\xi^{\varepsilon}(T)\right\rangle
d\theta.$

This, together with the definitions of
$\xi^{\varepsilon},y^{\varepsilon},z^{\varepsilon},\eta^{\varepsilon}$ and
$\zeta^{\varepsilon}$, yields that

$\displaystyle\mathcal{J}(u^{\varepsilon}(\cdot))-\mathcal{J}(\bar{u}(\cdot))$
$\displaystyle=$ $\displaystyle\textrm{Re}\int_{0}^{T}\Big{\\{}\delta
L(t)\chi_{E_{\varepsilon}}(t)+\left\langle\delta
L_{x}(t),\xi^{\varepsilon}(t)\right\rangle\chi_{E_{\varepsilon}}(t)+\left\langle
L_{x}(t),y^{\varepsilon}(t)+z^{\varepsilon}(t)\right\rangle+\left\langle
L_{x}(t),\zeta^{\varepsilon}(t)\right\rangle$
$\displaystyle+\int_{0}^{1}\left\langle(1-\theta)\left\\{L_{xx}(t,\bar{x}(t)+\theta\xi^{\varepsilon}(t),u^{\varepsilon}(t))-L_{xx}(t,\bar{x}(t),u^{\varepsilon}(t))\right\\}\xi^{\varepsilon}(t),\xi^{\varepsilon}(t)\right\rangle
d\theta$ $\displaystyle\indent+\frac{1}{2}\left\langle\delta
L_{xx}(t)\xi^{\varepsilon}(t),\xi^{\varepsilon}(t)\right\rangle\chi_{E_{\varepsilon}}(t)+\frac{1}{2}\left\langle
L_{xx}(t)\eta^{\varepsilon}(t),y^{\varepsilon}(t)\right\rangle+\frac{1}{2}\left\langle
L_{xx}(t)y^{\varepsilon}(t),y^{\varepsilon}(t)\right\rangle$
$\displaystyle\indent+\frac{1}{2}\left\langle
L_{xx}(t)\xi^{\varepsilon}(t),\eta^{\varepsilon}(t)\right\rangle\Big{\\}}dt+\textrm{Re}\left\langle
h_{x}(\bar{x}(T)),y^{\varepsilon}(T)+z^{\varepsilon}(T)\right\rangle$
$\displaystyle+\textrm{Re}\left\langle
h_{x}(\bar{x}(T)),\zeta^{\varepsilon}(T)\right\rangle+\frac{1}{2}\textrm{Re}\left\langle
h_{xx}(\bar{x}(T))\xi^{\varepsilon}(T),\eta^{\varepsilon}(T)\right\rangle$
$\displaystyle+\frac{1}{2}\textrm{Re}\left\langle
h_{xx}(\bar{x}(T))\eta^{\varepsilon}(T),y^{\varepsilon}(T)\right\rangle+\frac{1}{2}\textrm{Re}\left\langle
h_{xx}(\bar{x}(T))y^{\varepsilon}(T),y^{\varepsilon}(T)\right\rangle$
$\displaystyle+\textrm{Re}\int_{0}^{1}\left\langle(1-\theta)\left\\{h_{xx}(\bar{x}(T)+\theta\xi^{\varepsilon}(T))-h_{xx}(\bar{x}(T))\right\\}\xi^{\varepsilon}(T),\xi^{\varepsilon}(T)\right\rangle
d\theta.$

Similar to (3.45), for $t\in[0,T]$, we find that

$\displaystyle\left\|\int_{0}^{1}(1-\theta)\left\\{L_{xx}(t,\bar{x}(t)+\theta\xi^{\varepsilon}(t),u^{\varepsilon}(t))-L_{xx}(t,\bar{x}(t),u^{\varepsilon}(t))\right\\}d\theta\right\|_{\mathcal{L}(L^{p}(\mathscr{C});L^{p^{\prime}}(\mathscr{C}))}$
(3.54)
$\displaystyle\leq\mathcal{C}\int_{0}^{1}\left\|{L_{xx}(t,\bar{x}(t)+\theta\xi^{\varepsilon}(t),u^{\varepsilon}(t))-L_{xx}(t)}\right\|_{L(L^{p}(\mathscr{C});L^{p^{\prime}}(\mathscr{C}))}d\theta+\chi_{E_{\varepsilon}}(t).$

From (3.11)-(3.15) and (3.54), we deduce that

$\displaystyle\mathcal{J}(u^{\varepsilon}(\cdot))-\mathcal{J}(\bar{u}(\cdot))$
$\displaystyle=$ $\displaystyle\textrm{Re}\int_{0}^{T}\left\\{\delta
L(t)\chi_{E_{\varepsilon}}(t)+\left\langle
L_{x}(t),y^{\varepsilon}(t)+z^{\varepsilon}(t)\right\rangle+\frac{1}{2}\left\langle
L_{xx}(t)y^{\varepsilon}(t),y^{\varepsilon}(t)\right\rangle\right\\}dt$
$\displaystyle+\textrm{Re}\left\langle
h_{x}(\bar{x}(T)),y^{\varepsilon}(T)+z^{\varepsilon}(T)\right\rangle+\frac{1}{2}\textrm{Re}\left\langle
h_{xx}(\bar{x}(T))y^{\varepsilon}(T),y^{\varepsilon}(T)\right\rangle+\textit{{o}}(\varepsilon).$

∎

In order to establish the necessary conditions for an optimal pair
$(\bar{x}(\cdot),\bar{u}(\cdot))$ of Problem (QOC), we need to introduce the
following BQSDEs:

$\left\\{\begin{aligned}
d\phi(t)=&-\left\\{D_{x}(t)^{*}\phi(t)+\left(F^{e}_{x}(t)-F^{o}_{x}(t)+G_{x}(t)\right)^{*}\left(\Phi(t)_{e}-\Phi(t)_{o}\right)-L_{x}(t)\right\\}dt\\\
&+\Phi(t)dW(t),\ \rm{in}\ [0,T),\\\
\phi(T)=&-h_{x}(\bar{x}(T)),\end{aligned}\right.$ (3.55)

and

$\left\\{\begin{aligned}
dP(t)=&-\left\\{D_{x}(t)^{*}P(t)+P(t)D_{x}(t)+(F^{e}_{x}(t)-F^{o}_{x}(t)+G_{x}(t))^{*}Q(t)\right.\\\
&\indent+\left(F^{e}_{x}(t)-F^{o}_{x}(t)+G_{x}(t)\right)^{*}P(t)\left(F_{x}(t)+G^{e}_{x}(t)-G^{o}_{x}(t)\right)\\\
&\indent\left.+Q(t)(F^{e}_{x}(t)-F^{o}_{x}(t)+G_{x}(t))+\mathbb{H}_{xx}(t,\bar{x}(t),\bar{u}(t),\phi(t),\Phi(t))\right\\}dt\\\
&+Q(t)dW(t),\ \rm{in}\ [0,T),\\\
P(T)=&-h_{xx}(\bar{x}(T)),\end{aligned}\right.$ (3.56)

where the map $\Phi(\cdot):[0,T]\to\mathcal{H}^{p^{\prime}}(0,T)$ is adapted,
and $F_{x}^{e}(\cdot)-F_{x}^{o}(\cdot):=\Upsilon\circ F_{x}(\cdot)$,
$G_{x}^{e}(\cdot)-G_{x}^{o}(\cdot):=\Upsilon\circ G_{x}(\cdot)$, the map
$Q(\cdot):[0,T]\to\mathcal{L}(L^{p}(\mathscr{C});L^{p^{\prime}}(\mathscr{C}))$.
And the Hamiltonian function $\mathbb{H}(\cdot,\cdot,\cdot,\cdot,\cdot)$ is
defined by

$\displaystyle\mathbb{H}(t,x,u,\phi,\Phi)$ (3.57) $\displaystyle:$
$\displaystyle=\langle\phi,D(t,x,u)\rangle+\langle\Phi_{e}-\Phi_{o},F(t,x,u)_{e}-F(t,x,u)_{o}+G(t,x,u)\rangle-L(t,x,u),$
$\displaystyle\indent\indent(t,x,u,\phi,\Phi)\in[0,T]\times
L^{p}(\mathscr{C})\times U\times L^{p^{\prime}}(\mathscr{C})\times
L^{p^{\prime}}(\mathscr{C}).$

Hereafter we use $\langle\cdot,\cdot\rangle$ to represent the dual product of
$L^{p^{\prime}}(\mathscr{C})$ and $L^{p}(\mathscr{C})$, and $\langle
x,y\rangle=m(x^{*}y)$ for $x\in L^{p^{\prime}}(\mathscr{C}),y\in
L^{p}(\mathscr{C})$.

######  Theorem 3.3.

Suppose that Assumption 3.1 holds. Let $(\bar{x}(\cdot),\bar{u}(\cdot))$ be an
optimal pair of Problem (QOC). Let $(\phi(\cdot),\Phi(\cdot))$ be the solution
to (3.55), and $(P(\cdot),Q(\cdot))$ be the solution to (3.56). Then,

$\displaystyle{\rm{Re}}\mathbb{H}(t,\bar{x}(t),\bar{u}(t),\phi(t),\Phi(t))-{\rm{Re}}\mathbb{H}(t,\bar{x}(t),u,\phi(t),\Phi(t))$
(3.58)
$\displaystyle-\frac{1}{2}{\rm{Re}}\left\langle(P^{e}(t)-P^{o}(t))\\{(F(t,\bar{x}(t),\bar{u}(t))+G(t,\bar{x}(t),\bar{u}(t))_{e}-G(t,\bar{x}(t),\bar{u}(t)))_{o}\right.$
$\displaystyle\indent\indent-(F(t,\bar{x}(t),u)+G(t,\bar{x}(t),u)_{e}-G(t,\bar{x}(t),u)_{o})\\},$
$\displaystyle\indent\indent\\{(F(t,\bar{x}(t),\bar{u}(t))_{e}-F(t,\bar{x}(t),\bar{u}(t))_{o}+G(t,\bar{x}(t),\bar{u}(t)))$
$\displaystyle\indent\indent\left.-(F(t,\bar{x}(t),u)_{e}-F(t,\bar{x}(t),u)_{o}+G(t,\bar{x}(t),u))\\}\right\rangle\geq
0,\ {\rm a.e.}\ t\in[0,T],\ u\in U.$

###### Proof.

In view of the dual relation between (3.9) and (3.55), we have

$\displaystyle\langle\phi(T),y^{\varepsilon}(T)\rangle=-\left\langle
h_{x}(\bar{x}(T)),y^{\varepsilon}(T)\right\rangle$ (3.59) $\displaystyle=$
$\displaystyle-\int_{0}^{T}\Big{\\{}\left\langle
D_{x}(t)^{*}\phi(t),y^{\varepsilon}(t)\right\rangle+\left\langle\Phi(t)_{e}-\Phi(t)_{o},(F_{x}^{e}(t)-F_{x}^{o}(t)+G_{x}(t))y^{\varepsilon}(t)\right\rangle\Big{\\}}dt$
$\displaystyle+\int_{0}^{T}\Big{\\{}\langle\Phi(t)_{e}-\Phi(t)_{o},\delta(F(t)_{e}-F(t)_{o}+G(t))\rangle\chi_{E_{\varepsilon}}(t)+\langle\phi(t),D_{x}(t)y^{\varepsilon}(t)\rangle\Big{\\}}dt$
$\displaystyle+\int_{0}^{T}\left\\{\langle\Phi(t)_{e}-\Phi(t)_{o},(F_{x}^{e}(t)-F_{x}^{o}(t)+G_{x}(t))y^{\varepsilon}(t)\rangle+\left\langle
L_{x}(t),y^{\varepsilon}(t)\right\rangle\right\\}dt$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\Big{\\{}\left\langle
L_{x}(t),y^{\varepsilon}(t)\right\rangle+\langle\Phi(t)_{e}-\Phi(t)_{o},\delta(F(t)_{e}-F(t)_{o}+G(t))\rangle\chi_{E_{\varepsilon}}(t)\Big{\\}}dt,$

Similarly,

$\displaystyle\langle\phi(T),z^{\varepsilon}(T)\rangle=-\left\langle
h_{x}(\bar{x}(T)),z^{\varepsilon}(T)\right\rangle$ (3.60) $\displaystyle=$
$\displaystyle\int_{0}^{T}\Big{\\{}\langle\phi(t),\delta
D(t)\rangle+\langle\Phi(t)_{e}-\Phi(t)_{o},\delta(F_{x}^{e}(t)-F_{x}^{o}(t)+G_{x}(t))y^{\varepsilon}(t)\rangle\Big{\\}}\chi_{E_{\varepsilon}}(t)dt$
$\displaystyle+\int_{0}^{T}\left\langle
L_{x}(t),z^{\varepsilon}(t)\right\rangle+\frac{1}{2}\Big{\\{}\langle\phi(t),D_{xx}(t)(y^{\varepsilon}(t),y^{\varepsilon}(t))\rangle$
$\displaystyle\indent\indent\indent\indent+\langle\Phi(t)_{e}-\Phi(t)_{o},(F^{e}_{xx}(t)-F^{o}_{xx}(t)+G_{xx}(t))(y^{\varepsilon}(t),y^{\varepsilon}(t))\rangle\Big{\\}}dt,$

where $F^{e}_{xx}(t)-F^{o}_{xx}(t)=\Upsilon\circ F_{xx}(t)$. From (3.12),
(3.59) and (3.60), we obtain that

$\displaystyle\left\langle\phi(T),y^{\varepsilon}(T)+z^{\varepsilon}(T)\right\rangle=-\left\langle
h_{x}(\bar{x}(T)),y^{\varepsilon}(T)+z^{\varepsilon}(T)\right\rangle$ (3.61)
$\displaystyle=$ $\displaystyle\int_{0}^{T}\Big{\\{}\langle\phi(t),\delta
D(t)\rangle+\langle\Phi(t)_{e}-\Phi(t)_{o},\delta(F(t)_{e}-F(t)_{o}+G(t))\rangle\Big{\\}}\chi_{E_{\varepsilon}}(t)dt$
$\displaystyle+\int_{0}^{T}\frac{1}{2}\Big{\\{}\langle\Phi(t)_{e}-\Phi(t)_{o},(F^{e}_{xx}(t)-F^{o}_{xx}(t)+G_{xx}(t))(y^{\varepsilon}(t),y^{\varepsilon}(t))\rangle$
$\displaystyle\indent\indent+\langle\phi(t),D_{xx}(t)(y^{\varepsilon}(t),y^{\varepsilon}(t))\rangle\Big{\\}}+\left\langle
L_{x}(t),y^{\varepsilon}(t)+z^{\varepsilon}(t)\right\rangle
dt+\textit{{o}}(\varepsilon^{\frac{3}{2}}).$

By [32, Chapter 12.4], from (3.9) and (3.56), we have the following dual
product

$\displaystyle\langle
P(T)y^{\varepsilon}(T),y^{\varepsilon}(T)\rangle=-\left\langle
h_{xx}(\bar{x}(T))y^{\varepsilon}(T),y^{\varepsilon}(T)\right\rangle$ (3.62)
$\displaystyle=$
$\displaystyle\int_{0}^{T}\langle(P^{e}(t)-P^{o}(t))\delta(F(t)+G(t)_{e}-G(t)_{o}),\delta(F(t)_{e}-F(t)_{o}+G_{x}(t))\rangle\chi_{E_{\varepsilon}}(t)dt$
$\displaystyle-2\int_{0}^{T}\langle
Q^{e}(t)y^{\varepsilon}(t)_{o}+Q^{o}(t)y^{\varepsilon}(t)_{e},(F_{x}^{e}(t)-F_{x}^{o}(t)+G_{x}(t))y^{\varepsilon}(t)\rangle
dt$ $\displaystyle-2\int_{0}^{T}\langle
P^{o}(t)(F_{x}(t)+G_{x}^{e}(t)-G^{o}_{x}(t))y^{\varepsilon}(t),(F_{x}^{e}(t)-F_{x}^{o}(t)+G_{x}(t))y^{\varepsilon}(t)\rangle
dt$
$\displaystyle-\int_{0}^{T}\langle\mathbb{H}_{xx}(t,\bar{x}(t),\bar{u}(t),\phi(t),\Phi(t))y^{\varepsilon}(t),\
y^{\varepsilon}(t)\rangle dt+\textit{{o}}(\varepsilon^{\frac{3}{2}}).$

Substituting (3.62) into (3.16), and combining with (3.12) and (3.13), we have

$\displaystyle 0\leq$
$\displaystyle\mathcal{J}(u^{\varepsilon}(\cdot))-\mathcal{J}(\bar{u}(\cdot))$
(3.63) $\displaystyle=$
$\displaystyle-\textrm{Re}\int_{0}^{T}\chi_{E_{\varepsilon}}(t)\Big{\\{}-\delta
L(t)\langle\phi(t),\delta
D(t)\rangle+\langle\Phi(t)_{e}-\Phi(t)_{o},\delta(F(t)_{e}-F(t)_{o}+G(t))\rangle$
$\displaystyle-\frac{1}{2}\langle(P^{e}(t)-P^{o}(t))\delta(F(t)+G(t)_{e}-G(t)_{o}),\delta(F(t)_{e}-F(t)_{o}+G(t))\rangle\Big{\\}}dt+\textit{{o}}(\varepsilon).$

Therefore, we easily obtain

$\displaystyle\textrm{Re}\int_{0}^{T}\Bigg{\\{}\frac{1}{2}\left\langle(P^{e}(t)-P^{o}(t))\delta(F(t)+G(t)_{e}-G(t)_{o}),\delta(F(t)_{e}-F(t)_{o}+G(t))\right\rangle+\delta\mathbb{H}(t)\Bigg{\\}}\chi_{E_{\varepsilon}}(t)dt$
$\displaystyle\leq\textit{{o}}(\varepsilon),$

where

$\delta\mathbb{H}(t):=\mathbb{H}(t,\bar{x}(t),u(t),\phi(t),\Phi(t))-\mathbb{H}(t,\bar{x}(t),\bar{u}(t),\phi(t),\Phi(t)).$

Let $\varepsilon\to 0$, for a.e. $t\in[0,T]$, we have

$\displaystyle\textrm{Re}\mathbb{H}(t,\bar{x}(t),u(t),\phi(t),\Phi(t))-\textrm{Re}\mathbb{H}(t,\bar{x}(t),\bar{u}(t),\phi(t),\Phi(t))$
$\displaystyle+\frac{1}{2}\textrm{Re}\left\langle(P^{e}(t)-P^{o}(t))\delta(F(t)+G(t)_{e}-G(t)_{o}),\delta(F(t)_{e}-F(t)_{o}+G(t))\right\rangle\leq
0.$

Thus (3.58) is proved. ∎

######  Remark 3.1.

In (3.58),

$\langle(P^{e}(t)-P^{o}(t))\delta(F(t)+G(t)_{e}-G(t)_{o}),\delta(F(t)_{e}-F(t)_{o}+G(t))\rangle$
(3.64)

indicates that the operators do not commute with each other. If the maps
$F(\cdot,\cdot,\cdot)$, $G(\cdot,\cdot,\cdot):[0,T]\times
L^{p}(\mathscr{C})\times U\to L^{p}(\mathscr{C}_{e})$ are adapted, then
$F(\cdot,\cdot,\cdot),G(\cdot,\cdot,\cdot)$ can commute with $dW(t)$, and
(3.64) can be transformed into

$\langle(P^{e}(t)-P^{o}(t))\delta(F(t)+G(t)),\delta(F(t)+G(t))\rangle.$

Then, Theorem 3.3 is similar to the classical results [13, Theorem 2.6] and
[32, Theorem 12.17].

##  4 The Solutions to Backward Quantum Stochastic Differential Equations

In section 3, we present a necessary condition for quantum optimal control of
(1.3). That is, the solutions to BQSDEs is necessary for the Pontryagin-type
Maximum Principle. In this section, we investigate the solution to (3.55) in
$L^{p^{\prime}}(\mathscr{C})$ for $p^{\prime}\in(1,2]$. Let us introduce the
following semilinear BQSDE

$\left\\{\begin{aligned} dy(t)&=f(t,y(t),Y(t))dt+Y(t)dW(t),\ {\rm in}\
[0,T),\\\ y(T)&=y_{T},\end{aligned}\right.$ (4.1)

where $y_{T}\in L^{p^{\prime}}(\mathscr{C}_{T})$ is given, the map
$f(\cdot,\cdot,\cdot):[0,T]\times L^{p^{\prime}}(\mathscr{C})\times
L^{p^{\prime}}(\mathscr{C})\to L^{p^{\prime}}(\mathscr{C})$ is adapted,
$Y(\cdot)\in\mathcal{H}^{p^{\prime}}(0,T)$. To obtain the solution to (4.1),
we assume the following condition:

######  Assumption 4.1.

The map $f(\cdot,y(\cdot),Y(\cdot))$ is adapted for each $y(t)\in
L^{p^{\prime}}(\mathscr{C}_{t})$ and $Y(t)\in
L^{p^{\prime}}(\mathscr{C}_{t})$, $f(\cdot,0,0)\in
L^{p^{\prime}}_{\mathbb{A}}(\mathscr{C};L^{1}(0,T))$, and there exist
nonnegative functions $g_{1}(\cdot)\in L^{1}(0,T)$ and $g_{2}(\cdot)\in
L^{2}(0,T)$ such that

$\displaystyle\|f(t,y_{1},Y_{1})-f(t,y_{2},Y_{2})\|_{p^{\prime}}$
$\displaystyle\leq
g_{1}(t)\|y_{1}-y_{2}\|_{p^{\prime}}+g_{2}(t)\|Y_{1}-Y_{2}\|_{p^{\prime}},$
$\displaystyle\ \forall y_{1},\ y_{2}\in L^{p^{\prime}}(\mathscr{C}_{t}),$
$\displaystyle\ Y_{1},\ Y_{2}\in L^{p^{\prime}}(\mathscr{C}_{t}).$

######  Definition 4.1.

A pair $(y(\cdot),Y(\cdot))\in
C_{\mathbb{A}}(0,T;L^{p^{\prime}}(\mathscr{C}))\times\mathcal{H}^{p^{\prime}}(0,T)$
is called a solution to (4.1) if $y(\cdot)$ is an
$L^{p^{\prime}}(\mathscr{C})$-valued adapted, continuous process,
$Y(\cdot)\in\mathcal{H}^{p^{\prime}}(0,T)$, $f(\cdot,y(\cdot),Y(\cdot))\in
L^{p^{\prime}}_{\mathbb{A}}(\mathscr{C};L^{1}(0,T))$ a.s., and for any
$t\in[0,T]$,

$y(t)=y_{T}-\int_{t}^{T}f(\tau,y(\tau),Y(\tau))d\tau-\int_{t}^{T}Y(\tau)dW(\tau).$
(4.2)

######  Theorem 4.1.

Suppose that Assumption 4.1 holds. Then, there exists a unique pair
$(y(\cdot),Y(\cdot))\in
C_{\mathbb{A}}(0,T;L^{p^{\prime}}(\mathscr{C}))\times\mathcal{H}^{p^{\prime}}(0,T)$
which satisfies (4.2). Furthermore,

$\|(y(\cdot),Y(\cdot))\|_{C_{\mathbb{A}}(0,T;L^{p^{\prime}}(\mathscr{C}))\times\mathcal{H}^{p^{\prime}}(0,T)}\leq\mathcal{C}\left(\|y_{T}\|_{p^{\prime}}+\int_{0}^{T}\|f(\tau,0,0)\|_{p^{\prime}}d\tau\right).$
(4.3)

###### Proof.

We divide the proof into two steps.

Step 1. First, we claim that for any $f(\cdot)\in
L_{\mathbb{A}}^{p^{\prime}}(\mathscr{C};L^{1}(0,T))$ and $y_{T}\in
L^{p^{\prime}}(\mathscr{C})$, there is a pair $(y(\cdot),Y(\cdot))\in
C_{\mathbb{A}}(0,T;L^{p^{\prime}}(\mathscr{C}))\times\mathcal{H}^{p^{\prime}}(0,T)$
such that

$y(t)=y_{T}-\int_{t}^{T}f(\tau)d\tau-\int_{t}^{T}Y(\tau)dW(\tau),\quad
t\in[0,T).$

Let

$M(t):=\mathbb{E}\left(y_{T}-\int_{0}^{T}f(\tau)d\tau\bigg{|}\mathscr{C}_{t}\right),\quad
y(t):=\mathbb{E}\left(y_{T}-\int_{t}^{T}f(\tau)d\tau\bigg{|}\mathscr{C}_{t}\right).$
(4.4)

Thus, $M(0)=y(0)=\mathbb{E}(M(T))$. By the noncommutative martingales
representation theorem [35, Theorem 4.6], there exists
$Y(\cdot)\in\mathcal{H}^{p^{\prime}}(0,T)$ such that

$M(t)=M(0)+\int_{0}^{t}Y(\tau)dW(\tau),\quad t\in(0,T].$ (4.5)

Hence,

$\left\|\int_{0}^{T}Y(\tau)dW(\tau)\right\|_{p^{\prime}}^{2}\leq\|M(T)\|_{p^{\prime}}^{2}\leq
2\left(\|y_{T}\|_{p^{\prime}}^{2}+\left\|\int_{0}^{T}f(\tau)d\tau\right\|_{p^{\prime}}^{2}\right).$
(4.6)

From (4.5), we have

$\displaystyle y_{T}-\int_{0}^{T}f(\tau)d\tau$
$\displaystyle=M(0)+\int_{0}^{T}Y(\tau)dW(\tau)$
$\displaystyle=y(0)+\int_{0}^{T}Y(\tau)dW(\tau),$

and

$\displaystyle y(t)$ $\displaystyle=M(t)+\int_{0}^{t}f(\tau)d\tau$
$\displaystyle=y(0)+\int_{0}^{t}Y(\tau)dW(\tau)+\int_{0}^{t}f(\tau)d\tau$
$\displaystyle=y_{T}-\int_{t}^{T}f(\tau)d\tau-\int_{t}^{T}Y(\tau)dW(\tau).$

Then,
$y(\cdot)=y(0)+\int_{0}^{\cdot}f(\tau)d\tau+\int_{0}^{\cdot}Y(\tau)dW(\tau).$
It is clear that $y(\cdot)\in
C_{\mathbb{A}}(0,T;L^{p^{\prime}}(\mathscr{C}))$. From (4.4), we obtain that

$\|y(\cdot)\|^{2}_{C_{\mathbb{A}}(0,T;L^{p^{\prime}}(\mathscr{C}))}\leq\mathcal{C}\left(\|y_{T}\|_{p^{\prime}}^{2}+\left\|\int_{0}^{T}f(t)dt\right\|_{p^{\prime}}^{2}\right).$
(4.7)

On the other hand, from (4.6) and (4.7), we have

$\|(y(\cdot),Y(\cdot)\|^{2}_{C_{\mathbb{A}}(0,T;L^{p^{\prime}}(\mathscr{C}))\times\mathcal{H}^{p^{\prime}}(0,T)}\leq\mathcal{C}\left(\|y_{T}\|^{2}_{p^{\prime}}+\left\|\int_{0}^{T}f(t)dt\right\|_{p^{\prime}}^{2}\right).$
(4.8)

Step 2. We prove that the pair $(y(\cdot),Y(\cdot))\in
C_{\mathbb{A}}(0,T;L^{p^{\prime}}(\mathscr{C}))\times\mathcal{H}^{p^{\prime}}(0,T)$
is the unique solution to (4.1). For fixed $T_{1}\in[0,T)$ and any
$(z(\cdot),Z(\cdot))\in C_{\mathbb{A}}(T_{1},T;L^{p^{\prime}}(\mathscr{C}))$
$\times\mathcal{H}^{p^{\prime}}(T_{1},T)$. We consider the following BQSDE

$\left\\{\begin{aligned} dy(t)&=f(t,z(t),Z(t))dt+Y(t)dW(t),\ {\rm in}\
[T_{1},T),\\\ y(T)&=y_{T}.\end{aligned}\right.$ (4.9)

By the result in Step 1, the pair $(y(\cdot),Y(\cdot))\in
C_{\mathbb{A}}(T_{1},T;L^{p^{\prime}}(\mathscr{C}))\times\mathcal{H}^{p^{\prime}}(T_{1},T)$
is the unique solution to (4.9). Then, we define a map

$\Gamma:C_{\mathbb{A}}(T_{1},T;L^{p^{\prime}}(\mathscr{C}))\times\mathcal{H}^{p^{\prime}}(T_{1},T)\to
C_{\mathbb{A}}(T_{1},T;L^{p^{\prime}}(\mathscr{C}))\times\mathcal{H}^{p^{\prime}}(T_{1},T)$

by

$\Gamma(z(\cdot),Z(\cdot))=(y(\cdot),Y(\cdot)).$

Next, we claim that, for $T_{1}$ being sufficiently close to $T$,

$\displaystyle\|\Gamma(z,Z)-\Gamma(\bar{z},\bar{Z})\|_{C_{\mathbb{A}}(T_{1},T;L^{p^{\prime}}(\mathscr{C}))\times\mathcal{H}^{p^{\prime}}(T_{1},T)}$
(4.10)
$\displaystyle\indent\indent\leq\frac{1}{2}\|(z(\cdot)-\bar{z}(\cdot),Z(\cdot)-\bar{Z}(\cdot))\|_{C_{\mathbb{A}}(T_{1},T;L^{p^{\prime}}(\mathscr{C}))\times\mathcal{H}^{p^{\prime}}(T_{1},T)},$
$\displaystyle\indent\indent\indent\forall(z,Z),(\bar{z},\bar{Z})\in
C_{\mathbb{A}}(T_{1},T;L^{p^{\prime}}(\mathscr{C}))\times\mathcal{H}^{p^{\prime}}(T_{1},T).$

To show (4.10), let

$(\hat{y}(\cdot),\hat{Y}(\cdot)):=\Gamma(z,Z)-\Gamma(\bar{z},\bar{Z}),\quad\hat{f}(\cdot):=f(\cdot,z(\cdot),Z(\cdot))-f(\cdot,\bar{z}(\cdot),\bar{Z}(\cdot)).$

Then, $(\hat{y}(\cdot),\hat{Y}(\cdot))$ is solution to

$\left\\{\begin{aligned} d\hat{y}(t)&=\hat{f}(t)dt+\hat{Y}(t)dW(t),\ {\rm in}\
[T_{1},T),\\\ \hat{y}(T)&=0.\end{aligned}\right.$ (4.11)

By Assumption 4.1, the Hölder inequality and (2.2), we have

$\displaystyle\|(\hat{y}(\cdot),\hat{Y}(\cdot))\|^{2}_{C_{\mathbb{A}}(T_{1},T;L^{p^{\prime}}(\mathscr{C}))\times\mathcal{H}^{p^{\prime}}(T_{1},T)}\leq\mathcal{C}\left\|\int_{T_{1}}^{T}\hat{f}(t)dt\right\|^{2}_{p^{\prime}}$
(4.12)
$\displaystyle\leq\mathcal{C}\left(\int_{T_{1}}^{T}\left\|f(t,z(t),Z(t))-f(t,\bar{z}(t),\bar{Z}(t))\right\|_{p^{\prime}}dt\right)^{2}$
$\displaystyle\leq\mathcal{C}\|z(\cdot)-\bar{z}(\cdot)\|^{2}_{C_{\mathbb{A}}(T_{1},T;L^{p^{\prime}}(\mathscr{C}))}\left(\int_{T_{1}}^{T}|g_{1}(t)|dt\right)^{2}$
$\displaystyle\indent\indent\indent\indent+\mathcal{C}\|Z(\cdot)-\bar{Z}(\cdot)\|_{\mathcal{H}^{p^{\prime}}(T_{1},T)}^{2}\int_{T_{1}}^{T}|g_{2}(t)|^{2}dt$
$\displaystyle\leq\mathcal{C}\left\\{\left(\int_{T_{1}}^{T}|g_{1}(t)|dt\right)^{2}+\int_{T_{1}}^{T}|g_{2}(t)|^{2}dt\right\\}$
$\displaystyle\indent\indent\indent\indent\cdot\|(z(\cdot)-\bar{z}(\cdot),Z(\cdot)-\bar{Z}(\cdot))\|^{2}_{C_{\mathbb{A}}(T_{1},T;L^{p^{\prime}}(\mathscr{C}))\times\mathcal{H}^{p^{\prime}}(T_{1},T)},$

and

$\displaystyle\|(y(\cdot),Y(\cdot))\|^{2}_{C_{\mathbb{A}}(T_{1},T;L^{p^{\prime}}(\mathscr{C}))\times\mathcal{H}^{p^{\prime}}(T_{1},T)}\leq\mathcal{C}\left\\{\|y_{T}\|^{2}_{p^{\prime}}+\left(\int_{T_{1}}^{T}\|f(t,z(t),Z(t))\|_{p^{\prime}}dt\right)^{2}\right\\}$
(4.13)
$\displaystyle\leq\mathcal{C}\left\\{\|y_{T}\|_{p^{\prime}}^{2}+\left(\int_{T_{1}}^{T}\|f(t,0,0)\|_{p^{\prime}}+g_{1}(t)\|z(t)\|_{p^{\prime}}+g_{2}(t)\|Z(t)\|_{p^{\prime}}dt\right)^{2}\right\\}$
$\displaystyle\indent+\mathcal{C}\left\\{\left(\int_{T_{1}}^{T}|g_{1}(t)|dt\right)^{2}+\int_{T_{1}}^{T}|g_{2}(t)|^{2}dt\right\\}\left\|(z(\cdot),Z(\cdot))\right\|^{2}_{C_{\mathbb{A}}(T_{1},T;L^{p^{\prime}}(\mathscr{C}))\times\mathcal{H}^{p^{\prime}}(T_{1},T)}.$

Let us choose $T_{1}\in[0,T)$ such that

$\mathcal{C}\left\\{\left(\int_{T_{1}}^{T}|g_{1}(t)|dt\right)^{2}+\int_{T_{1}}^{T}|g_{2}(t)|^{2}dt\right\\}\leq\frac{1}{4}.$
(4.14)

Then, by (4.12), we obtain (4.10). This shows that the map $\Gamma$ is
contractive. Hence, there exists a unique fixed point, which is a solution to
(4.1) on $[T_{1},T]$. Moreover, from (4.13) and (4.14), we obtain that

$\|(y(\cdot),Y(\cdot))\|_{C_{\mathbb{A}}(T_{1},T;L^{p^{\prime}}(\mathscr{C}))\times\mathcal{H}^{p^{\prime}}(T_{1},T)}\leq\mathcal{C}\left(\|y_{T}\|_{p^{\prime}}+\int_{T_{1}}^{T}\|f(t,0,0)\|_{p^{\prime}}dt\right).$
(4.15)

Repeating the above argument, we obtain the existence of solution to (4.1).
Finally, the uniqueness and the estimate (4.3) follow from (4.15). ∎

By Assumption 3.1 and [35, Proposition 3.3], there exists constant
$\mathcal{C}$ such that

$\displaystyle\Big{\|}\left\\{D_{x}(t)^{*}\phi+\left(F^{e}_{x}(t)-F^{o}_{x}(t)+G_{x}(t)\right)^{*}\left(\Phi_{e}-\Phi_{o}\right)-L_{x}(t)\right\\}$
$\displaystyle\indent-\left\\{D_{x}(t)^{*}\widetilde{\phi}+\left(F^{e}_{x}(t)-F^{o}_{x}(t)+G_{x}(t)\right)^{*}\left(\widetilde{\Phi}_{e}-\widetilde{\Phi}_{o}\right)-L_{x}(t)\right\\}\Big{\|}_{p^{\prime}}$
$\displaystyle=$
$\displaystyle\left\|D_{x}(t)^{*}(\phi-\widetilde{\phi})+\left(F^{e}_{x}(t)-F^{o}_{x}(t)+G_{x}(t)\right)^{*}\left(\Phi_{e}-\Phi_{o}-\widetilde{\Phi}_{e}+\widetilde{\Phi}_{o}\right)\right\|_{p^{\prime}}$
$\displaystyle\leq$
$\displaystyle\mathcal{C}\left\|\phi-\widetilde{\phi}\right\|_{p^{\prime}}+\mathcal{C}\left\|\Phi_{e}-\widetilde{\Phi}_{e}\right\|_{p^{\prime}}+\mathcal{C}\left\|\widetilde{\Phi}_{o}-\Phi_{o}\right\|_{p^{\prime}}$
$\displaystyle\leq$
$\displaystyle\mathcal{C}\left\|\phi-\widetilde{\phi}\right\|_{p^{\prime}}+2\mathcal{C}\left\|\Phi-\widetilde{\Phi}\right\|_{p^{\prime}},$

for any $\phi,\widetilde{\phi}\in L^{p^{\prime}}(\mathscr{C})$,
$\Phi,\widetilde{\Phi}\in\mathcal{H}^{p^{\prime}}(0,T)$. Hence, by Theorem
4.1, (3.55) has a unique solution in
$C_{\mathbb{A}}(0,T;L^{p^{\prime}}(\mathscr{C}))\times\mathcal{H}^{p^{\prime}}(0,T)$.

######  Remark 4.1.

In the forthcoming paper [45], we will study the solution to (3.56) by the
relaxed transposition method [30, Chapter 6] and [32, Chapter 12.4].

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