Exact Controllability of the Heat Equation with Boundary Control and Boundary Noise

Abstract

We study an exact controllability problem for a system governed by the heat equation with Neumann boundary control and boundary noise. We reduce the control problem to a moment problem for which we establish sufficient conditions for its resolution.

1. Introduction

Let $(\mathrm{\Omega },\mathbb{F},P)$ be a complete probability space with filtration $\mathcal{F}={\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$, supporting a standard Brownian motion $W\left(t\right)$ in one-dimension. Let $L_{\mathcal{F}}^2(\mathrm{\Omega };L^2\left[0,1\right])$ the Hilbert space of all $\mathbb{F}$-measurable random variables $\varphi$ with values in $L^2\left[0,1\right]$ such that $E{\parallel \varphi \parallel }_{L^2[0,1]}^2<+\infty$ (E is the expectation) and by $L_{\mathcal{F}}^2(0,T;L^2\left[0,1\right])$ the space of the ${\mathcal{F}}_t-$ adapted, $L^2\left[0,1\right]$ valued processes $\phi$ such that $E{\int }_0^T{∥\phi \left(t,w\right)∥}^{2}dt<+\infty .$

The main propose of this paper is to study the exact controllability problem for the following stochastic heat equation:

$$\left\{\begin{array}{cccc}\hfill {\displaystyle \frac{\partial y\left(t,x\right)}{\partial t}}& ={\displaystyle \frac{{\partial }^{2}y\left(t,x\right)}{\partial x^2}}\hfill & \hfill & \mathrm{in}\left[0,T\right]\times \left(0,1\right),\hfill \\ \hfill {\displaystyle \frac{\partial y\left(t,0\right)}{\partial x}}& =\nu \left(t\right)+g\left(t\right){\displaystyle \frac{dW}{dt}}\hfill & \hfill & \mathrm{in}\left[0,T\right],\hfill \\ \hfill {\displaystyle \frac{\partial y\left(t,1\right)}{\partial x}}& =0\hfill & \hfill & \mathrm{in}\left[0,T\right],\hfill \\ \hfill y\left(0,x\right)& =y_0\left(x\right)\hfill & \hfill & \mathrm{in}\left(0,1\right).\hfill \end{array}\right.$$

(1)

Here the initial state $y_0\in L_{{\mathcal{F}}_0}^2(\mathrm{\Omega };L^2\left[0,1\right]),$ the controls $\nu \in L_{\mathcal{F}}^2(0,T;\mathbb{R})$ and $g\in L_{\mathcal{F}}^2(0,T;\mathbb{R}).$

• The exact controllability problem we consider may be stated as follows:

Let $y_0\in L_{{\mathcal{F}}_0}^2(\mathrm{\Omega };L^2\left[0,1\right]),$$y_T\in L_{{\mathcal{F}}_T}^2(\mathrm{\Omega };L^2\left[0,1\right]),$ are there controls $\nu (.),g\left(.\right)\in L_{\mathcal{F}}^2(0,T;\mathbb{R})$ such that the solution of (1) verifies

$$y\left(x,T\right)=y_T\left(x\right)in{L}_{{\mathcal{F}}_T}^2(\mathrm{\Omega };L^2\left[0,1\right]).$$

The controllability problems for deterministic partial differential equations have been extensively investigated in the literature (see, for example, [1,2] and the references therein). Mahmudov [3] established complete controllability of the one-dimensional wave equation with distributed control in the drift part and distributed noise. Lü [4] studied exact controllability problem for stochastic Schrödinger equations with a boundary control applied to the drift term and an internal control applied to the diffusion term. Regarding stochastic parabolic equations, various controllability results ranging from exact and approximate to null controllability have been obtained for systems driven by distributed controls and subject to distributed noise; see, e.g., [5,6,7] and the references cited therein. The case of stochastic parabolic equations with both boundary control and boundary noise has been considered, but for objectives different from exact controllability. For instance, the works in [8,9] investigated these systems in the context of optimal control, seeking to minimize a cost functional, whereas [10] addresses the related problem of stabilization. To the best of our knowledge, the exact controllability problem for a stochastic heat equation with this specific configuration (boundary control and boundary noise) studied in this paper has not been addressed in the existing literature.

The paper is organized as follows. In Section 2, we rewrite the controlled stochastic heat equation as an abstract linear stochastic system and recall some basic concepts on well-posedness of such systems. In Section 3, we show that the control system under consideration is well-posed. In Section 4, we reduce the controllability problem to a moment problem for which we establish sufficient conditions for its resolution.

2. Abstract Formulation and Background on Well-Posed Stochastic Systems

2.1. Abstract Formulation

We reformulate system (1) as an evolution equation in the infinite dimensional Hilbert space $H=L^2\left[0,1\right].$ For this purpose, we introduce the operators

$$\left\{\begin{array}{c}\mathrm{\Lambda }h={\displaystyle \frac{{\partial }^{2}h}{\partial x^2}}\\ D\left(\mathrm{\Lambda }\right)=\{h\in H:{\displaystyle \frac{{\partial }^{2}h}{\partial x^2}}\in H,{\displaystyle \frac{\partial h\left(1\right)}{\partial x}}={\displaystyle \frac{\partial h\left(0\right)}{\partial x}}=0\}\end{array}\right.,$$

(2)

where $D\left(\mathrm{\Lambda }\right)$ denotes the domain of $\mathrm{\Lambda }$. It is well known that $D\left(\mathrm{\Lambda }\right)$ is dense in $L^2\left[0,1\right]$ and that $\mathrm{\Lambda }$ is self-adjoint and nonpositive. Consequently, $\mathrm{\Lambda }$ is the infinitesimal generator of an analytic contraction semigroup $\mathcal{E}\left(t\right)$, see [11]. Moreover, for every fixed $\lambda >0$, one has

$$D\left({(\lambda -\mathrm{\Lambda })}^{\alpha }\right)=\left\{\begin{array}{c}{H}^{2\alpha }(0,1)\mathrm{if}\alpha \in (0,3/4),\\ \left\{h\in H^{2\alpha }(0,1):{\displaystyle \frac{\partial h}{\partial x}}\left(0\right)={\displaystyle \frac{\partial h}{\partial x}}\left(1\right)=0\right\}\mathrm{if}\alpha \in (3/4,1)\end{array}\right\}$$

(3)

where ${(\lambda -\mathrm{\Lambda })}^{\alpha }$ represents the fractional power of $\lambda -\mathrm{\Lambda }$. From [11], it follows that for any $\beta \ge 0$ and $\gamma ,\delta \in \mathbb{R}$:

(a)

the operator ${(\lambda -\mathrm{\Lambda })}^{\beta }$ is closed,

(b)

$\mathcal{E}\left(t\right):L^2(0,1)⟶D\left({(\lambda -\mathrm{\Lambda })}^{\beta }\right)$ if $t>0$,

(c)

$D\left({(\lambda -\mathrm{\Lambda })}^{\beta }\right)$ is dense in $L^2(0,1)$,

(d)

${(\lambda -\mathrm{\Lambda })}^{\beta }\mathcal{E}\left(t\right)h=\mathcal{E}\left(t\right){(\lambda -\mathrm{\Lambda })}^{\beta }h$, for all $h\in D\left({(\lambda -\mathrm{\Lambda })}^{\beta }\right)$,

(e)

There exists $M_{\beta }>0$. For $t>0$, the operator ${(\lambda -\mathrm{\Lambda })}^{\beta }\mathcal{E}\left(t\right)$ is bounded and satisfies, for every $h\in L^2(0,1)$

$${\parallel (\lambda -\mathrm{\Lambda })}^{\beta }{\mathcal{E}\left(t\right)h\parallel }_{L^2(0,1)}\le {\displaystyle \frac{M_{\beta }}{t^{\beta }}}{\parallel h\parallel }_{L^2(0,1)},$$

(4)

(f)

${(\lambda -\mathrm{\Lambda })}^{\gamma +\delta }h={(\lambda -\mathrm{\Lambda })}^{\gamma }{(\lambda -\mathrm{\Lambda })}^{\delta }h$ for all $h\in D\left(\left({(\lambda -\mathrm{\Lambda })}^{ϵ}\right)\right)$ where

$ϵ=max\left\{\delta ,\gamma ,\gamma +\delta \right\}$.

Next, we define the Neumann operator $N:\mathbb{R}\to L^2\left[0,1\right]$ by $\eta =Nf$ and

$$\begin{array}{cc}\hfill & \lambda \eta \left(x\right)-{\displaystyle \frac{{\partial }^2\eta \left(x\right)}{\partial x^2}}=0,x\in (0,1),\hfill \\ \hfill & {\displaystyle \frac{\partial \eta \left(0\right)}{\partial x}}=f,\hfill \\ \hfill & {\displaystyle \frac{\partial \eta \left(1\right)}{\partial x}}=0.\hfill \end{array}$$

(5)

Equation (1) can be reformulated as an abstract differential equation on $H_{-1}$

$$\left\{\begin{array}{ll}dX\left(t\right)=& \mathrm{\Lambda }X\left(t\right)dt+B\nu \left(t\right)dt+Bg\left(t\right)dW\left(t\right),t\in \left[0,T\right]\\ & X\left(0\right)=X_0.\end{array}\right.$$

(6)

where $X_0\in L_{{\mathcal{F}}_0}^2(\mathrm{\Omega };H)$ is the initial state, X is the state variable, $\nu \in L_{\mathcal{F}}^2(0,T;U)$ and $g\in L_{\mathcal{F}}^2(0,T;U)$ are the control variables. $H_{-1}$ is the completion of H with respect to the norm ${\left|x\right|}_{H_{-1}}={\left|{\left(\lambda I-\mathrm{\Lambda }\right)}^{-1}x\right|}_H$ and $B=\left(I-\mathrm{\Lambda }\right)N.$

2.2. Background on Well-Posed Stochastic Systems

We begin by considering a complete probability space $\left(\mathrm{\Omega },\mathbb{F},P\right)$ equipped with a filtration $\mathcal{F}={\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$ where ${\left\{W\left(t\right)\right\}}_{t\ge 0}$ is defined as real-valued Wiener process. Let $H,U$ denote Hilbert spaces endowed with inner products ${⟨.,.⟩}_H$ and ${⟨.,.⟩}_U,$ respectively. We use the notation $\left|.\right|$ for the norm in various spaces, adding a subscript when necessary to avoid ambiguity. We now define several key function spaces used throughout this paper:

• $L_{\mathcal{F}}^2\left(\mathrm{\Omega };V\right)$ is the Hilbert space of all $\mathcal{F}$-measurable square integrable variables with values in a Hilbert space V.
• $L_{\mathcal{F}}^2\left(0,T;V\right)$ is the Hilbert space of all $V$-valued processes $\varphi \left(·\right)$ that are ${\left\{{\mathcal{F}}_t\right\}}_{t_{\ge 0}}$-adapted and satisfy ${\int }_0^T\left(E{\left|\varphi \left(t\right)\right|}_V^2\right)dt<\infty .$
• $L_{\mathcal{F}}^{\infty }\left(0,T;L^2\left(\mathrm{\Omega },V\right)\right)$ the space encompasses all V-valued ${\left\{{\mathcal{F}}_t\right\}}_{t_{\ge 0}}$-adapted processes $\varphi \left(·\right)$ for which $\underset{t\in \left[0,T\right]}{esssup}$$E{\left|\varphi \left(t\right)\right|}_V^2<\infty .$
• $L_{Br}^2\left(0,T;V\right)$ consisting of all V-valued Borel measurable functions $\phi \left(·\right)$ such that ${\int }_0^T\left({\left|\phi \left(t\right)\right|}_V^2\right)dt<\infty .$
• $C_{\mathcal{F}}\left(\left[0,T\right];L^2\left(\mathrm{\Omega },V\right)\right)$ the space including all V-valued ${\left\{{\mathcal{F}}_t\right\}}_{t_{\ge 0}}$-adapted processes $\varphi \left(·\right)$ such that ${\left(E{\left|\varphi \left(t\right)\right|}_V^2\right)}^{{\displaystyle \frac{1}{2}}}$ is continuous.
• $\mathcal{L}\left(X,Y\right)$ the space of all linear bounded operators from a Hilbert space X to a Hilbert space Y.

The previously defined spaces are considered with their respective canonical norms. We first define the mild and weak solutions of (6).

Definition 1.

A process $X(.)$, which is H-valued and ${\mathcal{F}}_t$-adapted, is defined as a mild solution to (6) if

• $X(.)\in C_{\mathcal{F}}([0,T];L^2(\mathrm{\Omega };H))$;
• for all $t\in [0,T]$,

  $$X\left(t\right)=\mathcal{E}\left(t\right)X_0+{\int }_0^t\mathcal{E}(t-s)B\nu \left(s\right)ds+{\int }_0^t\mathcal{E}(t-s)Bg\left(s\right)dW\left(s\right),P-a.s.$$

  (7)

The classical use of Itô’s formula is generally not feasible in many situations. To avoid this problem, we will study the relation connecting mild and weak solutions.

Definition 2.

An H-valued $\mathcal{F}$-adapted process $X(.)\in C_{\mathcal{F}}([0,T];L^2(\mathrm{\Omega };H))$ is called a weak solution to (6) if for all $\zeta \in D\left({\mathrm{\Lambda }}^{*}\right)$ and all $t\in [0,T]$ we have

$$\begin{array}{ccc}\hfill {⟨X\left(t\right),\zeta ⟩}_H& =& {⟨x,\zeta ⟩}_H+{\int }_0^t{⟨X\left(s\right),{\mathrm{\Lambda }}^{*}\zeta ⟩}_{H}ds+{\int }_0^t{⟨\nu \left(s\right),B^{*}\zeta ⟩}_{H}ds\hfill \\ & & +{\int }_0^t{⟨g\left(s\right),B^{*}\zeta ⟩}_{H}dW\left(s\right),P\text{-}a.s.\hfill \end{array}$$

Admissible Control Operator

To define the concept of admissibility for a control operator, we introduce, for $t\in \left[0,T\right]$, the operator ${\mathrm{\Psi }}_t$ in $\mathcal{L}\left(L_{\mathcal{F}}^2(0,T;U)\times L_{\mathcal{F}}^2(0,T;U),L_{{\mathcal{F}}_t}^2(\mathrm{\Omega },H_{-1})\right)$ by

$${\mathrm{\Psi }}_t\left(\nu ,g\right)={\mathrm{\Psi }}_t^1\left(\nu \right)+{\mathrm{\Psi }}_t^2\left(g\right)={\int }_0^t\mathcal{E}(t-s)B\nu \left(s\right)ds+{\int }_0^t\mathcal{E}(t-s)Bg\left(s\right)dW\left(s\right),$$

(8)

where ${\mathrm{\Psi }}_t^1,{\mathrm{\Psi }}_t^2$ are in $\mathcal{L}\left(L_{\mathcal{F}}^2(0,T;U),L_{{\mathcal{F}}_t}^2(\mathrm{\Omega },H_{-1})\right).$

Definition 3.

If there exists a $t_0\in \left[0,T\right]$ such that Range $\left({\mathrm{\Psi }}_{t_0}\right)\subset L_{{\mathcal{F}}_{t_0}}^2(\mathrm{\Omega },H),$ then the operator $B\in \mathcal{L}\left(U,H_{-1}\right)$ is termed an admissible control operator for ${\left\{\mathcal{E}\left(t\right)\right\}}_{t\ge 0}.$

Definition 4.

System (6) is well-posed whenever its control operator $B\in \mathcal{L}\left(U,H_{-1}\right)$ is admissible.

The next result describes the conditions under which a control operator is admissible.

Proposition 1.

The admissibility of the control operator $B\in \mathcal{L}\left(U,H_{-1}\right)$ is equivalent to the existence of a constant $\kappa =\kappa \left(t_0\right)>0$ such that

$${\left|{\mathrm{\Psi }}_{t_0}\left(\nu ,g\right)\right|}_{L_{{\mathcal{F}}_{t_0}}^2(\mathrm{\Omega },H)}^2\le \kappa \left({\left|\nu \right|}_{L_{\mathcal{F}}^2(0,T;U)}^2+{\left|g\right|}_{L_{\mathcal{F}}^2(0,T;U)}^2\right),P\text{-}a.s.$$

(9)

for any control functions $\nu ,g\in L_{\mathcal{F}}^2(0,T;U)$.

Proof. 

The argument follows a line of reasoning comparable to that used in Proposition 2.1 in [12]. The first implication is obvious. Suppose, on the contrary, that there are sequences ${\left\{{\nu }_n\right\}}_{n=1}^{\infty },{\left\{g_n\right\}}_{n=1}^{\infty }$ in $L_{B_r}^2(0,T;U)\subset L_{\mathcal{F}}^2(0,T;U)$ with ${\left|{\nu }_n\right|}_{L_{B_r}^2(0,T;U)}=1$ and ${\left|g_n\right|}_{L_{B_r}^2(0,T;U)}=1$ such that

$${\left|{\mathrm{\Psi }}_{t_0}\left({\nu }_n,g_n\right)\right|}_{L_{{\mathcal{F}}_{t_0}}^2(\mathrm{\Omega },H)}^2\ge n.$$

(10)

Let $B_0={\left(\lambda I-\mathrm{\Lambda }\right)}^{-1}B$ with $\lambda \in \rho \left(\mathrm{\Lambda }\right).$ But $B\in \mathcal{L}\left(U,H_{-1}\right)$ so $B_0\in$$\mathcal{L}\left(U,H\right)$ and

$${\mathrm{\Psi }}_{t_0}\left({\nu }_n,g_n\right)=\left(\lambda I-\mathrm{\Lambda }\right){\int }_0^{t_0}\mathcal{E}(t-s)B_0{\nu }_n\left(s\right)ds+\left(\lambda I-\mathrm{\Lambda }\right){\int }_0^{t_0}\mathcal{E}(t-s)B_0{g}_n\left(s\right)dW\left(s\right).$$

Then, ${\mathrm{\Psi }}_{t_0}$ is closed. By the closed-graph theorem, a constant $\kappa \left(t_0\right)$ can be found, such that for any ${\nu }_n,g_n\in L_{B_r}^2(0,T;U),$

$${\left|{\mathrm{\Psi }}_{t_0}\left({\nu }_n,g_n\right)\right|}_{L_{{\mathcal{F}}_{t_0}}^2(\mathrm{\Omega },H)}^2\le \kappa \left(t_0\right)\left({\left|{\nu }_n\right|}_{L_{\mathcal{F}}^2(0,T;U)}^2+{\left|g_n\right|}_{L_{\mathcal{F}}^2(0,T;U)}^2\right),$$

and this contradicts (10). □

Proposition 2.

Let

$$\begin{array}{c}\mathrm{\Sigma }:\left[0,T\right]\times L_{\mathcal{F}}^2(0,T;U)\times L_{\mathcal{F}}^2(0,T;U)\to L_{\mathcal{F}}^2(\mathrm{\Omega },H),\\ \mathrm{\Sigma }\left(t,\nu ,g\right)={\mathrm{\Psi }}_t\left(\nu ,g\right).\end{array}$$

Under the admissibility condition on the control operator B, the continuity of the operator Σ holds.

Proof. 

The continuity of the mapping $\mathrm{\Sigma }\left(t,.,.\right),t\in \left[0,T\right]$ from $L_{\mathcal{F}}^2(0,T;U)\times L_{\mathcal{F}}^2(0,T;U)$ to $L_{{\mathcal{F}}_t}^2(\mathrm{\Omega },H)\subset L_{\mathcal{F}}^2(\mathrm{\Omega },H)$ can be deduced from Proposition 5.

We set $\varphi \left(t\right)=$${\mathrm{\Psi }}_t\left(\nu ,g\right)$ for fixed $\nu ,g\in L_{\mathcal{F}}^2(0,T;U).$

For $t_1,t_2\in \left[0,T\right]$ such that $t_1<t_2,$ we have

$$\begin{array}{cc}\hfill & \underset{t_2\to t_1^{+}}{lim}E{\left|\varphi \left(t_2\right)-\varphi \left(t_1\right)\right|}_H^2\hfill \\ \hfill & \le 2\underset{t_2\to t_1^{+}}{lim}E{\left|{\int }_{t_1}^{t_2}\mathcal{E}(t_2-s)B\nu \left(s\right)ds\right|}_H^2\hfill \\ \hfill & +2\underset{t_2\to t_1^{+}}{lim}E{\left|{\int }_0^{t_1}\left[\mathcal{E}(t_2-t_1)-I\right]\mathcal{E}(t_1-s)B\nu \left(s\right)ds\right|}_H^2\hfill \\ \hfill & +2\underset{t_2\to t_1^{+}}{lim}E{\left|{\int }_{t_1}^{t_2}\mathcal{E}(t_2-s)Bg\left(s\right)dW\left(s\right)\right|}_H^2\hfill \\ \hfill & +2\underset{t_2\to t_1^{+}}{lim}E{\left|{\int }_0^{t_1}\left[\mathcal{E}(t_2-t_1)-I\right]\mathcal{E}(t_1-s)Bg\left(s\right)dW\left(s\right)\right|}_H^2\hfill \\ \hfill & =0,\hfill \end{array}$$

and the conclusion of the right continuity of $\varphi \left(.\right)$ follows from the Lebesgue’s dominated convergence theorem. The left continuity is proved in a similar manner. Thus, the continuity of $\varphi \left(.\right)=\mathrm{\Sigma }\left(.,\nu ,g\right).$

Since

$${\mathrm{\Psi }}_t\left(\nu ,g\right)-{\mathrm{\Psi }}_s\left(v,h\right)={\mathrm{\Psi }}_t\left(\nu -v,g-h\right)+\left({\mathrm{\Psi }}_t-{\mathrm{\Psi }}_s\right)\left(v,h\right)$$

we deduce that $\mathrm{\Sigma }\left(.,.,.\right)$ is continuous. □

The existence and uniqueness of mild and weak solutions to (6) constitute the main objective of the forthcoming discussion.

Theorem 1.

Assuming that the control operator B is admissible, the corresponding mild solution $X(.)$ to (6) is unique and satisfies

$${\left|X\right|}_{C_{\mathcal{F}}\left(\left[0,T\right];L^2(\mathrm{\Omega },H)\right)}\le C\left(T\right)\left({\left|X_0\right|}_{L_{{\mathcal{F}}_0}^2(\mathrm{\Omega },H)}+{\left|\nu \right|}_{L_{\mathcal{F}}^2(0,T;U)}+{\left|g\right|}_{L_{\mathcal{F}}^2(0,T;U)}\right)$$

(11)

where $C\left(T\right)$ is positive constant.

Proof. 

Since B is admissible, then clearly system (6) is uniquely solvable, with the mild solution given by (7). We have

$$\begin{array}{cc}\hfill & {\left|X\left(t\right)\right|}_{L_{{\mathcal{F}}_t}^2(\mathrm{\Omega },H)}\hfill \\ \hfill & ={\left|\mathcal{E}\left(t\right)X_0+{\int }_0^t\mathcal{E}(t-s)B\nu \left(s\right)ds+{\int }_0^t\mathcal{E}(t-s)Bg\left(s\right)dW\left(s\right)\right|}_{L_{{\mathcal{F}}_t}^2(\mathrm{\Omega },H)}\hfill \\ \hfill & \le {\left|\mathcal{E}\left(t\right)X_0\right|}_{L_{{\mathcal{F}}_0}^2(\mathrm{\Omega },H)}+{\left|{\int }_0^t\mathcal{E}(t-s)B\nu \left(s\right)ds\right|}_{L_{{\mathcal{F}}_t}^2(\mathrm{\Omega },H)}\hfill \\ \hfill & +{\left|{\int }_0^t\mathcal{E}(t-s)Bg\left(s\right)dW\left(s\right)\right|}_{L_{{\mathcal{F}}_t}^2(\mathrm{\Omega },H)}\hfill \\ \hfill & \le C\left(t\right)\left({\left|X_0\right|}_{L_{{\mathcal{F}}_0}^2(\mathrm{\Omega },H)}+{\left|\nu \right|}_{L_{\mathcal{F}}^2(0,t;U)}+{\left|g\right|}_{L_{\mathcal{F}}^2(0,t;U)}\right)\hfill \\ \hfill & \le C\left(t\right)\left({\left|X_0\right|}_{L_{{\mathcal{F}}_0}^2(\mathrm{\Omega },H)}+{\left|\nu \right|}_{L_{\mathcal{F}}^2(0,T;U)}+{\left|g\right|}_{L_{\mathcal{F}}^2(0,T;U)}\right),\hfill \end{array}$$

thus

$${\left|X\right|}_{L_{\mathcal{F}}^{\infty }(0,T;L^2(\mathrm{\Omega },H))}\le C\left(T\right)\left({\left|X_0\right|}_{L_{{\mathcal{F}}_0}^2(\mathrm{\Omega },H)}+{\left|\nu \right|}_{L_{\mathcal{F}}^2(0,T;U)}+{\left|g\right|}_{L_{\mathcal{F}}^2(0,T;U)}\right).$$

(12)

For $0\le t_1\le t_2\le T,$ we have

$$\begin{array}{cc}\hfill & {\left|X\left(t_2\right)-X\left(t_1\right)\right|}_{L_{{\mathcal{F}}_{t_2}}^2(\mathrm{\Omega },H)}\hfill \\ \hfill & \le {\left|\left[\mathcal{E}(t_2-t_1)-I\right]\mathcal{E}\left(t_1\right)X_0\right|}_{L_{{\mathcal{F}}_{t_2}}^2(\mathrm{\Omega },H)}\hfill \\ \hfill & +{\left|{\int }_{t_1}^{t_2}\mathcal{E}(t_2-s)B\nu \left(s\right)ds\right|}_{L_{{\mathcal{F}}_{t_2}}^2(\mathrm{\Omega },H)}\hfill \\ \hfill & +{\left|{\int }_{t_1}^{t_2}\mathcal{E}(t_2-s)Bg\left(s\right)dW\left(s\right)\right|}_{L_{{\mathcal{F}}_{t_2}}^2(\mathrm{\Omega },H)}\hfill \\ \hfill & +{\left|{\int }_0^{t_1}\left[\mathcal{E}(t_2-t_1)-I\right]\mathcal{E}(t_1-s)B\nu \left(s\right)ds\right|}_{L_{{\mathcal{F}}_{t_2}}^2(\mathrm{\Omega },H)}\hfill \\ \hfill & +{\left|{\int }_0^{t_1}\left[\mathcal{E}(t_2-t_1)-I\right]\mathcal{E}(t_1-s)Bg\left(s\right)dW\left(s\right)\right|}_{L_{{\mathcal{F}}_{t_2}}^2(\mathrm{\Omega },H)}.\hfill \end{array}$$

From

$${\left|\left[\mathcal{E}(t_2-t_1)-I\right]\mathcal{E}\left(t_1\right)X_0\right|}_H\le C_1{\left|X_0\right|}_H,$$

$$\begin{array}{ccc}\hfill {\left|{\int }_0^{t_1}\mathcal{E}(t_1-s)B\nu \left(s\right)ds\right|}_{L_{{\mathcal{F}}_{t_2}}^2(\mathrm{\Omega },H)}& \le & C_2{\left|\nu \right|}_{L_{\mathcal{F}}^2(0,T;U)},\hfill \\ \hfill {\left|{\int }_0^{t_1}\mathcal{E}(t_1-s)Bg\left(s\right)dW\left(s\right)\right|}_{L_{{\mathcal{F}}_{t_2}}^2(\mathrm{\Omega },H)}& \le & C_3{\left|g\right|}_{L_{\mathcal{F}}^2(0,T;U)},\hfill \end{array}$$

and by applying Lebesgue’s dominated convergence theorem, we obtain

$$\underset{t_1\to t_2^{-}}{lim}{\left|\left[\mathcal{E}(t_2-t_1)-I\right]\mathcal{E}\left(t_1\right)X_0\right|}_{L_{{\mathcal{F}}_{t_2}}^2(\mathrm{\Omega },H)}=0,$$

(13)

$$\underset{t_1\to t_2^{-}}{lim}{\left|\left[\mathcal{E}(t_2-t_1)-I\right]{\int }_0^{t_1}\mathcal{E}(t_1-s)B\nu \left(s\right)ds\right|}_{L_{{\mathcal{F}}_{t_2}}^2(\mathrm{\Omega },H)}=0,$$

(14)

$$\underset{t_1\to t_2^{-}}{lim}{\left|\left[\mathcal{E}(t_2-t_1)-I\right]{\int }_0^{t_1}\mathcal{E}(t_1-s)Bg\left(s\right)dW\left(s\right)\right|}_{L_{{\mathcal{F}}_{t_2}}^2(\mathrm{\Omega },H)}=0.$$

(15)

Since B is admissible, we also have

$$\begin{array}{ccc}\hfill \underset{t_1\to t_2^{-}}{lim}{\left|{\int }_{t_1}^{t_2}\mathcal{E}(t_2-s)B\nu \left(s\right)ds\right|}_{L_{{\mathcal{F}}_{t_2}}^2(\mathrm{\Omega },H)}& \le & C_4\left(t_2\right)\underset{t_1\to t_2^{-}}{lim}{\int }_{t_1}^{t_2}{\left|\nu \left(s\right)\right|}_{L_{{\mathcal{F}}_{t_2}}^2(\mathrm{\Omega },U)}ds\hfill \\ & =& 0,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & \underset{t_1\to t_2^{-}}{lim}{\left|{\int }_{t_1}^{t_2}\mathcal{E}(t_2-s)Bg\left(s\right)dW\left(s\right)\right|}_{L_{{\mathcal{F}}_{t_2}}^2(\mathrm{\Omega },H)}\hfill \\ \hfill & \le C_5\left(t_2\right)\underset{t_1\to t_2^{-}}{lim}{\int }_{t_1}^{t_2}{\left|g\left(s\right)\right|}_{L_{{\mathcal{F}}_{t_2}}^2(\mathrm{\Omega },U)}ds\hfill \\ \hfill & =0.\hfill \end{array}$$

(17)

Thus, from (13)–(17), we deduce that

$$\underset{t_1\to t_2^{-}}{lim}{\left|X\left(t_2\right)-X\left(t_1\right)\right|}_{L_{{\mathcal{F}}_{t_2}}^2(\mathrm{\Omega },H)}=0.$$

Similarly, we can prove that

$$\underset{t_1\to t_2^{+}}{lim}{\left|X\left(t_2\right)-X\left(t_1\right)\right|}_{L_{{\mathcal{F}}_{t_2}}^2(\mathrm{\Omega },H)}=0.$$

So $X(.)\in C_{\mathcal{F}}\left(\left[0,T\right];L^2(\mathrm{\Omega },H)\right).$

From (12), we have

$$E{\left|X\right|}_H^2\le C\left(T\right)\left(E{\left|X_0\right|}_H^2+{\left|\nu \right|}_{L_{\mathcal{F}}^2(0,T;U)}+{\left|g\right|}_{L_{\mathcal{F}}^2(0,T;U)}\right),$$

then

$${\left|X\right|}_{C_{\mathcal{F}}\left(\left[0,T\right];L^2(\mathrm{\Omega },H)\right)}\le C\left(T\right)\left({\left|X_0\right|}_{L_{{\mathcal{F}}_0}^2(\mathrm{\Omega },H)}+{\left|\nu \right|}_{L_{\mathcal{F}}^2(0,T;U)}+{\left|g\right|}_{L_{\mathcal{F}}^2(0,T;U)}\right).$$

□

The next proposition clarifies how the mild solution relates to the weak solution of (6).

Proposition 3.

A solution to (6) is mild if and only if it is weak solution to (6).

Proof. 

First assume that $X(.)$ is weak solution to (6). For any $\xi \in D\left({\mathrm{\Lambda }}^{*}\right)$ and $r\in \left[0,T\right],$ we let

$$\begin{array}{ccc}\hfill {⟨X\left(r\right),\zeta ⟩}_H& =& {⟨X_0,\zeta ⟩}_H+{\int }_0^r{⟨X\left(s\right),{\mathrm{\Lambda }}^{*}\zeta ⟩}_{H}ds+{\int }_0^r{⟨\nu \left(s\right),B^{*}\zeta ⟩}_{H}ds\hfill \\ & & +{\int }_0^r{⟨g\left(s\right),B^{*}\zeta ⟩}_{H}dW\left(s\right),\mathrm{P}\text{-}\mathrm{a}.\mathrm{s}.\hfill \end{array}$$

If we take $\zeta ={\mathcal{E}}^{*}(t-r){\mathrm{\Lambda }}^{*}\phi$ for some $\phi \in D\left({\left({\mathrm{\Lambda }}^{*}\right)}^2\right)$ and $t\in \left[r,T\right].$ We find

$$\begin{array}{ccc}\hfill {⟨X\left(r\right),{\mathcal{E}}^{*}(t-r){\mathrm{\Lambda }}^{*}\phi ⟩}_H& =& {⟨X_0,{\mathcal{E}}^{*}(t-r){\mathrm{\Lambda }}^{*}\phi ⟩}_H\hfill \\ & & +{\int }_0^r{⟨X\left(s\right),{\mathrm{\Lambda }}^{*}{\mathcal{E}}^{*}(t-r){\mathrm{\Lambda }}^{*}\phi ⟩}_{H}ds\hfill \\ & & +{\int }_0^r{⟨\nu \left(s\right),B^{*}{\mathcal{E}}^{*}(t-r){\mathrm{\Lambda }}^{*}\phi ⟩}_{H}ds\hfill \\ & & +{\int }_0^r{⟨g\left(s\right),B^{*}{\mathcal{E}}^{*}(t-r){\mathrm{\Lambda }}^{*}\phi ⟩}_{H}dW\left(s\right).\hfill \end{array}$$

Integrating over $\left[0,t\right],$ we get

$$\begin{array}{ccc}\hfill {\int }_0^t{⟨X\left(t\right),{\mathcal{E}}^{*}(t-r){\mathrm{\Lambda }}^{*}\phi ⟩}_{H}dr& =& {⟨X_0,{\mathcal{E}}^{*}(t-r){\mathrm{\Lambda }}^{*}\phi ⟩}_H\hfill \\ & & +{\int }_0^t{\int }_0^r{⟨X\left(s\right),{\mathrm{\Lambda }}^{*}{\mathcal{E}}^{*}(t-r){\mathrm{\Lambda }}^{*}\phi ⟩}_{H}dsdr\hfill \\ & & +{\int }_0^t{\int }_0^r{⟨\nu \left(s\right),B^{*}{\mathcal{E}}^{*}(t-r){\mathrm{\Lambda }}^{*}\phi ⟩}_{H}dsdr\hfill \\ & & +{\int }_0^t{\int }_0^r{⟨g\left(s\right),B^{*}{\mathcal{E}}^{*}(t-r){\mathrm{\Lambda }}^{*}\phi ⟩}_{H}dW\left(s\right)dr.\hfill \end{array}$$

Using Fubini’s theorem, stochastic Fubini’s theorem (for more details, see [12]), we obtain

$${⟨X\left(t\right)-\mathcal{E}\left(t\right)X_0-{\int }_0^t\mathcal{E}(t-s)B\nu \left(s\right)ds-{\int }_0^t\mathcal{E}(t-s)Bg\left(s\right)dW\left(s\right),\phi ⟩}_H=0.$$

(18)

We know that $D\left({\left({\mathrm{\Lambda }}^{*}\right)}^2\right)$ is dense in H so (18) is also verified for each $\phi \in H.$ Then, X is mild solution to (6). Now we suppose that $X(.)$ is mild solution to (6). Following the same steps as in the first part, an integration over $\left[0,t\right]$ is performed, followed by an application of Fubini’s and stochastic Fubini’s theorem to get

$$\begin{array}{ccc}\hfill {⟨X\left(r\right),\zeta ⟩}_H& =& {⟨X_0,\zeta ⟩}_H+{\int }_0^r{⟨X\left(s\right),{\mathrm{\Lambda }}^{*}\zeta ⟩}_{H}ds+{\int }_0^r{⟨\nu \left(s\right),B^{*}\zeta ⟩}_{H}ds\hfill \\ & & +{\int }_0^r{⟨g\left(s\right),B^{*}\zeta ⟩}_{H}dW\left(s\right),\hfill \end{array}$$

thus, X is a weak solution to (6). □

3. Well-Posedness of System (1)

The current section is devoted to establishing the well-posedness of the linear system described by the stochastic heat Equation (1) with boundary control and boundary controlled noise with state space $H=L^2\left[0,1\right],$ and control space $U=\mathbb{R}$.

Proposition 4.

Let $T>0$ be given. Then, for any initial condition $y_0\in L_{{\mathcal{F}}_0}^2(\mathrm{\Omega };H)$ and any functions $u,g\in L_{\mathcal{F}}^2(0,T;U)$, there exists a unique mild solution $y\in C_{\mathcal{F}}([0,T];L^2(\mathrm{\Omega };H))$ of (1).

Proof. 

Based on the results of the previous subsection, it suffices to verify that the operator $B=\left(\lambda I-\mathrm{\Lambda }\right)N$ is an admissible control operator for the semigroup $\mathcal{E}\left(t\right)$ generated by $\mathrm{\Lambda }$ defined in (2). For (1), it follows that

$${\mathrm{\Psi }}_t\left(\nu ,g\right)={\int }_0^t\mathcal{E}\left(t-s\right)\left(\lambda I-\mathrm{\Lambda }\right)N\nu \left(s\right)ds+{\int }_0^t\mathcal{E}\left(t-s\right)\left(\lambda I-\mathrm{\Lambda }\right)Ng\left(s\right)dW\left(s\right).$$

From the definition $N_{\lambda }:={\left(\lambda I-\mathrm{\Lambda }\right)}^{{\displaystyle \frac{3}{4}}-ϵ}N\in \mathcal{L}\left(L_{\mathcal{F}}^2(0,T;H^{-{\displaystyle \frac{1}{2}}}\left(\mathbb{R}\right)),H\right)$. For all $\epsilon >0,$ we rewrite $\left(\lambda I-\mathrm{\Lambda }\right)N\nu \left(s\right)$ as ${\left(\lambda I-\mathrm{\Lambda }\right)}^{{\displaystyle \frac{1}{4}}+ϵ}{N}_{\lambda }\nu \left(s\right)$ and the stochastic term is rewritten similarly. So ${\mathrm{\Psi }}_t$ is reformulated as

$$\begin{array}{ccc}\hfill {\mathrm{\Psi }}_t\left(\nu ,g\right)& =& {\int }_0^t{\left(\lambda I-\mathrm{\Lambda }\right)}^{{\displaystyle \frac{1}{4}}+ϵ}\mathcal{E}\left(t-s\right)N_{\lambda }\nu \left(s\right)ds\hfill \\ & & +{\int }_0^t{\left(\lambda I-\mathrm{\Lambda }\right)}^{{\displaystyle \frac{1}{4}}+ϵ}\mathcal{E}\left(t-s\right)N_{\lambda }g\left(s\right)dW\left(s\right).\hfill \end{array}$$

We have

$$\begin{array}{cc}\hfill & E{\left|{\int }_0^t{\left(\lambda I-\mathrm{\Lambda }\right)}^{{\displaystyle \frac{1}{4}}+ϵ}\mathcal{E}\left(t-s\right)N_{\lambda }\nu \left(s\right)ds\right|}_H^2\hfill \\ \hfill & \le E{\left({\int }_0^t\left|{\left(\lambda I-\mathrm{\Lambda }\right)}^{{\displaystyle \frac{1}{4}}+ϵ}\mathcal{E}\left(t-s\right)N_{\lambda }\nu \left(s\right)\right|ds\right)}^2\hfill \\ \hfill & \le E{\left({\int }_0^t{\displaystyle \frac{M_{\frac{1}{4}+ϵ}{e}^{-w\left(t-s\right)}}{{\left(t-s\right)}^{\frac{1}{4}+ϵ}}}\left|N_{\lambda }\nu \left(s\right)\right|ds\right)}^2\hfill \\ \hfill & \le {\left|N_{\lambda }\right|}^2{M}_{{\displaystyle \frac{1}{4}}+ϵ}^{2}E{\left({\int }_0^t{\displaystyle \frac{e^{-w\left(t-s\right)}}{{\left(t-s\right)}^{\frac{1}{4}+ϵ}}}\left|\nu \left(s\right)\right|ds\right)}^2\hfill \\ \hfill & \le {\left|N_{\lambda }\right|}^2{M}_{{\displaystyle \frac{1}{4}}+ϵ}^2{\int }_0^t{\displaystyle \frac{e^{-2w\left(t-s\right)}}{{\left(t-s\right)}^{2\left(\frac{1}{4}+ϵ\right)}}}dsE\left({\int }_0^t{\left|\nu \left(s\right)\right|}^{2}ds\right)\hfill \\ \hfill & \le C_1{\left|\nu \right|}_{L_{\mathcal{F}}^2(0,T;U)}^2\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & E{\left|{\int }_0^t{\left(\lambda I-\mathrm{\Lambda }\right)}^{{\displaystyle \frac{1}{4}}+ϵ}\mathcal{E}\left(t-s\right)N_{\lambda }g\left(s\right)dW\left(s\right)\right|}^2\hfill \\ \hfill & \le E{\int }_0^t\left|{\left(\lambda I-\mathrm{\Lambda }\right)}^{{\displaystyle \frac{1}{4}}+ϵ}\mathcal{E}\left(t-s\right)N_{\lambda }g\left(s\right)\right|ds\hfill \\ \hfill & \le E{\int }_0^t{\left({\displaystyle \frac{M_{\frac{1}{4}+ϵ}}{{\left(t-s\right)}^{\frac{1}{4}+ϵ}}}{e}^{-w\left(t-s\right)}\right)}^2{\left|N_{\lambda }g\left(s\right)\right|}^{2}ds\hfill \\ \hfill & \le {\left|N_{\lambda }\right|}^2{M}_{{\displaystyle \frac{1}{4}}+ϵ}^{2}E{\int }_0^t{\displaystyle \frac{e^{-2w\left(t-s\right)}}{{\left(t-s\right)}^{2\left(\frac{1}{4}+ϵ\right)}}}{\left|g\left(s\right)\right|}^{2}ds\hfill \\ \hfill & \le {\left|N_{\lambda }\right|}^2{M}_{{\displaystyle \frac{1}{4}}+ϵ}^{2}E{\int }_0^T{\displaystyle \frac{e^{-2w\left(t-s\right)}}{{\left(t-s\right)}^{2\left(\frac{1}{4}+ϵ\right)}}}{\left|g\left(s\right)\right|}^{2}ds.\hfill \end{array}$$

Since

$${\int }_0^T{\displaystyle \frac{e^{-2w\left(t-s\right)}}{{\left(t-s\right)}^{2\left(\frac{1}{4}+ϵ\right)}}}ds\le {\displaystyle \frac{1}{1-2\left(\frac{1}{4}+ϵ\right)}}{T}^{1-2\left({\displaystyle \frac{1}{4}}+ϵ\right)}$$

if we put $h:\left[0,T\right]\to \mathbb{R},$$h\left(s\right)={\displaystyle \frac{e^{-w\left(t-s\right)}}{{\left(t-s\right)}^{\left(\frac{1}{4}+ϵ\right)}}}$ then $h\in L^2\left(\left[0,T\right];\mathbb{R}\right).$ The density of $C\left(\left[0,T\right];\mathbb{R}\right)$ in $L^2\left(\left[0,T\right];\mathbb{R}\right)$ ensures the existence of a sequence ${\left\{h_n\right\}}_{n\in \mathbb{N}}$ in $C\left(\left[0,T\right];\mathbb{R}\right)$ such that ${\displaystyle \underset{n\to \infty }{lim}}{h}_n=h.$ Then

$$E{\int }_0^T{\displaystyle \frac{e^{-2w\left(t-s\right)}}{{\left(t-s\right)}^{2\left(\frac{1}{4}+ϵ\right)}}}{\left|g\left(s\right)\right|}^{2}ds=\underset{n\to \infty }{lim}E{\int }_0^T{h}_n^2{\left|g\left(s\right)\right|}^{2}ds.$$

We have

$$\begin{array}{ccc}\hfill {\left|N_{\lambda }\right|}^2{M}_{{\displaystyle \frac{1}{4}}+ϵ}^{2}E{\int }_0^T{h}_n^2\left(s\right){\left|g\left(s\right)\right|}^{2}ds& \le & {\left|N_{\lambda }\right|}^2{M}_{{\displaystyle \frac{1}{4}}+ϵ}^2\underset{s\in \left[0,T\right]}{sup}{h}_n^2\left(s\right)E{\int }_0^T{\left|g\left(s\right)\right|}^{2}ds,\hfill \end{array}$$

so we conclude that

$$\begin{array}{ccc}\hfill E{\left|{\int }_0^t{\left(\lambda I-\mathrm{\Lambda }\right)}^{{\displaystyle \frac{1}{4}}+ϵ}\mathcal{E}\left(t-s\right)N_{\lambda }g\left(s\right)dW\left(s\right)\right|}^2& \le & {\left|N_{\lambda }\right|}^2{M}_{{\displaystyle \frac{1}{4}}+ϵ}^2{C}_{T}E{\int }_0^T{\left|g\left(s\right)\right|}^2\hfill \\ & \le & C_2{\left|g\right|}_{L_{\mathcal{F}}^2(0,T;U)}^2.\hfill \end{array}$$

Therefore,

$$\left|{\mathrm{\Psi }}_t\left(\nu ,g\right)\right|\le C_1{\left|\nu \right|}_{L_{\mathcal{F}}^2(0,T;U)}^2+C_2{\left|g\right|}_{L_{\mathcal{F}}^2(0,T;U)}^2.$$

So B is admissible. Consequently, we deduce from Theorem 1

$${\left|Y\right|}_{C_{\mathcal{F}}([0,T];L^2(\mathrm{\Omega };H))}\le C\left({\left|Y_0\right|}_{L_{{\mathcal{F}}_0}^2(\mathrm{\Omega };H)}+{\left|\nu \right|}_{L_{\mathcal{F}}^2(0,T;U)}+{\left|g\right|}_{L_{\mathcal{F}}^2(0,T;U)}\right).$$

According to Proposition 3, mild and weak solutions of (1) are equivalent. □

4. Reduction to Moment Problems

The eigenvalues of the operator $\mathrm{\Lambda }$ defined in (2) are ${\lambda }_n=-n^2{\pi }^2,n\ge 0$ and the corresponding normalized eigenfunctions $\left\{{\varphi }_n\right\}$ constitute an orthonormal basis of $L^2\left[0,1\right]$, and we note that

$$y_0\left(x\right)=\sum _{n=0}^{\infty }{\rho }_n{\varphi }_n\left(x\right),$$

(19)

$$y\left(t,x\right)=\sum _{n=0}^{\infty }{b}_n\left(t\right){\varphi }_n\left(x\right),$$

(20)

with ${\sum }_{n=0}^{\infty }E|{\rho }_n{|}^2<\infty ,{\sum }_{n=0}^{\infty }E{\int }_0^T{\left|b_n\left(t\right)\right|}^{2}dt<\infty .$ If the terminal state (20) is given by

$$y_T\left(x\right)=\sum _{n=0}^{\infty }{\gamma }_n{\varphi }_n\left(x\right),$$

(21)

the controls $\nu$ and g steer y to this terminal state if

$$b_n\left(T\right)={\gamma }_n,n=0,1,2,\dots P-a.s.$$

(22)

The backward stochastic differential equation, the so-called adjoint equation of (6) is

$$\left\{\begin{array}{cc}d{Y}_t=-\mathrm{\Lambda }{Y}_{t}dt+Z\left(t\right)dW\left(t\right)\hfill & \mathrm{in}t\in \left[0,T\right],\hfill \\ Y\left(T\right)=Y_T.\hfill \end{array}\right.$$

(23)

The study presented in [13] establishes the existence and regularity of weak solutions for (23).

For each $Y_T={\varphi }_k$, k fixed, the function

$$V\left(t,x\right)={\varphi }_k\left(x\right)e^{{\lambda }_k\left(t-T\right)}+\sum _{n=0}^{\infty }{\int }_t^T{e}^{{\lambda }_n\left(s-T\right)}⟨{\varphi }_n,Z\left(t\right)⟩dW\left(s\right),$$

(24)

is a solution of (23).

Computing $d_t⟨X\left(t\right),V(t,.)⟩$ by Itô’s rule, we get

$$\begin{array}{cc}\hfill & E{⟨X\left(T\right),V(T,.)⟩}_{L^2\left[0,1\right]}-E{⟨X\left(0\right),V(0,.)⟩}_{L^2\left[0,1\right]}=\hfill \\ \hfill & E{\int }_0^T{⟨\left(\lambda -\mathrm{\Lambda }\right)\left(N\nu \right)\left(t\right),V(t,.)⟩}_{L^2\left[0,1\right]}dt+E{\int }_0^T{⟨\left(\lambda -\mathrm{\Lambda }\right)\left(Ng\right)\left(t\right),Z\left(t\right)⟩}_{L^2\left[0,1\right]}dt.\hfill \end{array}$$

(25)

By using the definition of the operator $\mathrm{\Lambda }$ (2), we have

$$\begin{array}{cc}\hfill & {⟨\left(\lambda -\mathrm{\Lambda }\right)\left(N\nu \right)\left(t\right),V(t,.)⟩}_{L^2\left[0,1\right]}\hfill \\ \hfill & ={⟨\left(\lambda N\nu \right)\left(t\right),V\left(t,.\right)⟩}_{L^2\left[0,1\right]}-{⟨\mathrm{\Lambda }\left(N\nu \right)\left(t\right),V\left(t,.\right)⟩}_{L^2\left[0,1\right]}\hfill \\ \hfill & ={⟨\left(\lambda N\nu \right)\left(t\right),V\left(t,.\right)⟩}_{L^2\left[0,1\right]}-{⟨\left(N\nu \right)\left(t\right),\mathrm{\Lambda }V\left(t,.\right)⟩}_{L^2\left[0,1\right]}\hfill \\ \hfill & ={⟨\left(\lambda N\nu \right)\left(t\right),V\left(t,.\right)⟩}_{L^2\left[0,1\right]}-{\int }_0^1\left(N\nu \right)\left(t,x\right){\displaystyle \frac{{\partial }^{2}V\left(t,x\right)}{\partial x^2}}dx.\hfill \end{array}$$

(26)

The last term on the right-hand side of (26) is treated using integration by parts and taking into consideration that $N\nu (t,.)$ satisfies the boundary conditions (5), we obtain

$$\begin{array}{cc}\hfill & {⟨\left(\lambda -\mathrm{\Lambda }\right)\left(N\nu \right)\left(t\right),V(t,.)⟩}_{L^2\left[0,1\right]}\hfill \\ \hfill & ={\int }_0^1\left(\lambda N\nu \right)\left(t,x\right)V\left(t,x\right)dx-\nu \left(t\right)V\left(t,0\right)-{\int }_0^1{\displaystyle \frac{{\partial }^2\left(N\nu \right)\left(t,x\right)}{\partial x^2}}V\left(t,x\right)dx\hfill \\ \hfill & =-\nu \left(t\right)V\left(t,0\right).\hfill \end{array}$$

(27)

Similarly, by performing the same integration by parts to the term involving $Z(t,.)$ we obtain

$$⟨\left(\lambda -\mathrm{\Lambda }\right)\left(Ng\right)\left(t\right),Z\left(t\right)⟩=-g\left(t\right)Z\left(t,0\right).$$

(28)

Inserting (27) and (28) into (25), we get

$$E⟨X\left(T\right),V(T,.)⟩-E⟨X\left(0\right),V(0,.)⟩=-E{\int }_0^T\nu \left(t\right)V(t,0)dt-E{\int }_0^{T}g\left(t\right)Z\left(t,0\right)dt.$$

(29)

Employing (29) in conjunction with (19) and (21), it follows that

$$\begin{array}{cc}\hfill & E⟨\sum _{n=0}^{\infty }{\gamma }_n{\varphi }_n,{\varphi }_k⟩-E⟨\sum _{n=0}^{\infty }{\rho }_n{\varphi }_n,{\varphi }_k{e}^{-{\lambda }_{k}T}+\sum _{n=0}^{\infty }{\int }_0^T{e}^{{\lambda }_n\left(s-T\right)}⟨{\varphi }_n,Z\left(s\right)⟩{\varphi }_{n}dW\left(s\right)⟩\hfill \\ \hfill & =-E{\int }_0^T\nu \left(t\right)V(t,0)dt-E{\int }_0^{T}g\left(t\right)Z\left(t,0\right)dt.\hfill \end{array}$$

Recalling (24), we obtain

$$\begin{array}{cc}\hfill & E\left({\gamma }_k-{\rho }_k{e}^{-{\lambda }_{k}T}\right)-\sum _{n=0}^{\infty }E{\int }_0^T{\rho }_n{e}^{{\lambda }_n\left(s-T\right)}⟨{\varphi }_n,Z\left(s\right)⟩dW\left(s\right)\hfill \\ \hfill & =-E{\int }_0^T\nu \left(t\right)\left[{\varphi }_k\left(0\right)e^{{\lambda }_k\left(t-T\right)}+\sum _{n=0}^{\infty }{\int }_t^T{e}^{{\lambda }_n\left(s-T\right)}⟨{\varphi }_n,Z\left(s\right)⟩{\varphi }_n\left(0\right)dW\left(s\right)\right]dt\hfill \\ \hfill & -E{\int }_0^{T}g\left(t\right)Z(t,0)dt.\hfill \end{array}$$

(30)

We have

$$\sum _{n=0}^{\infty }E{\int }_0^T{\rho }_n{e}^{{\lambda }_n\left(s-T\right)}⟨{\varphi }_n,Z\left(s\right)⟩dW\left(s\right)=0.$$

(31)

Using Fubini’s Theorem on the first term on the right-hand side of (31), we obtain

$$\begin{array}{cc}\hfill & E{\int }_0^T\nu \left(t\right)\left[\sum _{n=0}^{\infty }{\int }_t^T{e}^{{\lambda }_n\left(s-T\right)}⟨{\varphi }_n,Z\left(s\right)⟩{\varphi }_n\left(0\right)dW\left(s\right)\right]dt\hfill \\ \hfill & =E\left[\sum _{n=0}^{\infty }{\int }_0^T{\int }_t^T\nu \left(t\right)e^{{\lambda }_n\left(s-T\right)}⟨{\varphi }_n,Z\left(s\right)⟩{\varphi }_n\left(0\right)dW\left(s\right)dt\right]\hfill \\ \hfill & =\sum _{n=0}^{\infty }E{\int }_0^T{\int }_0^S\nu \left(t\right)e^{{\lambda }_n\left(s-T\right)}⟨{\varphi }_n,Z\left(s\right)⟩{\varphi }_n\left(0\right)dtdW\left(s\right)\hfill \\ \hfill & =0.\hfill \end{array}$$

(32)

From (30), (31) and (32), one derives that

$$E\left({\gamma }_k-{\rho }_k{e}^{-{\lambda }_{k}T}\right)=-E{\int }_0^T\nu \left(t\right){\varphi }_k\left(0\right)e^{{\lambda }_k\left(t-T\right)}dt-E{\int }_0^{T}g\left(t\right)Z\left(t,0\right)dt.$$

(33)

which is verified for

$${\rho }_k{e}^{-{\lambda }_{k}T}-{\gamma }_k={\int }_0^T{e}^{-{\lambda }_{k}t}\left[\nu \left(t\right){\varphi }_k\left(0\right)e^{{\lambda }_k\left(2t-T\right)}+e^{{\lambda }_{k}t}g\left(t\right)Z\left(t,0\right)\right]dt,P-a.s.$$

(34)

We are led to find a function $h\in L_{\mathcal{F}}^2(0,T;\mathbb{R})$ that solves the following moment problem

$$\stackrel{T}{\underset{0}{\int }}{e}^{-{\lambda }_{k}t}h\left(t\right)dt=\left({\rho }_k{e}^{-{\lambda }_{k}T}-{\gamma }_k\right),k=0,1,2,\dots ,P-a.s.$$

(35)

Suppose that a sequence $\left\{{\psi }_k\right\}$ of functions can be constructed that is biorthogonal to the set $\left\{e^{-{\lambda }_{k}t}\right\}$ in $L^2\left[0,T\right],$ i.e, such that

$$\stackrel{T}{\underset{0}{\int }}{\psi }_k\left(t\right)e^{-{\lambda }_{m}t}dt={\delta }_{km},k,m=0,1,2,\dots$$

then the moment problem (35) has a formal solution

$$h\left(t\right)=\sum _{k=1}^{\infty }{h}_k\left(t\right),\mathrm{with}{h}_k=\left({\rho }_k{e}^{-{\lambda }_{k}T}-{\gamma }_k\right){\psi }_k\left(t\right),P-a.s.$$

(36)

provided that the series in (36) is convergent in $L_{\mathcal{F}}^2(0,T;\mathbb{R}).$

We characterize all sequences $\left\{h_k\right\}$ for which (36) is absolutely convergent. We have

$${\left|h_k\right|}_{L_{\mathcal{F}}^2(0,T;\mathbb{R})}^2\le E{\left({\rho }_k{e}^{-{\lambda }_{n}T}-{\gamma }_k\right)}^2{\left|{\psi }_k\right|}_{L^2\left[0,T\right]}^2,$$

From Theorem IV.1.9 in [14], one can find positive constants C and $\eta$ satisfying

$${\left|h_k\right|}_{L_{\mathcal{F}}^2(0,T;\mathbb{R})}^2\le CE{\left({\rho }_k{e}^{-{\lambda }_{k}T}-{\gamma }_k\right)}^2{e}^{\eta \left({\lambda }_k+1\right)}.$$

Consequently, if

$$\sum _{k=0}^{\infty }E{\left({\rho }_k{e}^{-{\lambda }_{k}T}-{\gamma }_k\right)}^2{e}^{\eta \left({\lambda }_k+1\right)}<\infty .$$

(37)

then the series given in (36) is absolutely convergent. This completes the proof and establishes the stated result.

Theorem 2.

Let $y_0\in L_{{\mathcal{F}}_0}^2(\mathrm{\Omega };L^2\left[0,1\right]),$$y_T\in L_{{\mathcal{F}}_T}^2(\mathrm{\Omega };L^2\left[0,1\right]),$ be given respectively by (19) and (21). If

$$E{\left({\rho }_n{e}^{-{\lambda }_{n}T}-{\gamma }_k\right)}^2\le M{e}^{-\eta \left({\lambda }_n+1\right)},k=0,1,2,\dots ,$$

where ${\lambda }_n=-n^2{\pi }^2,$ for some $M,\eta >0,$ then there exist controls $\nu ,$$g\in L_{\mathcal{F}}^2(0,T;\mathbb{R})$ which steer the state of (1) from $y_0$ to $y_T$ at time $T>0.$

5. Conclusions

In this paper, we considered an exact controllability problem for the one-dimensional parabolic equation with both boundary control and boundary noise. We first established the existence and uniqueness of a mild solution. Then, we derived sufficient conditions ensuring the exact controllability of the system. This work extends the existing results on exact controllability of parabolic equations with distributed control and distributed noise to the case of boundary control and boundary noise. Moreover, in order to prove the existence and the uniqueness of the mild solution of the considered PDE, we are led to the proof of the existence and the uniqueness of the mild solution for general class of linear infinite dimensional systems with unbounded control operator in both the drift and diffusion terms.

This study opens perspectives for further research. Future directions may include the analysis of more general class of stochastic parabolic systems, the development of numerical schemes to implement the control in real systems. For instance, in aerospace engineering, our results could inform the design of control systems for high-speed vehicles where skin temperature must be regulated precisely despite random aerodynamic heating [15].