# A semigroup approach to stochastic systems with input delay at the boundary

S. Hadd  Department of Mathematics, Faculty of Sciences, Hay Dakhla, BP. 8106,
80000–Agadir, Morocco<EMAIL_ADDRESS>and  F.Z. Lahbiri  Department of
Mathematics, Faculty of Sciences, Hay Dakhla, BP. 8106, 80000–Agadir, Morocco
<EMAIL_ADDRESS>
###### Abstract.

This work focuses on the well-posedness of abstract stochastic linear systems
with boundary input delay and unbounded observation operators. We use product
spaces and a semigroup approach to reformulate such delay systems into free-
delay distributed stochastic systems with unbounded control and observation
operators. This gives us the opportunity to use the concept of admissible
control and observation operators as well as the concept of Yosida extensions
to prove the existence and uniqueness of the solution process and provide an
estimation of the observation process in relation to initial conditions and
control process. As an example, we consider a stochastic Schrödinger system
with input delay.

###### Key words and phrases:

Stochastic systems and control, admissible operators, Boundary input delay,
semigroup

###### 2020 Mathematics Subject Classification:

Primary: 93E03, 93C23; Secondary: 93B28

##  1\. Introduction

The theory of infinite dimensional systems is now a well established field,
see e.g. [6], [18], [29], [30], [31], [34]. Recently, the author of [25]
extended some parts of this theory to stochastic linear systems of infinite
dimension. In addition, the work [23] extends in a way the Salamon–Weiss
systems to Stochastic Port-Hamiltonian Systems. More recently, in [22] we
introduced a theoretical approach to a large class of stochastic infinite
dimensional systems. This approach uses the concept of Yosida extensions of
admissible observation operators to give a representation to the output
function of the system. For other works dealing with stochastic infinite
dimensional control systems we refer to [1] and [2], [9], [10], [12], [13],
[26], [27].

This work is a natural continuation of the work [22] which considers a
stochastic linear system with input delays at the boundary conditions and a
boundary observation of the form

(1.1) $\displaystyle\begin{cases}dX(t)=A_{m}X(t)dt+M(X(t))dW(t),\quad
X(0)=\xi,&t>0,\\\
GX(t)=\displaystyle\int_{-r}^{0}d\nu(\theta)U(t+\theta),&t>0,\\\
U(t)=\varphi(t),\quad\mathbb{P}-a.s.&a.e.\;t\in[-r,0],\cr
Y(t)=\mathscr{C}X(t),&t>0.\end{cases}$

Here $A_{m}:D(A_{m}):=\mathscr{Z}\subset H\to H$ is a linear closed operator
on a Hilbert space $H$, $M:H\to H$ is a linear bounded operator,
$G:\mathscr{Z}\to\mathscr{U}$ and $\mathscr{C}:\mathscr{Z}\to\mathscr{Y}$ are
linear (unbounded) operators, the trace and observation operator,
respectively, with $\mathscr{U}$ and $\mathscr{Y}$ are Hilbert spaces,
$\nu:[-r,0]\to{\mathcal{L}}(\mathscr{U})$ is a function of bounded variation
continuous at zero with total variation $|\nu|$ (a positive Borel measure)
such that $|\nu|([-\alpha,0])$ goes to zero as $\alpha\to 0$. On the other
hand, we will work with a standard one-dimensional Brownian motion
$(W(t))_{t\geq 0}$ together with a filtered probability space
($\Omega,{\mathcal{F}},\mathbb{F},\mathbb{P}$) with a natural filtration
$\mathbb{F}=\\{\mathscr{F}_{t}\\}_{t\geq 0}$ generated by $(W(t))_{t\geq 0}$.
For $t<0$, $\mathscr{F}_{t}$ is taken to be $\mathscr{F}_{0}$. Finally,
$X:[0,+\infty)\times\Omega\to H,$ denotes the solution process of the system,
$U:[-r,+\infty)\times\Omega\to\mathscr{U},$ the control process, and
$Y:[0,+\infty)\times\Omega\to\mathscr{Y},$ the observation process of the
system.

The main purpose of this work is to prove the well-posedness of the system
(1.1). We first notice that the case $\nu=B_{1}\mathbbm{1}_{\\{0\\}}$ with
$B_{1}\in\mathcal{L}(\mathscr{U})$ ( so that the boundary condition takes the
form $GX(t)=U(t)$), is recently investigated in [22] and [25], [26], [27].

In the absence of the noise (i.e. in the deterministic case), input delay
systems are well-studied using a natural feedback theory and/or a direct
computation for some particular cases such as discrete delays, that is
$\nu=\sum_{k=1}^{n}B_{k}\mathbbm{1}_{[-r_{k},0]}$ for $r_{k}\in[0,1]$ and
$B_{k}\in\mathcal{L}(\mathscr{U})$, see e.g. [5] [7], [8], [16], [35] and the
references therein. We also mention that, another easy way to prove the well-
posedness of deterministic case of (1.1) is the use of the work [20, Section
5]. In fact, with some effort one can use product spaces to reformulate the
systems as a Cauchy problem with unbounded perturbation of the generator. In
this case, the well-posedness is heavily related to the feedback theory of
regular linear systems [34].

In the presence of the noise the situation is quite different due to
sensitivity of the stochastic convolution. In this case, the lack of a natural
feedback theory for stochastic linear systems (see [22]) makes the problem
difficult. To overcome a such problem we will use a pure operator theory
approach and boundary value problem (see e.g. [4], [14]) to convert the delay
stochastic system (1.1) into a free-boundary input delay system (see the
system (3.5)). This allows us to use the recent work [22] to prove the well-
posedness of the system (1.1) (see Theorem 3.3).

In this work we will use the following notation. For any Hilbert space
${\mathcal{X}}$ and any real number $\tau\in(0,\infty)$, we denote by
$L^{2}_{\mathbb{F}}([0,\tau],{\mathcal{X}})$ the set of all
$\mathbb{F}$-adapted (${\mathcal{X}}$ valued) processes
$\zeta:[0,\tau]\times\Omega\rightarrow{\mathcal{X}}$ such that
$\mathbb{E}\|\zeta\|^{2}_{L^{2}([0,\tau],{\mathcal{X}})}<\infty.$ We denote by
$L^{2}_{{\mathcal{F}}}(\Omega,H)$ the Hilbert space of all
${\mathcal{F}}$-measurable ($H$-valued) random variables
$\xi:\Omega\rightarrow H$ satisfying $\mathbb{E}\|\xi\|_{H}^{2}<\infty$. With
these notations, the initial conditions of the problem (1.1) are supposed as
follow: $\xi\in L^{2}_{\mathscr{F}_{0}}(\Omega,H)$ and
$\varphi:[-r,0]\times\Omega\to\mathscr{U}$ a $\mathscr{F}_{0}$-measurable
process such that

$\displaystyle\mathbb{E}\int^{0}_{-r}\|\varphi(s)\|^{2}ds<\infty.$

Some classical background on semigroup theory is also required. Let
$A:D(A)\subset H\to H$ be the generator of a strongly continuous semigroup
$(T(t))_{t\geq 0}$ on $H$. We denote by $\rho(A)$ the resolvent set of $A$,
that is, the set of all $\lambda\in\mathbb{C}$ such that the inverse $(\lambda
I-A)^{-1}:=R(\lambda,A)$ exists (which is linear bounded due to the closed
graph theorem). The spectrum of $A$ is the set
$\sigma(A)=\mathbb{C}\backslash\rho(A)$. The type of the semigroup
$(T(t))_{t\geq 0}$ is defined by
$\omega_{0}(A):=\inf_{t>0}\frac{1}{t}\log\|T(t)\|$. According to [11, chap.II,
Section 5], if $\beta>\omega_{0}(A)$, then $\\{\lambda\in\mathbb{C}:{\rm
Re}\lambda>\beta\\}\subset\rho(A)$ and there exists $M\geq 1$ such that
$\|T(t)\|\leq Me^{\beta t}$ for any $t\geq 0$. On one hand, the domain $D(A)$
endowed with the following graph norm $\|x\|_{1}:=\|x\|+\|Ax\|,\;x\in D(A),$
is a Banach space. On the other hand, for $\alpha\in\rho(A),$ we define a new
norm $\|x\|_{-1}=\|R(\lambda,A)x\|$ for $x\in X$. This norm is independent of
$\alpha$. We now define $X_{-1}$ as the completion of $X$ with respect to the
norm $\|\cdot\|_{-1}$, which means that
$X_{-1}=\overline{X}^{\|\cdot\|_{-1}}$. This space is called the extrapolation
space associated with $A$ and $X$. As $H$ is a Hilbert space, then it is known
that $X_{-1}$ is isomorphic to the topological dual $D(A^{\ast})^{\prime},$
where $A^{\ast}$ is the adjoint operator of $A$. Furthermore, the semigroup
$(T(t))_{t\geq 0}$ is extended to another strongly continuous semigroup
$(T_{-1}(t))_{t\geq 0}$ on $X_{-1},$ whose generator $A_{-1}:X\to X_{-1}$ is
the extension of $A$ to $X$, see [11, Chap.II, Theorem 5.5].

The organization of this work is as follow: In Section 2, we recall some facts
about deterministic well-posed and regular linear systems. In Section 3, we
state and prove the main result of the work about the well-posedness of the
system (1.1). The last section is concerned with an example of the well-
posedness of a stochastic Schrödinger system with input delay.

##  2\. A concise background on infinite dimensional linear systems

The theory of infinite-dimensional linear systems with unbounded control and
observation operators is now a well established theory which plays a key role
in control engineering, see e.g. [6], [30], [31]. Recently, parts of this
theory has been extended to stochastic systems of infinite dimension, see
[22], [23], [25], [26],[27].

Throughout this section we use the following notations:
$(\mathfrak{X},|\cdot|)$ and
$(\partial\mathfrak{X},\|\cdot\|_{\partial\mathfrak{X}})$ are Hilbert space,
$\mathfrak{A}_{m}:D(\mathfrak{A}_{m})\subset\mathfrak{X}\to\mathfrak{X}$ is a
closed linear operator, and
$\Theta:D(\mathfrak{A}_{m})\to\partial\mathfrak{X}$ is a surjective linear
operator.

###  2.1. Admissible observation operators

Consider the observed system

(2.1) $\displaystyle\begin{cases}\dot{z}(t)=\mathfrak{A}_{m}z(t),\quad
z(0)=x,&t>0,\cr\Theta z(t)=0,&t\geq 0,\cr y(t)=\Upsilon z(t),&t\geq
0,\end{cases}$

where $\Upsilon:D(\mathfrak{A}_{m})\to\partial\mathfrak{X}$ is a linear
operator (not necessarily closed or closeable). In what follow, we assume that
the linear operator

(2.2)
$\displaystyle\mathfrak{A}:=\left(\mathfrak{A}_{m}\right)_{|D(\mathfrak{A})}\quad\text{with
domain}\quad D(\mathfrak{A}):=\ker\Theta$

generates a strongly continuous semigroup $(\mathfrak{T}(t))_{t\geq 0}$ on
$\mathfrak{X}$. We consider the linear operator

(2.3) $\displaystyle\mathfrak{C}:=\Upsilon\quad\text{with}\quad
D(\mathfrak{C}):=D(\mathfrak{A}).$

The boundary system (2.1) can be reformulated as the following distributed
system

(2.4) $\displaystyle\begin{cases}\dot{z}(t)=\mathfrak{A}z(t),\quad
z(0)=x,&t>0,\cr y(t)=\mathfrak{C}z(t),&t\geq 0,\end{cases}$

Observe that for the initial condition $x\in\mathfrak{X}$, the function
$z(t)=\mathfrak{T}(t)x,\;t\geq 0,$ is the mild solution of the differential
equation in (2.4). However, the expression
$y(t;x)=\mathfrak{C}\mathfrak{T}(t)x$ is well defined only if $x\in
D(\mathfrak{A})$. The observed linear system (2.4) (or (2.1)) is called well-
posed if for any $x\in\mathfrak{X},$ the output function $t\mapsto y(t;x)$ can
be extended to a function (denoted by the same symbol) $y(\cdot;x)\in
L^{2}_{loc}([0,+\infty),\partial\mathfrak{X})$ such that
$\|y(\cdot;x)\|_{L^{2}([0,\alpha],\partial\mathfrak{X})}\leq\gamma|x|,$ for
any $x\in\mathfrak{X}$ and $\alpha>0,$ where $\gamma:=\gamma(\alpha)>0$ is a
constant.

In the following we give conditions for which the system (2.4) is well-posed.
To this end, we need the following definition.

######  Definition 2.1.

The operator
$\mathfrak{C}\in{\mathcal{L}}(D(\mathfrak{A}),\partial\mathfrak{X})$ is called
an admissible observation operator for $\mathfrak{A},$ if for some (hence all)
$\alpha>0,$ there exists a constant $\gamma:=\gamma(\alpha)>0$ such that

(2.5)
$\displaystyle\int^{\alpha}_{0}\|\mathfrak{C}\mathfrak{T}(t)x\|^{2}_{\partial\mathfrak{X}}dt\leq\gamma^{2}|x|^{2}$

for any $x\in D(\mathfrak{A})$. In this case, we also say that
$(\mathfrak{C},\mathfrak{A})$ is admissible.

Now if $(\mathfrak{C},\mathfrak{A})$ is admissible then the system (2.4) is
well-posed. In fact, the estimate (2.5) together with the density of
$D(\mathfrak{A})$ in $\mathfrak{X}$ imply that the following map

(2.6) $\displaystyle\Psi^{\mathfrak{A},\mathfrak{C}}:D(\mathfrak{A})\to
L^{2}([0,\alpha],\partial\mathfrak{X}),\quad
x\mapsto\Psi^{\mathfrak{A},\mathfrak{C}}x:=\mathfrak{C}\mathfrak{T}(\cdot)x=y(\cdot;x)$

has a unique linear bounded extension to $\mathfrak{X}$ (denoted by the same
symbol)
$\Psi^{\mathfrak{A},\mathfrak{C}}\in{\mathcal{L}}(\mathfrak{X},L^{2}([0,\alpha],\partial\mathfrak{X}))$.
In order to give a representation of $\Psi^{\mathfrak{A},\mathfrak{C}},$ let
us consider the following operator (called the Yosida extension of
$\mathfrak{C}$ for $\mathfrak{A}$),

(2.7)
$\displaystyle\begin{split}D(\mathfrak{C}_{\Lambda})&=\left\\{z\in\mathfrak{X}:\lim_{\lambda\to+\infty}\mathfrak{C}\lambda
R(\lambda,\mathfrak{A})z\;\text{exists
in}\;\partial\mathfrak{X}\right\\}\cr\mathfrak{C}_{\Lambda}z&:=\lim_{\lambda\to+\infty}\mathfrak{C}\lambda
R(\lambda,\mathfrak{A})z.\end{split}$

It is shown in [32] that if $\mathfrak{C}$ is an admissible observation
operator for $\mathfrak{A}$, then

(2.8) $\displaystyle\mathfrak{T}(t)x\in
D(\mathfrak{C}_{\Lambda}),\quad\text{and}\quad(\Psi^{\mathfrak{A},\mathfrak{C}}x)(t)=\mathfrak{C}_{\Lambda}\mathfrak{T}(t)x,\quad\forall
x\in\mathfrak{X},\;a.e.\;t>0.$

Thus if $(\mathfrak{C},\mathfrak{A})$ is well-posed, then the extended output
function of the system (2.4) is

$\displaystyle y(t;x)=\mathfrak{C}_{\Lambda}\mathfrak{T}(t)x$

for any $x\in X,$ and a.e. $t>0$.

###  2.2. Admissible control operators

Consider the following boundary control system

(2.9) $\displaystyle\begin{cases}\dot{z}(t)=\mathfrak{A}_{m}z(t),\quad
z(0)=x,&t>0,\cr\Theta z(t)=u(t),&t\geq 0,\end{cases}$

where $u:[0,+\infty)\to\partial\mathfrak{X}$ is the control function assumed
to be a $2$-integrable function. According to [14], the fact that $\Theta$ is
surjective implies that the domain $D(\mathfrak{A}_{m})$ can be viewed as the
direct sum of $D(\mathfrak{A})$ and $\ker(\lambda-\mathfrak{A}_{m})$ for any
$\lambda\in\rho(\mathfrak{A})$. In particular, the following inverse exists

$\mathfrak{D}_{\lambda}:=(\Theta_{|\ker(\lambda-\mathfrak{A}_{m})})^{-1}:\partial\mathfrak{X}\rightarrow\ker(\lambda-\mathfrak{A}_{m}),\qquad\lambda\in\rho(\mathfrak{A}).$

The operator $\mathfrak{D}_{\lambda}$ is called the Dirichlet operator
associated with $\mathfrak{A}_{m}$ and $\Theta$, and satisfies
$\mathfrak{D}_{\lambda}\in{\mathcal{L}}(\partial\mathfrak{X},\mathfrak{X})$
for any $\lambda\in\rho(\mathfrak{A})$. We select the following operator

(2.10)
$\displaystyle\mathfrak{B}:=(\lambda-\mathfrak{A}_{-1})\mathfrak{D}_{\lambda}\in{\mathcal{L}}(\partial\mathfrak{X},\mathfrak{X}_{-1}),$

where $\mathfrak{A}_{-1}:\mathfrak{X}\to\mathfrak{X}$ is the generator of the
extrapolation semigroup extension of $(\mathfrak{T}(t))_{t\geq 0}$ (see
notations in the introductory section). As for any $v\in\partial\mathfrak{X}$
and $\lambda\in\rho(\mathfrak{A}),$ we have
$\mathfrak{A}_{m}\mathfrak{D}_{\lambda}v=\lambda\mathfrak{D}_{\lambda}v,$ then
$(\mathfrak{A}_{m}-\mathfrak{A}_{-1})\mathfrak{D}_{\lambda}v=(\lambda-\mathfrak{A}_{-1})\mathfrak{D}_{\lambda}v=\mathfrak{B}v$.
As $\mathfrak{D}_{\lambda}$ is the inverse of $\Theta,$ we obtain

(2.11)
$\displaystyle\mathfrak{A}_{m}=(\mathfrak{A}_{-1}+\mathfrak{B}\Theta)_{|D(\mathfrak{A}_{m})}.$

Using the equality (2.11), the boundary system (2.13) can be reformulated as

(2.12) $\displaystyle\dot{z}(t)=\mathfrak{A}z(t)+\mathfrak{B}u(t),\quad
z(0)=x,\;t>0.$

The mild (or integral) solution of the nonhomogeneous equation (2.12) is given
by

$\displaystyle
z(t)=\mathfrak{T}(t)x+\int^{t}_{0}\mathfrak{T}_{-1}(t-s)\mathfrak{B}u(s)ds$

for any $t\geq 0,\;x\in\mathfrak{X}$ and $u\in
L^{2}([0,+\infty),\partial\mathfrak{X})$. This solution satisfies
$z(t)\in\mathfrak{X}_{-1}$ for $t>0$. Usually, we are interested in having
$\mathfrak{X}$-valued solution of the equation (2.12). To that purpose, we
need the following definition.

######  Definition 2.2.

The operator
$\mathfrak{B}\in{\mathcal{L}}(\partial\mathfrak{X},\mathfrak{X}_{-1})$ is
called an admissible control operator for $\mathfrak{A}$ if there exists
$\tau>0$ such that

$\displaystyle\Phi^{\mathfrak{A},\mathfrak{B}}_{\tau}u:=\int^{\tau}_{0}\mathfrak{T}_{-1}(\tau-s)\mathfrak{B}u(s)ds\in\mathfrak{X}$

for any $u\in L^{2}([0,+\infty,\partial\mathfrak{X})$. In this case, we also
say that $(\mathfrak{A},\mathfrak{B})$ is admissible.

Using the closed graph theorem, it is shown in [33] that if
$(\mathfrak{A},\mathfrak{B})$ is admissible, then

$\displaystyle\Phi^{\mathfrak{A},\mathfrak{B}}_{t}\in{\mathcal{L}}(L^{2}([0,+\infty),\partial\mathfrak{X}),\mathfrak{X}),\qquad\forall
t\geq 0.$

Thus if $\mathfrak{B}$ is an admissible control operator for $\mathfrak{A},$
the solution of (2.12) is continuous function from $[0,+\infty)$ to
$\mathfrak{X}$.

###  2.3. Well-posed linear systems

The object of this part it to define the concept of well-posedess for boundary
control systems with a point observation. In fact, we consider the augmented
system

(2.13) $\displaystyle\begin{cases}\dot{z}(t)=\mathfrak{A}_{m}z(t),\quad
z(0)=x,&t>0,\cr\Theta z(t)=u(t),&t\geq 0,\cr y(t)=\Upsilon z(t),&t\geq
0,\end{cases}$

where the operators $\mathfrak{A}_{m},\Theta$ and $\Upsilon$ are defined
before as above. According to the previous two subsection, the system (2.13)
can be reformulated as the following input-output distributed linear system

(2.14)
$\displaystyle\begin{cases}\dot{z}(t)=\mathfrak{A}z(t)+\mathfrak{B}u(t),\quad
z(0)=x,&t>0,\cr y(t)=\mathfrak{C}z(t),&t\geq 0,\end{cases}$

where the operators $\mathfrak{A},\mathfrak{C}$ and $\mathfrak{B}$ are given
by (2.2), (2.3) and (2.10), respectively.

######  Definition 2.3.

We say that the system (2.14) (or (2.13)) is well-posed if it admits a
continuous mild solution $z:[0,+\infty)\to\mathfrak{X}$ and an output function
$t\mapsto y(t;x,u)$ that can be extended to a locally $2$-integrable function
on $[0,+\infty)$ such that

$\displaystyle\|y(\cdot;x,u)\|_{L^{2}([0,\alpha],\partial\mathfrak{X})}\leq\kappa\left(|x|+\|u\|_{L^{2}([0,\alpha],\partial\mathfrak{X})}\right)$

for any $x\in X,\alpha>0$ and $u\in L^{2}([0,\alpha],\partial\mathfrak{X})$,
where $\kappa:=\kappa(\alpha)>0$ is a constant.

######  Remark 1.

(i) As discussed in Subsection 2.2, a sufficient condition for the existence
of a continuous solution $z:[0,+\infty)\to\mathfrak{X}$ of the differential
equation in (2.14) is that $(\mathfrak{A},\mathfrak{B})$ is well-posed.
(ii) Even if $(\mathfrak{A},\mathfrak{B})$ is well-posed, the expression
$"\mathfrak{C}z(t)"$ is not well defined for a general initial condition
$x\in\mathfrak{X}$ and all control $u\in
L^{2}([0,+\infty),\partial\mathfrak{C})$. To overcome this obstacle, for any
$t>0,$ we define the space

$\displaystyle u\in W^{2,2}_{0,t}(\partial\mathfrak{X}):=\\{u\in
W^{2,2}([0,t],\partial\mathfrak{X}):u(0)=\dot{u}(0)=0\\}.$

This space is dense in $L^{2}([0,t],\partial\mathfrak{X})$. Now assume
(without loss of generality) $0\in\rho(\mathfrak{A})$ and take $u\in
W^{2,2}_{0,t}(\partial\mathfrak{X})$. Then an integration by parts implies

$\displaystyle\Phi^{\mathfrak{A},\mathfrak{B}}_{t}u$
$\displaystyle=\int^{t}_{0}\mathfrak{T}_{-1}(t-s)(-\mathfrak{A}_{-1})\mathfrak{D}_{0}u(s)ds$
$\displaystyle=\left[\mathfrak{T}(t-s)\mathfrak{D}_{0}u(s)\right]^{s=t}_{s=0}-\int^{t}_{0}\mathfrak{T}(t-s)\mathfrak{D}_{0}\dot{u}(s)ds$
$\displaystyle=\mathfrak{D}_{0}u(t)-\int^{t}_{0}\mathfrak{T}(t-s)\mathfrak{D}_{0}\dot{u}(s)ds.$

Thus $\Phi^{\mathfrak{A},\mathfrak{B}}_{t}u\in D(\mathfrak{A}_{m})$ for $t\geq
0$. Moreover, for any $(x,u)\in D(\mathfrak{A})\times
W^{2,2}_{0,\alpha}(\partial\mathfrak{X}),$ for any $t\in[0,\alpha],$ we have

$\displaystyle y(t)$
$\displaystyle=\Upsilon(\mathfrak{T}(t)x+\Phi^{\mathfrak{A},\mathfrak{B}}_{t}u)$
$\displaystyle=\mathfrak{C}\mathfrak{T}(t)x+\Upsilon\Phi^{\mathfrak{A},\mathfrak{B}}_{t}u$
$\displaystyle=\left(\Psi^{\mathfrak{A},\mathfrak{C}}x+{\bf
F}^{\mathfrak{A},\mathfrak{B},\mathfrak{C}}u\right)(t),$

where $\Psi^{\mathfrak{A},\mathfrak{A}}$ is defined in (2.6) and

(2.15) $\displaystyle\left({\bf
F}^{\mathfrak{A},\mathfrak{B},\mathfrak{C}}u\right)(t)=\Upsilon\Phi^{\mathfrak{A},\mathfrak{B}}_{t}u.$

By combining Definition 2.3 and Remark 1 one can easily prove the following
result.

######  Proposition 1.

The input-output system (2.14) (or (2.13)) is well-posed if and only if the
following assertions hold

  * (i)

$(\mathfrak{A},\mathfrak{B})$ is admissible.

  * (ii)

$(\mathfrak{C},\mathfrak{A})$ is admissible.

  * (iii)

There exists a constant $\delta>0$ such that

(2.16) $\displaystyle\left\|{\bf
F}^{\mathfrak{A},\mathfrak{B},\mathfrak{C}}u\right\|_{L^{2}([0,\alpha],\partial\mathfrak{X})}\leq\kappa\|u\|_{L^{2}([0,\alpha],\partial\mathfrak{X})}$

for any $u\in W^{2,2}_{0,\alpha}(\partial\mathfrak{X})$, where ${\bf
F}^{\mathfrak{A},\mathfrak{B}}$ is the linear operator define in (2.15). In
this case we also say that $(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is well-
posed triple.

If the system (2.14) is well-posed, then the density of
$W^{2,2}_{0,\alpha}(\partial\mathfrak{X})$ in
$L^{2}([0,\alpha],\partial\mathfrak{X})$ and the inequality (2.16) imply that
the operator ${\bf F}^{\mathfrak{A},\mathfrak{B},\mathfrak{C}}$ is extended to
a linear bounded operator (denoted by the same symbol) ${\bf
F}^{\mathfrak{A},\mathfrak{B},\mathfrak{C}}\in{\mathcal{L}}(L^{p}([0,\alpha],\partial\mathfrak{X}))$
for any $\alpha>0$. Furthermore, the extended output function of the system
(2.14) is given by

(2.17)
$\displaystyle\begin{split}y(t;x,u)&=\left(\Psi^{\mathfrak{A},\mathfrak{C}}x+\textbf{F}^{\mathfrak{A},\mathfrak{B},\mathfrak{C}}u\right)(t)\cr&=\mathfrak{C}_{\Lambda}\mathfrak{T}(t)x+\left(\textbf{F}^{\mathfrak{A},\mathfrak{B},\mathfrak{C}}u\right)(t)\end{split}$

for a.e. $t>0,$ and all $(x,u)\in\mathfrak{X}\times
L^{2}_{loc}([0,+\infty),\partial\mathfrak{X})$.

Next, in order to give a complete representation of the output function
$y(\cdot)$ of the system (2.14), we need the following subclass of well-posed
systems.

######  Definition 2.4.

A well-posed triple $(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is called
regular if

$\displaystyle\lim_{t\to 0^{+}}\frac{1}{t}\int^{t}_{0}\left({\bf
F}^{\mathfrak{A},\mathfrak{B},\mathfrak{C}}\mathbbm{1}_{[0,+\infty)}v_{0}\right)(s)ds=0$

for any $v_{0}\in\partial\mathfrak{X}$.

This definition is also equivalent to ${\rm
Range}(\mathfrak{D}_{\lambda})\subset D(\mathfrak{C}_{\Lambda})$ for some
(hence all) $\lambda\in\rho(\mathfrak{A})$.

According to [20, Lem.3.6], if $(\mathfrak{A},\mathfrak{B},\mathfrak{C})$ is
regular, then

(2.18) $\displaystyle D(\mathfrak{A}_{m})\subset
D(\mathfrak{C}_{\Lambda})\quad\text{and}\quad(\mathfrak{C}_{\Lambda})_{|D(\mathfrak{A}_{m})}=\Upsilon.$

Furthermore, due to [34], we also have

$\displaystyle\Phi^{\mathfrak{A},\mathfrak{B}}_{t}u\in
D(\mathfrak{C}_{\Lambda})\quad\text{and}\quad\left({\bf
F}^{\mathfrak{A},\mathfrak{B},\mathfrak{C}}u\right)(t)=\mathfrak{C}_{\Lambda}\Phi^{{\mathcal{A}},{\mathcal{B}}}_{t}u$

for a.e. $t\geq 0$ and all $u\in L^{2}([0,+\infty),\partial\mathfrak{X})$.
Thus, in this case, the state trajectory of the system (2.14) (equivalently of
the system (2.13))
$z(t)=\mathfrak{T}(t)x+\Phi^{\mathfrak{A},\mathfrak{B}}_{t}u$ satisfies
$z(t)\in D(\mathfrak{C}_{\Lambda})$ and $y(t;x,u)=\mathfrak{C}_{\Lambda}z(t)$
for a.e. $t>0$ and all $(x,u)\in\mathfrak{X}\times
L^{2}_{loc}([0,+\infty),\partial\mathfrak{X})$, due to (2.17).

In the following example we provide a regular system associated with the left
shift semigroup on a Lebesque space. This example is important in the
reformulation of the input delay stochastic system (1.1) to a free-delay
stochastic system in product spaces (see Section 3).

######  Example 1.

Let $\mathscr{U}$ be a Hilbert space. On the Lebesgue space
$L^{2}([-r,0],\mathscr{U})$ (with $r>0$ is a fixed real number), we consider
the operator $Q_{m}g=g^{\prime}$ (the fist derivative) for $g\in
W^{1,2}([-r,0],\mathscr{U})$. This operator is closed. We consider the dirac
operator at zero $\delta_{0}:W^{1,2}([-r,0],\mathscr{U})\to\mathscr{U}$ such
that $\delta_{0}g=g(0),$ which is a surjective operator. Now consider the
operator

$\displaystyle Qg=g^{\prime},\quad D(Q)=\ker\delta_{0}=\left\\{g\in
W^{1,2}([-r,0],\mathscr{U}):g(0)=0\right\\}.$

It is well known (see e.g. [11, Chap.II]) that $Q$ generates the left shift
semigroup $S:=(S(t))_{t\geq 0}$ on $L^{2}([-r,0],\mathscr{U})$ defined by

$\displaystyle(S(t)g)(\theta)=\begin{cases}0,&-t\leq\theta\leq 0,\cr
g(t+\theta),&-r\leq\theta<-t.\end{cases}$

The Dirichlet operator associated with $Q_{m}$ and $\delta$ is
$d_{\lambda}=e_{\lambda}$ for $\lambda\in\rho(Q)=\mathbb{C},$ where
$e_{\lambda}:\mathscr{U}\to L^{2}([-r,0],\mathscr{U})$ is such that
$(e_{\lambda}v)(\theta)=e^{\lambda\theta}v$ for any $\theta\in[-r,0]$ and
$v\in\mathscr{U}$. We select $\beta:=(\lambda-
Q_{-1})d_{\lambda}\in{\mathcal{L}}\left(\mathscr{U},(L^{2}([-r,0],\mathscr{U}))_{-1}\right)$.
It is known that $\beta$ is an admissible control operator for $Q$ and the
associated control maps are given by

$\displaystyle\left(\Phi^{Q,\beta}_{t}u\right)(\theta)=\begin{cases}u(t+\theta),&-t\leq\theta\leq
0,\cr 0,&-r\leq\theta<-t,\end{cases}$

for any $u\in L^{2}([0,+\infty),\mathscr{U})$, see [19]. Now we define

(2.19) $\displaystyle Lg=\int^{0}_{-r}d\mu(\theta)g(\theta),\qquad g\in
W^{1,2}([-r,0],\mathscr{U}),$

where $\mu:[-r,0]\to{\mathcal{L}}(\mathscr{U})$ is a function of bounded
variation with total variation $|\mu|$ satisfying $|\mu|([-\varepsilon,0])\to
0$ as $\varepsilon\to 0$. If we denote by $\mathbb{L}:=L_{|D(Q)}$ then
$\mathbb{L}$ is an admissible observation operator for $Q,$ see [17, Lem.6.2].
Further more, according to [19], the triple $(Q,\beta,\mathbb{L})$ is regular.

##  3\. Well-posedness of boundary input delay stochastic systems

Throughout this section assume the following setting for the system (1.1). The
operators $A_{m},G,\mathscr{M}$ and $\mathscr{C}$ are as in the introductory
section. In addition, we suppose that $A:=(A_{m})_{|D(A)}$ with $D(A)=\ker G$
generates a strongly continuous semigroup $T:=(T(t))_{t\geq 0}$ on the space
$H$ and the boundary operator $G:\mathscr{Z}\to\mathscr{U}$ is surjective.
Similarly to Section 2, we consider the following Dirichlet operator
associated with $A_{m}$ and $G,$

$\mathbb{D}_{\lambda}:=(G_{|\ker(\lambda-
A_{m})})^{-1}:\mathscr{U}\rightarrow\ker(\lambda-
A_{m}),\qquad\lambda\in\rho(A).$

Thus we have $\mathbb{D}_{\lambda}\in{\mathcal{L}}(\mathscr{U},H)$ for any
$\lambda\in\rho(A)$. We select the following operators

(3.1) $\displaystyle B:=(\lambda-
A_{-1})\mathbb{D}_{\lambda}\in{\mathcal{L}}(\mathscr{U},H_{-1})\quad\text{and}\quad
C:=\mathscr{C}_{|D(A)}\in{\mathcal{L}}(D(A),\mathscr{Y}).$

With these notation we can write

(3.2) $\displaystyle A_{m}=\left(A_{-1}+BG\right)_{|\mathscr{Z}}.$

For the control process $U:[-r,\infty)\times\Omega\to\mathscr{U}$ and for each
$t\geq 0,$ we denote by $U_{t}:[-r,0]\times\Omega\to\mathscr{U}$ the history
of $U$ defined by $U_{t}(\theta,\omega)=U(t+\theta,\omega)$ for a.e.
$\theta\in[-r,0]$ and $\mathbb{P}$-a.s $\omega\in\Omega$. Now we set
$V(t,\theta,\omega):=U_{t}(\theta,\omega)$ for $t\geq 0,\;\theta\in[-r,0]$ and
$\mathbb{P}$-a.s $\omega\in\Omega$. This process satisfies the following
boundary control problem

(3.3) $\displaystyle\begin{split}&\frac{\partial}{\partial
t}V(t,\cdot)=\frac{\partial}{\partial\theta}V(t,\cdot),\quad
V(0,\cdot)=\varphi,\quad t>0,\cr&V(t,0)=U(t),\quad t>0.\end{split}$

In the sequel we will transform the input delay system (1.1) to a free delay
boundary control system. For this, we define the product space

$\displaystyle\mathscr{H}=H\times L^{2}([-r,0],\mathscr{U})\quad\text{with
norm}\quad\left\|\left(\begin{smallmatrix}x\\\
g\end{smallmatrix}\right)\right\|=\|x\|+\|g\|_{2}.$

We introduce the new state

(3.4) $\displaystyle Z(t)=\begin{pmatrix}X(t)\\\ U_{t}\end{pmatrix},\qquad
t\geq 0.$

We also define the following operators

$\displaystyle\mathscr{A}_{m}:=\begin{pmatrix}A_{m}&0\\\
0&Q_{m}\end{pmatrix},\quad
D(\mathscr{A}_{m}):=\left\\{\left(\begin{smallmatrix}x\\\
g\end{smallmatrix}\right)\in\mathscr{Z}\times
W^{1,2}([-r,0],\mathscr{U}):Gx=Lg\right\\},$
$\displaystyle\mathscr{G}=(0\;\;\delta_{0}):D(\mathscr{A}_{m})\to\mathscr{U},$
$\displaystyle\mathscr{M}(\begin{smallmatrix}x\\\
g\end{smallmatrix})=\left(\begin{smallmatrix}Mx\\\
0\end{smallmatrix}\right),\quad(\begin{smallmatrix}x\\\
g\end{smallmatrix})\in\mathscr{H},$
$\displaystyle\mathscr{N}:=(\mathscr{C}\;\;0):D(\mathscr{A}_{m})\to\mathscr{Y},$

where the operators $Q_{m}$ and $L$ are defined in Example 1. Using these
operators and the system (3.3), the initial input delay system (1.1) becomes

(3.5)
$\displaystyle\begin{cases}dZ(t)=\mathscr{A}_{m}Z(t)dt+\mathscr{M}(Z(t))dW(t),\quad
Z(0)=(\begin{smallmatrix}\xi\\\
\varphi\end{smallmatrix}),&t>0,\cr\mathscr{G}Z(t)=U(t),&t\geq 0,\cr
Y(t)=\mathscr{N}Z(t),&t\geq 0.\end{cases}$

In order to master the system (3.5), it is more convenient to reformulate it
as a distributed system on $\mathscr{H},\mathscr{U}$ and $\mathscr{Y}$. We
will use the work [14]. We need the following lemmas.

######  Lemma 3.1.

The operator $\mathscr{G}:D(\mathscr{A}_{m})\to\mathscr{U}$ is surjective.

###### Proof.

Let $v\in\mathscr{U}$ and set $g=e_{\lambda}v$, so $g\in
W^{1,2}([-r,0],\mathscr{U})$ and $\delta_{0}g=v$. On the other hand, as $G$ is
surjective and $Lg\in U$ then there exists $x\in\mathscr{Z}$ such that
$Gx=Lg$. This proves that $(\begin{smallmatrix}x\\\ g\end{smallmatrix})\in
D(\mathscr{A}_{m})$ and $\mathscr{G}(\begin{smallmatrix}x\\\
g\end{smallmatrix})=v$. ∎

######  Lemma 3.2.

The operator $\mathscr{A}_{m}$ is closed.

###### Proof.

Let $(\begin{smallmatrix}x_{n}\\\ g_{n}\end{smallmatrix})\in
D(\mathscr{A}_{m})$ and $(\begin{smallmatrix}x\\\
g\end{smallmatrix}),(\begin{smallmatrix}\overline{x}\\\
\overline{g}\end{smallmatrix})\in\mathscr{H}$ such that
$\|x_{n}-x\|+\|g_{n}-g\|_{2}\to 0$ and
$\|A_{m}x_{n}-\overline{x}\|+\|g^{\prime}_{n}-\overline{g}\|_{2}\to 0$ as
$n\to\infty$. As $(A_{m},\mathscr{Z})$ and
$(\frac{d}{d\theta},W^{1,2}([-r,0],\mathscr{U}))$ are closed then
$x\in\mathscr{Z},$ $g\in W^{1,2}([-r,0],\mathscr{U}),$ and
$A_{m}x=\overline{x},$ $g^{\prime}=\overline{g}$. Moreover, for any $n$
$Gx_{n}=Lg_{n}$ and we can write $x_{n}=x_{n}^{0}+x_{n}^{1}$ with
$x_{n}^{0}\in D(A)$ and $x_{n}^{1}\in\ker(\lambda-A_{m})$. Thus
$Gx_{n}^{1}=Lg_{n},$ so that $x_{n}^{1}=\mathbb{D}_{\lambda}Lg_{n}$ for any
$n$. Let show that $Lg_{n}\to Lg$ as $n\to\infty$. In fact, we know that there
exists a set $N\subset[-r,0]$ with Lebsegue measure $\lambda(N)=0$ and
$g_{n_{k}}(\theta)\to g(\theta)$ for any $\theta\in[-r,0]\backslash N$ as
$k\to\infty$. Now choose $\theta_{0}\in[-r,0]\backslash N$ and write

(3.6) $\displaystyle
g_{n_{k}}=e_{0}g_{n_{k}}(\theta_{0})+h_{n_{k}}\quad\text{and}\quad
g=e_{0}g(\theta_{0})+h$

where

$\displaystyle
h_{n}(\theta)=\int^{\theta_{0}}_{\theta}(-g_{n}^{\prime})(\sigma)d\sigma\quad\text{and}\quad
h(\theta)=\int^{\theta_{0}}_{\theta}(-g^{\prime})(\sigma)d\sigma.$

According to (3.6),

$\displaystyle\|Lg_{n_{k}}-Lg\|\leq|\mu|([-r,0])\left(\|g_{n_{k}}(\theta_{0})-g(\theta_{0})\|+\|h_{n_{k}}-h\|_{\infty}\right).$

Using Hölder inequality, we obtain

$\displaystyle\|h_{n}-h\|_{\infty}\leq
r^{\frac{1}{2}}\|g_{n}^{\prime}-g^{\prime}\|_{L^{2}([-r,0],\mathscr{U})}\to
0\quad(n\to\infty).$

This implies that $Lg_{n_{k}}\to Lg$ as $k\to\infty$, so that
$\mathbb{D}_{\lambda}Lg_{n}\to\mathbb{D}_{\lambda}Lg$. Thus
$x_{n_{k}}^{1}\to\mathbb{D}_{\lambda}Lg$, so that $x_{n_{k}}^{0}\to
x-\mathbb{D}_{\lambda}Lg$ as $k\to\infty$. On the other hand,
$Ax_{n_{k}}^{0}=A_{m}x_{n_{k}}-A_{m}x_{n_{k}}^{1}=A_{m}x_{n_{k}}-\lambda
x_{n_{k}}^{1}$ for any $k$. This implies that $Ax_{n_{k}}^{0}\to
A_{m}x-\lambda\mathbb{D}_{\lambda}Lg$. As $A$ is closed then in particular
$x-\mathbb{D}_{\lambda}Lg\in D(A)=\ker G,$ with means that $Gx=Lg$. Then
$(\begin{smallmatrix}x\\\ g\end{smallmatrix})\in D(\mathscr{A}_{m})$ and
$\mathscr{A}_{m}(\begin{smallmatrix}x\\\
g\end{smallmatrix})=(\begin{smallmatrix}\overline{x}\\\
\overline{g}\end{smallmatrix})$. This shows that $\mathscr{A}_{m}$ is closed.
∎

Let $\mathbb{L}$ as in Example 1.

######  Proposition 2.

Assume that $B$ is an admissible control operator for $A$. The following
operator

(3.7)
$\displaystyle\mathscr{A}:=\left(\mathscr{A}_{m}\right)_{|D(\mathscr{A})}\quad\text{with}\quad
D(\mathscr{A}):=\ker\mathscr{G}$

generates a strongly continuous semigroup
$\mathscr{T}:=(\mathscr{T}(t))_{t\geq 0}$ on $\mathscr{H}$ given by

(3.8) $\displaystyle\mathscr{T}(t)=\begin{pmatrix}T(t)&R(t)\\\
0&S(t)\end{pmatrix},\quad t\geq 0,$

where $R(t):L^{2}([-r,0],\mathscr{U})\to H$ is defined by

$\displaystyle R(t)g=\int^{t}_{0}T_{-1}(t-s)B\mathbb{L}_{\Lambda}S(s)gds.$

###### Proof.

Let us prove that the family of operators $(\mathscr{T}(t))_{t\geq 0}$ define
a strongly continuous semigroup on $\mathscr{H}$. Clearly
$\mathscr{T}(0)=I_{\mathscr{H}}$. On one hand, the strong continuity of this
family follows from the fact that $R(t)f\to 0$ as $t\to 0^{+}$, for any $f\in
L^{2}([-r,0],\mathscr{U})$, and the fact that $T$ and $S$ are
$C_{0}$–semigroups on $H$ and $L^{2}([-r,0],\mathscr{U})$, respectively. On
the other hand, for any $t,s\geq 0,$ we have

$\displaystyle\mathscr{T}(t)\mathscr{T}(s)=\begin{pmatrix}T(t+s)&T(t)R(s)+R(t)S(s)\\\
0&S(t+s)\end{pmatrix}.$

Moreover, a simple change of variables shows that $T(t)R(s)+R(t)S(s)=R(t+s)$.
Thus $\mathscr{T}(t)\mathscr{T}(s)=\mathscr{T}(t+s)$. Let
$E:D(E)\subset\mathscr{H}\to\mathscr{H}$ be the generator of $\mathscr{T}$ and
prove that $E=\mathscr{A}$. Let $\lambda>0$ be sufficiently large. By taking
Laplace Transform in both sides of (3.8), we obtain

(3.9) $\displaystyle
R(\lambda,E)=\begin{pmatrix}R(\lambda,A)&\mathbb{D}_{\lambda}\mathbb{L}R(\lambda,Q)\\\
0&R(\lambda,Q)\end{pmatrix}.$

Let now $(\begin{smallmatrix}x\\\ f\end{smallmatrix})\in D(\mathscr{A})$ and
$(\begin{smallmatrix}x_{0}\\\ f_{0}\end{smallmatrix})\in\mathscr{H}$ such that

$\displaystyle(\lambda-\mathscr{A})(\begin{smallmatrix}x\\\
f\end{smallmatrix})=(\begin{smallmatrix}x_{0}\\\ f_{0}\end{smallmatrix}).$

In particular, we have $(\lambda-A_{m})x=x_{0}$ and $f=R(\lambda,Q)f_{0}$. As
$x\in\mathscr{Z},$ then by (3.2), we have $(\lambda-A_{-1})x-BGx=x_{0}$. But
$Gx=Lf=LR(\lambda,Q)f_{0}$ and $\mathbb{D}_{\lambda}=R(\lambda,A_{-1})B$. Then
$x=R(\lambda,A)x_{0}+\mathbb{D}_{\lambda}LR(\lambda,Q)f_{0}$. This implies
that $\lambda-\mathscr{A}$ is invertible and
$(\lambda-\mathscr{A})^{-1}=R(\lambda,E),$ due to (3.9). Hence
$\mathscr{A}=E$. This ends the proof. ∎

The following result is needed in the proof of the main result of this work.

######  Proposition 3.

Assume that the triple $(A,B,C)$ is regular. The following operator

$\displaystyle\mathscr{P}:=\mathscr{N}_{|D(\mathscr{A})}\in{\mathcal{L}}(D(\mathscr{A}),\mathscr{U})$

is an admissible observation operator for $\mathscr{A}$. Furthermore, if
$\mathscr{P}_{\Lambda},\,C_{\Lambda}$ and $\mathbb{L}_{\Lambda}$ denote Yosida
extension of $\mathscr{P},C$ and $\mathbb{L}$, for $\mathscr{A},$ $A$ and $Q,$
respectively, then

(3.10) $\displaystyle D(C_{\Lambda})\times D(\mathbb{L}_{\Lambda})\subset
D(\mathscr{P}_{\Lambda})\quad\text{and}\quad\left(\mathscr{P}\right)_{|D(C_{\Lambda})\times
D(\mathbb{L}_{\Lambda})}=(C_{\Lambda}\;\;0).$

###### Proof.

First, we give some information about the Yosida extension of $\mathscr{P}$
for $\mathscr{A}$. Let $\lambda>0$ be sufficiently large and let
$(\begin{smallmatrix}x\\\ g\end{smallmatrix})\in D(C_{\Lambda})\times
D(\mathbb{L}_{\Lambda})$. According to (3.9), we have

$\displaystyle\mathscr{P}\lambda
R(\lambda,\mathscr{A})(\begin{smallmatrix}x\\\ g\end{smallmatrix})=C\lambda
R(\lambda,A)+C\mathbb{D}_{\lambda}\mathbb{L}\lambda R(\lambda,Q)g,$
$\displaystyle\|C\mathbb{D}_{\lambda}\mathbb{L}\lambda
R(\lambda,Q)g\|\leq\|C\mathbb{D}_{\lambda}\|\left(\|\mathbb{L}\lambda
R(\lambda,Q)g-\mathbb{L}_{\Lambda}g\|+\|\mathbb{L}_{\Lambda}g\|\right).$

As the transfer function of the regular system $(A,B,C)$ satisfies
$C\mathbb{D}_{\lambda}\to 0$ as $\lambda\to+\infty$, the property (3.10)
follows. Second, we show that $\mathscr{P}$ is an admissible observation
operator for $\mathscr{A}$. In fact, let $(\begin{smallmatrix}x\\\
g\end{smallmatrix})\in D(\mathscr{A})$ and $\alpha>0$. Then

$\displaystyle\int^{\alpha}_{0}\|\mathscr{P}\mathscr{T}(t)(\begin{smallmatrix}x\\\
g\end{smallmatrix})\|^{2}dt=\int^{\alpha}_{0}\|\mathscr{P}_{\Lambda}\mathscr{T}(t)(\begin{smallmatrix}x\\\
g\end{smallmatrix})\|^{2}dt$

As $(A,B,C)$ is regular and $\mathbb{L}$ is an admissible observation operator
for $Q,$ then

$\displaystyle T(t)x\in D(C_{\Lambda}),\quad
R(t)g=\Phi^{A,B}_{t}\mathbb{L}_{\Lambda}S(\cdot)g\in D(C_{\Lambda}),\quad
S(t)g\in D(\mathbb{L}_{\Lambda}),$

for a.e. $t\geq 0$. Thus $\mathscr{T}(t)(\begin{smallmatrix}x\\\
g\end{smallmatrix})\in D(C_{\Lambda})\times D(\mathbb{L}_{\Lambda})$ for a.e.
$t\geq 0$, due to (3.8). Now according to (3.10), we have

$\displaystyle\int^{\alpha}_{0}\|\mathscr{P}\mathscr{T}(t)(\begin{smallmatrix}x\\\
g\end{smallmatrix})\|^{2}dt$
$\displaystyle=\int^{\alpha}_{0}\|C_{\Lambda}T(t)x+C_{\Lambda}\Phi_{t}^{A,B}\mathbb{L}_{\Lambda}S(\cdot)g\|^{2}dt$
$\displaystyle\leq
2\gamma\|x\|^{2}+2\kappa\|\mathbb{L}_{\Lambda}S(\cdot)g\|^{2}_{L^{2}([0,\alpha],\mathscr{U})}$
$\displaystyle\leq 2(\gamma+\kappa
c)\left(\|x\|^{2}+\|g\|^{2}_{L^{2}([-r,0],\mathscr{U})}\right)$
$\displaystyle\leq\tilde{c}\left\|(\begin{smallmatrix}x\\\
g\end{smallmatrix})\right\|^{2},$

where $\gamma,\kappa,c$ and $\tilde{c}$ are constant, some of them depend on
$\alpha$. ∎

The following is the main result of this section.

######  Theorem 3.3.

Assume that $B$ is an admissible control operator for $A$. Then there exists
an admissible control operator
$\mathscr{B}\in{\mathcal{L}}(\mathscr{U},\mathscr{H}_{-1})$ for $\mathscr{A}$
such that the system (3.5) can be rewritten as

(3.11)
$\displaystyle\begin{cases}dZ(t)=\left(\mathscr{A}Z(t)+\mathscr{B}U(t)\right)dt+\mathscr{M}(Z(t))dW(t),\quad
Z(0)=(\begin{smallmatrix}\xi\\\ \varphi\end{smallmatrix}),&t>0,\cr
Y(t)=\mathscr{P}Z(t),&t>0.\end{cases}$

Moreover this system has a unique mild solution
$Z\in{\mathcal{C}}_{\mathbb{F}}(0,+\infty;L^{2}(\Omega,\mathscr{H}))$
satisfying

(3.12) $\displaystyle
Z^{\xi,\varphi,U}(t)=\mathscr{T}(t)(\begin{smallmatrix}\xi\\\
\varphi\end{smallmatrix})+\int^{t}_{0}\mathscr{T}(t-s)\mathscr{M}(Z^{\xi,\varphi,0}(s))dW(s)+\Phi^{W}_{t}U$

for any $U\in L^{2}_{\mathbb{F}}(0,+\infty;\mathscr{U})$, where
$\Phi^{W}_{t}\in{\mathcal{L}}(L^{2}_{\mathbb{F}}(0,+\infty;\mathscr{U}),L^{2}_{\mathscr{F}_{t}}(\Omega,\mathscr{H}))$
for any $t\geq 0$ and

(3.13)
$\displaystyle\Phi^{W}_{t}U=\Phi^{\mathscr{A},\mathscr{B}}_{t}U+\int^{t}_{0}\mathscr{T}(t-s)\mathscr{M}(\Phi^{W}_{s}U)dW(s),$

where $(\Phi^{\mathscr{A},\mathscr{B}}_{t})_{t\geq 0}$ is the family of
control maps associated with the pair $(\mathscr{A},\mathscr{B})$.
Furthermore, if the triple $(A,B,C)$ is regular and if $X^{\xi,\varphi,U}(t)$
is the first component of $Z^{\xi,\varphi,U}(t)$, then
$X^{\xi,\varphi,U}(t)\in D(C_{\Lambda})$ for a.e. $t>0$ and $\mathbb{P}$-a.s.
and

$\displaystyle\|C_{\Lambda}X(\cdot)\|_{L^{2}_{\mathbb{F}}(0,\alpha;\mathscr{Y})}\leq
c(\alpha)\left(\|\xi\|_{L^{2}_{\mathscr{F}_{0}}(\Omega,\mathscr{H})}+\|U\|_{L^{2}_{\mathbb{F}}(0,\alpha;\mathscr{U})}\right)$

for any $\alpha>0,$ $\xi\in L^{2}_{\mathscr{F}_{0}}(\Omega,H),$ $\varphi\in
L^{2}_{\mathscr{F}_{0}}(\Omega,L^{2}([-r,0],\mathscr{U})),$ $U\in
L^{2}_{\mathbb{F}}(0,\alpha;\mathscr{U})$ and some constant $c(\alpha)>0$.

###### Proof.

Using the technique of Section 2, we will reformulate the stochastic boundary
problem (3.5) to a distributed linear input-output stochastic system. In fact,
let $\mathscr{D}_{\lambda},\;\lambda\in\rho(\mathscr{A})=\rho(A),$ be the
Dirichlet operator associated with $\mathscr{A}_{m}$ and $\mathscr{G}$, and
select

(3.14)
$\displaystyle\mathscr{B}:=(\lambda-\mathscr{A}_{-1})\mathscr{D}_{\lambda},\quad\lambda\in\rho(\mathscr{A}).$

This implies $\mathscr{A}_{m}=\mathscr{A}_{-1}+\mathscr{B}\mathscr{G}$ on
$D(\mathscr{A}_{m})$ (see Section 2). Thus the system (3.5) is transformed
into the system (3.11). Let us prove that $\mathscr{B}$ is an admissible
control operator for $\mathscr{A}$. To this end, we first compute the
expression of $\mathscr{D}_{\lambda}$ for $\lambda\in\rho(\mathscr{A})$. Let
$(\begin{smallmatrix}x\\\
g\end{smallmatrix})\in\ker(\lambda-\mathscr{A}_{m})$. This means that
$x\in\mathscr{Z}$ and $g\in W^{1,2}([-r,0],\mathscr{U})$ are such that $Gx=Lg$
$(\lambda-\mathscr{A}_{m})x=0$ and $\lambda g-g^{\prime}=0$. This proves that
$x=\mathscr{D}_{\lambda}v$ with $v\in\mathscr{U}$ and $g=e_{\lambda}g(0)$. We
also have, $v=G\mathscr{D}_{\lambda}v=Le_{\lambda}g(0)$. We then deduce that

(3.15)
$\displaystyle\mathscr{D}_{\lambda}v=\begin{pmatrix}\mathbb{D}_{\lambda}Le_{\lambda}v\\\
e_{\lambda}v\end{pmatrix},\quad\lambda\in\rho(A).$

Let $(Q,\beta,\mathbb{L})$ be the regular system studied in Example 1 with
control maps $(\Phi^{Q,\beta}_{t})_{t\geq 0}$ and extended input operator
${\bf F}^{Q,\beta,\mathbb{L}}:={\bf F}^{L}$. Using (3.1), (3.14), (3.15) and
the injectivity of the Laplace transform, we have

(3.16)
$\displaystyle\Phi^{\mathscr{A},\mathscr{B}}_{t}u:=\int^{t}_{0}\mathscr{T}_{-1}(t-s)\mathscr{B}u(s)ds=\begin{pmatrix}\Phi^{A,B}_{t}{\bf
F}^{L}u\\\ \Phi^{Q,\beta}_{t}u\end{pmatrix}$

for any $t\geq 0$ and $u\in L^{2}([0,+\infty),\mathscr{U})$. The admissibility
of $\mathscr{B}$ for $\mathscr{A}$ follows immediately from those of $B$ and
$\beta$ for $A$ and $Q,$ respectively. Now according to [22, Theorem 2.7] (see
also [25]), the system (3.11) has a unique mild solution
$Z\in{\mathcal{C}}_{\mathbb{F}}(0,+\infty,L^{2}(\Omega,\mathscr{H}))$
satisfying (3.13). According to Proposition 3, the operator $\mathscr{P}$ is
an admissible observation operator for $\mathscr{A}$. Next we show that the
triple $(\mathscr{A},\mathscr{B},\mathscr{P})$ is regular. In fact, the
regularity of the triple $(A,B,C)$ is regular implies that $\Phi^{A,B}_{t}u\in
D(C_{\Lambda})$ for a.e. $t\geq 0$ and $u\in
L^{2}_{loc}(\mathbb{R}^{+},\mathscr{U})$. Similarly for the system associated
with the left shift semigroup, $\Phi^{Q,\beta}u\in D(\mathbb{L}_{\Lambda})$
for a.e. $t\geq 0$ and $u\in L^{2}_{loc}(\mathbb{R}^{+},\mathscr{U})$. Now
according to (3.10) and (3.16), we have
$\Phi^{\mathscr{A},\mathscr{B}}_{t}u\in D(\mathscr{P}_{\Lambda})$ for a.e.
$t\geq 0$ and $u\in L^{2}_{loc}(\mathbb{R}^{+},\mathscr{U})$ and the operator

$\displaystyle\left({\bf
F}^{\mathscr{A},\mathscr{B},\mathscr{P}}u\right)(t)=\mathscr{P}_{\Lambda}\Phi^{\mathscr{A},\mathscr{B}}_{t}u=\left({\bf
F}^{A,B,C}{\bf F}^{L}u\right)(t),$

is linear bounded from $L^{2}([0,\alpha],\mathscr{U})$ to
$L^{2}([0,\alpha],\mathscr{U})$ for any $\alpha>0$. This shows that the triple
$(\mathscr{A},\mathscr{B},\mathscr{P})$ is well-posed. Again by regularity of
triples $(A,B,C)$ and $(Q,\beta,\mathbb{L})$, we have ${\rm
Range}(\mathbb{D}_{\lambda})\subset D(C_{\Lambda})$ and ${\rm
Range}(e_{\lambda})\subset D(\mathbb{L}_{\Lambda})$ for some (hence all)
$\lambda\in\rho(A)$. This shows that ${\rm
Range}(\mathscr{D}_{\lambda})\subset D(\mathscr{P}_{\Lambda})$ for
$\lambda\in\rho(\mathscr{A})$, due to (3.10). Thus
$(\mathscr{A},\mathscr{B},\mathscr{P})$ is regular. Now, by [22, Theorem
2.15], we have $Z(t):=Z^{\xi,\varphi,U}(t)\in D(\mathscr{P}_{\Lambda})$ for
a.e. $t>0$ and $\mathbb{P}$-a.s., and

(3.17)
$\displaystyle\|\mathscr{P}_{\Lambda}Z^{\xi,\varphi,U}(\cdot)\|_{L^{2}_{\mathbb{F}}(0,\alpha;\mathscr{U})}\leq
c(\alpha)\left(\|(\begin{smallmatrix}\xi\\\
\varphi\end{smallmatrix})\|_{L^{2}_{\mathscr{F}_{0}}(\Omega,\mathscr{H})}+\|U\|_{L^{2}_{\mathbb{F}}(0,\alpha;\mathscr{U})}\right)$

for any $\alpha>0,$ $\xi\in L^{2}_{\mathscr{F}_{0}}(\Omega,H),$ $\varphi\in
L^{2}_{\mathscr{F}_{0}}(\Omega,L^{2}([-r,0],\mathscr{U})),$ $U\in
L^{2}_{\mathbb{F}}(0,\alpha;\mathscr{U})$ and some constant $c(\alpha)>0$. We
denote by

$\displaystyle Z^{\xi,\varphi,U}(t)=\begin{pmatrix}X^{\xi,\varphi,U}(t)\\\
V(t,\cdot)\end{pmatrix},\qquad t\geq 0.$

Then, by using Proposition 2, the equations (3.12), (3.13) and the relation
(3.16), we have

$\displaystyle
X^{\xi,\varphi,U}(t)=T(t)\xi+\int^{t}_{0}T_{-1}(t-s)B\mathbb{L}_{\Lambda}U_{s}ds+\int^{t}_{0}T(t-s)M(X^{\xi,\varphi,U}(s))dW(s),$
$\displaystyle V(t,\cdot)=U_{t}$

for a.e. $t\geq 0$ and $\mathbb{P}$-a.s. Again by [22], we have
$X^{\xi,\varphi,U}(t)\in D(C_{\Lambda})$ and $V(t,\cdot)\in
D(\mathbb{L}_{\Lambda})$ for a.e. $t>0$ and $\mathbb{P}$-a.s. This implies
that $Z^{\xi,\varphi,U}(t)\in D(C_{\Lambda})\times D(\mathbb{L}_{\Lambda})$
for a.e. $t>0$ and $\mathbb{P}$-a.s. Thus the results follow by (3.10) and
(3.17). ∎

######  Example 2.

Consider the following input-output partial differential stochastic system

(3.18) $\displaystyle\begin{cases}\partial_{t}X(t,x)=\Delta
X(t,x)+\displaystyle\int_{\mathscr{O}}k(x,y)X(t,y)dy\partial_{t}W(t),&x\in\mathscr{O},\quad
t>0,\cr X(0,x)=\xi(x),&x\in\mathscr{O},\cr\nabla
X(t,x)|\nu(x)=U(t-r,x),&x\in\partial\mathscr{O},\quad t>0,\cr
U(t,x)=\varphi(t,x),&t\in[-r,0],\;x\in\mathscr{O},\cr
Y(t,x)=c(x)X(t,x),&x\in\partial\mathscr{O},\quad t\geq 0.\end{cases}$

Here, $\mathscr{O}$ is a bounded open subset of $\mathbb{R}^{n}$ with a
$C^{2}$-boundary $\partial\mathscr{O}$ and outer unit normal $\nu(x)$, the
initial condition $\xi\in L^{2}(\Omega\times\mathscr{O}),$ $r>0$ is a real
number, $k\in L^{2}(\mathscr{O}\times\mathscr{O}),$ and $c(\cdot)\in
C_{b}(\partial\mathscr{O})$. We set

$\displaystyle
H=L^{2}(\mathscr{O}),\quad\mathscr{Z}=H^{2}(\mathscr{O}),\quad\mathscr{U}=\mathscr{Y}=L^{2}(\partial\mathscr{O}).$

For any $f\in H,$ we define

$\displaystyle\left[M(f)\right](x)=\int_{\mathscr{O}}k(x,y)f(y)dy,\qquad
x\in\mathscr{O}.$

Using Hölder’s inequality, we get
$\|M(f)\|_{H}\leq\|k\|_{L^{2}(\mathscr{O}\times\mathscr{O})}\|f\|_{H}$. We
also define the delay operator

$\displaystyle Lg=g(-r),\qquad g\in W^{1,2}([-r,0],\mathscr{U}).$

Let us now select the following operators

$\displaystyle A_{m}:=\Delta,\quad D(A_{m}):=\mathscr{Z},$ $\displaystyle
G:\mathscr{Z}\to\mathscr{U},\quad(G\psi)(x)=\nabla\psi(x)|\nu(x),$
$\displaystyle\mathscr{C}:\mathscr{Z}\to\mathscr{Y},\quad(\mathscr{C}\psi)(x)=c(x)\psi(x).$

We know that the operator

$\displaystyle A:=A_{m},\qquad
D(A):=\\{f\in\mathscr{Z}:(G\psi)(x)=\nabla\psi(x)|\nu(x)=0,\;x\in\partial\mathscr{O}\\}$

generates an analytic semigroup $T:=(T(t))_{t\geq 0}$ on $H$. In what follows,
we collect from [24, Chap.3] some facts about fractional operators and their
relation to Neumann boundary conditions. We denote by
$\phi=\mathbb{D}_{1}\psi\in W^{3/2,2}(\mathscr{O})$ the solution of the
elliptic boundary-value problem $\Delta\phi=0$ on $\mathscr{O}$ and
$\nabla\phi|\nu=\psi$ on $\partial\mathscr{O}$, where $\psi\in
L^{2}(\partial\mathscr{O})$. Then $\mathbb{D}_{1}$ maps
$L^{2}(\partial\mathscr{O})$ continuously into $D((I-A)^{\beta})$ for
$\beta\in(0,\frac{3}{4})$. On the other hand, by using the analyticity of the
semigroup $T,$ we obtain

(3.19) $\displaystyle\|(I-A)^{\alpha}T(t-s)(I-A_{-1})\mathbb{D}_{1}v\|_{H}\leq
c_{\beta}(t-s)^{\beta-\alpha-1}\|v\|_{L^{2}(\partial\mathscr{O})}$

for any $v\in L^{2}(\partial\mathscr{O}),\;\alpha\geq 0$ and
$\beta\in(0,\frac{3}{4}),$ where $c_{\beta}>0$ is a constant. In particular
for $\alpha=0$ and $\beta\in(\frac{1}{2},\frac{3}{4})$ the control maps
$\Phi^{A,B}_{t}$ associated to the control operator
$B:=(I-A_{-1})\mathbb{D}_{1}$ satisfy
$\Phi^{A,B}_{t}:L^{2}([0,+\infty),\mathscr{U})\to H$ and
$\|\Phi^{A,B}_{t}\|\leq\delta_{\beta}t^{2\beta-1}$ for any $t\geq 0$ and a
constant $\delta_{\beta}>0$. This shows that $(A,B)$ is well-posed. On the
other hand, the observation operator $\mathscr{C}\varphi=c(\cdot){\rm
tr}\varphi$, where ${\rm tr}$ is the trace operator, is uniformly bounded from
$D((I-A)^{\alpha})$ to $L^{2}(\partial\mathscr{O})$ for $\alpha>\frac{1}{4}$.
Now if we set $C:=\mathscr{C}$ with domain $D(C)=D(A)$, then for any
$\varphi\in D(A)$, $\|CT(t)\varphi\|\leq\gamma t^{-\alpha}$ for a constant
$\gamma>0$. Thus by choosing $\alpha\in(\frac{1}{4},\frac{1}{2})$, we deduce
that $(C,A)$ is well-posed. To apply theorem 3.3, we need to prove that the
triple $(A,B,C)$ is regular. In fact, for
$\alpha\in(\frac{1}{4},\frac{1}{2}),$ $\beta\in\frac{1}{2},\frac{3}{4})$ and
$v\in W^{2,2}([0,t],\mathscr{U})$ with $v(0)=0$, we have

(3.20) $\displaystyle\|{\bf
F}^{A,B,C}v=\mathscr{C}\Phi^{A,B}_{\cdot}v\|_{L^{2}([0,t],\mathscr{U})}\leq\kappa_{\beta}t^{\beta-\alpha}\|v\|_{L^{2}([0,t],\mathscr{U})}$

for a constant $\kappa_{\beta}>0,$ due to (3.19). Now by Proposition 1, the
triple $(A,B,C)$ is well-posed. On the other hand, let $v_{0}\in\mathscr{U}$.
By Hölder’s inequality and the estimate (3.20), we obtain

$\displaystyle\left\|\frac{1}{t}\int^{t}_{0}\left({\bf
F}^{A,B,C}\mathbbm{1}_{[0,+\infty)}v_{0}\right)(s)ds\right\|_{\mathscr{Y}}\leq\kappa_{\beta}t^{\beta-\alpha}\|v_{0}\|_{\mathscr{U}}\underset{t\to
0^{+}}{\longrightarrow}0.$

Thus $(A,B,C)$ is regular. By a simple computation one can verify that the
operator $L$ is an admissible observation operator for the left shift
semigroup on $L^{2}([-r,0],H)$. Thus the assumption of Theorem 3.3 are
satisfied for the stochastic input delay system (3.18), so it is a well-posed
system.

##  4\. Application: a stochastic Schrödinger system

In this section, we are concerned with the well-posedness of a stochastic
Schrödinger equation with a partial Dirichlet input delay control and a
collocated observation. We first need some notations and definitions in order
to describe the system that will be studied.

Let $\mathscr{O}\subset\mathbb{R}^{n}$ ($n\geq 2$) be an open bounded region
with $C^{3}$-boundary
$\partial\mathscr{O}=\overline{\Gamma}_{0}\cup\overline{\Gamma}_{1}$, where
$\Gamma_{0}$ and $\Gamma_{1}$ are disjoint parts of the boundary relatively
open in $\partial\mathscr{O}$ and ${\rm int}(\Gamma_{0})\neq\emptyset$. Let
$H=H^{-1}(\mathscr{O})$ (the dual space of the Sobolev space
$H^{1}_{0}(\mathscr{O})$ with respect to the pivot space
$L^{2}(\mathscr{O})$). We define a bilinear form $a(\cdot,\cdot)$ on
$H^{1}_{0}(\mathscr{O})$ by

$\displaystyle a(f,g):=\int_{\mathscr{O}}\nabla f(x)\overline{\nabla
g(x)}dx,\qquad f,g\in H^{1}_{0}(\mathscr{O}).$

To this form, we associate the following linear operator
${\mathbb{A}}:D({\mathbb{A}}):=H^{1}_{0}(\mathscr{O})\to H,$

$\displaystyle\langle{\mathbb{A}}f,g\rangle=a(f,g),\qquad f,g\in
H^{1}_{0}(\mathscr{O}),$

see e.g. [3], [28]. Clearly ${\mathbb{A}}$ defines a canonical isomorphism
from $D({\mathbb{A}})$ to $H$. Now if we denote by
$-\Delta:H^{2}(\mathscr{O})\cap H^{1}_{0}(\mathscr{O})\to L^{2}(\mathscr{O})$
the Laplacian operator, then ${\mathbb{A}}$ coincides with $-\Delta$ on
$H^{2}(\mathscr{O})\cap H^{1}_{0}(\mathscr{O})$. This implies that
${\mathbb{A}}^{-1}f=(-\Delta)^{-1}f$ for any $f\in L^{2}(\mathscr{O})$. We
also recall (see e.g. [21]) that
$D({\mathbb{A}}^{\frac{1}{2}})=L^{2}(\mathscr{O})$ and
${\mathbb{A}}^{\frac{1}{2}}$ is a canonical isomorphism from
$L^{2}(\mathscr{O})$ to $H$. We also have the following continuous embedding:

$\displaystyle D({\mathbb{A}})\subset
D({\mathbb{A}}^{\frac{1}{2}})=L^{2}(\mathscr{O})\hookrightarrow
H=H^{\prime}\hookrightarrow(D({\mathbb{A}}^{\frac{1}{2}}))^{\prime}\subset
D({\mathbb{A}})^{\prime}.$

Now consider the system

(4.1) $\displaystyle\begin{cases}dX(t,x)=-i\Delta
X(t,x)dt+q(x)X(t,x)dW(t),&t>0,\;x\in\mathscr{O},\cr
X(0,x)=\xi(x),&x\in\mathscr{O},\cr X(t,x)=0,&t\geq 0,\;x\in\Gamma_{1},\cr
X(t,x)=\displaystyle\int^{0}_{-r}d\mu(\theta,x)U(t+\theta,x),&t\geq
0,\;x\in\Gamma_{0},\cr U(t,x)=\varphi(t,x),&-r\leq t\leq
0,\;x\in\Gamma_{0},\cr
Y(t,x)=i\displaystyle\frac{\partial(\Delta^{-1}X)}{\partial\nu},&t\geq
0,\;x\in\Gamma_{0},\end{cases}$

where $\nu$ is the unit normal of $\partial\mathscr{O}$ pointing towards the
exterior of $\mathscr{O}$, $q\in L^{\infty}(\mathscr{O})$, and for
$\\{\mu(\theta,x):\theta\in[-r,0],\;x\in\Gamma_{0}\\}\subset{\mathcal{L}}(\mathbb{R}^{n})$
and $\mu(\theta,\cdot)\in{\mathcal{L}}(L^{2}(\Gamma_{0}))$ is a function of
bounded variation on $\theta$. Denote $\mathscr{U}:=L^{2}(\Gamma_{0})$. We
consider the following closed operator

$\displaystyle A_{m}:=i(-\Delta),\quad D(A_{m})=\\{f\in
H^{2}(\mathscr{O}):f_{|\Gamma_{1}}=0\\}.$

We define the following operators

$\displaystyle\mathscr{M}g=qg,\qquad g\in H^{-1}(\mathscr{O}),$ $\displaystyle
Gf:=f_{|\Gamma_{0}},\qquad f\in D(A_{m}),$
$\displaystyle\mathscr{C}f=i\displaystyle\frac{\partial(\Delta^{-1}f)}{\partial\nu},\qquad
f\in D(A_{m}).$

With these notations, the system (4.1) takes the form of our model system
(1.1). Thus to show the well-posednees of the system (4.1) it suffices to
prove that the assumptions of Theorem 3.3 hold. In fact, we define
$A:=(A_{m})_{|D(A)}$ with $D(A)=\ker G$. Thus we have

$\displaystyle A=i{\mathbb{A}},\qquad D(A)=\\{f\in
H^{2}(\mathscr{O}):f_{|\partial\mathscr{O}}=0\\}.$

Let $\mathbb{D}_{0}$ (we take $\lambda=0$) be the Dirichlet operator
associated with $A_{m}$ and $G$. It is determined as follow:
$\mathbb{D}_{0}u=v$ if and only if $\Delta v=0$ in $\mathscr{O},$ $v=0$ on
$\Gamma_{1}$ and $v=u$ on $\Gamma_{0}$. We have
$\mathbb{D}_{0}\in{\mathcal{L}}(\mathscr{U},L^{2}(\mathscr{O}))$. We select
$B=(-A_{-1})\mathbb{D}_{0}=-i{\mathbb{A}}_{-1}\mathbb{D}_{0}$, where
${\mathbb{A}}_{-1}\in{\mathcal{L}}\left(D({\mathbb{A}}^{\frac{1}{2}}),(D({\mathbb{A}}^{\frac{1}{2}})^{\prime})\right)$
is the extension of ${\mathbb{A}}$. Let us show that
$B\in{\mathcal{L}}(\mathscr{U},(D({\mathbb{A}}^{\frac{1}{2}})^{\prime})$ is an
admissible control operator for $A$. To this end, it suffice to prove that its
adjoint $B^{\ast}$ is an admissible observation operator for $A$. As shown in
[15], we have

$\displaystyle
B^{\ast}f=\left(i\displaystyle\frac{\partial(\Delta^{-1}f)}{\partial\nu}\right)_{|\Gamma_{0}}.$

Otherwise, $B^{\ast}=\mathscr{C}_{|D(A)}$. According to [15, Theorem 1.2], the
triple $(A,B,B^{\ast})$ is regular. Thus, by Theorem 3.3, the stochastic
system (4.1) is well-posed.

######  Remark 2.

We can choose the potential $q$ in (4.1) to be a stochastic process. In this
case, the result of this section remains the same if we assume that $q\in
L^{\infty}(\Omega\times\mathscr{O})$. In fact, we define a multiplication
operator on $L^{2}(\Omega,H)$ by
$(\tilde{\mathscr{M}}f)(\omega,x)=q(\omega,x)f(\omega)$ for $x\in\mathscr{O}$
and $\mathbb{P}$-a.e. $\omega\in\Omega$. As our main results is based on [22,
Theorems 2.7 and 2.15] and the proofs of these theorems are not affected by
this choice, the result of this section will also not be affected.

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