# Proof of Proposition 3.1 in the paper titled “Backstepping control of a
class of space-time-varying linear parabolic PDEs via time invariant kernel
functions”

Qiaoling Chen1<EMAIL_ADDRESS>Jun Zheng1,2
<EMAIL_ADDRESS>Guchuan Zhu2<EMAIL_ADDRESS>1School of
Mathematics, Southwest Jiaotong University, Chengdu 611756, Sichuan, China
2Department of Electrical Engineering, Polytechnique Montréal, P.O. Box 6079,
Station Centre-Ville, Montreal, QC, Canada H3T 1J4

###### Abstract

We provide a detailed proof of Proposition 3.1 in the paper titled
“Backstepping control of a class of space-time-varying linear parabolic PDEs
via time invariant kernel functions”.

In the paper titled “Backstepping control of a class of space-time-varying
linear parabolic PDEs via time invariant kernel functions”, we addressed the
problem of exponential stabilization and continuous dependence of solutions on
initial data in different norms for a class of $1$-D linear parabolic PDEs
with space-time-varying coefficients under backstepping boundary control. In
order to stabilize the system without involving a Gevrey-like condition or the
event-triggered scheme, a boundary feedback controller was designed via a time
invariant kernel function. By using the approximative Lyapunov method, the
exponential stability of the closed-loop system was established in the spatial
$L^{p}$-norm and $W^{1,p}$-norm, respectively, whenever $p\in[1,+\infty]$. It
was also shown that the solution to the considered system depends continuously
on the spatial $L^{p}$-norm and $W^{1,p}$-norm, respectively, of the initial
data.

###### keywords:

Backstepping control, parabolic PDEs, approximative Lyapunov method,
exponential stabilization, continuous dependence.

, ,

##  1 Brief review on the considered problem

Before presenting the proof of Proposition 3.1 in the paper titled
“Backstepping control of a class of space-time-varying linear parabolic PDEs
via time invariant kernel functions”, i.e., [1], we would like to restate the
considered problem. Specifically, given certain initial data $w_{0}$, we study
the problem of stabilization and continuous dependence of solutions on initial
data in different norms for a class of linear parabolic PDEs with space and
time dependent coefficients:

$\displaystyle w_{t}(x,t)=$ $\displaystyle
w_{xx}(x,t)+c(x,t)w(x,t)+\int_{0}^{x}w(y,t)f(x,y)\text{d}y,(x,t)\in(0,1)\times{\mathbb{R}_{>0}},$
(1a) $\displaystyle w_{x}(0,t)=$ $\displaystyle 0,t\in{\mathbb{R}_{>0}},$ (1b)
$\displaystyle w_{x}(1,t)=$ $\displaystyle U(t),t\in{\mathbb{R}_{>0}},$ (1c)
$\displaystyle w(x,0)=$ $\displaystyle w_{0}(x),x\in(0,1),$ (1d)

where $c:[0,1]\times{\mathbb{R}_{\geq 0}}\rightarrow\mathbb{R}$ and
$f:[0,1]\times[0,1]\rightarrow\mathbb{R}$ are given functions, and $U$ is the
control input to be determined to stabilize the system.

We assume that $p\in[1,+\infty)$ is either an arbitrary constant, or
$p=+\infty$, $\theta\in\left(0,\frac{1}{2}\right]$ is a constant, $f\in
C([0,1]\times[0,1])$, and the reaction coefficient
$c\in\mathcal{H}^{\theta,\frac{\theta}{2}}\left({\overline{Q}_{\infty}}\right)$
is given by

$\displaystyle c(x,t):=$ $\displaystyle c_{1}(x)+c_{2}(t)+c_{3}(x,t),$

with some functions $c_{1}\in C([0,1]),c_{2}\in C(\mathbb{R}_{\geq 0})$, and
$c_{3}\in{W^{2,1}_{2}(Q_{T})}$ for all $T\in\mathbb{R}_{>0}.$ Assume further
that

$\displaystyle\sup_{(x,t)\in{\overline{Q}_{\infty}}}c(x,t)<+\infty,$

and

$\displaystyle|c_{3}(x,t)-c_{3}(y,t)|\leq$ $\displaystyle L(t)\omega(|x-y|),$

holds true for all $x,y\in[0,1]$ and all $t\in\mathbb{R}_{\geq 0}$, where
$\omega\in C([0,1])$ and $L\in C(\mathbb{R}_{\geq 0})$ are nonnegative
functions satisfying

$\displaystyle\lim_{t\rightarrow+\infty}L(t)=0~{}~{}\text{and}~{}~{}\omega(s)\leq
1,\forall s\in[0,1].$

Let $\lambda_{0}>0$ be an arbitrary constant and satisfy
$\lambda_{0}>\sup_{(x,t)\in\overline{Q}_{\infty}}c(x,t).$ Let
$D:=\\{(x,y)\in\mathbb{R}^{2}|~{}0\leq y\leq x\leq 1\\}$ and

$\displaystyle{\mu(x,y):=\lambda_{0}-c_{1}(x)+c_{1}(y)},\forall(x,y)\in D.$

Consider the following equation defined over $D$:

$\displaystyle k_{xx}(x,y)-k_{yy}(x,y)=$
$\displaystyle{\mu(x,y)}k(x,y)+f(x,y)+\int_{y}^{x}k(x,z)f(z,y)\text{d}z,$ (2a)
$\displaystyle 2\frac{\text{d}}{\text{d}x}\left(k(x,x)\right)=$
$\displaystyle{\lambda_{0}},$ (2b) $\displaystyle k_{y}(x,0)=$ $\displaystyle
0,$ (2c) $\displaystyle k(0,0)=$ $\displaystyle 0,$ (2d)

where
$\frac{\text{d}}{\text{d}x}\left(k(x,x)\right):=k_{x}(x,y)|_{y=x}+k_{y}(x,y)|_{y=x}$.

Then, by means of such a kernel function $k$, we define the boundary feedback
control law as

$\displaystyle U(t):=-k(1,1)w(1,t)-\int_{0}^{1}k_{x}(1,y)w(y,t)\text{d}y.$ (3)

The existence, regularity and maximum estimate of a solution $k$ to the
equation (2) is guaranteed by the following proposition, which is Proposition
3.1 of [1]. The proof will be given in Section 2.

######  Proposition 1 ([1, Proposition 3.1]).

The equation (2) admits a unique solution $k$ that belongs to
$C^{2}\left(D\right)$ and satisfies

$\displaystyle|k(x,y)|\leq Me^{2M},\forall(x,y)\in D,$ (4)

where $M:=\frac{\overline{f}+{\overline{\Lambda}}}{2}$ with
${\overline{f}}:=\max_{(x,y)\in[0,1]\times[0,1]}|f(x,y)|,$ and
${{\overline{\Lambda}}}:={\max\left\\{\lambda_{0},\max_{(x,y)\in\mathcal{D}}|\mu(x,y)|\right\\}}.$

Now, we define the set

$\displaystyle W_{0}:=\bigg{\\{}$ $\displaystyle
w|~{}w\in\mathcal{H}^{2+\theta}([0,1]),w_{x}(0)=0,w_{x}(1)=-k(1,1)w(1)-\int_{0}^{1}k_{x}(1,y)w(y)\text{d}y\bigg{\\}},$

for the initial data of system (1), and denote
$\lambda(x,t):=\lambda_{0}-c\left(x,t\right),$ and
${\underline{\lambda}}:=\inf_{(x,t)\in{\overline{Q}_{\infty}}}\lambda(x,t).$

The main result obtained in [1] is the following theorem, whose proof is given
in [1].

######  Theorem 2.

Given initial data $w_{0}\in W_{0}$ and considering system (1) under the
boundary feedback control law (3), the following statements hold true:

  1. (i)

The system (1) admits a unique classical solution $w$, which belongs to
$C^{2,1}(\overline{Q}_{T})$ for any $T\in\mathbb{R}_{>0}$.

  2. (ii)

The system (1) is exponentially stable in the spatial $L^{p}$-norm, having the
estimate for all $t\in{\mathbb{R}_{>0}}$:

$\displaystyle\|w[t]\|_{L^{p}(0,1)}\leq
C_{0}e^{-{\sigma_{0}}t}\|w_{0}\|_{L^{p}(0,1)},$ (5)

where $\sigma_{0}$ and $C_{0}$ are positive constants depending only on
$\underline{\lambda},k$ and $\max_{t\in\mathbb{R}_{\geq 0}}L(t)$ whenever
$p\in[1,+\infty]$.

In addition, the solution to system (1) is continuously dependent on the
spatial $L^{p}$-norm of the initial data, having the estimate for all
$T\in\mathbb{R}_{>0}$:

$\displaystyle\|w_{1}-w_{2}\|_{C([0,T];L^{p}(0,1))}\leq
C_{0}\|w_{01}-w_{02}\|_{L^{p}(0,1)},$

where $w_{i}$ denotes the solution to system (1) corresponding to the initial
data $w_{0i}\in W_{0}(i=1,2)$.

  3. (iii)

Assume further that $c_{1}\in C^{1}([0,1])$ and

$\displaystyle\omega(s)=s^{\gamma_{0}},\forall s\in[0,1],$

with some constant $\gamma_{0}\geq 1$. Then, system (1) is exponentially
stable in the spatial $W^{1,p}$-norm, having the estimate for all
$t\in{\mathbb{R}_{>0}}$:

$\displaystyle\|w[t]\|_{W^{1,p}(0,1)}\leq\mathcal{C}_{0}e^{-\frac{\sigma_{0}}{2}t}\|w_{0}\|_{W^{1,p}(0,1)},$

where $\sigma_{0}$ is the same as in (5), and $\mathcal{C}_{0}$ is a positive
constant depending only on $p,c_{x},\underline{\lambda},k$ and
$\max_{t\in\mathbb{R}_{\geq 0}}L(t)$ for $p\in[1,+\infty)$, and only on
$c_{x},\underline{\lambda}$ and $k$ for $p=+\infty$, respectively.

In addition, the solution to system (1) is continuously dependent on the
spatial $W^{1,p}$-norm of the initial data, having the estimate for all
$T\in\mathbb{R}_{>0}$:

$\displaystyle\|w_{1}-w_{2}\|_{C([0,T];W^{1,p}(0,1))}\leq$
$\displaystyle\mathcal{C}_{0}\|w_{01}-w_{02}\|_{W^{1,p}(0,1)},$

where $w_{i}$ denotes the solution to system (1) corresponding to the initial
data $w_{0i}\in W_{0}(i=1,2)$.

As a special case of Theorem 2, the following result indicates that, for
certain parabolic PDEs with time-varying coefficients, a boundary feedback
control can be designed via modified Bessel functions, which are often used to
design boundary controls for stabilizing parabolic PDEs with constant
coefficients. The proof is also given in [1].

######  Corollary 3.

Let $c(x,t)\equiv c_{2}(t)$ and $f(x,y)\equiv 0$. Under the following boundary
control law111Note that $\lim_{s\rightarrow
0}\frac{\mathcal{I}_{1}(s)}{s}=\frac{1}{2}$ and $\lim_{s\rightarrow
0}\frac{\mathcal{I}_{2}(s)}{s^{2}}=\frac{1}{8}$; thus (6) is well-defined.

$\displaystyle U(t):=$
$\displaystyle-\frac{\lambda_{0}}{2}w(1,t)-\lambda_{0}\int_{0}^{1}\frac{\mathcal{I}_{1}\left(\sqrt{\lambda_{0}(1-y^{2})}\right)}{\sqrt{\lambda_{0}(1-y^{2})}}w(y,t)\text{d}y-\lambda_{0}\int_{0}^{1}\frac{\mathcal{I}_{2}\left(\sqrt{\lambda_{0}(1-y^{2})}\right)}{{1-y^{2}}}w(y,t)\text{d}y,$
(6)

system (1) is exponentially stable in the spatial $L^{p}$-norm (and
$W^{1,p}$-norm under the further assumption that $c_{2}\in
C^{1}(\mathbb{R}_{\geq 0})$), where $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$
are modified Bessel functions (see, e.g., [7]) of order one and order two,
respectively.

##  2 Proof of Proposition 1

In this section we provide a detailed proof for Proposition 1, i.e., [1,
Proposition 3.1].

We prove by using the method of successive approximations as in, e.g., [2, 3,
4, 5, 6, 7, 8]. Indeed, let $\xi:=x+y$ and $\eta:=x-y.$ It follows that
$\eta\in[0,1]$ and $\xi\in[\eta,2-\eta]$. Let

$\displaystyle
G(\xi,\eta):=k(x,y)=k\left(\frac{\xi+\eta}{2},\frac{\xi-\eta}{2}\right).$

Then (2) is changed into

$\displaystyle 4G_{\xi\eta}(\xi,\eta)=$
$\displaystyle{\widetilde{\mu}(\xi,\eta)}G(\xi,\eta)+f\left(\frac{\xi+\eta}{2},\frac{\xi-\eta}{2}\right)+\int_{\frac{\xi-\eta}{2}}^{\frac{\xi+\eta}{2}}\widetilde{\mathcal{G}}(z,\xi,\eta)\text{d}z,$
(7a) $\displaystyle G(\xi,0)=$ $\displaystyle{\frac{\lambda_{0}}{4}}\xi,$ (7b)
$\displaystyle G_{\xi}(\xi,\xi)=$ $\displaystyle G_{\eta}(\xi,\xi),$ (7c)
$\displaystyle G(0,0)=$ $\displaystyle 0,$ (7d)

where

$\displaystyle{\widetilde{\mu}(\xi,\eta):=\mu(x,y)=\lambda_{0}-c_{1}\left(\frac{\xi+\eta}{2}\right)+c_{1}\left(\frac{\xi-\eta}{2}\right)},$
$\displaystyle\widetilde{\mathcal{G}}(z,\xi,\eta):=f\left(z,\frac{\xi-\eta}{2}\right)G\left(\frac{\xi+\eta}{2}+z,\frac{\xi+\eta}{2}-z\right).$

Integrating (7a) w.r.t. $\eta$ from $0$ to $\eta$ and differentiating (7b)
w.r.t. $\xi$ give

$\displaystyle G_{\xi}(\xi,\eta)=$
$\displaystyle{\frac{\lambda_{0}}{4}}+{{\frac{{1}}{4}}\int_{0}^{\eta}\widetilde{\mu}(\xi,s)G(\xi,s)\text{d}s}+\frac{1}{4}\int_{0}^{\eta}f\left(\frac{\xi+s}{2},\frac{\xi-s}{2}\right)\text{d}s+\frac{1}{4}\int_{0}^{\eta}\int_{\xi}^{\xi+\eta-s}\widehat{G}(\tau,s,\xi)\text{d}\tau\text{d}s,$
(8)

where
$\widehat{G}(\tau,s,\xi):=f\left(\frac{\tau-s}{2},\xi-\frac{\tau+s}{2}\right)G(\tau,s)$.

Integrating (8) w.r.t. $\xi$ from $\eta$ to $\xi$ gives

$\displaystyle G(\xi,\eta)=$ $\displaystyle
G(\eta,\eta)+{\frac{\lambda_{0}}{4}}(\xi-\eta)+{{\frac{{1}}{4}}\int_{\eta}^{\xi}\int_{0}^{\eta}\widetilde{\mu}(\tau,s)G(\tau,s)\text{d}s\text{d}\tau}+\frac{1}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}f\left(\frac{\tau+s}{2},\frac{\tau-s}{2}\right)\text{d}s\text{d}\tau$
$\displaystyle+\frac{1}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}\int_{z}^{z+\eta-s}\widehat{G}(\tau,s,z)\text{d}\tau\text{d}s\text{d}z.$
(9)

By (7c), we have

$\displaystyle\frac{\text{d}}{\text{d}\xi}\left(G(\xi,\xi)\right)=2G_{\xi}(\xi,\xi).$
(10)

Substituting $\eta=\xi$ into (8), then integrating it over $[0,\xi]$ and
applying (7d) and (10), we obtain

$\displaystyle G(\xi,\xi)=$
$\displaystyle{\frac{\lambda_{0}}{2}}\xi+{{\frac{{1}}{2}}\int_{0}^{\xi}\int_{0}^{\tau}\widetilde{\mu}(\tau,s)G(\tau,s)\text{d}s\text{d}\tau}+\frac{1}{2}\int_{0}^{\xi}\int_{0}^{\tau}f\left(\frac{\tau+s}{2},\frac{\tau-s}{2}\right)\text{d}s\text{d}\tau$
$\displaystyle+\frac{1}{2}\int_{0}^{\xi}\int_{0}^{z}\int_{z}^{2z-s}\widehat{G}(\tau,s,z)\text{d}\tau\text{d}s\text{d}z,$

which gives

$\displaystyle G(\eta,\eta)=$
$\displaystyle{\frac{\lambda_{0}}{2}}\eta+{{\frac{{1}}{2}}\int_{0}^{\eta}\int_{0}^{\tau}\widetilde{\mu}(\tau,s)G(\tau,s)\text{d}s\text{d}\tau}+\frac{1}{2}\int_{0}^{\eta}\int_{0}^{\tau}f\left(\frac{\tau+s}{2},\frac{\tau-s}{2}\right)\text{d}s\text{d}\tau$
$\displaystyle+\frac{1}{2}\int_{0}^{\eta}\int_{0}^{z}\int_{z}^{2z-s}\widehat{G}(\tau,s,z)\text{d}\tau\text{d}s\text{d}z.$
(11)

We deduce by (2) and (2) that

$\displaystyle G(\xi,\eta)=$
$\displaystyle{\frac{\lambda_{0}}{4}}\left(\xi+\eta\right)+\frac{1}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}f\left(\frac{\tau+s}{2},\frac{\tau-s}{2}\right)\text{d}s\text{d}\tau+\frac{1}{2}\int_{0}^{\eta}\int_{0}^{\tau}f\left(\frac{\tau+s}{2},\frac{\tau-s}{2}\right)\text{d}s\text{d}\tau$
$\displaystyle+{{\frac{{1}}{4}}\int_{\eta}^{\xi}\int_{0}^{\eta}\widetilde{\mu}(\tau,s)G(\tau,s)\text{d}s\text{d}\tau}+{{\frac{{1}}{2}}\int_{0}^{\eta}\int_{0}^{\tau}\widetilde{\mu}(\tau,s)G(\tau,s)\text{d}s\text{d}\tau}+\frac{1}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}\int_{z}^{z+\eta-s}\widehat{G}(\tau,s,z)\text{d}\tau\text{d}s\text{d}z$
$\displaystyle+\frac{1}{2}\int_{0}^{\eta}\int_{0}^{z}\int_{z}^{2z-s}\widehat{G}(\tau,s,z)\text{d}\tau\text{d}s\text{d}z.$
(12)

We rewrite the integral equation (2) as

$\displaystyle G(\xi,\eta)=G_{0}(\xi,\eta)+\Phi_{G}(\xi,\eta),$ (13)

where we defined

$\displaystyle G_{0}(\xi,\eta):=$
$\displaystyle{\frac{\lambda_{0}}{4}}\left(\xi+\eta\right)+\frac{1}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}f\left(\frac{\tau+s}{2},\frac{\tau-s}{2}\right)\text{d}s\text{d}\tau+\frac{1}{2}\int_{0}^{\eta}\int_{0}^{\tau}f\left(\frac{\tau+s}{2},\frac{\tau-s}{2}\right)\text{d}s\text{d}\tau,$

and $\Phi_{G}(\xi,\eta)$ via

$\displaystyle{\Phi_{H}(\xi,\eta)}:=$
$\displaystyle{{\frac{{1}}{4}}\int_{\eta}^{\xi}\int_{0}^{\eta}\widetilde{\mu}(\tau,s)H(\tau,s)\text{d}s\text{d}\tau}+{{\frac{{1}}{2}}\int_{0}^{\eta}\int_{0}^{\tau}\widetilde{\mu}(\tau,s)H(\tau,s)\text{d}s\text{d}\tau}+\frac{1}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}\int_{z}^{z+\eta-s}\widehat{H}(\tau,s,z)\text{d}\tau\text{d}s\text{d}z$
$\displaystyle+\frac{1}{2}\int_{0}^{\eta}\int_{0}^{z}\int_{z}^{2z-s}\widehat{H}(\tau,s,z)\text{d}\tau\text{d}s\text{d}z,$
$\displaystyle\widehat{H}(\xi,\eta,z):=$ $\displaystyle
f\left(\frac{\xi-\eta}{2},z-\frac{\xi+\eta}{2}\right)H(\xi,\eta),$ (14)

for any $\eta\in[0,1]$, $\xi\in[\eta,2-\eta]$, $z\in[\eta,2-\eta]$, and any
function $H$.

Define the sequence $\\{G_{n}(\xi,\eta)\\}$ for $n\in\mathbb{N}_{0}$ via

$\displaystyle G_{n+1}(\xi,\eta):=G_{0}(\xi,\eta)+{\Phi_{G_{n}}}(\xi,\eta),$
(15)

and let $\Delta G_{n}(\xi,\eta):=G_{n+1}(\xi,\eta)-G_{n}(\xi,\eta)$. It holds
that

$\displaystyle\Delta G_{n+1}(\xi,\eta)={\Phi_{\Delta G_{n}}}(\xi,\eta)$

and

$\displaystyle G_{n+1}{(\xi,\eta)}=G_{0}{(\xi,\eta)}+\sum_{j=0}^{n}\Delta
G_{j}(\xi,\eta).$ (16)

From (15), we see that if $\\{G_{n}(\xi,\eta)\\}$ converges uniformly w.r.t.
$(\xi,\eta)$ when $n\rightarrow\infty$, then
$G(\xi,\eta):=\lim_{n\to\infty}G_{n}(\xi,\eta)$ is the solution to the
integral equation (13), and therefore the existence of a kernel function $k$
given by (2) is guaranteed. In addition, in view of (16), the convergence of
$\\{G_{n}(\xi,\eta)\\}$ is equivalent to the convergence of the series
$\sum_{n=0}^{\infty}\Delta G_{n}(\xi,\eta)$. Therefore, it suffices to show
that the series $\sum_{n=0}^{\infty}\Delta G_{n}(\xi,\eta)$ is uniformly
convergent w.r.t. $(\xi,\eta)$. Furthermore, recalling the Weierstrass M-test,
it suffices to show that the following inequality

$\displaystyle\left|\Delta
G_{n}(\xi,\eta)\right|\leq\frac{M^{n+2}}{(n+1)!}\left(\xi+\eta\right)^{n+1}$
(17)

holds true for all $n\in\mathbb{N}_{0}$, where

$\displaystyle M:=$
$\displaystyle\frac{\overline{f}+{\overline{\Lambda}}}{2},{\overline{f}}:=\max_{(x,y)\in[0,1]\times[0,1]}|f(x,y)|,{{\overline{\Lambda}}}:={\max\left\\{\lambda_{0},\max_{(x,y)\in\mathcal{D}}|\lambda_{0}-c_{1}(x)+c_{1}(y)|\right\\}}.$

Now we prove (17) by induction. First of all, since

$\displaystyle\left|G_{0}(\xi,\eta)\right|$
$\displaystyle\leq{\frac{\lambda_{0}}{4}}\left(\xi+\eta\right)+\frac{{\overline{f}}}{4}(\xi\eta-\eta^{2})+\frac{{\overline{f}}}{4}\eta^{2}\leq
M,$ (18)

it follows that

$\displaystyle\left|\Delta G_{0}(\xi,\eta)\right|=$
$\displaystyle\left|\Phi_{G_{0}}(\xi,\eta)\right|$ $\displaystyle\leq$
$\displaystyle\frac{{{{\overline{\Lambda}}}}}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}M\text{d}s\text{d}\tau+{\frac{{{{\overline{\Lambda}}}}}{2}}\int_{0}^{\eta}\int_{0}^{\tau}M\text{d}s\text{d}\tau+\frac{\overline{f}}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}\int_{z}^{z+\eta-s}M\text{d}\tau\text{d}s\text{d}z+\frac{\overline{f}}{2}\int_{0}^{\eta}\int_{0}^{z}\int_{z}^{2z-s}M\text{d}\tau\text{d}s\text{d}z$
$\displaystyle=$
$\displaystyle{M\left(\frac{{{{\overline{\Lambda}}}}}{4}\xi\eta+\frac{{\overline{f}}}{8}\xi\eta^{2}-\frac{{\overline{f}}}{24}\eta^{3}\right)}$
$\displaystyle\leq$ $\displaystyle{M^{2}\left(\xi+\eta\right),}$ (19)

which shows that (17) holds true for $n=0$.

Supposing that the inequality (17) holds true for a general $n\in\mathbb{N}$,
we need to show that (17) also holds true for $n+1$. Indeed, we get

$\displaystyle\left|\Delta G_{n+1}(\xi,\eta)\right|=$
$\displaystyle\left|{\Phi_{\Delta
G_{n}}}(\xi,\eta)\right|=|I_{1}+I_{2}+I_{3}+I_{4}|,$

where

$\displaystyle I_{1}:=$
$\displaystyle{\frac{{1}}{4}}\int_{\eta}^{\xi}\int_{0}^{\eta}{\widetilde{\mu}(\tau,s)}\Delta
G_{n}(\tau,s)\text{d}s\text{d}\tau,$ $\displaystyle I_{2}:=$
$\displaystyle{\frac{{1}}{2}}\int_{0}^{\eta}\int_{0}^{\tau}{\widetilde{\mu}(\tau,s)}\Delta
G_{n}(\tau,s)\text{d}s\text{d}\tau,$ $\displaystyle I_{3}:=$
$\displaystyle\frac{1}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}\int_{z}^{z+\eta-s}\widehat{\Delta
G_{n}}(\tau,s,z)\text{d}\tau\text{d}s\text{d}z,$ $\displaystyle I_{4}:=$
$\displaystyle\frac{1}{2}\int_{0}^{\eta}\int_{0}^{z}\int_{z}^{2z-s}\widehat{\Delta
G_{n}}(\tau,s,z)\text{d}\tau\text{d}s\text{d}z,$

with $\widehat{\Delta G_{n}}$ defined via (14).

Since (17) holds true for $n$, it follows that

$\displaystyle|I_{1}|\leq$
$\displaystyle\frac{{{\overline{\Lambda}}}M^{n+2}}{4(n+1)!}\int_{\eta}^{\xi}\int_{0}^{\eta}(\tau+s)^{n+1}\text{d}s\text{d}\tau\leq\frac{{{\overline{\Lambda}}}M^{n+2}}{4(n+1)!}\frac{1}{(n+2)(n+3)}\left((\xi+\eta)^{n+3}-(2\eta)^{n+3}\right),$
$\displaystyle|I_{2}|\leq$
$\displaystyle\frac{{{\overline{\Lambda}}}M^{n+2}}{2(n+1)!}\int_{0}^{\eta}\int_{0}^{\tau}(\tau+s)^{n+1}\text{d}s\text{d}\tau\leq\frac{{{\overline{\Lambda}}}M^{n+2}}{4(n+1)!}\frac{1}{(n+2)(n+3)}(2\eta)^{n+3},$
$\displaystyle|I_{3}|\leq$
$\displaystyle\frac{\overline{f}M^{n+2}}{4(n+1)!}\int_{\eta}^{\xi}\int_{0}^{\eta}\int_{z}^{z+\eta-s}(\tau+s)^{n+1}\text{d}\tau\text{d}s\text{d}z\leq\frac{\overline{f}M^{n+2}}{4(n+1)!}\frac{1}{(n+2)(n+3)}\left((\xi+\eta)^{n+3}-(2\eta)^{n+3}\right),$
$\displaystyle|I_{4}|\leq$
$\displaystyle\frac{\overline{f}M^{n+2}}{2(n+1)!}\int_{0}^{\eta}\int_{0}^{z}\int_{z}^{2z-s}(\tau+s)^{n+1}\text{d}\tau\text{d}s\text{d}z\leq\frac{\overline{f}M^{n+2}}{2(n+1)!}\frac{1}{(n+2)(n+3)}(2\eta)^{n+3}.$

Then we get

$\displaystyle|\Delta G_{n+1}(\xi,\eta)|\leq$
$\displaystyle\left(|I_{1}|+|I_{2}|\right)+\left(|I_{3}|+|I_{4}|\right)$
$\displaystyle\leq$
$\displaystyle\frac{{{{\overline{\Lambda}}}}}{4(n+3)}\frac{M^{n+2}}{(n+2)!}(\xi+\eta)^{n+3}{+}\frac{{\overline{f}}}{2(n+3)}\frac{M^{n+2}}{(n+2)!}(\xi+\eta)^{n+3}$
$\displaystyle\leq$
$\displaystyle\left(\frac{{{{\overline{\Lambda}}}}}{2(n+3)}+\frac{{\overline{f}}}{n+3}\right)\frac{M^{n+2}}{(n+2)!}(\xi+\eta)^{n+2}$
$\displaystyle\leq$
$\displaystyle\frac{M^{n+3}}{(n+2)!}\left(\xi+\eta\right)^{n+2},$

which implies that (17) holds for $n+1$. Therefore, (17) holds true for
$n\in\mathbb{N}_{0}$.

We conclude that the series $\sum_{n=0}^{\infty}\Delta G_{n}(\xi,\eta)$
converges absolutely and uniformly in $\eta\in[0,1]$ and
$\xi\in[\eta,2-\eta]$, and that

$\displaystyle G(\xi,\eta):=\lim_{n\to\infty}G_{n}(\xi,\eta)$ (20)

is a solution to (13). Moreover, $G(\xi,\eta)$ is continuous in $(\xi,\eta)$.
Furthermore, in view of (2) with $f\in C\left([0,1]\times[0,1]\right)$ and
$c_{1}\in C\left([0,1]\right)$, we deduce that $G$ is $C^{2}$-continuous. By
using the change of variables, we deduce that the problem (2) admits a
solution $k\in C^{2}\left(D\right)$.

Regarding the uniqueness of solution to (2), which is equivalent to the
uniqueness of solution to the integral equation (2), it suffices to show that
any twice continuously differentiable solution of (2) can be approximated by
the sequence $\\{G_{n}(\xi,\eta)\\}$ constructed via (15). More precisely, for
any twice continuously differentiable solution $\overline{G}(\xi,\eta)$ of
(2), we intend to show that the estimate

$\displaystyle|G_{n}(\xi,\eta)-\overline{G}(\xi,\eta)|\leq\frac{\delta
M^{n}}{n!}\left(\xi+\eta\right)^{n}$ (21)

holds true for all $n\in\mathbb{N}_{0}$, where

$\displaystyle\delta:=\max_{\eta\in[0,1],\xi\in[\eta,2-\eta]}|\Phi_{\overline{G}}(\xi,\eta)|.$

This is because (21) implies the uniform convergence of
$\\{G_{n}(\xi,\eta)\\}$ to $\overline{G}(\xi,\eta)$, while the uniqueness of
$G(\xi,\eta):=\lim_{n\to\infty}G_{n}(\xi,\eta)$ leads to
$\overline{G}(\xi,\eta)\equiv G(\xi,\eta)$. Thus, the solution to (2) is
unique, and consequently, the solution to (2) is also unique.

Now we prove (21) by induction. First, since $\overline{G}(\xi,\eta)$ is a
solution to (2), using the definition of $\Phi_{\overline{G}}$, we obtain

$\displaystyle|G_{0}(\xi,\eta)-\overline{G}(\xi,\eta)|=|G_{0}(\xi,\eta)-\left(G_{0}(\xi,\eta)+\Phi_{\overline{G}}(\xi,\eta)\right)|=|\Phi_{\overline{G}}(\xi,\eta)|\leq\delta,$

which shows that (21) holds true for $n=0$.

Suppose that (21) holds true for a general $n\in\mathbb{N}$. Then, by this
assumption and the linearity of $\Phi_{H}(\xi,\eta)$ w.r.t. $H$, we have

$\displaystyle|G_{n+1}(\xi,\eta)-\overline{G}(\xi,\eta)|=\left|\Phi_{G_{n}}(\xi,\eta)-\Phi_{\overline{G}}(\xi,\eta)\right|=\left|\Phi_{G_{n}-\overline{G}}(\xi,\eta)\right|\leq
I_{5}+I_{6}+I_{7}+I_{8},$

where

$\displaystyle I_{5}:=$
$\displaystyle\delta\frac{{{\overline{\Lambda}}}M^{n}}{4n!}\int_{\eta}^{\xi}\int_{0}^{\eta}(\tau+s)^{n}\text{d}s\text{d}\tau,$
$\displaystyle I_{6}:=$
$\displaystyle\delta\frac{{{\overline{\Lambda}}}M^{n}}{2n!}\int_{0}^{\eta}\int_{0}^{\tau}(\tau+s)^{n}\text{d}s\text{d}\tau,$
$\displaystyle I_{7}:=$
$\displaystyle\delta\frac{\overline{f}M^{n}}{4n!}\int_{\eta}^{\xi}\int_{0}^{\eta}\int_{z}^{z+\eta-s}(\tau+s)^{n}\text{d}\tau\text{d}s\text{d}z,$
$\displaystyle I_{8}:=$
$\displaystyle\delta\frac{\overline{f}M^{n}}{2n!}\int_{0}^{\eta}\int_{0}^{z}\int_{z}^{2z-s}(\tau+s)^{n}\text{d}\tau\text{d}s\text{d}z.$

Analogous to the estimates of $I_{1},I_{2},I_{3}$ and $I_{4}$, it holds that

$\displaystyle I_{5}\leq$
$\displaystyle\delta\frac{{{\overline{\Lambda}}}M^{n}}{4n!}\frac{1}{(n+1)(n+2)}\left((\xi+\eta)^{n+2}-(2\eta)^{n+2}\right),$
$\displaystyle I_{6}\leq$
$\displaystyle\delta\frac{{{\overline{\Lambda}}}M^{n}}{4n!}\frac{1}{(n+1)(n+2)}(2\eta)^{n+2},$
$\displaystyle I_{7}\leq$
$\displaystyle\delta\frac{\overline{f}M^{n}}{4n!}\frac{1}{(n+1)(n+2)}\left((\xi+\eta)^{n+2}-(2\eta)^{n+2}\right),$
$\displaystyle I_{8}\leq$
$\displaystyle\delta\frac{\overline{f}M^{n}}{2n!}\frac{1}{(n+1)(n+2)}(2\eta)^{n+2}.$

Then, we obtain

$\displaystyle|G_{n+1}(\xi,\eta)-\overline{G}(\xi,\eta)|\leq\delta\left(\frac{{{{\overline{\Lambda}}}}}{2(n+2)}+\frac{{\overline{f}}}{n+2}\right)\frac{M^{n}}{(n+1)!}(\xi+\eta)^{n+1}\leq{\frac{\delta
M^{n+1}}{(n+1)!}\left(\xi+\eta\right)^{n+1}},$

which implies that (21) holds true for $n+1$. Thus, (21) holds true for all
$n\in\mathbb{N}_{0}$. We conclude that the solution to (2) is unique.

Finally, the estimate (4) follows immediately from (20), (18), (16), (17) and
the change of variables. The proof is completed.

######  Remark 1.

For the stabilization problem of linear parabolic PDEs with time invariant
coefficients, the “uniqueness of kernel functions” was claimed in [7] and
proved by estimating

$\displaystyle|\Delta
G(\xi,\eta)|:=|G^{\prime}(\xi,\eta)-G^{\prime\prime}(\xi,\eta)|\leq\frac{C_{0}}{n!}(\xi+\eta)^{n}$
(22)

with some constant $C_{0}>0$, where $G^{\prime},G^{\prime\prime}$ are two
kernel functions; see (39) and (40) of [7].

It is worth noting that a gap exists in the proof given in [7]. On one hand,
if $G^{\prime}$ and $G^{\prime\prime}$ are given by (see (39) of [7])

$\displaystyle
G^{\prime}(\xi,\eta)=\sum_{n=0}^{\infty}G_{n}(\xi,\eta),~{}G^{\prime\prime}(\xi,\eta)=\sum_{n=0}^{\infty}G_{n}(\xi,\eta),$
(23)

as indicated in [7], the estimate (22) can be obtained in the same way as in
(38) of [7]. Therefore the equality of $G^{\prime}=G^{\prime\prime}$ can be
concluded by letting $n\rightarrow\infty$. However, since the limit of
$\sum_{i=0}^{n}G_{i}(\xi,\eta)$ is unique, which is known in the limit theory,
due to (23), the equality of $G^{\prime}=G^{\prime\prime}$ holds true
trivially. Therefore, the proof presented in [7] is essentially for the
uniqueness of the limit of $\\{G_{n}\\}$ rather than the uniqueness of the
solution.

On the other hand, if $G^{\prime}$ and $G^{\prime\prime}$ are not given by
(23), without imposing more conditions, the estimate (22) cannot be obtained
in the same way as in (38) of [7].

In order to address the aforementioned issues, as presented in this paper, a
feasible idea for proving the uniqueness of kernel functions is to prove that
all solutions to integral equations (e.g., (2) or (13)) can be approximated by
the sequence $\\{G_{n}\\}$. That is to say, $G^{\prime}$ and
$G^{\prime\prime}$ in [7] have to be with a form of (23). Then the uniqueness
of the solution is determined by the uniqueness of the limit of $\\{G_{n}\\}$.
Note that, by using this method, the estimate considered in this paper (see
(21)) is different from (22). In particular, (21) implies (22).

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