Design of boundary stabilizers for the non-autonomous cubic semilinear heat equation, driven by a multiplicative noise

TL;DR

This paper develops explicit boundary feedback controllers for stabilizing non-autonomous cubic stochastic heat equations using eigenfunction-based feedback, transforming the problem into a deterministic form for analysis.

Contribution

It introduces a simple, explicit linear boundary feedback stabilizer for a class of stochastic heat equations with cubic non-linearity, supported on boundary subsets.

Findings

Successfully stabilizes unbounded trajectories.

Transforms stochastic equations into deterministic forms for analysis.

Provides explicit feedback control formulas.

Abstract

Here we design boundary feedback stabilizers to unbounded trajectories, for semi-linear stochastic heat equation with cubic non-linearity. The feedback controller is linear, given in a simple explicit form and involves only the eigenfunctions of the Laplace operator. It is supported in a given open subset of the boundary of the domain. Via a rescaling argument, we transform the stochastic equation into a random deterministic one. Then, the simple form of the feedback, we propose here, allows to write the solution, of the random equation, in a mild formulation via a kernel. Appealing to a fixed point argument the existence \& stabilization result is proved.

Equations

$$\left\{\begin{array}[]{l}\begin{aligned} dY(t,x)=(\Delta Y(t,x)+cY(t,x)&+f(t,x,Y(t,x)))dt+\vartheta Y(t,x)dW(t),\\ &\text{for }t>0,\ x\in\mathcal{O},\end{aligned}\\ \\ \frac{\partial}{\partial\mathbf{n}}Y(t,x)=u(t,x),\text{ on }t\geq 0,\ x\in\Gamma_{1},\\ \\ \frac{\partial}{\partial\mathbf{n}}Y(t,x)=0,\ \text{ on }t\geq 0,\ x\in\Gamma_{2},\\ \\ Y(0,x)=y_{o}(x),\ x\in\mathcal{O}.\end{array}\right.\$$

$$f(t,x,y)=a_2(t,x)y^2+a_3(t,x)y^3.$$

$$0\le m_1^i\le m_2^i\le ...\le m_{S_i}^i,$$

$$(H_0)\underset{x\in \mathcal{O}}{\sup }|a_i(t,x)|\le C_a\left(\sum _{k=1}^{S_i}{t}^{m_k^i}+1\right),\forall t\ge 0,i=2,3.$$

$$(H_{1})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \,\frac{1}{2}\vartheta^{2}-m_{S}-\frac{1}{100}=\vartheta_{1}>0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$$

$$e^{\alpha t}{\int }_{\mathcal{O}}{Y}^2(t,x)dx<const.,\forall t\ge 0,\mathbb{P}-\text{a.s.},$$

$$Y(t)=\Gamma (t)y(t),t\in [0,\infty ),$$

$$d\Gamma (t)=\vartheta \Gamma (t)dW(t),t\ge 0,\Gamma (0)=1,$$

$$\Gamma (t)=e^{\vartheta W(t)-\frac{t}{2}{\vartheta }^2},t\ge 0.$$

$$e^{-at}\le t^{-a},\forall t>0,a\ge 0.$$

$$\Gamma (t)=e^{\vartheta W(t)-{\vartheta }_{1}t}{e}^{-(m_S+\frac{1}{100})t}\le C_{\Gamma }{e}^{-(m_S+\frac{1}{100})t},\forall t>0,\mathbb{P}-\text{a.s.},$$

$$\Gamma (t)\underset{x\in \mathcal{O}}{\sup }|a_i(t,x)|$$

$$(\text{since 0≤m1i≤m2i≤...≤mSii≤mS})$$

$$\le C\left(\sum _{k=1}^{S_i}{t}^{m_k^i}{e}^{-(m_k^i+\frac{1}{100})t}+e^{-\frac{1}{100}t}\right)$$

$$\le C\left(\sum _{k=1}^{S_i}{t}^{m_k^i}{t}^{-(m_k^i+\frac{1}{100})}+t^{-\frac{1}{100}}\right)$$

$$\le C{t}^{-\frac{1}{100}},\forall t>0.$$

$$\left\{\begin{array}[]{l}\begin{aligned} \partial_{t}y(t,x)=\Delta y(t,x)+cy(t,x)&+\Gamma^{-1}(t)f(t,x,\Gamma(t)y(t,x)),\\ &\text{for }t>0,\ x\in\mathcal{O},\end{aligned}\\ \\ \frac{\partial}{\partial\mathbf{n}}y(t,x)=u(t,x),\text{ on }t\geq 0,\ x\in\Gamma_{1},\\ \\ \frac{\partial}{\partial\mathbf{n}}y(t,x)=0,\ \text{ on }t\geq 0,\ x\in\Gamma_{2},\\ \\ y(0,x)=y_{o}(x),\ x\in\mathcal{O}.\end{array}\right.\$$

$$W^{s,p}(\mathcal{O}):=\left\{y\in L^p(\mathcal{O}):\frac{|y(x)-y(\xi )|}{|x-\xi {|}^{\frac{2}{p}+s}}\in L^p(\mathcal{O}\times \mathcal{O})\right\},$$

$$\Vert y{\Vert }_{s,p}:={\left({\int }_{\mathcal{O}}|y{|}^{p}dx+{\int }_{\mathcal{O}}{\int }_{\mathcal{O}}\frac{|y(x)-y(\xi ){|}^p}{|x-\xi {|}^{2+sp}}dxd\xi \right)}^{\frac{1}{p}}.$$

$$\Vert \cdot {\Vert }_s:=\Vert \cdot {\Vert }_{s,2}.$$

$$|y{|}_4\le C\Vert f{\Vert }_{\frac{1}{2}},\forall f\in H^{\frac{1}{2}}(\mathcal{O}),$$

$$\mathcal{A}y=-(\Delta y+cy),\forall y\in \mathcal{D}(\mathcal{A}),$$

$$\mathcal{D}(\mathcal{A})=\left\{y\in H^2(\mathcal{O}):\frac{\partial }{\partial \mathbf{n}}y=0\text{ on }\partial \mathcal{O}\right\}.$$

$$-\Delta {\phi }_j={\lambda }_j{\phi }_j\text{ in }\mathcal{O},\frac{\partial }{\partial \mathbf{n}}{\phi }_j=0\text{ on }\partial \mathcal{O},\forall j=1,2,3,...,$$

$$⟨{\phi }_i,{\phi }_j⟩={\delta }_{ij},\forall i,j=1,2,3,...,$$

$$y=\sum _{j=1}^{\infty }⟨y,{\phi }_j⟩{\phi }_j$$

$$|y{|}_2={\left(\sum _{j=1}^{\infty }|⟨y,{\phi }_j⟩{|}^2\right)}^{\frac{1}{2}},$$

$$C_1{\left(\sum _{j=1}^{\infty }{\lambda }_j^{\frac{1}{2}}|⟨y,{\phi }_j⟩{|}^2\right)}^{\frac{1}{2}}\le \Vert y{\Vert }_{\frac{1}{2}}\le C_2{\left(\sum _{j=1}^{\infty }{\lambda }_j^{\frac{1}{2}}|⟨y,{\phi }_j⟩{|}^2\right)}^{\frac{1}{2}},\forall y\in W^{\frac{1}{2},2}.$$

$$(H_{1})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sum_{j=2}^{\infty}\frac{1}{\lambda^{\frac{5}{3}}_{j}}<\infty.\ \ \ \ \ \ \ \ \ \ \ \$$

$$|{\phi }_j{|}_{\infty }\le C{\lambda }_j^{\frac{1}{4}},\forall j=2,3,...,$$

$${\Vert {\phi }_j\Vert }_{L^2(\partial \mathcal{O})}\le C{\lambda }_j^{\frac{1}{6}},j=2,\mathrm{3....}$$

$$\sum _{j=M}^{\infty }{e}^{(-2{\lambda }_j+C)t}{\phi }_j^2(\xi )\le C_0\frac{1}{t},\forall t>0,\xi \in \mathcal{O}.$$

Full text

MSC 2010 Classification:

Keywords:

Ionut Munteanu

e-mail: ionut.munteanu@uaic.ro

Alexandru Ioan Cuza University, Department of Mathematics

and Octav Mayer Institute of Mathematics (Romanian Academy)

700506 Iaşi, Romania

1 Presentation of the model

Let $\mathcal{O}\subset\mathbb{R}^{2}$ be a bounded domain, with its smooth boundary $\partial\mathcal{O}$ split in two parts as $\partial\mathcal{O}=\Gamma_{1}\cup\Gamma_{2}$, such that $\Gamma_{1}$ has non-zero surface measure. We consider the following boundary controlled semi-linear heat equation, with cubic non-linearity, driven by a multiplicative noise

[TABLE]

Here, $dW$ denotes a Gaussian time noise, that is usually understood as the distribution derivative of the Brownian sheet $W(t)$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with normal filtration $(\mathcal{F}_{t})_{t\geq 0}$. $c$ and $\vartheta$ are some positive constants. $f$ is a cubic polynomial with time-space coefficients, of the form

[TABLE]

On the functions $a_{i},\ i=2,3,$ we assume that: there exist $C_{a}>0$ and

[TABLE]

for some $S_{i}\in\mathbb{N}$, $i=2,3,$ for which

[TABLE]

Moreover, we assume that $m^{2}_{S_{2}}$, $m^{3}_{S_{3}}$ and $\vartheta$ are such that:

[TABLE]

where $m_{S}:=\max\left\{m^{2}_{S_{2}},m^{3}_{S_{3}}\right\}$.

(We remark that, when $a_{2}\equiv 0$, then we stumble exactly on the non-autonomous Chafee-Infante equation.)

Finally, $\mathbf{n}$ stands for the outward unit normal to the boundary $\partial\mathcal{O}$, and $u$ is the control. The initial data $y_{o}$ is $\mathcal{F}_{0}$-adapted.

The aim of the present paper is to find a feedback law $u$ such that, once inserted into the equation (1.1), the corresponding solution of the closed-loop equation (1.1) satisfies

[TABLE]

for a prescribed positive constant $\alpha$, provided that the initial data $y_{o}$ is small enough in the $L^{2}-$norm (that is the main result stated in Theorem 3.1 below). Note that this is an almost sure path-wise local boundary stabilization type result. Besides this, since the coefficients are time-dependent, our considerations are related, in fact, to the problem of stabilization to trajectories (i.e., non-steady states). In the existing literature there are only few results on this problem. Regarding the internal stabilization we refer to [5], while for the the boundary case we cite [23, 24, 16]. In any case, the time-dependent coefficients are assumed to be bounded, while here, we let them explode when $t$ goes to infinity. This, together with the noise perturbations, makes our task a lot more difficult. Note that even the well-posedness is not known for our example. Anyway, the simple form of the controller, which we shall introduce below, allows us to write the equations in an integral formulation, via a kernel. Then, via a fixed point argument and a proper choose of the spaces, the three raised problems, i.e., existence, uniqueness and stabilization, will be solved.

It is worth to mention that the work [9] studies the effect of noise on the Chafee-Infante equation, and the conclusions there state that a single multiplicative Ito noise, of sufficient intensity, will stabilize the origin of the system. However, we remark that the coefficients there are assumed to be bounded, and then, the ”sufficient intensity” of the noise is related to their bounds. While here, due to the unboundedness of the coefficients, those arguments cannot be applied. Anyway, the presence of the noise is mandatory. This can be seen from the imposed hypothesis (1.3). But, even if the level of the noise, $\vartheta$, is large, it cannot ensure the stability of the system. A boundary stabilizer is needed. In conclusion, the result of this paper is first in this general framework.

It is clear that, due to the general form of the nonlinearity $f$, the results presented here can be applied not only to the Chafee-Infante equation, since, many examples of cubic semi-linear equation arise from biology, chemistry or physics, such as the FitzHugh-Nagumo model [10](in neuroscience) or the Fischer-Kolmogorov model [11] (in evolution of population dynamics).

The method to design the feedback controller $u$ relies on the ideas in [15], where a proportional type law was proposed to stabilize, in mean, the stochastic heat equation. We emphasize that, unlike to the equation in [15], which is linear and evolves in a bounded interval, now we deal with a nonlinear one of order three, evolving in the 2-D domain $\mathcal{O}$. In order to overcome this complexity, we further develop the ideas in [15]. Roughly saying, we design a similar feedback as in [15]: linear, of finite-dimensional structure, given in a very simple form, being easy to manipulate from the computational point of view, involving only the eigenfunctions of the Neumann-Laplace operator (see relations (3.16)-(3.18) below). Then, we plug it into the equations, and show that it achieves the stability by using the estimates on the magnitude of the controller and a fixed point argument in a properly chosen space (see Theorem 3.1 below). The idea to use fixed point arguments in order to show the stability of deterministic or stochastic equations has been previously used in papers like [8, 14]. Proportional type feedback, similar to that one we design here, has its origins in the works [3, 17], while in the papers [18, 19, 20, 21, 22], it has been used to stabilize other important parabolic-type equations, such as the Navier-Stokes equations (also with delays), the Magnetohydrodynaimc equations, or the phase field equations. Besides the method of proportional-type controllers, the backstepping technique has been developed with lots of important results. Even if, at a first glance, the two methods seem to be very similar, conceptually they are totally different. For more details, we refer to the works [1, 6], while in [12] it is provided also a stabilization result for the stochastic Burgers equation.

2 The random equation

There is a well-known trick, by now, on how to avoid to deal with stochastic equations. Namely, to equivalently rewrite them as random deterministic ones via a rescaling argument. This is explained in full details in the work [2]. To this end, in (1.1), let us consider the transformation

[TABLE]

where $\Gamma(t):L^{2}(\mathcal{O})\rightarrow L^{2}(\mathcal{O})$ is the linear continuous operator defined by the equations

[TABLE]

that can be equivalently expressed as

[TABLE]

Frequently below we shall use the obvious inequality

[TABLE]

By the law of the iterated logarithm and arguing similarly as in Lemma 3.4 in [4], it follows that there exists a constant $C_{\Gamma}>0$ such that

[TABLE]

where we have used that $\frac{1}{2}\vartheta^{2}=m_{S}+\frac{1}{100}+\vartheta_{1}$ assumed in (1.3). Then, taking advantage of (1.2), we deduce that, for $i=2,3$, we have

[TABLE]

Next, applying Itö’s formula in (1.1) , we obtain that $y$ satisfies the following random partial differential equation

[TABLE]

3 The boundary feedback stabilizer and the main result of the work

Let $(X,\|\cdot\|_{X})$ stand for some normed space. We set $C_{b}([0,\infty),X)$ for the space of all continues $X$-valued functions, that are $\|\cdot\|_{X}-$bounded on $[0,\infty)$. Next, we denote by $L^{p}$, $1\leq p\leq\infty$, the Lebesgue space $L^{p}(\mathcal{O})$ consisting of all power $p$ integrable functions, endowed with the standard norm $|\cdot|_{p}$; we denote by $W^{s,p},\ s\in(0,1)$ the corresponding fractional Sobolev space, i.e.

[TABLE]

which is an intermediary Banach space between $L^{p}$ and $W^{1,p}(\mathcal{O})$, endowed with the natural norm

[TABLE]

For the particular case $p=2$, we set $H^{s}:=W^{s,2}$ and

[TABLE]

By interpolation, one can extend the definition of $H^{s}$ for each $s>0$. We set $H_{0}^{1}(\mathcal{O})$ for the completion of the $C_{0}^{\infty}(\mathcal{O})$ ( the set of $C^{\infty}$-compact supported functions in $\mathcal{O}$) in the $H^{1}$-norm.

It is well known the following fractional Sobolev embedding (for details see [7])

[TABLE]

where $C=C(\mathcal{O})$ is some positive constant, depending on the domain $\mathcal{O}$.

Finally, we set $\left<\cdot,\cdot\right>$ for the natural scalar product in $L^{2}$; and $\left<\cdot,\cdot\right>_{N}$, the euclidean scalar product in $\mathbb{R}^{N},\ N\in\mathbb{N}$. We shall denote by $C$ different constants that may change from line to line, though we keep denote them by the same letter $C$, for the sake of the simplicity of the writing.

Let us denote by

[TABLE]

[TABLE]

Here, $-\Delta$ is the Neumann-Laplace operator on $\mathcal{O}.$ It is well known that it has a discrete spectrum, i.e., it has a countable set of semi-simple non-negative eigenvalues $\left\{\lambda_{j}\right\}_{j=1}^{\infty}$ with $\lambda_{1}=0$. We assume that the eigenvalues set is arranged as an increasing sequence with $\lambda_{j}\rightarrow\infty$ when $j\rightarrow\infty$. We denote by $\left\{\varphi_{j}\right\}_{j=1}^{\infty}$, the corresponding eigenfunctions, which form an orthonormal basis in $L^{2}$. More precisely, we have

[TABLE]

and

[TABLE]

$\delta_{ij}$ being the Kronecker symbol. Besides this, by the Parseval’s identity, we have for a function $y\in L^{2}$, the following decomposition

[TABLE]

and

[TABLE]

where $\left<y,\varphi_{j}\right>,\ j=1,2,3,...,$ are called the (Fourier) modes of $y$. Moreover, since $\mathcal{O}$ is bounded with smooth boundary, it is also known that the norm $\|\cdot\|_{\alpha}$ is equivalent with $|(-\Delta)^{\frac{\alpha}{2}}\cdot|_{2},$ for all $\alpha>0.$ Thus, one can find some constants $C_{1},C_{2}>0$ such that

[TABLE]

In this work, we shall assume that the eigenvalues system $\left\{\lambda_{j}\right\}_{j}$ obeys

[TABLE]

In the Appendix below, we shall verify that, when $\mathcal{O}$ is a square, then $(H_{1})$ holds true. But one can easily find many more examples of domains $\mathcal{O}$ for which assumption $(H_{1})$ is full-filed.

We go on and recall the well-known $L^{\infty}$-bounds of the Laplace eigenfunctions

[TABLE]

that hold true without making any geometric assumption on the domain $\mathcal{O}\subset\mathbb{R}^{2}$. We are also aware of Tataru’s trace estimates

[TABLE]

For latter purpose, let us show that, for an arbitrary constant $C>0$ and a sufficiently large $M>0$, we have that

[TABLE]

Here, $C_{0}>0$ is some constant. Indeed, since $M$ is large enough and $\lim_{j\rightarrow\infty}\lambda_{j}=\infty$, we have that

[TABLE]

Thus

[TABLE]

The latter term is the the rest of order $M$ of the well-known Neumann heat kernel, which is known to be less or equal of some constant times $\frac{1}{t}$. From this, our claim (3.6) follows immediately.

Now, let us come back to the above defined operator $\mathcal{A}$. It is clear that it has as-well discrete semi-simple spectrum, namely

[TABLE]

with the corresponding eigenfunctions $\left\{\varphi_{j}\right\}_{j=1}^{\infty}.$

Let some $\rho>1$. Then, there exists $N\in\mathbb{N}$ such that

[TABLE]

The first $N$ eigenvalues are usually called the unstable eigenvalues.

It is obvious that, given any prescribed $\alpha>0$, if we take $\rho$ and $N$ large enough, we may suppose that the following relations hold true:

[TABLE]

for all $i,j=1,2,...,N$.

Although it would be possible to treat the case of semi-simple unstable eigenvalues following [17], for the sake of simplicity, we assume that

[TABLE]

i.e., we have

[TABLE]

Now, since we are set with the theoretical results and the hypotheses of the paper, we may proceed to apply the approach from [15],[17]. Firstly, in order to lift the boundary control into the equations (to obtain an internal control-type problem), we introduce the so-called Neumann operator as: given $g\in L^{2}(\Gamma_{1})$ and $\gamma>0$, we denote by $D_{\gamma}g:=y$, the solution to the equation

[TABLE]

For $\gamma$ large enough, equation (3.9) has a unique solution, defining so the map $D_{\gamma}\in L(L^{2}(\Gamma_{1}),H^{\frac{1}{2}})$(for further details check e.g. [13, p. 6]). Also, appealing to Green formula (see the computations in [17, Eqs. (4.1)-(4.2)]), we deduce that

[TABLE]

Here, $d\sigma$ is the surface measure on $\Gamma_{1}$.

Next, we choose

[TABLE]

$N$ constants, large enough, such that equation (3.9) is well-posed for each of them, and denote by $D_{\gamma_{i}},\ i=1,2,...,N$, the corresponding Neumann maps.

Following the ideas in [17], we denote by B the Gram matrix of the system $\left\{\varphi_{i}|_{\Gamma_{1}}\right\}_{i=1}^{N}$ in the Hilbert space $L^{2}(\Gamma_{1})$, with the standard scalar product

[TABLE]

More precisely,

[TABLE]

Further, we introduce the matrices

[TABLE]

[TABLE]

and

[TABLE]

where

[TABLE]

We recall the Appendix in [17] where it is shown that the sum $B_{1}+B_{2}+\dots+B_{N}$ is an invertible matrix, and consequently, the matrix $A$ is well-defined.

Now, let us introduce the feedback laws:

[TABLE]

for $t\geq 0,\ x\in\Gamma_{1}$, and $k=1,2,...,N$. Then, define $u=u(y)$ as

[TABLE]

which, in a condensed form, can be written as

[TABLE]

We claim that, once inserted this feedback form $u$ into the equation (2.5) it yields the local exponential asymptotic stability of the corresponding closed-loop system (2.5). More exactly, we will show that

Theorem 3.1**.**

Let $\eta>0$ be sufficiently small. Under $(H_{0})$-$(H_{4})$, for each $y_{o}\in L^{2}$ with $|y_{0}|_{2}<\eta$, once plugged the feedback law $u$, given by (3.18), into the equation (2.5), there exists a unique solution $y$ to the closed-loop equation (2.5), which belongs to the space

[TABLE]

Consequently, $Y(t)=\Gamma(t)y(t)$ is the unique solution of the stochastic cubic equation (1.1), which satisfies

[TABLE]

4 Proof of the main result

In order to ease our problem, we shall equivalently rewrite equation (2.5) as an internal control-type problem, by using similar arguments as in [15, Eqs. (17)-(19)]. We arrive to:

[TABLE]

The following result is related to the linear operator which governs equation (4.1), i.e.

[TABLE]

$\forall y\in\mathcal{D}(\mathbb{A})=\mathcal{D}(\mathcal{A}).$ It says that the semigroup generated by it can be written in a mild formulation via a kernel $p$, as

[TABLE]

Its proof is given in the Appendix.

Lemma 4.1**.**

The solution $z$ of

[TABLE]

can be written in a mild formulation as

[TABLE]

where

[TABLE]

for $t\geq 0,x,\xi\in\mathcal{O}$. Here

[TABLE]

[TABLE]

and

[TABLE]

The quantities $q_{ji}(t)$ and $w_{i}^{j}(t)$, involved in the definition of $p$, obey the estimates: for some $C_{q}>0$,

[TABLE]

for all $i,j=1,2,...,N$, and for some $C_{w}>0$

[TABLE]

for all $i=1,2,...,N$ and $j=N+1,N+2,...$. (Recall that we denoted by $\lambda_{j}$ the eigenvalues of the Laplace operator. )

In particular, we have that $\mathbb{A}$ generates a $C_{0}-$semigroup in $L^{2}$, which is exponentially decaying, i.e.

[TABLE]

Besides this, we also have that

[TABLE]

Relying on the above lemma, we may now proceed to prove the main existence & stabilization result of the present work.

[TABLE]

Proof of Theorem 3.1. The space

[TABLE]

is endowed with the norm

[TABLE]

It is clear that, for all $y\in\mathcal{Y}$, we have

[TABLE]

For $r>0$, we denote by $B_{r}(0)$ the ball of radius $r$, centered at the origin, of the space $\mathcal{Y}$, i.e.

[TABLE]

Now, let us introduce the map $\mathcal{G}:\mathcal{Y}\rightarrow\mathcal{Y}$, as

[TABLE]

where

[TABLE]

Clearly seen, if there exists a solution $y$ to (4.1), then necessarily it has to be a fixed point of the map $\mathcal{G}$. Thus, in what follows we aim to show that $\mathcal{G}$ is a contraction, which maps the ball $B_{r}(0)$ into itself, for $r>0$ properly chosen. Then, via the contraction mappings theorem, we deduce that $\mathcal{G}$ has a unique fixed point $y$, which is, in fact, the mild solution to the equation (4.1) (or, equivalently to (2.5)). Then, easily, one arrives to the wanted conclusion claimed by the theorem.

Let us first take care of the term $\mathcal{F}y$. For $i\in\mathbb{N}\setminus\left\{0\right\},$ we denote by

[TABLE]

Taking advantage of Lemma 4.1, where it is described the form of the kernel $p$, and notation (4.9), we equivalently rewrite the term $\mathcal{F}y$ as

[TABLE]

About $\mathcal{P}_{i},\ i\in\mathbb{N}\setminus\left\{0\right\}$, we have the following result, which will be proved in the Appendix.

Lemma 4.2**.**

With respect to the notations in (4.9), for all $\mu>0$, $i,j\in\mathbb{N}\setminus\left\{0\right\}$ and $0<s<t$, we have, concerning $A_{2}^{i}$:

[TABLE]

and

[TABLE]

and

[TABLE]

and

[TABLE]

Next, concerning $A_{3}^{i}$:

[TABLE]

and

[TABLE]

and

[TABLE]

and

[TABLE]

Relations (4.11)-(4.18), given in Lemma 4.2, are the key bounds used for estimating the term $\mathcal{F}(y)$, in the $|\cdot|_{2}$ and $\|\cdot\|_{\frac{1}{2}}$-norm, respectively. Indeed, we have, in virtue of Parseval’s identity, relation (4.4) and the notations in (4.9) and (4.10), that

[TABLE]

since, in virtue of (3.7), we have that

[TABLE]

and

[TABLE]

for all $i=1,2,...,N$. The above leads to

[TABLE]

where $B(x,y)$ is the classical beta function, which is finite for $x,y>0$.

We go on with $\mathcal{F}_{2}$. We appeal again to Parseval’s identity, to deduce that

[TABLE]

Thus, by (4.8), the latter implies that

[TABLE]

We move on to the term $\mathcal{F}_{3}(y)$. Taking advantage of the relation (4.5), we have, via Parseval’s formula, that

[TABLE]

Recall that $\mu_{j}=\lambda_{j}-c$, and so, the series

[TABLE]

has the same nature as

[TABLE]

which, by (3.3), is convergent. Hence, (4.23) yields that

[TABLE]

It then follows by (4.20), (4.22) and (4.24), that

[TABLE]

which, together with (4.6), drives us to the following estimate

[TABLE]

for all $y\in\mathcal{Y}$.

Next, the effort is to obtain similar estimates for the $\|\cdot\|_{\frac{1}{2}}$-norm as-well. To this end, we start again with the term $\mathcal{F}(y)$ introduced in (4.10)-(4.9). We proceed in a similar manner as in (4.19)

[TABLE]

Therefore, in virtue of (4.8), we are lead to

[TABLE]

Here we used relation (3.7), namely

[TABLE]

and

[TABLE]

It then follows by (4.28), that,

[TABLE]

Finally, with similar arguments as above and from (4.22) and (4.23), via Lemma 4.2, we may deduce as-well that

[TABLE]

and

[TABLE]

respectively.

We conclude by (4.26), (4.29)-(4.31) and (4.7), that

[TABLE]

Of course, a similar procedure may be applied to the difference $\mathcal{G}y-\mathcal{G}\overline{y}$, for some $y,\overline{y}\in\mathcal{Y}$, to deduce that

[TABLE]

Recall that $|y_{o}|_{2}<\eta.$ Hence, (4.32) yields that, if we take $\eta=r^{2}$, then for all $y\in B_{r}(0)$, we have

[TABLE]

Hence, if $r$ is close enough to zero, one has

[TABLE]

and, by (4.33),

[TABLE]

for some $q<1$. Thus, $\mathcal{G}$ maps the ball $B_{r}(0)$ into itself, and it is a contraction on $B_{r}(0)$, as claimed. The conclusion of the theorem follows immediately. Other details are omitted. ∎

5 Conclusions

In this work, based on the ideas of constructing proportional type stabilizing feedbacks in [15] together with a fixed point argument, we managed to obtain a first result of boundary stabilization of the stochastic nonautonomous cubic heat equation. In comparison to [15] and [16], in this work we managed to pass from the $1-D$ case to the $2-D$ case domain $\mathcal{O}$, based on the $L^{\infty}$-estimations and of $L^{2}-$estimates of the eigenfunctions of the Laplacean. As a future work, we intend to solve the $3-D$ case as-well.

Acknowledgment

This work was supported by a grant of the ”Alexandru Ioan Cuza” University of Iasi, within the Research Grants program, Grant UAIC, code GI-UAIC-2018-03.

Appendix

Before we give the details for the proofs of Lemmas 4.1 and 4.2, we first show that, in case $\mathcal{O}=[0,\pi]\times[0,\pi]$, relation (3.3) $(H_{1})$ holds true. Indeed, in this case, it is known that the nonzero eigenvalues of the Laplace operator are of the precise form

[TABLE]

So, the summation in (3.3), reads as

[TABLE]

since the series $\sum_{i=1}^{\infty}\frac{1}{(i^{2}+1)^{\frac{2}{3}}}$ is convergent.

Next, we go on with the two proofs.

Proof of Lemma 4.1. In equation (4.2), we decompose $z$ as

[TABLE]

where $z_{j}(t)=\left<z(t),\varphi_{j}\right>,\ j=1,2,....$

Scalarly multiplying equation (4.2) by $\varphi_{j},\ j=1,...,N,$ and arguing as in [17, Eqs.(4.11)- (4.13)], we get that the first $N$ modes of the solution $z$ satisfy

[TABLE]

where we have denoted by

[TABLE]

This yields that, there exist continuous functions $\left\{q_{ij}:[0,\infty)\rightarrow\mathbb{R}\right\}_{i,j=1}^{N}$ such that

[TABLE]

Besides this, scalarly multiplying (5.1) by $A\mathcal{Z}$ we get as in [17] that

[TABLE]

Here, $\|\cdot\|_{N}$ is the euclidean norm in $\mathbb{R}^{N}$.

Thus, (5.2), (5.3), and the fact that $\gamma_{1}>\rho$ yield

[TABLE]

Since, by (3.16), the feedback forms $u_{i},\ i=1,...,N,$ are some linear combinations of the modes $z_{1},...,z_{N}$, we get from (5.2) that there exist continuous functions $\left\{r_{ik}:[0,\infty)\times\Gamma_{1}\rightarrow\mathbb{R}\right\}_{i,k=1}^{N}$ such that

[TABLE]

where, simple computations, involving (3.16) and (5.4), imply that

[TABLE]

We move on to the modes $z_{j},\ j>N$. Scalarly multiplying equation (4.2) by $\varphi_{j},\ j>N$, we get

[TABLE]

where using (3.10) we arrive to

[TABLE]

Then, the variation of constants formula gives

[TABLE]

which by (5.5) becomes

[TABLE]

Setting

[TABLE]

$k=1,2,...,N$ and $j>N$, the previous relation can be rewritten as

[TABLE]

In virtue of (3.5) and (5.6), taking into account the form of $w_{k}^{j}$ one can easily show that

[TABLE]

Now, it is clear that, by (5.2) and (5.7), (5.4) and (5.8), all the relations (4.3)-(4.6) are proved. To conclude, we notice that, scalarly multiplying equation (4.2) by $z$, integrating over time, and using the $|\cdot|_{2}$-exponential decay (4.6), one may show that relation (4.7) holds true as-well. ∎

[TABLE]

Proof of Lemma 4.2. We have, in virtue of (1.2), (3.4) and Schwarz inequality that

[TABLE]

Here and below, we shall frequently use the next two simple but useful inequalities:

[TABLE]

and

[TABLE]

By the first inequality, we have that

[TABLE]

While, by the second inequality, we have that

[TABLE]

Having these in mind, in yields by (5.9) that

[TABLE]

Analogously, by Schwarz’s inequality

[TABLE]

So, relations (4.11) and (4.15) are proved.

As seen in the proof of Theorem 3.1, the above estimates can be used to bound the terms $\mathcal{F}_{1}(y)$ and $\mathcal{F}_{3}(y)$, while, for the term $\mathcal{F}_{2}(y)$, relations (4.12) and (4.16) are involved. We show them below. We use Schwarz’s inequality and (1.2), to deduce that

[TABLE]

Likewise

[TABLE]

by the fractional Sobolev inequality (3.1).

We move to the estimates containing the $\lambda_{j}$’s (which correspond to the $\|\cdot\|_{\frac{1}{2}}$-estimates). In a similar manner as above, we also have that, for each $j\in\mathbb{N}\setminus\left\{0\right\}$,

[TABLE]

Similarly,

[TABLE]

Thus, relations (4.13) and (4.17) are proved.

We conclude by showing the last two bounds. We have, as in (5.12),

[TABLE]

and, as in (5.13)

[TABLE]

The proof is complete. ∎