Global Stabilization of 2D Forced Viscous Burgers' Equation Around Nonconstant Steady State Solution by Nonlinear Neumann Boundary Feedback Control:Theory and Finite Element Analysis

TL;DR

This paper develops a nonlinear boundary feedback control method to achieve global stabilization of the 2D forced viscous Burgers' equation around a nonconstant steady state, with finite element analysis and numerical validation.

Contribution

It introduces a novel nonlinear Neumann boundary feedback control for stabilizing the 2D Burgers' equation around nonconstant steady states, including error estimates and convergence analysis.

Findings

Global stabilization achieved under smallness condition.

Optimal error estimates in $L^ abla(L^2)$ and $L^ abla(H^1)$ norms.

Numerical experiments confirm theoretical results.

Abstract

Global stabilization of viscous Burgers' equation around constant steady state solution has been discussed in the literature. The main objective of this paper is to show global stabilization results for the 2D forced viscous Burgers' equation around a nonconstant steady state solution using nonlinear Neumann boundary feedback control law, under some smallness condition on that steady state solution. On discretizing in space using $C^0$ piecewise linear elements keeping time variable continuous, a semidiscrete scheme is obtained. Moreover, global stabilization results for the semidiscrete solution and optimal error estimates for the state variable in $L^\infty(L^2)$ and $L^\infty(H^1)$-norms are derived. Further, optimal convergence result is established for the boundary feedback control law. All our results in this paper preserve exponential stabilization property. Finally, some numerical experiments are documented to confirm our theoretical findings.

Figures

Tables

Table 1. Table 1: Errors and convergence rate for the state variable w 𝑤 w with c 0 = 1 subscript 𝑐 0 1 c_{0}=1 and t = t n = 1 𝑡 subscript 𝑡 𝑛 1 t=t_{n}=1

$h$	$\parallel w\left(t_n\right)-W^n\parallel$	Conv. Rate	${\parallel w\left(t_n\right)-W^n\parallel }_1$	Conv. Rate
$\frac{1}{4}$	$0.0214813$		$0.153906$
$\frac{1}{8}$	$0.0059996$	1.8401	$0.0777889$	0.9844
$\frac{1}{16}$	$0.00157007$	1.934	$0.0383679$	1.019
$\frac{1}{32}$	$0.00041675$	1.9135	$0.0185611$	1.047

Table 2. Table 2: Errors and convergence rate for the feedback control variable v 𝑣 v when c 0 = 1 subscript 𝑐 0 1 c_{0}=1 and t n = 1 subscript 𝑡 𝑛 1 t_{n}=1

$h$	$\left|v\left(t_n\right)-v_h\left(t_n\right)\right|$	Conv. Rate
$\frac{1}{4}$	$0.13599$
$\frac{1}{8}$	$0.0412476$	1.7211
$\frac{1}{16}$	$0.0115142$	1.8409
$\frac{1}{32}$	$0.00309629$	1.8595

Equations

$$u_t-\nu \Delta u+u(\nabla u\cdot 1)=f^{\infty }\text{in}(x,t)\in \Omega \times (0,\infty ),$$

$$\frac{\partial u}{\partial n}(x,t)=v_2(x,t)\text{on}(x,t)\in \partial \Omega \times (0,\infty ),$$

$$u(x,0)=u_0(x)x\in \Omega ,$$

$$-\nu \Delta u^{\infty }+u^{\infty }(\nabla u^{\infty }\cdot 1)$$

$$\frac{\partial u^{\infty }}{\partial n}$$

$$\underset{t\to \infty }{\lim }u(x,t)=u^{\infty }\text{a. e.}x\in \Omega ,$$

$$w_t-\nu \Delta w+u^{\infty }(\nabla w\cdot 1)+w(\nabla u^{\infty }\cdot 1)+w(\nabla w\cdot 1)=0\text{in}(x,t)\in \Omega \times (0,\infty ),$$

$$\frac{\partial w}{\partial n}(.,t)=v_2(x,t),\text{on}\partial \Omega \times (0,\infty ),$$

$$w(0)=u_0-u^{\infty }=w_0(\text{say})\text{in}\Omega .$$

$${\Vert v\Vert }_{L^p((0,T);X)}:={\left({\int }_0^T{\Vert v(t)\Vert }_X^{p}dt\right)}^{\frac{1}{p}}<\infty \text{for}1\le p<\infty ,$$

$${\Vert v\Vert }_{L^{\infty }((0,T);X)}:=\underset{0\le t\le T}{\mathrm{esssup}}{\Vert v(t)\Vert }_X<\infty .$$

$$T:H^1(\Omega )\hookrightarrow L^q(\partial \Omega )\text{for}2\le q<\infty$$

$${\Vert Ty\Vert }_{L^q(\partial \Omega )}\le C{\Vert y\Vert }_{H^1(\Omega )},$$

$$\left\lVert y\right\rVert^{2}\leq C_{F}\Big{(}\left\lVert\nabla y\right\rVert^{2}+\left\lVert y\right\rVert^{2}_{L^{2}(\partial\Omega)}\Big{)},$$

$${\int }_{\Omega }{y}^{2}dx\le \underset{x\in \partial \Omega }{\sup }|x{|}^2{\int }_{\Omega }|\nabla y{|}^{2}dx+\underset{x\in \partial \Omega }{\sup }|x|{\int }_{\partial \Omega }{y}^{2}d\Gamma .$$

$$\displaystyle\left\lVert w\right\rVert_{L^{4}}\leq C\Big{(}\left\lVert w\right\rVert^{1/2}\left\lVert\nabla w\right\rVert^{1/2}+\left\lVert w\right\rVert\Big{)},\quad\text{and for $w\in H^{2}(\Omega)$}\quad\left\lVert\nabla w\right\rVert_{L^{4}}\leq C\Big{(}\left\lVert w\right\rVert^{1/4}\left\lVert\Delta w\right\rVert^{3/4}+\left\lVert w\right\rVert\Big{)}.$$

$$\left\lVert z\right\rVert_{L^{\infty}}\leq C\Big{(}\left\lVert z\right\rVert^{\frac{1}{2}}\left\lVert\Delta z\right\rVert^{\frac{1}{2}}+\left\lVert z\right\rVert\Big{)}.$$

$${\Vert f^{\infty }\Vert }_{-1}\le \frac{3{\nu }^2}{16N},$$

$$\Vert \nabla u^{\infty }\Vert \le \frac{\nu }{4N}.$$

$$\nu {\Vert \nabla u^{\infty }\Vert }^2$$

$$\displaystyle\leq\Big{(}\left\lVert f^{\infty}\right\rVert_{-1}+N\left\lVert\nabla u^{\infty}\right\rVert^{2}\Big{)}\left\lVert\nabla u^{\infty}\right\rVert.$$

$$\displaystyle\left\lVert\nabla u^{\infty}\right\rVert\Big{(}\nu-N\left\lVert\nabla u^{\infty}\right\rVert\Big{)}\leq\left\lVert f^{\infty}\right\rVert_{-1}.$$

$$N{\Vert \nabla u^{\infty }\Vert }^2-\nu \Vert \nabla u^{\infty }\Vert +\frac{3{\nu }^2}{16N}\ge 0,$$

$$(\Vert \nabla u^{\infty }\Vert -\frac{3\nu }{4N})(\Vert \nabla u^{\infty }\Vert -\frac{\nu }{4N})\ge 0.$$

$$\nu {\Vert \Delta u^{\infty }\Vert }^2$$

$$\le \frac{\nu }{4}{\Vert \Delta u^{\infty }\Vert }^2+\frac{1}{\nu }{\Vert f^{\infty }\Vert }^2+{\Vert u^{\infty }\Vert }_{L^4}{\Vert \nabla u^{\infty }\Vert }_{L^4}\Vert \Delta u^{\infty }\Vert$$

$$\displaystyle\leq\frac{\nu}{2}\left\lVert\Delta u^{\infty}\right\rVert^{2}+\frac{1}{\nu}\left\lVert f^{\infty}\right\rVert^{2}+C_{GN}\Big{(}\left\lVert u^{\infty}\right\rVert^{6}\left\lVert\nabla u^{\infty}\right\rVert^{4}+\left\lVert u^{\infty}\right\rVert^{3}\left\lVert\nabla u^{\infty}\right\rVert+\left\lVert u^{\infty}\right\rVert^{10}+\left\lVert u^{\infty}\right\rVert^{4}\Big{)},$$

$$\nu(\nabla u^{\infty},\nabla v)+B\big{(}u^{\infty};u^{\infty},v\big{)}=(f^{\infty},v)\quad\forall v\in H^{1}(\Omega)/\mathbb{R}.$$

$$\nu(\nabla u^{\infty}_{m},\nabla v_{m})+B\big{(}u^{\infty}_{m};u^{\infty}_{m},v_{m}\big{)}=(f^{\infty},v_{m})\quad\forall v_{m}\in V^{m}.$$

$$\displaystyle(F(\alpha))_{i}=\nu(\nabla v_{m},\nabla\phi_{i})+B\big{(}v_{m};v_{m},\phi_{i}\big{)}-(f^{\infty},\phi_{i}).$$

$$(F(\alpha ),\alpha )$$

$$\displaystyle=\nu\left\lVert\nabla v_{m}\right\rVert^{2}-B\big{(}v_{m};v_{m},v_{m}\big{)}-(f^{\infty},v)$$

$$\displaystyle\geq\left\lVert\nabla v_{m}\right\rVert\Big{(}\nu\left\lVert\nabla v_{m}\right\rVert-N\left\lVert\nabla v_{m}\right\rVert^{2}-\left\lVert f^{\infty}\right\rVert_{-1}\Big{)}.$$

$$\displaystyle(F(\alpha),\alpha)\geq\left\lVert\nabla v_{m}\right\rVert\Big{(}\frac{3\nu}{4}\left\lVert\nabla v_{m}\right\rVert-\left\lVert f^{\infty}\right\rVert_{-1}\Big{)}$$

$$\Big{(}F(\alpha),\alpha\Big{)}>0\quad\forall\quad|\alpha|=R.$$

Full text

Keywords: Forced viscous Burgers’ equation, boundary feedback control, stabilization, finite element method, error estimate, numerical experiments

AMS subject classification: 35B37, 65M60, 65M15, 93B52, 93D15

1 Introduction

Consider the following two-dimensional forced viscous Burgers’ equation with Neumann boundary control : seek $u=u(x,t),$ $t>0$ which satisfies

[TABLE]

where $\Omega$ is a bounded subset in $\mathbb{R}^{2}$ with smooth boundary, $\nu>0$ is a constant, $v_{2}$ is a scalar control input, ${\bf{1}}=(1,1)$, $\nu\Delta u$ is the diffusive term, $u(\nabla u\cdot{\bf{1}})=u\sum_{i=1}^{2}u_{x_{i}}$ is the nonlinear convective term, forcing function $f^{\infty}=f^{\infty}(x)$ is independent of the time and $u_{0}$ is a given function. When $\nu=0$, it is called inviscid Burgers’ equation, which is studied in nonlinear wave propagation.

For one dimensional Burgers’ equation, local stabilization results are documented in [5], [6] for distributed and Dirichlet boundary control and in [7] for Neumann boundary control with small initial data. For more details, see [11], [12] and [18]. Using linear feedback control law for the linearized part as in Navier-Stokes equations [20], local stabilizability results can be proved for the two dimensional Burgers’ equation. In [23], the authors have shown local stabilization results for the two dimensional Burgers’ equation directly through a nonlinear feedback control law and several numerical experiments are also reported in their article conforming their theoretical results. Subsequently Buchot et al. [4] have derived local stabilization results in the case of partial information for the two dimensional Burgers’ type equation.

Based on Lyapunov type functional, global stabilization result around constant steady state solution for one dimensional Burgers’ equation is derived in [13] and [3] for both Dirichlet and Neumann boundary control laws. For more detailed references, we refer to [17], [21] and [22]. In [8], authors implement Dirichlet boundary feedback control law on the obstacles for two dimensional Burgers’ equation by solving both Riccati equation and Chandrasekhar equations. In [14], it is shown that using a nonlinear Neumann boundary feedback control laws, solution of 1D Burgers’ equation converges exponentially to its constant steady state solution in $L^{\infty}(L^{2})$, $L^{\infty}(L^{\infty})$, $L^{\infty}(H^{1})$ and $L^{\infty}(H^{2})$ norms. Then, an application of finite element method in spatial direction yields a semidiscrete system and global stabilization results are proved for the semidiscrete approximation. Finally, optimal error estimates for the state variable and superconvergence result for the control laws are established. This analysis is then extended to 2D Burgers’ equation in [15], and stabilization results depicting convergence of the solution to its constant steady state solution are derived. Moreover, convergence result for the nonlinear feedback control law is also documented.

But when the steady state solution is non constant in the case of forced viscous Burgers’ equation, it is not known whether global stabilization results still holds or not. Also, when applying finite element method, no result is available in the literature on rate of convergence. Hence, in this paper, an attempt has been made to fill this gap.

Now the corresponding equilibrium or steady state problem becomes: find $u^{\infty}$ as a solution of

[TABLE]

To achieve

[TABLE]

it is enough to consider $\lim_{t\to\infty}w=0,$ where $w=u-u^{\infty}$ and $w$ satisfies

[TABLE]

The motivation to choose Neumann boundary control comes from the physical situation. For instance, in thermal problem, one can not actuate the temperature $w$, but the heat flux $\frac{\partial w}{\partial n}$. Here, the control variable $v_{2}$ is to be chosen as a function of $w$ appropriately. In this article, we first prove global stabilization results in $L^{\infty}(L^{2})$, $L^{\infty}(H^{1})$- norms for the problem (1.6)-(1.8) with control law (3.1). Then, $C^{0}$- conforming finite element method is applied to discretize the spatial variable, keeping time variable continuous and global stabilization results are proved for the semidiscrete solution. Further, optimal error estimates are derived for the state variable and superconvergence results are obtained for the nonlinear Neumann boundary feedback control law.

The rest of the paper is organized as follows. Section $2$ contains notations, preliminary results and properties of the steady state solution. In Section $3,$ we focus on global stabilization results using nonlinear feedback control law. Section $4$ deals with finite element approximation and global stabilization results for the semidiscrete system. Further, optimal error estimates are obtained for the state variable and convergence result is derived for the feedback control law. Finally, Section $4$ concludes with some numerical experiments.

2 Notations and properties of steady state solution

In this section, we discuss some properties for the steady state problem (1.4)-(1.5) along with some preliminary results.

Throughout this paper, we use standard Sobolev space $H^{m}(\Omega)=W^{m,2}(\Omega)$ with norm $\left\lVert\cdot\right\rVert_{m},$ and seminorm $|\cdot|_{m}$. For $m=0,$ it corresponds to the usual $L^{2}$ norm and is denoted by $\left\lVert\cdot\right\rVert$. Moreover, $(\cdot,\cdot)$ and $\langle\cdot,\cdot\rangle$ denote the innerproducts in $L^{2}(\Omega)$ and $L^{2}(\partial\Omega)$ respectively. Also we use the Sobolev space $H^{s}(\partial\Omega)$ of order $s$. The space $L^{p}((0,T);X)$ $1\leq p\leq\infty,$ consists of all strongly measurable functions $v:[0,T]\rightarrow X$ with norm

[TABLE]

and

[TABLE]

When there is no confusion we denote $L^{p}((0,T);X)$ by $L^{p}(X)$. For a trilinear form \Big{(}v\big{(}\nabla w\cdot{\bf{1}}\big{)},\phi\Big{)}, denote by B\big{(}v;w,\phi\big{)}:=\Big{(}v\big{(}\nabla w\cdot{\bf{1}}\big{)},\phi\Big{)}.

Now we present a few well known theorems and inequalities which are crucial for our analysis.

Theorem 2.1**.**

([25]) (Brouwer’s fixed point theorem): Let $H$ be a finite dimensional Hilbert space with inner product $(\cdot,\cdot)$ and norm $\left\lVert.\right\rVert.$ Let ${\bf{F}}:H\rightarrow H$ be a continuous function. If there is a real number $R>0$ such that $({\bf{F}}(z),z)>0$ $\forall~{}z$ with $\left\lVert z\right\rVert=R,$ then there exists $z^{*}\in H$ such that $\left\lVert z^{*}\right\rVert\leq R$ and ${\bf{F}}(z^{*})=0.$

The following trace embedding result holds for 2D.

**Boundary Trace Embedding Theorem (page $164,$ [1]): ** There exists a bounded linear map

[TABLE]

such that

[TABLE]

for each $y\in H^{1}(\Omega)$. Below, we recall the following inequalities for our subsequent use

Friedrichs’s inequality: (See [15]) For $y\in H^{1}(\Omega),$ there holds

[TABLE]

where $C_{F}>0$ is Friedrichs’s constant. More explicitly

[TABLE]

Hence the Friedrichs’s constant can be taken as $C_{F}=\max\{\sup_{x\in\partial\Omega}|x|^{2},\sup_{x\in\partial\Omega}|x|\}$.

Gagliardo-Nireberg inequality (see [19]): For $w\in H^{1}(\Omega)$

[TABLE]

Agmon’s inequality (see [2]): For $z\in H^{2}(\Omega),$ there holds

[TABLE]

2.1 Some properties of the steady state problem

Let $H^{1}/\mathbb{R}$ be the quotient space. Infact, $\left\lVert\nabla v\right\rVert$ is a norm on this space. We now make the following assumption:

Assumption ${\bf(A1)}$

[TABLE]

where $\left\lVert f^{\infty}\right\rVert_{-1}=\sup_{v\in H^{1}/\mathbb{R}}\frac{(f^{\infty},v)}{\left\lVert\nabla v\right\rVert}$ and $N=\max(N_{1},N_{2}),$ with N_{1}=\sup_{v,\hskip 2.84544ptz,\hskip 2.84544pt\phi\in H^{1}/\mathbb{R}}\frac{B\big{(}v;z,\phi\big{)}}{\left\lVert\nabla v\right\rVert\left\lVert\nabla z\right\rVert\left\lVert\nabla\phi\right\rVert} and N_{2}=\sup_{v\in H^{1}/\mathbb{R},\hskip 2.84544pt(z,\phi)\in(H^{1}(\Omega))^{2}}\frac{B\big{(}v;z,\phi\big{)}}{\left\lVert\nabla v\right\rVert\left|\!\left|\!\left|z\right|\!\right|\!\right|\left|\!\left|\!\left|\phi\right|\!\right|\!\right|}.

The assumption ${\bf(A1)}$ provides bound for the steady state problem (1.4)-(1.5).

Lemma 2.1**.**

Under the assumption ${\bf(A1)}$, there exists a solution $u^{\infty}\in H^{1}/\mathbb{R}$ of (1.4)-(1.5) satisfying the following estimate:

[TABLE]

In addition, if $u^{\infty}\in H^{2}(\Omega)$ and $f^{\infty}\in L^{2}(\Omega)$, then $\left\lVert\Delta u^{\infty}\right\rVert\leq C(\nu,\left\lVert f^{\infty}\right\rVert)$.

Proof.

Multiply (1.4) by $u^{\infty}$ to obtain

[TABLE]

Therefore

[TABLE]

Using assumption ${\bf(A1)}$, it follows that

[TABLE]

that is,

[TABLE]

Hence, when both the factor is negative, we obtain $\left\lVert\nabla u^{\infty}\right\rVert\leq\frac{3\nu}{4N}$ and $\left\lVert\nabla u^{\infty}\right\rVert\leq\frac{\nu}{4N}$.

Multiply (1.4) by $-\Delta u^{\infty}$ to arrive using the bounds of $\left\lVert u^{\infty}\right\rVert$ and $\left\lVert\nabla u^{\infty}\right\rVert$ at

[TABLE]

where the constant $C_{GN}$ is appeared in the Gagliardo Nirenberg inequality.

Hence when $f^{\infty}\in L^{2}(\Omega)$, \left\lVert\Delta u^{\infty}\right\rVert\leq C\Big{(}\nu,\left\lVert f^{\infty}\right\rVert\Big{)}.

Now below, we provide the proof for the existence of a weak solution $u^{\infty}$ which satisfy (2.3). Corresponding weak formulation for the steady state solution (1.4)-(1.5) is to seek $u^{\infty}\in H^{1}(\Omega)/\mathbb{R}$ such that

[TABLE]

Let $\{\phi_{i}\}_{i=1}^{\infty}$ be an orthogonal basis for $H^{1}(\Omega)/\mathbb{R}$. Denote $V^{m}=span\{\phi_{1},\phi_{2},...,\phi_{m}\}$ and finite linear combination of $\phi_{i}$ are dense in $H^{1}(\Omega)/\mathbb{R}$. We look for $u^{\infty}_{m}\in V^{m}$ such that

[TABLE]

It is sufficient to verify (2.6) for $v_{m}=\phi_{i}$, $1\leq i\leq m$. Let $\alpha=(\alpha_{i})_{i=1}^{m}\in\mathbb{R}^{m}$. For such $\alpha$, we relate a unique element $v_{m}\in V^{m}$ by $v_{m}=\sum_{i=1}^{m}\alpha_{i}\phi_{i}$. This establishes a linear bijection between $\mathbb{R}^{m}$ and $V^{m}$. Since $\{\phi_{i}\}$ are orthogonal in $H^{1}(\Omega)/\mathbb{R}$, $\left\lVert v\right\rVert^{2}_{1}=\sum_{i=1}^{m}|\alpha_{i}|^{2}=|\alpha|^{2}$. Now we define $F:\mathbb{R}^{m}\mapsto\mathbb{R}^{m}$ by

[TABLE]

Hence (2.6) has a solution if there exists a $\alpha$ such that $F(\alpha)=0$. It is valid that

[TABLE]

Using the bound for the steady state solution $\left\lVert\nabla v_{m}\right\rVert\leq\frac{\nu}{4N}$, we get

[TABLE]

Hence if $|\alpha|=R$ is chosen large enough such that $\frac{3\nu}{4}R>\left\lVert f^{\infty}\right\rVert_{-1}$, we have

[TABLE]

From the condition $\left\lVert f^{\infty}\right\rVert_{-1}\leq\frac{3\nu^{2}}{16N}$, infact we can take $R=\frac{\nu}{4N}$. Further by (2.6), $\left\lVert u^{\infty}_{m}\right\rVert_{1}\leq R$ and $R$ is independent of $m$. Since $\{u^{\infty}_{m}\}$ is uniformly bounded in $H^{1}(\Omega)/\mathbb{R}$, so there exists a convergent subsequence still denoted by $\{u^{\infty}_{m}\}$ such that $u^{\infty}_{m}\rightharpoonup u^{\infty}$ in $H^{1}(\Omega)/\mathbb{R}$. Since $H^{1}(\Omega)/\mathbb{R}$ is compactly embedded in $L^{2}$, then there exists a convergent subsequence still denoted by $\{u^{\infty}_{m}\}$ such that $u^{\infty}_{m}\to u^{\infty}$ in $L^{2}$.

Hence by Brouwer’s fixed point theorem 2.1, there exists a $\alpha^{*}$ such that $\alpha^{*}\leq R$ and $F(\alpha^{*})=0$.

Let $m>i$. From (2.6), we obtain

[TABLE]

Clearly

[TABLE]

For the nonlinear term, we can rewrite

[TABLE]

Boundedness of $\left\lVert u^{\infty}_{m}\right\rVert_{1}$ and $\left\lVert u^{\infty}_{m}\right\rVert_{L^{\infty}}$, strong convergence of $u^{\infty}_{m}\to u^{\infty}$ in $L^{2}$ norm and compact embedding of $H^{1}(\Omega)$ onto $H^{s}(\partial\Omega)$ with $s<1$ implies that right hand side goes to zero in the above inequality. Hence when $i$ goes to infinity, we obtain

[TABLE]

By density, (2.7) is true for all $v\in H^{1}(\Omega)/\mathbb{R}$. ∎

3 Stabilization results

Before proceeding, let us first construct the feedback control law $v_{2}$. To obtain $v_{2}$, we consider Lyapunov energy functional $V(t)=\frac{1}{2}\int_{\Omega}w(x,t)^{2}\;dx$. Then

[TABLE]

Using the notation B\big{(}\cdot;\cdot,\cdot\big{)} for trilinear term, we can bound the right hand term as

[TABLE]

and

[TABLE]

Hence, we get

[TABLE]

Now, choose the Neumann boundary feedback control law as

[TABLE]

to obtain

[TABLE]

C_{LYP}=\frac{2}{C_{F}}\min\Big{\{}\frac{7\nu}{8},\frac{c_{0}}{2}+\frac{7\nu}{8}\Big{\}}. Now, $w$ satisfies the weak formulation of (1.6)-(1.8) as

[TABLE]

with $w(0)=w_{0},$ where $\langle v,w\rangle_{\partial\Omega}:=\int_{\partial\Omega}vw\;d\Gamma$. Throughout the paper $C=C(\left\lVert w_{0}\right\rVert_{3},\nu,f^{\infty})$ is a generic positive constant independent of the discretizing parameter $h$.

Further we make the following assumption

Assumption ${\bf(A2)}$

• •

Let $u^{\infty}\in H^{2}(\Omega)$, $f^{\infty}\in L^{2}(\Omega)$.

• •

Compatibility conditions at $t=0$ \Big{(}\frac{\partial w_{0}}{\partial n}=v_{2}(x,0),\frac{\partial w_{t}}{\partial n}(x,0)=v_{2t}(x,0)\Big{)} are satisfied.

• •

There exists a unique weak solution $w$ of (3.2) satisfying the following regularity result

[TABLE]

Our main objective in this section is to establish global stabilization results for the state variable $w(t)$ of the continuous problem (3.2). Throughout the paper, all the results hold under the assumptions ${\bf(A1)}$ and ${\bf(A2)}$ with the same decay rate $\alpha$

[TABLE]

Lemma 3.1**.**

Assume that assumption ${\bf(A1)}$ is satisfied and $u^{\infty}$ is a steady state solution satisfying (2.3) of (1.4)-(1.5). Let $w_{0}\in L^{2}(\Omega)$. Then, there holds

[TABLE]

where \delta=\min\{\Big{(}\frac{7\nu}{4}-2\alpha C_{F}\Big{)},\Big{(}c_{0}+\frac{7\nu}{4}-2\alpha C_{F}\Big{)}\}>0, and $C_{F}>0$ is the constant in the Friedrichs’s inequality (2.2).

Proof.

Set $v=e^{2\alpha t}w$ in (3.2) to obtain

[TABLE]

For the first term on the right hand side of (3.4), we use integration by parts for the first sub-term and then bound it as follows

[TABLE]

Similarly, using the Young’s inequality, the second term on the right hand side of (3.4) is bounded by

[TABLE]

Now, using the Friedrichs’s inequality (2.2), it follows that

[TABLE]

Hence, from (3.4), we arrive using (3.5), (3.6) and (3.7) at

[TABLE]

Choose $\alpha$ as (3.3), so that the coefficients on the left hand side of (3) are non-negative. Integrate (3) from [math] to $t,$ and then, multiply the resulting inequality by $e^{-2\alpha t}$ to obtain

[TABLE]

This completes the proof. ∎

Remark 3.1**.**

The above Lemma also holds for $\alpha=0,$ that is,

[TABLE]

*Moreover, using the Friedrichs’s inequality, it follows that *

[TABLE]

Remark 3.2**.**

If we define the before mentioned Neumann control on some part of the boundary denoted by $\Gamma_{N}$ where measure of $\Gamma_{N}$ is non zero with remaining part zero Dirichlet boundary condition, still the stabilization result holds. For instance, consider $\partial\Omega=\Gamma_{D}\cup\Gamma_{N}$ with $\Gamma_{D}\cap\Gamma_{N}=\phi$, where $\Gamma_{D}$ and $\Gamma_{N}$ are sufficiently smooth. Hence, from (3.4), we arrive at

[TABLE]

Using the Friedrichs’s inequality \left\lVert v\right\rVert^{2}\leq C_{F}\Big{(}\left\lVert\nabla v\right\rVert^{2}+\left\lVert v\right\rVert^{2}_{L^{2}(\Gamma_{N})}\Big{)}, it follows that

[TABLE]

This complete the rest of the proof for $L^{2}$- stabilization. Stabilization result also holds similarly in higher order norms when control works on some part of the boundary.

Lemma 3.2**.**

Assume that assumption ${\bf(A1)}$ is satisfied and $u^{\infty}$ is a steady state solution satisfying (2.3) of (1.4)-(1.5). Let $w_{0}\in H^{1}(\Omega).$ Then, there holds

[TABLE]

Proof.

Form an $L^{2}$- inner product between (1.6) and $-e^{2\alpha t}\Delta w$ to obtain

[TABLE]

The fourth term on the left hand side of (3.11) can be rewritten as

[TABLE]

The terms on the right hand side of (3.11) are bounded by

[TABLE]

and using Lemma 3.1

[TABLE]

Finally, using the bounds of $\left\lVert u^{\infty}\right\rVert_{2}$, we arrive from (3.11) at

[TABLE]

Integrate the above inequality from [math] to $t,$ and then use the Grönwall’s inequality with Lemma 3.1 to obtain

[TABLE]

Use Remark 3.1 for the integral term under the exponential sign, and then multiply the resulting inequality by $e^{-2\alpha t}$ to complete the rest of the proof. ∎

Lemma 3.3**.**

Assume that assumption ${\bf(A1)}$ is satisfied and $u^{\infty}$ is a steady state solution satisfying (2.3) of (1.4)-(1.5). Let $w_{0}\in H^{1}(\Omega)$. Then, the following estimate holds

[TABLE]

Proof.

Choose $v=e^{2\alpha t}w_{t}$ in (3.2) to obtain

[TABLE]

The terms on the right hand side of (3.13) are bounded by

[TABLE]

and using Lemma 3.1

[TABLE]

Hence, rewriting the boundary integral term in (3.13) as in previous Lemma 3.2, we arrive from (3.13) at

[TABLE]

Apply Lemmas 3.1 and 3.2, and the Grönwall’s inequality to the above inequality to complete the rest of the proof. ∎

Lemma 3.4**.**

Assume that assumptions ${\bf(A1)}$ and ${\bf(A2)}$ are satisfied and $u^{\infty}$ is a steady state solution satisfying (2.3) of (1.4)-(1.5). Then, there holds

[TABLE]

Proof.

Differentiate (1.6) with respect to $t$ and then take the inner product with $e^{2\alpha t}w_{t}$ to obtain

[TABLE]

The first right hand side term in (3.14) is bounded by

[TABLE]

The other right hand term in (3.14) is bounded by

[TABLE]

Therefore, from (3.14), we obtain

[TABLE]

To calculate $\left\lVert w_{t}(0)\right\rVert,$ take the inner product between (1.6) and $w_{t}$ to obtain at $t=0$

[TABLE]

Integrate the inequality (3.15) from [math] to $t$ and then use Lemmas 3.1-3.3 to complete the rest of the proof. ∎

Lemma 3.5**.**

Assume that assumptions ${\bf(A1)}$ and ${\bf(A2)}$ are satisfied and $u^{\infty}$ is a steady state solution satisfying (2.3) of (1.4)-(1.5). Then the following estimate holds

[TABLE]

Proof.

Differentiate (1.6) with respect to $t$ and then take inner product with $-e^{2\alpha t}\Delta w_{t}$ to obtain

[TABLE]

The first three terms on the right hand side of (3.16) are bounded by

[TABLE]

and using Lemma 3.1

[TABLE]

The boundary term on the right hand side of (3.16) is bounded by

[TABLE]

Hence, from (3.16), we arrive at

[TABLE]

Integrate the above inequality from [math] to $t$ and then apply the Grönwall’s inequality along with Lemmas 3.1-3.4 to obtain

[TABLE]

Differentiate (1.6) with respect to $x_{1}$ and $x_{2}$ to get $\left\lVert\nabla w_{t}(0)\right\rVert\leq C\left\lVert w_{0}\right\rVert_{3}$. Further, by boundary trace embedding theorem, $\left\lVert w_{t}(0)\right\rVert_{L^{2}(\partial\Omega)}\leq C\left\lVert w_{t}(0)\right\rVert^{2}_{1}$ and

[TABLE]

Again, using of Lemmas 3.1, 3.2 and 3.4 to (3.17) completes the proof. ∎

4 Finite element method

In this section, we discuss semidiscrete Galerkin approximation keeping time variable continuous and prove optimal error estimates for both state variable and feedback controller.

Given a regular triangulation $\mathcal{T}_{h}$ of $\overline{\Omega},$ let $h_{K}=\text{diam}(K)$ for all $K\in\mathcal{T}_{h}$ and $h=\displaystyle\max_{K\in\mathcal{T}_{h}}h_{K}$.

Set

[TABLE]

The discrete weak formulation of the corresponding steady state solution is to seek some approximation of $u^{\infty}$ as $u^{\infty}_{h}\in V_{h}$ such that

[TABLE]

Corresponding steady state solution for the discrete problem satisfies

[TABLE]

where discrete Laplacian $-\Delta_{h}u^{\infty}_{h}:V_{h}\longrightarrow V_{h}$ is defined by

[TABLE]

Note that second term in (4.4) is zero, but we keep it to attach a meaning of discrete Laplacian in (4.8) for general nonhomogeneous boundary condition. For $u^{\infty}_{h}=\sum_{j=1}^{N}\alpha_{j}\phi_{j}$ where $\{\phi_{j}\}_{j=1}^{N_{h}}$ are the basis functions for finite element space $V_{h}$, the discrete normal derivative $\frac{\partial u^{\infty}_{h}}{\partial n}$ is defined as $\frac{\partial u^{\infty}_{h}}{\partial n}=\sum_{j=1}^{N}\alpha_{j}(\nabla\phi_{j}\cdot n)$. In terms of a basis function $\{\phi_{j}\}_{j=1}^{N_{h}}$, the above analogue (4.4) of Green’s formula defines $-\Delta_{h}u^{\infty}_{h}=\sum_{j=1}^{N_{h}}d_{j}\phi_{j}$ by

[TABLE]

See [24] for more details. As in continuous case similarly the following bound holds for $u^{\infty}_{h}$ under the assumption ${\bf(A1)}$.

Lemma 4.1**.**

Under the assumption ${\bf(A1)}$, there exists a solution $u^{\infty}_{h}\in V_{h}/\mathbb{R}$ to the problem (4.2)-(4.3) satisfying

[TABLE]

Moreover, when $\Delta_{h}u^{\infty}_{h}\in V_{h}$ and $f^{\infty}\in L^{2}$, then $\left\lVert\Delta_{h}u^{\infty}_{h}\right\rVert\leq C(\nu,\left\lVert f^{\infty}\right\rVert)$.

The semidiscrete approximation corresponding to the problem (3.2) is to seek $w_{h}(t)=w_{h}(\cdot,t)\in V_{h}$ such that

[TABLE]

with $w_{h}(0)=P_{h}u_{0}-u^{\infty}_{h}=w_{0h}$ (say), an approximation of $w_{0},$ where, $P_{h}u_{0}$ is the $H^{1}$ projection of $u_{0}$ onto $V_{h}$ such that

[TABLE]

Since $V_{h}$ is finite dimensional, (4.6) leads to a system of nonlinear ODEs. Therefore due to Picard’s theorem there exists a unique solution $w_{h}$ locally i.e for $t\in(0,t_{n})$. Moreover applying Lemmas 4.2-4.3 from below, the continuation arguments yields existence of a unique solution for all $t>0$. Following stabilization results hold for the semidiscrete solution in a similar approach as in continuous case.

Lemma 4.2**.**

Assume that assumption $\bf(A1)$ is satisfied and discrete steady state solution $u^{\infty}_{h}$ of (4.2)-(4.3) satisfy (4.5). Let $w_{0}\in L^{2}(\Omega)$. Then, there holds

[TABLE]

where $\delta$ and $\beta$ are same as in the continuous case.

Now the corresponding semidiscrete problem satisfies

[TABLE]

Lemma 4.3**.**

Assume that assumption $\bf(A1)$ is satisfied and discrete steady state solution $u^{\infty}_{h}$ of (4.2)-(4.3) satisfy (4.5). Let $w_{0}\in H^{1}(\Omega).$ Then, there holds

[TABLE]

Proof.

Forming an $L^{2}$- inner product between (4.8) and $-e^{2\alpha t}\Delta_{h}w_{h},$ we obtain

[TABLE]

Finally bounding the right hand side term, from (4.11), it follows that

[TABLE]

Integrate from [math] to $t,$ and then use the Grönwall’s inequality with Lemma 4.2 to obtain

[TABLE]

Apply Lemma 4.2 for $\alpha=0$ to the integral term under the exponential sign, and then multiply the resulting inequality by $e^{-2\alpha t}$ to complete the rest of the proof. ∎

Lemma 4.4**.**

Assume that assumption $\bf(A1)$ is satisfied and discrete steady state solution $u^{\infty}_{h}$ of (4.2)-(4.3) satisfy (4.5). Let $w_{0}\in H^{1}(\Omega)$. Then, there holds

[TABLE]

Proof.

Proof follows as in continuous case, namely; Lemma 3.3. ∎

Lemma 4.5**.**

Assume that assumption $\bf(A1)$ is satisfied and discrete steady state solution $u^{\infty}_{h}$ of (4.2)-(4.3) satisfy (4.5). Let $w_{0}\in H^{2}(\Omega)$. Then, we get

[TABLE]

Proof.

Differentiating (4.8) with respect to t and then taking inner product with $e^{2\alpha t}w_{ht},$ we obtain

[TABLE]

Bounding the right hand side term of (4.12), it follows that

[TABLE]

To obtain $\left\lVert w_{ht}(0)\right\rVert,$ take the inner product between (4.8) and $w_{ht}$ to arrive at

[TABLE]

Integrate the inequality (4.13) from [math] to $t$ and then use Lemmas 4.2-4.4 to complete the proof. ∎

Lemma 4.6**.**

Assume that assumption $\bf(A1)$ is satisfied and discrete steady state solution $u^{\infty}_{h}$ of (4.2)-(4.3) satisfy (4.5). Let $w_{0}\in H^{3}(\Omega).$ Then, we obtain

[TABLE]

Proof.

Differentiate (4.8) with respect to $t$ and then take inner product with $-e^{2\alpha t}\Delta_{h}w_{ht}$ to obtain

[TABLE]

Therefore in a similar fashion as in continuous case, from (4.14), we arrive at

[TABLE]

Integrate the above inequality from [math] to $t$ and then apply the Grönwall’s inequality with Lemmas 4.2-4.4, 4.5 to obtain

[TABLE]

As in continuous case, we can find the value $\left\lVert\nabla w_{ht}(0)\right\rVert$. The other two terms namely $\left\lVert w_{ht}(0)\right\rVert_{L^{2}(\partial\Omega)}$ and $\left\lVert w_{h}(0)w_{ht}(0)\right\rVert^{2}_{L^{2}(\partial\Omega)}$ are bounded respectively by $C\left\lVert w_{ht}(0)\right\rVert^{2}_{1}$ and $C\left\lVert w(0)\right\rVert^{2}_{L^{4}(\Omega)}\left\lVert w_{ht}(0)\right\rVert^{2}_{1}$. Again, a use of Lemmas 4.2, 4.3, and 4.5 to the above inequality completes the rest of the proof. ∎

4.1 Error estimates

Define an auxiliary projection $\tilde{w}_{h}\in V_{h}$ of $w$ through the following form

[TABLE]

where $\lambda\geq 1$ is some fixed positive number. For a given $w,$ the existence of a unique $\tilde{w}_{h}$ follows by the Lax-Milgram lemma. Let $\eta:=w-\tilde{w}_{h}$ be the error involved in the auxiliary projection. Then, the following error estimates hold:

[TABLE]

For a proof, we refer to Thomée [24]. Following Lemma 4.7 is needed to establish error estimates.

Lemma 4.7**.**

Let $F\in H^{3/2+\epsilon}(\Omega)$, for some $\epsilon>0$, and $G\in H^{1/2}(\partial\Omega).$ Then $FG\in H^{1/2}(\partial\Omega)$ and

[TABLE]

Proof.

For a proof see [10]. ∎

In addition, for proving error estimates for state variable and feedback controllers, we need the following estimate of $\eta$ and $\eta_{t}$ at boundary which are proved in [15].

Lemma 4.8**.**

For smooth $\partial\Omega,$ there holds

[TABLE]

Proof.

for a proof see [15]. ∎

With $e:=w-w_{h},$ decompose $e:=(w-\tilde{w}_{h})-(w_{h}-\tilde{w}_{h})=:\eta-\theta,$ where $\eta=w-\tilde{w}_{h}$ and $\theta=w_{h}-\tilde{w}_{h}$.

Since estimates of $\eta$ are known from (4.1) and Lemma 4.8, it is sufficient to estimate $\theta$. Subtracting the weak formulation (3.2) from (4.6), a use of (4.15) yields

[TABLE]

Further, (4.1) can be rewritten as

[TABLE]

Before proving the main error estimate theorem, we first estimate for $\left\lVert u^{\infty}-u^{\infty}_{h}\right\rVert_{j}$ for $j=0,\hskip 2.84544pt1$. For steady state error estimates, subtracting the corresponding steady state weak formulation to obtain

[TABLE]

Let $\tilde{u}^{\infty}_{h}\in V_{h}$ be the elliptic projection of $u^{\infty}\in H^{1}(\Omega)$ defined by

[TABLE]

where $\lambda\geq 1$ is some fixed positive number. and $\tilde{u}^{\infty}_{h}$ coincides with $u^{\infty}_{h}$ on the boundary. The steady state error is splitted into two parts $\tilde{e}:=u^{\infty}-u^{\infty}_{h}=(u^{\infty}-\tilde{u}^{\infty}_{h})-(u^{\infty}_{h}-\tilde{u}^{\infty}_{h})=:\tilde{\eta}-\tilde{\theta}$, where $\tilde{\eta}$ satisfies

[TABLE]

Now equation in $\tilde{\theta}$ becomes

[TABLE]

The following steady state error estimates hold

Theorem 4.1**.**

For $u^{\infty}\in H^{2}$, there holds

[TABLE]

Proof.

Setting $\chi=\tilde{\theta}$ in (4.21), it follows that

[TABLE]

Now using the bound of $\left\lVert\nabla u^{\infty}\right\rVert$, we obtain N\Big{(}\left\lVert\nabla u^{\infty}\right\rVert+\left\lVert\nabla u^{\infty}_{h}\right\rVert\Big{)}\leq\frac{\nu}{2}. Hence we arrive at

[TABLE]

Using estimate of $\left\lVert\nabla\tilde{\eta}\right\rVert$ and a use of triangle inequality completes the first part of the proof. For $L^{2}$-error estimate, we use Aubin-Nitsche technique. Consider the problem

[TABLE]

where $\phi^{\infty}$ satisfies $\left\lVert\phi^{\infty}\right\rVert_{2}\leq C\left\lVert\tilde{e}\right\rVert$. Now for proving error estimates, it is enough to estimate for $\left\lVert\theta(t)\right\rVert$. In its weak formulation, it satisfies

[TABLE]

Set $v=\tilde{e}$ to obtain

[TABLE]

Also subtracting the corresponding steady state weak formulation we obtain

[TABLE]

Hence from (4.24) and (4.25), we arrive at

[TABLE]

Set $\chi=\tilde{\phi^{\infty}_{h}}$ satisfying $\left\lVert\nabla\phi^{\infty}-\nabla\tilde{\phi^{\infty}_{h}}\right\rVert\leq Ch\left\lVert\phi^{\infty}\right\rVert_{2}\leq Ch\left\lVert\tilde{e}\right\rVert$. Therefore

[TABLE]

Hence

[TABLE]

∎

Lemma 4.9**.**

[TABLE]

Proof.

Proof follows similarly as to show estimate $\left\lVert\eta\right\rVert_{H^{-1/2}(\partial\Omega)}$ in Lemma 4.8. Since $\tilde{u}^{\infty}_{h}=u^{\infty}_{h}$ on the boundary, so it completes the proof. ∎

Theorem 4.2**.**

Under the assumptions $\bf(A1)$ and $\bf(A2)$, there holds

[TABLE]

where $\beta_{1}=\min\{(\frac{\nu}{2}-2\alpha C_{F}),(2c_{0}+2\nu-2\alpha C_{F})\}$.

Proof.

Setting $\chi=\theta$ in (4.1), we obtain

[TABLE]

The first term $I_{1}(\theta)$ on the right hand side of (4.1) is bounded by

[TABLE]

where the positive number $\epsilon>0$ is to be chosen later. The first subterm in the second term $I_{2}(\theta)$ on the right hand side is bounded by

[TABLE]

For other subterms in the second term $I_{2}(\theta),$ we note that

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

For the third term $I_{3}(\theta)$ first we bound the following sub-terms as

[TABLE]

The other sub-term in $I_{3}(\theta)$ is bounded by

[TABLE]

For the fourth term $I_{4}(\theta)$, bound the subterm as in third term

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Now, contribution from the fourth term $I_{4}(\theta)$ becomes

[TABLE]

Finally, using Lemmas 3.1-3.4, 4.2-4.3 and 4.8, we arrive from (4.1) at

[TABLE]

Multiply (4.27) by $e^{2\alpha t}$. Using the Friedrichs’s inequality

[TABLE]

and Lemmas 3.1-3.2, 4.3, 4.5 and 4.8 in (4.27), it follows choosing $\epsilon=\frac{\beta_{1}}{2C_{F}}$ that

[TABLE]

Integrate the above inequality from [math] to $t$ and then use the Grönwall’s inequality to obtain

[TABLE]

A use of Lemmas 3.1-3.5 and 4.2-4.3 to the above inequality with a multiplication of $e^{-2\alpha t}$ completes the the proof. ∎

Theorem 4.3**.**

Under the assumptions $\bf(A1)$ and $\bf(A2)$, there holds

[TABLE]

Proof.

Set $\chi=\theta_{t}$ in (4.1) to obtain

[TABLE]

The first term $I_{1}(\theta_{t})$ on the right hand side of (4.1) is bounded by

[TABLE]

For the second term $I_{2}(\theta_{t})$ on the right hand side of (4.1), first bound the subterms as

[TABLE]

For the other subterms in $I_{2}(\theta_{t})$ on the right hand side of (4.1), the bounds are as follows

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

For the third term $I_{3}(\theta_{t})$ on the right hand side of (4.1), first we rewrite the sub terms as

[TABLE]

and using integration by parts

[TABLE]

and hence it follows that

[TABLE]

Similarly,

[TABLE]

The other two sub-terms in the fourth term are bounded by

[TABLE]

For the fourth term $I_{4}(\theta_{t})$ on the right hand side of (4.1), subterms can be rewritten as

[TABLE]

[TABLE]

[TABLE]

and hence adding all the subterms

[TABLE]

Finally, from (4.1), we arrive at

[TABLE]

Multiply the above inequality by $e^{2\alpha t}$ and use Lemmas 3.1, 3.4 and4.2- 4.4 to obtain

[TABLE]

Integrate the above inequality from [math] to $t$. Then multiply the resulting inequality by $e^{-2\alpha t}$ and use Lemmas 3.2, 3.4- 3.5 and 4.3-4.6 and Theorem 4.2 to arrive at

[TABLE]

Finally, it follows that

[TABLE]

This completes the proof. ∎

Theorem 4.4**.**

Under the assumptions $\bf(A1)$ and $\bf(A2)$, there holds

[TABLE]

and

[TABLE]

Proof.

From estimates (4.1) and Theorems 4.2-4.3 with a use of triangle inequality completes the first part of the proof.

For the second part, we note that

[TABLE]

Therefore,

[TABLE]

A use of Lemmas 3.2, 4.3 and 4.8 and Theorem 4.3 completes the proof. ∎

5 Numerical experiments

Now in this section, our aim is to conduct some numerical experiments to show the order of convergence for the state variable and for the feedback control law. In addition, stabilization result is also shown numerically. For complete discrete scheme the time variable is discretized by replacing the time derivative with difference quotient. Based on backward Euler method, we now discretize the semidiscrete solution. Let $W^{n}$ be an approximation of $w(t)$ in $V_{h}$ at $t=t_{n}=nk$ where $0<k<1$ denote the time step size and $t_{n}=nk,$ $n$ is nonnegative integer. For smooth function $\phi$ defined on $[0,\infty),$ set $\phi^{n}=\phi(t_{n})$ and $\bar{\partial}_{t}\phi^{n}=\frac{(\phi^{n}-\phi^{n-1})}{k}$.

Now the backward Euler method applied to (1.6) determines a sequence of functions $\{{W^{n}}\}_{n\geq 1}\in V_{h}$ as a solution of

[TABLE]

with $W^{0}=w_{0h}$. To compute this, we use Freefem++ which provides an interpolation operator convect to calculate the convective term and for final plots, Matlab has been used.

Example 5.1**.**

In this example, we consider $w_{0}=\sin(\pi x_{1})\times\sin(\pi x_{2})-(-0.2x_{1})$ , where $u^{\infty}=-0.2x_{1}$ is the steady state solution with $\nu=0.1$ and $c_{0}=1$ in $\Omega=[0,1]\times[0,1]$ with time step $k=0.0005$. We take zero Neumann boundary conditions which is without control and denoted it as ”Uncontrolled solution” in Figure 2. For controlled solution, we choose the control (3.1) and denoted it as ”controlled solution with cubic nonlinear law” in Figure 2.

In Figure 2, it is observable that with control (3.1), the solution for the problem (3.2) in $L^{2}$ norm goes to zero exponentially. From Table LABEL:table:6.1, it follows that $L^{2}$ and $H^{1}$ orders of convergence for state variable are $2$ and $1$, respectively, which confirms our theoretical results, in Theorem 4.4. Take very refined mesh solution as exact solution and derive the order of convergence. In Table LABEL:table:6.2, it is noted that the order of convergence of nonlinear Neumann feedback control law (3.1) is $2$, which verify our theoretical result in Theorem 4.4.

Example 5.2**.**

In this example, we take the initial condition $w_{0}=sin(\pi x_{1})sin(\pi x_{2})-0.2x_{1}-(-0.2x_{1})=sin(\pi x_{1})sin(\pi x_{2})$ with $u^{\infty}=-0.2$ as the steady state solution in the domain $\Omega=[0,1]\times[0,1]$ with $\nu=0.1$ and time step $k=0.0005$. We choose the uncontrolled solution as the solution of (1.6) with some part on the boundary zero Dirichlet condition namely on $\Gamma_{1}=\{1\}\times[0,1]$ and on the remaining part ($\Gamma-\Gamma_{1}$) zero Neumann boundary condition and denoted it as ”Uncontrolled solution ” in Figure 4. For the controlled solution, we take the solution of (1.6) with feedback control law (3.1) on the remaining Neumann boundary part $\Gamma-\Gamma_{1}$ with $c_{0}=1$ and denoted it as ”controlled solution using nonlinear law with $c0=1$” in Figure 4.

From Figure 4, it is documented that as time increases the uncontrolled solution does not go to zero. But with feedback control law (3.1), the solution of (1.6) goes to zero. Corresponding feedback control law settle down at zero as time increases which is documented in Figure 4.

Acknowledgments

The first author was supported by the ERC advanced grant 668998 (OCLOC) under the EU’s H2020 research program. Jean-Pierre Raymond is gratefully acknowledged for his constructive remarks and suggestions which complete the paper during first author’s visit to him in Universitat paul Sabatier, Toulouse.