all $n\in\mathds{N}$, $\lim\limits_{n\rightarrow\infty}\frac{j_{n}}{n}=0$, and

$\displaystyle\lim\limits_{n\rightarrow\infty}\max\left\\{\left|\sqrt{\eta^{(n,\lambda,\mu)}_{j}}-\sqrt{\eta^{(\lambda,\mu)}\left(\frac{2\pi
j}{n}\right)}\right|:\,j\in\\{j_{n},\ldots,n\\}\right\\}=0\,.$ (S104)

Since $\\{\eta_{i}^{(n,\lambda,\mu)}\\}_{i=1,2,\ldots,n}$ and
$\eta^{(\lambda,\mu)}$ are both bounded, with lower bound $0$ and upper bound
$1$, the limit in (S95) follows. ∎

In the forthcoming Theorem S13 we study the parameter region of zero
capacities of our model of optical fibre with memory effects.

######  Theorem S14.

Let $\lambda\in(0,1]$, $\mu\in[0,1)$, and $\nu\geq 0$. Let
$C(\lambda,\mu,\nu)$ be one of the following capacities of the quantum memory
channel $\\{\Phi_{\lambda,\mu,\nu}^{(n)}\\}_{n\in\mathds{N}}$: quantum
capacity $Q$, two-way quantum capacity $Q_{2}$, or secret key capacity $K$. It
holds that

$\displaystyle C(\lambda,\mu,\nu)>0\qquad\Longleftrightarrow\qquad
C(\mathcal{E}_{\eta^{(\lambda,\mu)}(2\pi),\nu})>0\,,$ (S105)

where $\eta^{(\lambda,\mu)}$ is reported in (S96). In particular, in the
absence of thermal noise, i.e. $\nu=0$, it holds that

$\displaystyle
Q(\lambda,\mu,\nu=0)>0\qquad\Longleftrightarrow\qquad\sqrt{\mu}>\frac{\log_{2}\left(\frac{1}{\lambda}\right)-1}{\log_{2}\left(\frac{1}{\lambda}\right)+1}\,.$
(S106)

In addition, for all $\nu\geq 0$ it holds that

$\displaystyle
K(\lambda,\mu,\nu),\,Q_{2}(\lambda,\mu,\nu)>0\qquad\Longleftrightarrow$
$\displaystyle\sqrt{\mu}>\frac{\ln\left(\frac{1}{\lambda}\right)-\ln(1+\frac{1}{\nu})}{\ln\left(\frac{1}{\lambda}\right)+\ln(1+\frac{1}{\nu})}\,,$
(S107) $\displaystyle Q(\lambda,\mu,\nu)>0\qquad\Longrightarrow$
$\displaystyle\sqrt{\mu}>\frac{\ln\left(\frac{1}{\lambda}\right)-\ln\left(1+\frac{1}{2\nu+1}\right)}{\ln\left(\frac{1}{\lambda}\right)+\ln\left(1+\frac{1}{2\nu+1}\right)}\,,$
$\displaystyle Q(\lambda,\mu,\nu)>0\qquad\Longleftarrow$
$\displaystyle\sqrt{\mu}>\frac{\ln\left(\frac{1}{\lambda}\right)-\ln\left(1+2^{-g(\nu)}\right)}{\ln\left(\frac{1}{\lambda}\right)+\ln\left(1+2^{-g(\nu)}\right)}\,.$

###### Proof.

Theorem S8 establishes that the quantum memory channel
$\\{\Phi_{\lambda,\mu,\nu}^{(n)}\\}_{n\in\mathds{N}}$ is unitarily equivalent
to the quantum memory
$\left\\{\bigotimes_{i=1}^{n}\mathcal{E}_{\eta_{i}^{(n,\lambda,\mu)},\nu}\right\\}_{n\in\mathbb{N}}$.
Hence, their capacity $C$ is identical. Let us begin with the proof of the
following implication:

$\displaystyle C(\lambda,\mu,\nu)>0\Longrightarrow
C(\mathcal{E}_{\eta^{(\lambda,\mu)}(2\pi),\nu})>0\,.$ (S108)

(S98) establishes that all the transmissivities
$\\{\eta^{(n,\lambda,\mu)}_{n}\\}_{i=1,2,\ldots,n}$ are upper bounded by
$\eta^{(\lambda,\mu)}(2\pi)$. Hence, since the composition rule (S17) holds,
Alice and Bob can simulate
$\bigotimes_{i=1}^{n}\mathcal{E}_{\eta_{i}^{(n,\lambda,\mu)},\nu}$ by using
$n$ uses of the memoryless channel
$\mathcal{E}_{\eta^{(\lambda,\mu)}(2\pi),\nu}$. This implies that
$C(\mathcal{E}_{\eta^{(\lambda,\mu)}(2\pi),\nu})\geq C(\lambda,\mu,\nu)$,
which constitutes a proof of (S108). Now, let us show that

$\displaystyle C(\lambda,\mu,\nu)>0\Longleftarrow
C(\mathcal{E}_{\eta^{(\lambda,\mu)}(2\pi),\nu})>0\,.$ (S109)

Assume $C(\mathcal{E}_{\eta^{(\lambda,\mu)}(2\pi),\nu})>0$. Since the function
$x\longmapsto C(\mathcal{E}_{\eta^{(\lambda,\mu)}(2\pi x),\nu})$ is continuous
in $x\in[0,1]$, there exists $\bar{x}\in[0,1]$ such that
$C(\mathcal{E}_{\eta^{(\lambda,\mu)}(2\pi x),\nu})>0$ for all
$x\in[\bar{x},1]$. Observe that Theorem S13 implies that

$\displaystyle\lim\limits_{n\rightarrow\infty}\eta^{(n,\lambda,\mu)}_{\left\lfloor\left(\frac{1+\bar{x}}{2}\right)n\right\rfloor}$
$\displaystyle=\eta^{(\lambda,\mu)}\left(2\pi\left(\frac{1+\bar{x}}{2}\right)\right)>\eta^{(\lambda,\mu)}(2\pi\bar{x})\,,$
(S110)

where $\left\lfloor\cdot\right\rfloor$ is the floor function. Consequently, if
$n$ is sufficiently large it holds that

$\displaystyle\eta^{(n,\lambda,\mu)}_{j}>\eta^{(\lambda,\mu)}(2\pi\bar{x})\qquad\forall\,j\in\left\\{\left\lfloor\left(\frac{1+\bar{x}}{2}\right)n\right\rfloor,\ldots,n\right\\}\,.$
(S111)

Hence, since the composition rule (S17) holds, Alice and Bob can simulate
$n-\left\lfloor\left(\frac{1+\bar{x}}{2}\right)n\right\rfloor+1$ uses of the
memoryless channel $\mathcal{E}_{\eta^{(\lambda,\mu)}(2\pi)}$ by using the
channel $\bigotimes_{i=1}^{n}\mathcal{E}_{\eta_{i}^{(n,\lambda,\mu)},\nu}$,
for $n$ sufficiently large. Hence, since

$\displaystyle\lim\limits_{n\rightarrow\infty}\frac{n-\left\lfloor\left(\frac{1+\bar{x}}{2}\right)n\right\rfloor+1}{n}=\frac{1-\bar{x}}{2}\,,$
(S112)

it holds that

$\displaystyle
C(\lambda,\mu,\nu)\geq\left(\frac{1-\bar{x}}{2}\right)C(\mathcal{E}_{\eta^{(\lambda,\mu)}(2\pi)})>0\,,$
(S113)

i.e. the implication in (S109) is proved. Hence, we have proved that
$C(\lambda,\mu,\nu)>0$ if and only if
$C(\mathcal{E}_{\eta^{(\lambda,\mu)}(2\pi),\nu})>0$. As a consequence of this
fact, by using that

$\displaystyle\eta^{(\lambda,\mu)}(2\pi)=\lambda^{\frac{1-\sqrt{\mu}}{1+\sqrt{\mu}}}\,,$
(S114)

and by exploiting the bounds on the parameter region of zero capacities
reported in (S21), (S22), and (S23), one can show the validity of (S106) and
(S107). ∎

Now, let us discuss why Theorem 5 is interesting. In the absence of memory
effects ($\mu=0$), it is known that no quantum communication tasks can be
achieved when the transmissivity is sufficiently low ($Q=0$ for
$\lambda\leq\frac{1}{2}$, and $Q_{2}=K=0$ for $\lambda\leq\frac{\nu}{\nu+1}$).
However, in Theorem 5 we established that for any transmissivity $\lambda>0$
and thermal noise $\nu\geq 0$, there exists a critical value of the memory
parameter $\mu$ above which, it becomes possible to achieve qubit distribution
($Q>0$), entanglement distribution ($Q_{2}>0$), and secret-key distribution
($K>0$). This result is particularly intriguing as it demonstrates that memory
effects provide an advantage, enabling quantum communication tasks in highly
noisy regimes that were previously deemed impossible. Specifically, this
phenomenon bears resemblance to the “die-hard quantum communication”
phenomenon observed in [10, 11] within their toy model of optical fibre with
memory effects, which was limited to the analysis of $Q$ only and does not
consider $K$ and $Q_{2}$. In our paper, we not only demonstrate the
persistence of such a phenomenon in a more realistic model, accounting for
$Q$, $Q_{2}$, and $K$, but we also derive an analytical expression for the
critical value of the memory parameter $\mu$, which goes beyond the findings
of [10, 11]. Note that, by relating $\mu$ to the temporal interval $\delta t$
between subsequent input signals (e.g. $\mu=e^{-\delta t/t_{E}}$ with $t_{E}$
being the thermalisation timescale), Theorem 5 can also be expressed in terms
of the critical $\delta t$ below which the above mentioned quantum
communication tasks can be achieved.

In the forthcoming Theorem S15 we calculate the quantum capacity $Q$, the two-
way quantum capacity $Q_{2}$, and the secret-key capacity $K$ of our model in
the absence of thermal noise ($\nu=0$). It is worth noting that in the
presence of thermal noise ($\nu>0$), the capacities remain unknown even in the
absence of memory effects ($\mu=0$), as the capacities of the thermal
attenuator are currently unknown.

######  Theorem S15.

Let $\lambda\in[0,1)$, $\mu\in[0,1)$, $\nu\geq 0$. Let $C$ be one of the
following capacities: quantum capacity $Q$, two-way quantum capacity $Q_{2}$,
or secret key capacity $K$. In addition, let $C(\lambda,\mu,\nu)$ be the
capacity $C$ of the quantum memory channel
$\\{\Phi_{\lambda,\mu,\nu}^{(n)}\\}_{n\in\mathds{N}}$. In absence of thermal
noise, i.e. $\nu=0$, it holds that

$\displaystyle
C(\lambda,\mu,\nu=0)=\int_{0}^{2\pi}\frac{\mathrm{d}x}{2\pi}C\left(\mathcal{E}_{\eta^{(\lambda,\mu)}(x),0}\right)\,.$
(S115)

where $\eta^{(\lambda,\mu)}(x)$ is the effective transmissivity function
expressed in (S96). In particular,

$\displaystyle Q\left(\lambda,\mu,\nu=0\right)$
$\displaystyle=\int_{0}^{2\pi}\frac{\mathrm{d}x}{2\pi}\,\max\left\\{0,\log_{2}\left(\frac{\eta^{(\lambda,\mu)}(x)}{1-\eta^{(\lambda,\mu)}(x)}\right)\right\\}\,,$
(S116) $\displaystyle Q_{2}\left(\lambda,\mu,\nu=0\right)$
$\displaystyle=K\left(\lambda,\mu,\nu=0\right)$
$\displaystyle=\int_{0}^{2\pi}\frac{\mathrm{d}x}{2\pi}\,\log_{2}\left(\frac{1}{1-\eta^{(\lambda,\mu)}(x)}\right)\,.$

Moreover, in the presence of thermal noise, i.e. $\nu>0$, it holds that

$\displaystyle
C(\lambda,\mu,\nu)\geq\int_{0}^{2\pi}\frac{\mathrm{d}x}{2\pi}C\left(\mathcal{E}_{\eta^{(\lambda,\mu)}(x),\nu}\right)\,.$
(S117)

###### Proof.

Let us fix $P\in\mathds{N}$ with $P\geq 2$. In addition, for all
$n\in\mathds{N}$ and all $i\in\mathds{N}$ with $i\geq n$, let us define

$\displaystyle\eta^{(n,\lambda,\mu)}_{i}\coloneqq\eta^{(n,\lambda,\mu)}_{n}\,.$
(S118)

By theorem S13, there exists a sequence
$\\{j_{k}\\}_{k\in\mathds{N}}\subseteq\mathds{N}$ such that $j_{k}\leq k$ for
all $k\in\mathds{N}$, $\lim\limits_{k\rightarrow\infty}\frac{j_{k}}{k}=0$, and

$\displaystyle\max\left\\{\left|\eta^{(n,\lambda,\mu)}_{j}-\eta^{(\lambda,\mu)}\left(\frac{2\pi
j}{n}\right)\right|:\,j\in\\{j_{n},\ldots,n\\}\right\\}<\frac{1}{P}$ (S119)

for $n$ sufficiently large. In particular,

$\displaystyle\eta^{(n,\lambda,\mu)}_{\left\lceil\frac{n}{P}\right\rceil p}$
$\displaystyle<\eta^{(\lambda,\mu)}\left(\frac{2\pi
p}{P}\right)+\frac{1}{P}\,\qquad\forall\,p\in\\{1,2,\ldots,P\\}\,,$ (S120)
$\displaystyle\eta^{(n,\lambda,\mu)}_{\left\lfloor\frac{n}{P}\right\rfloor
p+1}$ $\displaystyle>\eta^{(\lambda,\mu)}\left(\frac{2\pi
p}{P}\right)-\frac{1}{P}\,\qquad\forall\,p\in\\{1,2,\ldots,P-1\\}\,,$

for $n$ sufficiently large, where $\left\lceil\cdot\right\rceil$ and
$\left\lfloor\cdot\right\rfloor$ denote the ceil and floor function,
respectively. It holds that

$\displaystyle C(\lambda,\mu,\nu)$
$\displaystyle\stackrel{{\scriptstyle\mathclap{\scriptsize\mbox{(i)}}}}{{=}}C\left(\left\\{\bigotimes_{i=1}^{n}\mathcal{E}_{\eta_{i}^{(n,\lambda,\mu)},\nu}\right\\}_{n\in\mathbb{N}}\right)$
(S121)
$\displaystyle\stackrel{{\scriptstyle\mathclap{\scriptsize\mbox{(ii)}}}}{{\leq}}C\left(\left\\{\bigotimes_{i=1}^{\left\lceil\frac{n}{P}\right\rceil
P}\mathcal{E}_{\eta_{i}^{(n,\lambda,\mu)},\nu}\right\\}_{n\in\mathbb{N}}\right)$
(S122)
$\displaystyle\stackrel{{\scriptstyle\mathclap{\scriptsize\mbox{(iii)}}}}{{\leq}}C\left(\left\\{\left(\bigotimes_{p=1}^{P}\mathcal{E}_{\eta_{\left\lceil\frac{n}{P}\right\rceil
p}^{(n,\lambda,\mu)},\nu}\right)^{\otimes\left\lceil\frac{n}{P}\right\rceil}\right\\}_{n\in\mathbb{N}}\right)$
(S123)
$\displaystyle\stackrel{{\scriptstyle\mathclap{\scriptsize\mbox{(iv)}}}}{{\leq}}C\left(\left\\{\left(\bigotimes_{p=1}^{P}\mathcal{E}_{\min\\{1,\eta^{(\lambda,\mu)}(2\pi\frac{p}{P})+\frac{1}{P}\\},\nu}\right)^{\otimes\left\lceil\frac{n}{P}\right\rceil}\right\\}_{n\in\mathbb{N}}\right)$
(S124)
$\displaystyle\stackrel{{\scriptstyle\mathclap{\scriptsize\mbox{(v)}}}}{{=}}\frac{1}{P}C\left(\bigotimes_{p=1}^{P}\mathcal{E}_{\min\\{1,\eta^{(\lambda,\mu)}(2\pi\frac{p}{P})+\frac{1}{P}\\},\nu}\right)\,,$
(S125)

where: in (i) we applied Theorem S8; (ii) is a consequence of the fact that

$\displaystyle\bigotimes_{i=1}^{n}\mathcal{E}_{\eta_{i}^{(n,\lambda,\mu)},\nu}$
(S126)

can be simulated via

$\displaystyle\bigotimes_{i=1}^{\left\lceil\frac{n}{P}\right\rceil
P}\mathcal{E}_{\eta_{i}^{(n,\lambda,\mu)},\nu}\,,$ (S127)

by using only the first $n$ channel uses and discarding the remaining
$\left\lceil\frac{n}{P}\right\rceil P-n$ ones; in (iii) we exploited the
composition rule in (S17) and the fact that the transmissivity
$\eta_{i}^{(Pl,\lambda,\mu)}$ is monotonically non-decreasing in $i$; in (iv)
we made the assumption that $n$ is sufficiently large, which can be done
without loss of generality because the capacities are defined in the limit as
the number of channel uses $n$ approaches infinity, and in addition we
employed the inequality stated in (S120); in (v) we exploited (S19) and the
notation introduced in Section S19 to denote the capacities of memory and
memoryless quantum channels. Analogously, it holds that

$\displaystyle C(\lambda,\mu,\nu)$
$\displaystyle=C\left(\left\\{\bigotimes_{i=1}^{n}\mathcal{E}_{\eta_{i}^{(n,\lambda,\mu)},\nu}\right\\}_{n\in\mathbb{N}}\right)$
(S128)
$\displaystyle\stackrel{{\scriptstyle\mathclap{\scriptsize\mbox{(vi)}}}}{{\geq}}C\left(\left\\{\bigotimes_{i=1}^{\left\lfloor\frac{n}{P}\right\rfloor
P}\mathcal{E}_{\eta_{i}^{(n,\lambda,\mu)},\nu}\right\\}_{n\in\mathbb{N}}\right)$
(S129) $\displaystyle\geq
C\left(\left\\{\left(\bigotimes_{p=1}^{P-1}\mathcal{E}_{\eta_{\left\lfloor\frac{n}{P}\right\rfloor
p+1}^{(n,\lambda,\mu)},\nu}\right)^{\otimes\left\lfloor\frac{n}{P}\right\rfloor}\right\\}_{n\in\mathbb{N}}\right)$
(S130) $\displaystyle\geq
C\left(\left\\{\left(\bigotimes_{p=1}^{P}\mathcal{E}_{\max\\{0,\eta^{(\lambda,\mu)}(2\pi\frac{p}{P})-\frac{1}{P}\\},\nu}\right)^{\otimes\left\lfloor\frac{n}{P}\right\rfloor}\right\\}_{n\in\mathbb{N}}\right)$
(S131)
$\displaystyle=\frac{1}{P}C\left(\bigotimes_{p=1}^{P}\mathcal{E}_{\max\\{0,\eta^{(\lambda,\mu)}(2\pi\frac{p}{P})-\frac{1}{P}\\},\nu}\right)\,,$
(S132)

where (vi) is a consequence of the fact that

$\displaystyle\bigotimes_{i=1}^{\left\lfloor\frac{n}{P}\right\rfloor
P}\mathcal{E}_{\eta_{i}^{(n,\lambda,\mu)},\nu}$ (S133)

can be simulated via

$\displaystyle\bigotimes_{i=1}^{n}\mathcal{E}_{\eta_{i}^{(n,\lambda,\mu)},\nu}$
(S134)

by using only the first $\left\lfloor\frac{n}{P}\right\rfloor P$ channel uses
and discarding the remaining $n-\left\lfloor\frac{n}{P}\right\rfloor P$ ones.
Consequently, the capacity $C(\lambda,\mu,\nu)$ can be bounded as

$\displaystyle\frac{1}{P}C\left(\bigotimes_{p=1}^{P-1}\mathcal{E}_{\max(0,\eta^{(\lambda,\mu)}\left(2\pi\frac{p}{P}\right)-\frac{1}{P}),\nu}\right)\leq
C(\lambda,\mu,\nu)\leq\frac{1}{P}C\left(\bigotimes_{p=1}^{P}\mathcal{E}_{\min(1,\eta^{(\lambda,\mu)}\left(2\pi\frac{p}{P}\right)+\frac{1}{P}),\nu}\right)$
(S135)

for all $P\in\mathds{N}$ with $P\geq 2$. In particular

$\displaystyle\limsup_{P\rightarrow\infty}\frac{1}{P}C\left(\bigotimes_{p=1}^{P-1}\mathcal{E}_{\max(0,\eta^{(\lambda,\mu)}\left(2\pi\frac{p}{P}\right)-\frac{1}{P}),\nu}\right)\leq
C(\lambda,\mu,\nu)\leq\liminf_{P\rightarrow\infty}\frac{1}{P}C\left(\bigotimes_{p=1}^{P}\mathcal{E}_{\min(1,\eta^{(\lambda,\mu)}\left(2\pi\frac{p}{P}\right)+\frac{1}{P}),\nu}\right)$
(S136)

Consequently, it holds that

$\displaystyle C(\lambda,\mu,\nu)$
$\displaystyle\geq\limsup_{P\rightarrow\infty}\frac{1}{P}C\left(\bigotimes_{p=1}^{P-1}\mathcal{E}_{\max(0,\eta^{(\lambda,\mu)}\left(2\pi\frac{p}{P}\right)-\frac{1}{P}),\nu}\right)$
(S137)
$\displaystyle\stackrel{{\scriptstyle\mathclap{\scriptsize\mbox{(i)}}}}{{\geq}}\limsup_{P\rightarrow\infty}\frac{1}{P}\sum_{p=1}^{P-1}C\left(\mathcal{E}_{\max(0,\eta^{(\lambda,\mu)}\left(2\pi\frac{p}{P}\right)-\frac{1}{P}),\nu}\right)$
(S138)
$\displaystyle\stackrel{{\scriptstyle\mathclap{\scriptsize\mbox{(ii)}}}}{{=}}\int_{0}^{2\pi}\frac{\mathrm{d}x}{2\pi}C\left(\mathcal{E}_{\eta^{(\lambda,\mu)}(x),\nu}\right)\,,$
(S139)

where: (i) is a consequence of the fact that Alice and Bob can independently
employ the optimal communication strategy for each of the $P$ single-mode
channels that define the $P$-mode attenuator
$\bigotimes_{p=1}^{P-1}\mathcal{E}_{\max(0,\eta^{(\lambda,\mu)}\left(2\pi\frac{p}{P}\right)-\frac{1}{P}),\nu}$;
in (ii) we exploited the fact that the function $\lambda\longmapsto
C(\mathcal{E}_{\lambda,0})$ is continuous in $\lambda\in[0,1)$ and that

$\displaystyle\lambda^{\frac{1}{1-\mu}}\leq\eta^{(\lambda,\mu)}(x)\leq\lambda^{\frac{1-\sqrt{\mu}}{1+\sqrt{\mu}}}<1\,,$
(S140)

as established by (S98). Hence, we have proved (S117).

Now, let us assume that the thermal noise is zero, i.e. $\nu=0$. Under this
assumption, we can exploit the additivity of the capacity $C$ of the pure-loss
channels, i.e. $C$ is such that for all transmissivities
$\\{\lambda_{p}\\}_{p=1,2,\ldots,P}$ and all $P\in\mathds{N}$ it holds that

$\displaystyle
C\left(\bigotimes_{p=1}^{P}\mathcal{E}_{\lambda_{p},\nu}\right)=\sum_{p=1}^{P}C\left(\mathcal{E}_{\lambda_{p},\nu}\right)\,.$
(S141)

Indeed, the capacities $Q$ [41], $Q_{2}$ [28], and $K$ [28] of the pure-loss
channel are additive. Hence, (S136) implies that:

$\displaystyle\limsup_{P\rightarrow\infty}\frac{1}{P}\sum_{p=1}^{P-1}C\left(\mathcal{E}_{\max(0,\eta^{(\lambda,\mu)}\left(2\pi\frac{p}{P}\right)-\frac{1}{P}),0}\right)\leq
C(\lambda,\mu,0)\leq\liminf_{P\rightarrow\infty}\frac{1}{P}\sum_{p=1}^{P}C\left(\mathcal{E}_{\min(1,\eta^{(\lambda,\mu)}\left(2\pi\frac{p}{P}\right)+\frac{1}{P}),0}\right)\,.$
(S142)

By exploiting that $\eta^{(\lambda,\mu)}(2\pi)<1$ and the fact that the
function $\lambda\longmapsto C(\mathcal{E}_{\lambda,0})$ is continuous in
$\lambda\in[0,1)$, we have that the left-hand side and the right-hand side of
(S142) converge to the same quantity:

$\displaystyle\liminf_{P\rightarrow\infty}\frac{1}{P}\sum_{p=1}^{P}C\left(\mathcal{E}_{\min(1,\eta^{(\lambda,\mu)}\left(2\pi\frac{p}{P}\right)+\frac{1}{P}),0}\right)$
$\displaystyle=\int_{0}^{2\pi}\frac{\mathrm{d}x}{2\pi}C(\mathcal{E}_{\eta^{(\lambda,\mu)}(x),0})\,,$
(S143)
$\displaystyle\limsup_{P\rightarrow\infty}\frac{1}{P}\sum_{p=1}^{P-1}C\left(\mathcal{E}_{\max(0,\eta^{(\lambda,\mu)}\left(2\pi\frac{p}{P}\right)-\frac{1}{P}),0}\right)$
$\displaystyle=\int_{0}^{2\pi}\frac{\mathrm{d}x}{2\pi}C(\mathcal{E}_{\eta^{(\lambda,\mu)}(x),0})\,.$

Hence, in the absence of thermal noise, the capacity $C(\lambda,\mu,0)$ of our
quantum memory channel is given by

$\displaystyle
C(\lambda,\mu,0)=\int_{0}^{2\pi}\frac{\mathrm{d}x}{2\pi}C(\mathcal{E}_{\eta^{(\lambda,\mu)}(x),0})\,.$
(S144)

Consequently, by leveraging the expression of the capacities $Q$ [41], $Q_{2}$
[28], and $K$ [28] of the pure-loss channel reported in (S20), we obtain
(S116).

∎

##  VI Capacities of the Localised Interaction Model

In this section, we provide an expression for the capacities of the "Localised
Interaction Model" (LIM) of references [16, 17, 18], which has been reviewed
both in the main text and in Section II.

Let $Q^{\text{LIM}}(\lambda,\mu,\nu)$, $Q_{2}^{\text{LIM}}(\lambda,\mu,\nu)$,
and $K^{\text{LIM}}(\lambda,\mu,\nu)$ be the quantum capacity, two-way quantum
capacity, and secret-key capacity of the LIM with transmissivity $\lambda$,
memory parameter $\mu$, and thermal noise $\nu$. The effective transmissivity
function of the LIM is given by [16, Eq. (5)]:

$\displaystyle\eta_{\text{LIM}}^{(\lambda,\mu)}(x)=\frac{\mu+\lambda-2\sqrt{\mu\lambda}\cos\left(\frac{x}{2}\right)}{1+\mu\lambda-2\sqrt{\mu\lambda}\cos\left(\frac{x}{2}\right)}\qquad\forall\,x\in[0,2\pi]\,.$
(S145)

Note that $\eta_{\text{LIM}}^{(\lambda,\mu)}(x)$ is monotonically increasing
in $x$ and thus achieves is maximum in $x=2\pi$:

$\displaystyle\max_{x\in[0,2\pi]}\eta_{\text{LIM}}^{(\lambda,\mu)}(x)=\eta_{\text{LIM}}^{(\lambda,\mu)}(x=2\pi)=\left(\frac{\sqrt{\mu}+\sqrt{\lambda}}{1+\sqrt{\mu\lambda}}\right)^{2}\,.$
(S146)

We can apply the same reasoning used in Theorem S14 in order to determine the
parameter region where the capacities are strictly positive. Specifically, in
the absence of thermal noise ($\nu=0$), the quantum capacity
$Q^{\text{LIM}}(\lambda,\mu,\nu=0)$ is strictly positive if and only if
$\eta_{\text{LIM}}^{(\lambda,\mu)}(x=2\pi)>\frac{1}{2}$. This condition can be
expressed as follows:

$\displaystyle
Q^{\text{LIM}}(\lambda,\mu,\nu=0)>0\qquad\Longleftrightarrow\qquad\sqrt{\mu}>\frac{1-\sqrt{2\lambda}}{\sqrt{2}-\sqrt{\lambda}}\,.$
(S147)

Additionally, the two-way quantum capacity
$Q_{2}^{\text{LIM}}(\lambda,\mu,\nu)$ and secret-key capacity
$K^{\text{LIM}}(\lambda,\mu,\nu)$ are strictly positive if and only if
$\eta_{\text{LIM}}^{(\lambda,\mu)}(x=2\pi)>\frac{\nu}{\nu+1}$. This condition
can be expressed as follows:

$\displaystyle
Q^{\text{LIM}}(\lambda,\mu,\nu),K^{\text{LIM}}(\lambda,\mu,\nu)>0\qquad\Longleftrightarrow\qquad\sqrt{\mu}>\frac{\sqrt{\nu}-\sqrt{\lambda(\nu+1)}}{\sqrt{\nu+1}-\sqrt{\lambda\nu}}\,.$
(S148)

The quantum capacity $Q^{\text{LIM}}(\lambda,\mu,0)$ of the LIM in the absence
of thermal noise ($\nu=0$) is given by [16]:

$\displaystyle
Q^{\text{LIM}}(\lambda,\mu,0)=\int_{0}^{2\pi}\frac{\mathrm{d}x}{2\pi}\max\left(0,\log_{2}\left(\frac{\eta_{\text{LIM}}^{(\lambda,\mu)}(x)}{1-\eta_{\text{LIM}}^{(\lambda,\mu)}(x)}\right)\right)\,.$
(S149)

One can numerically verify that $Q^{\text{LIM}}(\lambda,\mu,\nu=0)$ is an
increasing function of $\mu$ for each assigned transmissivity value $\lambda$,
implying that memory effects can improve quantum communication.

Furthermore, by exploiting the same reasoning used in Theorem S15, we can
directly derive an expression for the two-way quantum and the secret-key
capacity of the LIM in the absence of thermal noise. Namely, they are given
by:

$\displaystyle
K^{\text{LIM}}(\lambda,\mu,0)=Q^{\text{LIM}}_{2}(\lambda,\mu,0)=\int_{0}^{2\pi}\frac{\mathrm{d}x}{2\pi}\log_{2}\left(\frac{1}{1-\eta_{\text{LIM}}^{(\lambda,\mu)}(x)}\right)\,.$
(S150)

##  VII Transversal attenuation

A possible limitation of the DIM introduced before is the fact that the
environments are taken to be single-mode. This would not be an issue in
absence of memory effects, since one could always select a single-mode
component of the environment as the effective state interacting with the
input. Instead, memory effects allow for more complex dynamics. One could
easily extend our model to capture a memoryless contribution to the noise, by
considering the case where between each interaction with $E_{1}^{(j)}$ and
$E_{1}^{(j+1)}$ there is an additional thermal loss of attenuation
$\gamma^{1/M}$ and environment state $\tau_{\nu}$, but without memory effects,
as depicted in Fig. 7. In this way, we are allowing for the presence of an
additional mechanism of interaction with the environment which happens on a
faster timescale. By taking $\gamma^{1/M}$ to be uniform along the several
uses of the transmission line, the action on the annihilation operators of the
input modes turns out to be a simple rescaling by $\sqrt{\gamma^{1/M}}$, which
commutes with the original interaction. The net effect is to place identical
thermal attenuators at each mode, before or after the memory channel.

Figure 7: Depiction of the quantum memory channel which models both memory and
memoryless noise. In contrast to the model presented in Figure 6, here we have
also the red beam splitters of transmissivity $\gamma^{1/M}$ which models the
presence of memoryless contribution to the noise. This addition, while it does
not substantially affect the mathematical structure of the model, captures the
possibility of having different kinds of noise effects on the fibre at the
same time, such as scattering, which could enable memory effects, and
absorption, which may be memoryless.

This simple model captures the possibility of having the communication line
interacting with a multi-mode system, where effectively there are two
independent noise effects acting on different timescales. For example, one may
think of the memory effects to be due to scattering and the memoryless noise
to be due to absorption. Since at the end we will see that the memory channel
can be written as a tensor product of attenuators, by the composition rules
for attenuators, the capacities of this enhanced model are also completely
computable. In particular, it follows that the advantage on the quantum
capacity might remain, if $\gamma$ is sufficiently large. If $\nu=0$, the
advantage due to memory effects on one-way qubit transmission remains if and
only if $\gamma>\frac{1}{2}$, while the two-way quantum capacity can be non-
zero as soon as $\gamma\geq\frac{\nu}{\nu+1}$.

In concrete, we modify the model by introducing new modes $\mathbf{f}$ as
depicted in Fig. 7, i.e.

$\displaystyle\mathbf{a}$
$\displaystyle\coloneqq(a_{1},\,a_{2},\ldots,\,a_{n})^{T}\,,$ (S151)
$\displaystyle\mathbf{a_{M}}$
$\displaystyle\coloneqq(a_{1,M},\,a_{2,M},\ldots,\,a_{n,M})^{T}\,,$
$\displaystyle\mathbf{b}$
$\displaystyle\coloneqq(b_{1}^{(1)},\,b_{1}^{(2)},\ldots,\,b_{1}^{(M)},\,b_{2}^{(1)},\,b_{2}^{(2)},\ldots,\,b_{2}^{(M)},\ldots,\,b_{n}^{(1)},\,b_{n}^{(2)},\ldots,\,b_{n}^{(M)})^{T}\,,$
$\displaystyle\mathbf{f}$
$\displaystyle\coloneqq(f_{1,1},f_{1,2},\ldots,\,f_{1,M-1},f_{1,M},\ldots,\,f_{n-1,1},\,f_{n,2},\ldots,\,f_{n,M-1},f_{n,M})^{T}\,,$

one can modify (S39) as

$\displaystyle\mathbf{a_{M}}=A^{(M,n,\lambda,\mu,\gamma)}\mathbf{a}+E^{(M,n,\lambda,\mu,\gamma)}\mathbf{b}+F^{(M,n,\lambda,\mu,\gamma)}\mathbf{f}\,,$
(S152)

where $A^{(M,n,\lambda,\mu,\gamma)}$ is an $n\times n$ real matrix,
$E^{(M,n,\lambda,\mu,\gamma)}$ is an $n\times nM$ real matrix, and
$F^{(M,n,\lambda,\mu,\gamma)}$ is an $n\times nM$ real matrix which satisfy

$\displaystyle
A^{(M,n,\lambda,\mu,\gamma)}{A^{(M,n,\lambda,\mu,\gamma)}}^{T}+E^{(M,n,\lambda,\mu,\gamma)}{E^{(M,n,\lambda,\mu,\gamma)}}^{T}+F^{(M,n,\lambda,\mu,\gamma)}{F^{(M,n,\lambda,\mu,\gamma)}}^{T}=\mathbb{1}_{n\times
n}\,,$ (S153)

thanks to (S37). By following the same steps of the proof of Theorem S8, we
get again

$\displaystyle\chi_{\Phi_{\lambda,\mu,\nu,\gamma}^{(M,n)}(\rho^{(n)})}(\mathbf{z})$
$\displaystyle\coloneqq\operatorname{Tr}\left[\Phi_{\lambda,\mu,\nu,\gamma}^{(M,n)}(\rho^{(n)})\,e^{\mathbf{a}^{\dagger}\mathbf{z}-\mathbf{z}^{\dagger}\mathbf{a}}\right]$
(S154)
$\displaystyle=\chi_{\rho^{(n)}}\left({A^{(M,n,\lambda,\mu,\gamma)}}^{T}\mathbf{z}\right)\chi_{\tau_{\nu}^{\otimes
nM}}\left({E^{(M,n,\lambda,\mu,\gamma)}}^{T}\mathbf{z}\right)\chi_{\tau_{\nu}^{\otimes
nM}}\left({F^{(M,n,\lambda,\mu,\gamma)}}^{T}\mathbf{z}\right)$
$\displaystyle=\chi_{\rho^{(n)}}\left({A^{(M,n,\lambda,\mu,\gamma)}}^{T}\mathbf{z}\right)e^{-(\nu+\frac{1}{2})\mathbf{z}^{\dagger}(\mathbb{1}_{n\times
n}-A^{(M,n,\lambda,\mu,\gamma)}{A^{(M,n,\lambda,\mu,\gamma)}}^{T})\mathbf{z}}\qquad\forall\,\mathbf{z}\in\mathbb{C}^{n}\,,$

And the analysis continues as in the proof of Theorem S8. We are left to
specify $A^{(M,n,\lambda,\mu,\gamma)}$ in the modified model. By exploiting
(S11) and the notation in Fig. 7, we derive the following passive
transformation:

$\displaystyle a_{i,j}$
$\displaystyle=\sqrt{\gamma^{1/M}}\sqrt{\lambda^{1/M}}\,a_{i,j-1}+\sqrt{\gamma^{1/M}}\sqrt{1-\lambda^{1/M}}\,m_{i-1,j}+\sqrt{1-\gamma^{1/M}}\,f_{i,j-1}\,,$
(S155) $\displaystyle m_{i,j}$
$\displaystyle=-\sqrt{1-\mu}\,b_{i+1}^{(j)}+\sqrt{\mu\lambda^{1/M}}\,m_{i-1,j}-\sqrt{\mu(1-\lambda^{1/M})}\,a_{i,j-1}\,,$

where the transmissivity $\gamma$ is associated to memoryless noise. By
defining $\tilde{a}_{i,j}\coloneqq\gamma^{-j/M}a_{i,j}$, and
$\tilde{m}_{i,j}\coloneqq\gamma^{-(j-1)/M}m_{i,j}$ we obtain

$\displaystyle\tilde{a}_{i,j}$
$\displaystyle\simeq\sqrt{\lambda^{1/M}}\,\tilde{a}_{i,j-1}+\sqrt{1-\lambda^{1/M}}\,\tilde{m}_{i-1,j}\,,$
(S156) $\displaystyle\tilde{m}_{i,j}$
$\displaystyle\simeq\sqrt{\mu\lambda^{1/M}}\,\tilde{m}_{i-1,j}-\sqrt{\mu(1-\lambda^{1/M})}\,\tilde{a}_{i,j-1}\,,$

which are the recurrence relations for the model at $\gamma=1$. We immediately
get

$\displaystyle
A^{(M,n,\lambda,\mu,\gamma)}=\sqrt{\gamma}A^{(M,n,\lambda,\mu,1)}=\sqrt{\gamma}A^{(M,n,\lambda,\mu)}$
(S157)

and therefore

$\displaystyle\bar{A}_{i,h}^{(n,\lambda,\mu,\gamma)}\coloneqq\lim\limits_{M\rightarrow\infty}A_{i,h}^{(M,n,\lambda,\mu,\gamma)}=\sqrt{\gamma}\bar{A}_{i,h}^{(n,\lambda,\mu)}\,.$
(S158)

This means that this modification of the model is equivalent to placing a
thermal attenuator with transmissivity $\gamma$ before or after the memory
channel. Mathematically, the square of the absolute values of the singular
values of $\bar{A}_{i,h}^{(n,\lambda,\mu,\gamma)}$ are obtained by rescaling
those of $\bar{A}_{i,h}^{(n,\lambda,\mu)}$ by $\gamma$. It follows that as
long as $\gamma>1/2$ and for $\nu=0$, there are values of $\mu$ and $\lambda$
such that the quantum capacity is still non-zero. In particular
$\gamma\lambda$, which can be thought as the effective attenuation on the
fibre, could be less than $1/2$ while the quantum capacities are still non
zero. A similar observation holds for the two-way capacities. If there were no
memory effects, the two-way quantum capacity would be zero if and only if
$\gamma\lambda\leq\frac{\nu}{\nu+1}$, while in our model it can be still non-
zero even if the inequality is satisfied.


# Boundary output feedback stabilisation for 2-D and 3-D parabolic equations

Hugo Lhachemi  Université Paris-Saclay, CNRS, CentraleSupèlec, Laboratoire des
signaux et systèmes, 91190, Gif-sur-Yvette, France e-mail:
<EMAIL_ADDRESS>Ionuţ Munteanu  Faculty of Mathematics, Al.
I. Cuza University, Bd. Carol I, 11, Iasi 700506, Romania  O. Mayer Institute
of Mathematics, Romanian Academy, Bd. Carol I, 8, Iasi 700505, Romania  e-mail
of the corresponding author<EMAIL_ADDRESS>Christophe Prieur
Université Grenoble Alpes, CNRS, Grenoble-INP, GIPSA-lab, F-38000, Grenoble,
France  e-mail<EMAIL_ADDRESS>
###### Abstract

The present paper addresses the topic of boundary output feedback
stabilization of parabolic-type equations, governed by linear differential
operators which can be diagonalized by the introduction of adequate weighting
functions (by means of the Sturm-Liouville method), and which evolve in
bounded spatial domains that are subsets of $\mathbb{R}^{d},\ d=1,2,3$.
Combining ideas inspired by [19] for the boundary output feedback control of
1-D parabolic PDEs and [22] for the state feedback control of multi-D
parabolic PDEs, we report in this paper an output feedback boundary
stabilizing control with internal Dirichlet measurements designed by means of
a finite-dimensional observer. The reported control design procedure is shown
to be systematic for 2-D and 3-D parabolic equations.

Keywords: second order parabolic equations, exponential asymptotic
stabilization, observer design, eigenvalues and eigenfunctions, spectral
decomposition, proportional feedback control

MSC2020: 35K10, 93D15, 93B53, 93B52.

##  1 Introduction

There is now a large amount of literature dealing with boundary control of
dynamical infinite-dimensional systems. In particular, various techniques have
been developed for the design of control strategies for 1-D partial
differential equations (PDEs). These include: Lyapunov methods [5],
backstepping design [16], linear quadratic control methods [8], characteristic
analysis [6], among other approaches. Even if some of these approaches have
been generalized to multi-D PDEs, such as LQR methods (see e.g., [29]),
constructive methods for multi-D PDEs are not so developed. Among the
contributions, one can find in [4] the design of simple proportional-type
boundary stabilizing controllers for multi-D parabolic-type equations under a
restrictive assumption concerning the linear independence of the traces of the
normal derivatives of the eigenfunctions. Extensions of this approach while
removing the aforementioned restrictive assumption have been reported in the
recent textbook [22]. Boundary control and observation of heat equations on
multi-dimensional domains has also been solved in [9] under the restrictive
assumption that all the unstable modes are simple.

In this paper, we study the problem of output feedback stabilization of $2$-D
and $3$-D parabolic PDEs using spectral reduction methods. Spectral
decomposition techniques consist first of the projection of the PDE into a
finite-dimensional unstable system plus a residual stable infinite-dimensional
system. In this framework, the control strategy is designed on the unstable
finite-dimensional part of the plant. Although simple in their basic concepts,
spectral-based control design methods are challenging because one must ensure
that the control strategy does not introduce any spillover effect, that is:
the control strategy originally designed on a finite-dimensional approximation
of the PDE plant may actually fail to stabilize the infinite-dimensional
system due to the interconnection of the controller with the infinite-
dimensional residual dynamics (see, e.g., [2, 1]). These type of approaches
originate back to the 60s in the works of Rusell [24] and to the 80s in
particular in the work [27]. These ideas were further developed later in
various directions: see e.g. [7, 21, 23] for the one-dimensional case and e.g.
[3, 22, 28, 30] for the multi-dimensional case.

Spectral reduction-based methods are very attractive in practice because they
allow the design of finite-dimensional control strategies for parabolic PDEs;
see for instance the seminal works [27, 25] in the context of output feedback.
They are particularly relevant because they allow the computation of reduced
order models, making numerical computations and practical implementations much
easier to handle compared to infinite-dimensional control and observation
strategies [15]. Adopting spectral reduction-based representation of parabolic
PDEs [7, 24] and leveraging on pioneer works [27, 25] augmented with Linear
Matrix Inequalities (LMIs) procedures [14], stabilization problems for 1-D
parabolic PDEs have been solved in a systematic manner for various boundary
control and boundary output [19], including the possibility to handle systems
of 1-D parabolic PDEs [12]. In this paper, we further develop and generalize
these methods to the case of multi-D parabolic PDEs.

The aim of this paper is to report a constructive control design procedure for
the output feedback boundary stabilization of multi-D parabolic PDEs that is
systemic in the 2-D and 3-D cases. Extending procedures reported in [19] for
1-D PDEs and combining this latter work with analyses on the eigenspaces
inspired by [22], we succeed to design a finite-dimensional observer-based
output feedback control strategy for multi-D parabolic-type equations. More
precisely, we consider in this work the following boundary-controlled
parabolic-type equation evolving in $\mathcal{O}$, an open and connected
subset of $\mathbb{R}^{d}$ with $d\in\\{1,2,3\\}$, with smooth boundary
$\partial\mathcal{O}$ split into two disjoint parts
$\partial\mathcal{O}=\Gamma_{1}\cup\Gamma_{2},$ such that $\Gamma_{1}$ has
non-zero Lebesgue measure. Then, the system is described by

(1a)
$\displaystyle\partial_{t}z(x,t)+\sum_{i,j=1}^{d}a_{ij}(x)\partial_{ij}z(x,t)+\sum_{i=1}^{d}b_{i}(x)\partial_{i}z(x,t)+c(x)z(x,t)=0,\
t>0,\ x\in\mathcal{O};$ (1b) $\displaystyle z(x,t)=u(x,t),\ x\in\Gamma_{1},\
z(x,t)=0\ x\in\Gamma_{2},\ t>0;$ (1c) $\displaystyle z(0,x)=z_{o}(x),\
x\in\mathcal{O}.$

The system output consists of $M\in\mathbb{N}^{*}$ in-domain measurements:

(2) $y(t)=\left(z(\xi_{1},t),z(\xi_{2},t),\ldots,z(\xi_{M},t)\right),$

with $\xi_{i}\in\mathcal{O}$ that are 2 by 2 distinct. As we shall see, the
number of measurements $M$ to be selected will depend on the maximum
multiplicity of the unstable eigenvalues of the plant to be stabilized by the
control strategy. In this context, our only assumption in the present work is
that the second order governing differential operator can be diagonalized in a
suitable Riesz-basis (this will be described in details in Subsection 2
below). The main result of this paper can be informally stated as follows (see
Theorem 11 for a precise statement)

Theorem: Assuming that the governing linear operator of equation (1) is in
divergence form (as written in (4) below), there exists an explicit output
feedback controller (see (20) below with $U$ provided by (47)) which
exponentially stabilizes the reaction-diffusion equation (1) based on the sole
internal measurement (2).

The outline of the paper is as follows. Various notations, assumptions, and
preliminary properties are summarized in Section 2. Then, the proposed control
stategy along with the main stability result, that rigorously formalizes the
above informal theorem, is reported in Section 3. Concluding remarks are
formulated in Section 4.

##  2 Notation and preliminary properties

###  2.1 Notation and basic definitions

Spaces $\mathbb{R}^{n}$ are endowed with the Euclidean scalar product
$\left<\cdot,\cdot\right>_{n}$ and norm $\|\cdot\|_{n}$. The associated
induced norms of matrices are denoted by $\|\cdot\|$. $L^{2}(\mathcal{O})$
stands for the space of square Lebesgue integrable functions on $\mathcal{O}$
and is endowed with the inner product
$\left<f,g\right>=\int_{\mathcal{O}}f(x)g(x)dx$ with associated norm denoted
by $\|\cdot\|_{L^{2}}.$ In addition, we denote by
$\left<\cdot,\cdot\right>_{L^{2}(\Gamma_{1})}$ the scalar product in
$L^{2}(\Gamma_{1})$ with the Lebesgue surface measure. For an integer $m\geq
1$, the $m-$order Sobolev space is denoted by $H^{m}(\mathcal{O})$ and is
endowed with its usual norm denoted by $\|\cdot\|_{H^{m}}.$ We set
$H_{0}^{1}(\mathcal{O})$ for the completion of the space of infinitely
differentiable functions, which are nonzero only on a compact subset of
$\mathcal{O},$ with respect to the Sobolev norm $\|\cdot\|_{H^{1}}$. For a
symmetric matrix $P\in\mathbb{R}^{n\times n},\ P\succeq 0$ (resp. $P\succ 0$)
means that $P$ is positive semi-definite (resp. positive definite).

Let $\left\\{\varphi_{n}\right\\},\ n\in\mathbb{N}^{*}$ be a sequence in a
Hilbert space $(H,\left<\cdot,\cdot\right>_{H},\|\cdot\|_{H}).$ It is called a
Riesz basis if: (i) $\overline{\text{span}\left\\{\varphi_{n}\right\\}}=H$;
and (ii) there exists constants $0<c\leq C<\infty$ such that

$c\sum_{n\geq 1}|\alpha_{n}|^{2}\leq\left\|\sum_{n\geq
1}\alpha_{n}\varphi_{n}\right\|_{H}^{2}\leq C\sum_{n\geq 1}|\alpha_{n}|^{2},$

for all sequences of scalars $\left\\{\alpha_{n}\right\\}_{n}$ so that
$\sum_{n\geq 1}|\alpha_{n}|^{2}<\infty.$

For any given function $\mu\in C(\mathcal{O})$, we introduce the weighted
Lebesgue space

$L^{2}_{\mu}(\mathcal{O})=\left\\{f:\mathcal{O}\rightarrow\mathbb{R}\;\mathrm{measurable}\,:\,\int_{\mathcal{O}}f^{2}(x)\mu(x)dx<\infty\right\\}.$

If there exist constants $0<\mu_{m},\mu_{M}<\infty$ such that
$0<\mu_{m}\leq\mu(x)\leq\mu_{M}$ almost everywhere, then the two spaces
$L^{2}(\mathcal{O})$ and $L_{\mu}^{2}(\mathcal{O})$ are both algebraically and
topologically equivalent. This implies, in particular, that a Riesz basis in
$L^{2}(\mathcal{O})$ is also a Riesz basis in $L_{\mu}^{2}(\mathcal{O})$ and
vice versa.

###  2.2 Differential operator in divergence form

Let us denote by $\mathcal{A}:H^{2}(\mathcal{O})\rightarrow
L^{2}(\mathcal{O})$ the second order differential operator:

(3)
$\mathcal{A}f=\sum_{i,j=1}^{d}a_{ij}(x)\partial_{ij}f+\sum_{i=1}^{d}b_{i}(x)\partial_{i}f+c(x)f.$

###### Assumption 1

We assume that there exists a multiplier $\mu\in
C^{2}(\overline{\mathcal{O}})$, with $0<\mu_{m}\leq\mu(x)\leq\mu_{M}$ for all
$x\in\mathcal{O}$ for some constants $0<\mu_{m},\mu_{M}<\infty$, such that
$\mu\mathcal{A}$ can be rewritten in divergence form:

(4)
$\mu\mathcal{A}f=-\sum_{i=1}^{d}\partial_{i}(\tilde{a}_{i}(x)\partial_{i}f)+\tilde{c}(x)f$

with $C^{1}(\overline{\mathcal{O}})$-smooth coefficients $\tilde{a}_{i},\
\tilde{c}$, for which there exists constants $0<\tilde{a}_{m}<\tilde{a}_{M}$
and $\tilde{c}_{m}<\tilde{c}_{M}$ such that
$0<\tilde{a}_{m}\leq\tilde{a}_{i}(x)\leq\tilde{a}_{M}$ and
$\tilde{c}_{m}\leq\tilde{c}(x)\leq\tilde{c}_{M}$ for all
$x\in\overline{\mathcal{O}}$ and all $1\leq i\leq d$.

Previous assumption implies that
$-\mathcal{A}_{0}:=-\mathcal{A}|_{H^{2}(\mathcal{O})\cap
H_{0}^{1}(\mathcal{O})}$ is the generator of a $C_{0}$-analytic semigroup in
$L^{2}(\mathcal{O})$. To show this, we apply the well-known Hille-Yosida
theorem. First, $\mathcal{D}(\mathcal{A}_{0})=H^{2}(\mathcal{O})\cap
H_{0}^{1}(\mathcal{O})$ is dense in $L^{2}(\mathcal{O})$. Then, for
$\lambda>0$, we consider the equation

$(\lambda+\mathcal{A}_{0})f=g.$

It follows, by scalarly multiplying this equation by $\mu f$ that

$\displaystyle\left<g,\mu f\right>$
$\displaystyle=\left<(\lambda+\mathcal{A}_{0})f,\mu f\right>$
$\displaystyle=\lambda\left<\mu f,f\right>+\left<\mu\mathcal{A}_{0}f,f\right>$
$\displaystyle\geq(\lambda\mu_{m}+\tilde{c}_{m})\|f\|_{L^{2}(\mathcal{O})}^{2}+\tilde{a}_{m}\|\nabla
f\|_{L^{2}(\mathcal{O})}^{2}.$

Hence, for $\lambda$ large enough, we have $\lambda\mu_{m}+\tilde{c}_{m}>0$
and

$(\lambda\mu_{m}+\tilde{c}_{m})\|f\|_{L^{2}(\mathcal{O})}^{2}\leq\mu_{M}\|g\|_{L^{2}(\mathcal{O})}\|f\|_{L^{2}(\mathcal{O})}$

or, equivalently

$\|(\lambda+\mathcal{A}_{0})^{-1}g\|_{L^{2}(\mathcal{O})}=\|f\|_{L^{2}(\mathcal{O})}\leq\frac{\mu_{M}}{\lambda\mu_{m}+\tilde{c}_{m}}\|g\|_{L^{2}(\mathcal{O})}.$

Thus, $-\mathcal{A}_{0}$ is the generator of a $C_{0}$-analytic semigroup
$\left\\{e^{-t\mathcal{A}_{0}}:\ t\geq 0\right\\}$ in $L^{2}(\mathcal{O}).$

Besides this, the above inequalities, together with the compact embedding of
$H_{0}^{1}(\mathcal{O})$ in $L^{2}(\mathcal{O})$, implies the compactness of
the resolvent of $-\mathcal{A}_{0}$. Therefore, $-\mathcal{A}_{0}$ has a
countable set of eigenvalues, which accumulates at infinity, and for which the
corresponding eigenfunctions form a Riesz basis in $L^{2}(\mathcal{O}).$ More
exactly, in view of the divergence form (4), we consider the following
weighted eigenvalue problem ($\lambda\in\mathbb{C}$):

$-\sum_{i=1}^{d}\partial_{i}(\tilde{a}_{i}\partial_{i}\varphi)+\tilde{c}\varphi=\mu\lambda\varphi,\
x\in\mathcal{O};\ \varphi=0\text{ on }\partial\mathcal{O}.$

Owing to classical theory on spectral properties of elliptic self-adjoint
operators (see, e.g. [11, Chapter 8]) we know that the above problem has a
countable set of solutions formed by an increasing sequence of real
eigenvalues $\left\\{\lambda_{n}\right\\}_{n\in\mathbb{N}^{*}}$ which
accumulate to infinity and with corresponding eigenfunctions
$\left\\{\varphi_{n}\right\\}_{n\in\mathbb{N}^{*}}$ that form an orthonormal
basis in $L^{2}_{\mu}(\mathcal{O}).$ By the above discussions, we know that
$\left\\{\varphi_{n}\right\\}_{n\in\mathbb{N}^{*}}$ forms a Riesz basis in
$L^{2}(\mathcal{O})$ as well (but is not necessarily orthonormal). Let us
define $\psi_{n}:=\mu\varphi_{n},\ n\geq 1.$ Then
$\left\\{\varphi_{n}\right\\}_{n\in\mathbb{N}^{*}}$ and
$\left\\{\psi_{n}\right\\}_{n\in\mathbb{N}^{*}}$ are bi-orthonormal in
$L^{2}(\mathcal{O})$, i.e.,

$\left<\varphi_{i},\psi_{j}\right>=\delta_{i,j},\ i,j\geq 1,$

where $\delta_{i,j}$ is the Kronecker symbol. In particular, there exist
constants $c_{1},c_{2}>0$ such that

(5) $c_{1}\sum_{n\geq
1}\left<f,\psi_{n}\right>^{2}\leq\|f\|_{L^{2}(\mathcal{O})}^{2}\leq
c_{2}\sum_{n\geq 1}\left<f,\psi_{n}\right>^{2},\quad\forall f\in
L^{2}(\mathcal{O}).$

The following lemma, which is the key ingredient for the introduction of the
Lyapunov functional in the proof of our main result stated in Theorem 11,
provides a direct relation between the $H_{0}^{1}$-norm and the coefficients
of projection onto the Riesz basis $\\{\phi_{n}\\}_{n}$.

###### Lemma 2

Let us fix $\nu\geq 0$ so that $\tilde{c}_{m}+\nu\mu_{m}>0$. Then, there exist
constants $c_{3},c_{4}>0$ such that

(6) $c_{3}\|f\|_{H_{0}^{1}(\mathcal{O})}^{2}\leq\sum_{n\geq
1}(\lambda_{n}+\nu)\left<f,\psi_{n}\right>^{2}\leq
c_{4}\|f\|_{H_{0}^{1}(\mathcal{O})}^{2},$

for all $f\in H^{2}(\mathcal{O})\cap H_{0}^{1}(\mathcal{O})$.

Proof. Recall that, thanks to the Poincaré inequality, we have that
$\|\nabla\cdot\|_{L^{2}(\mathcal{O})}$ is an equivalent norm in
$H_{0}^{1}(\mathcal{O})$. In particular, there exists a constant
$\mathcal{C}>0$ such that

$\|\nabla
f\|^{2}_{L^{2}(\mathcal{O})}\geq\mathcal{C}\|f\|^{2}_{H_{0}^{1}(\mathcal{O})},\
\forall f\in H_{0}^{1}(\mathcal{O}).$

Next, we observe for any $f\in H^{2}(\mathcal{O})\cap H_{0}^{1}(\mathcal{O})$
the following:

$\displaystyle\left<(\mathcal{A}_{0}+\nu)f,f\right>_{L_{\mu}^{2}(\mathcal{O})}$
$\displaystyle=\int_{\mathcal{O}}\mu\left\\{(\mathcal{A}_{0}f)f+\nu
f^{2}\right\\}dx$
$\displaystyle=-\sum_{i=1}^{d}\int_{\mathcal{O}}\partial_{i}(\tilde{a}_{i}\partial_{i}f)fdx+\int_{\mathcal{O}}(\tilde{c}+\nu\mu)f^{2}dx$
$\displaystyle=\sum_{i=1}^{d}\int_{\mathcal{O}}\tilde{a}_{i}(\partial_{i}f)^{2}dx+\int_{\mathcal{O}}(\tilde{c}+\nu\mu)f^{2}dx$

where we have applied integration by parts. This shows that

$\left<(\mathcal{A}_{0}+\nu)f,f\right>_{L_{\mu}^{2}(\mathcal{O})}\geq(\tilde{c}_{m}+\nu\mu_{m})\|f\|^{2}_{L^{2}(\mathcal{O})},$

and so $\lambda_{n}>-\nu$ for all $n\geq 1$; and that

$\mathcal{C}\tilde{a}_{m}\|f\|^{2}_{H_{0}^{1}(\mathcal{O})}\leq\|(\mathcal{A}_{0}+\nu)^{1/2}f\|_{L_{\mu}^{2}(\mathcal{O})}^{2}\leq\max(\tilde{a}_{M},\tilde{c}_{M}+\nu\mu_{M})\|f\|^{2}_{H_{0}^{1}(\mathcal{O})}.$

Then it can be seen from (5) that

$\displaystyle\sum_{n\geq 1}(\lambda_{n}+\nu)\left<f,\psi_{n}\right>^{2}$
$\displaystyle\leq\frac{1}{c_{1}}\|(\mathcal{A}_{0}+\nu)^{1/2}f\|_{L^{2}(\mathcal{O})}^{2}$
$\displaystyle\leq\frac{1}{c_{1}\mu_{m}}\|(\mathcal{A}_{0}+\nu)^{1/2}f\|_{L_{\mu}^{2}(\mathcal{O})}^{2}$
$\displaystyle\leq\frac{\max(\tilde{a}_{M},\tilde{c}_{M}+\nu\mu_{M})}{c_{1}\mu_{m}}\|f\|_{H_{0}^{1}(\mathcal{O})}^{2}$

while

$\displaystyle\sum_{n\geq 1}(\lambda_{n}+\nu)\left<f,\psi_{n}\right>^{2}$
$\displaystyle\geq\frac{1}{c_{2}}\|(\mathcal{A}_{0}+\nu)^{1/2}f\|_{L^{2}(\mathcal{O})}^{2}$
$\displaystyle\geq\frac{1}{c_{2}\mu_{M}}\|(\mathcal{A}_{0}+\nu)^{1/2}f\|_{L_{\mu}^{2}(\mathcal{O})}^{2}$
$\displaystyle\geq\frac{\mathcal{C}\tilde{a}_{m}}{c_{2}\mu_{M}}\|f\|_{H_{0}^{1}(\mathcal{O})}^{2}.$

This concludes the proof of Lemma 2.

We fix the integer $N_{0}\in\mathbb{N}^{*}$ such that
$\lambda_{j}\geq\lambda_{N_{0}+1}>0$ for all $j\geq N_{0}+1$; and let
$N\in\mathbb{N}$ be large enough such that $N\geq N_{0}$ and $\lambda_{n}\geq
1,\ \forall n\geq N+1$. A key element in the application of the control
strategy reported in this paper relies on the asymptotic behavior of the
eigenfunctions $\varphi_{n}$ evaluated at the measurement locations $\xi_{i}$.
This asymptotic behavior is assessed through the following lemma.

###### Lemma 3

The following holds:

(7) $\sum_{n\geq
N+1}\frac{\varphi^{2}_{n}(\xi_{i})}{\lambda_{n}^{2}}<\infty,\qquad\forall
i\in\\{1,2,\ldots,M\\}.$

Proof. Considering $\mathcal{O}$ as a manifold with the Riemannian metric $\mu
dx$, it follows that $\mathcal{A}$ is a self-adjoint operator in the Lebesgue
$L^{2}$ space associated to this manifold. Hence, one may argue as in [13] (or
as in [26] for the particular case of the Laplace operator) to deduce the
existence of a constant $C>0$ such that, for any $\lambda\geq 1$, we have

(8) $\sum_{\sqrt{\lambda_{j}}\in[\lambda,\lambda+1)}|\varphi_{j}(\xi)|^{2}\leq
C\lambda^{d-1},\ \forall\xi\in\overline{\mathcal{O}}.$

Let $0<\frac{1}{5-d}<\beta<1$, where we recall that $d\in\\{1,2,3\\}$. We note
that $[1,\infty)=\cup_{m\geq 1}[m^{\beta},m^{\beta}+1).$ Therefore, taking
$\lambda=m^{\beta}$ in (8), we infer for any $\xi\in\overline{\mathcal{O}}$
that

$\displaystyle\sum_{n\geq N+1}\frac{\varphi^{2}_{n}(\xi)}{\lambda_{n}^{2}}$
$\displaystyle\leq\sum_{m=1}^{\infty}\left(\sum_{j\geq
N+1,\,\sqrt{\lambda_{j}}\in[m^{\beta},m^{\beta}+1)}\frac{\varphi^{2}_{j}(\xi)}{\lambda_{j}^{2}}\right)\leq\sum_{m=1}^{\infty}\frac{1}{m^{4\beta}}\left(\sum_{j\geq
N+1,\,\sqrt{\lambda_{j}}\in[m^{\beta},m^{\beta}+1)}\varphi^{2}_{j}(\xi)\right)$
$\displaystyle\leq
C\sum_{m=1}^{\infty}\frac{1}{m^{4\beta}}m^{(d-1)\beta}=C\sum_{m=1}^{\infty}\frac{1}{m^{(5-d)\beta}}<\infty,$

since $\beta(5-d)>1.$

###### Remark 4

It is worth being noted that for dimensions $d\geq 4$ the quantities
$\sum_{n\geq N+1}\frac{\varphi^{2}_{n}(\xi_{i})}{\lambda_{n}^{2}},\
i\in\left\\{1,2,...,M\right\\},$ might not be finite, in general. This is the
key point that restricts the application of the proposed control strategy for
multi-D equations with $d\geq 4$.

For latter purpose, we need to show a unique continuation property of the
eigenfunctions of the operator $\mathcal{A}$, as stated in the following
lemma.

###### Lemma 5

Let any $\varphi\not\equiv 0$ satisfying

$-\sum_{i=1}^{d}\partial_{i}(\tilde{a}_{i}(x)\partial_{i}\varphi)+\tilde{c}(x)\varphi-\mu(x)\lambda\varphi=0\text{
in }\mathcal{O};\ \varphi=0\text{ on }\partial\mathcal{O}.$

Then,
$\sum_{i=1}^{d}n_{i}(\cdot)\tilde{a}_{i}(\cdot)\partial_{i}\varphi(\cdot)$ is
not identically zero on $\Gamma_{1}$. Here, $n_{i}$ are the components of the
unit outward normal to the boundary of $\mathcal{O}$.

Proof. This property holds true because the principal part of the differential
operator is uniformly elliptic (which, usually, is called the elliptic
continuation principle). Let us assume by contradiction that
$\sum_{i=1}^{d}n_{i}(\cdot)\tilde{a}_{i}(\cdot)\partial_{i}\varphi(\cdot)\equiv
0\text{ on }\Gamma_{1}.$ Choose some $x_{0}\in\Gamma_{1}$, and choose
coordinates $x=(x^{\prime},x_{d})$ so that $x_{0}=0$ and for some $r>0$

$\mathcal{O}\cap B(0,r)=\left\\{x\in B(0,r);\ x_{d}>g(x^{\prime})\right\\},$

where $g:\mathbb{R}^{d-1}\rightarrow\mathbb{R}$ is a $C^{\infty}-$function. We
extend the domain near $x_{0}$ by choosing $\psi\in
C_{c}^{\infty}(\mathbb{R}^{d-1})$ with $\psi=0$ for $\|x^{\prime}\|_{d-1}\geq
r/2$ and $\psi=1$ for $\|x^{\prime}\|_{d-1}\leq r/4,$ and by letting

$\mathcal{O}^{*}=\mathcal{O}\cup\left\\{x\in B(0,r);\
x_{d}>g(x^{\prime})-\varepsilon\psi(x^{\prime})\right\\}.$

Here, $\varepsilon>0$ is chosen so small that $\left\\{(x^{\prime},x_{d});\
\|x^{\prime}\|_{d-1}\leq r/2,\
x_{d}=g(x^{\prime})-\varepsilon\psi(x^{\prime})\right\\}$ is contained in
$B(0,r)$. Then, $\mathcal{O}^{*}$ is connected open set with smooth boundary.

Define the functions

$\varphi^{*}(x)=\left\\{\begin{array}[]{ll}\varphi(x)&\text{ if
}x\in\mathcal{O},\\\ 0&\text{ if
}x\in\mathcal{O}^{*}\setminus\mathcal{O},\end{array}\right.\ $
$\tilde{a}_{i}^{*}(x)=\left\\{\begin{array}[]{ll}\tilde{a}_{i}(x)&\text{ if
}x\in\mathcal{O},\\\ 0&\text{ if
}x\in\mathcal{O}^{*}\setminus\mathcal{O},\end{array}\right.\ $
$(\tilde{c}-\mu\lambda)^{*}(x)=\left\\{\begin{array}[]{ll}(\tilde{c}-\mu(x)\lambda)(x)&\text{
if }x\in\mathcal{O},\\\ 0&\text{ if
}x\in\mathcal{O}^{*}\setminus\mathcal{O}.\end{array}\right.\ $

Then, $\varphi^{*}\in H^{2}(\mathcal{O}^{*})$ is solution to

$-\sum_{i=1}^{d}\partial_{i}(\tilde{a}^{*}_{i}(x)\partial_{i}\varphi^{*}(x))+(\tilde{c}-\mu\lambda)(x)^{*}\varphi^{*}(x)=0\text{
a.e. in }\mathcal{O^{*}},$

with $\varphi^{*}\equiv 0$ in some open ball contained in
$\mathcal{O}^{*}\setminus\mathcal{O}.$ Invoking the result in [10], we
immediately get that $\varphi\equiv 0$ in $\mathcal{O}$, which is in
contradiction with the hypothesis. It concludes the proof of Lemma 5.

With all these elements in hands, we are now in position to introduce the
output feedback control strategy that exponentially stabilizes the reaction-
diffusion equation (1) based on the sole internal measurement (2). This is
discussed in the next section.

##  3 Design of the control strategy

Let us denote by
$A_{0}:=\left(\left<-\mathcal{A}\varphi_{i},\psi_{j}\right>\right)_{1\leq
i,j\leq N_{0}}.$ It is seen that
$A_{0}=-\text{diag}(\lambda_{1},\lambda_{2},\dots,\lambda_{N_{0}}).$ For
dimension $d\geq 2$, it is highly possible to have multiple eigenvalues. Note
however that the first eigenvalue is always simple. Therefore, the number of
scalar measurements $M$ from the system output (2), as well as the structure
of the control strategy, needs to be slightly adapted in function of the
multiplicity of the unstable eigenvalues. To fix the ideas and to keep the
presentation as concise as possible, we make the following choice for the
spectrum structure (other possible choices can be treated in a similar manner
as below without any supplementary effort): the second eigenvalue is of
multiplicity equal two, while the rest of the first $N_{0}$ eigenvalues are
simple, i.e., we have

$\lambda_{1}<\lambda_{2}=\lambda_{3}<\lambda_{4}<\ldots<\lambda_{N_{0}}.$

This configuration leads us to consider $M=2$ scalar outputs, i.e.,

(9) $y(t)=(z(\xi_{1},t),z(\xi_{2},t))^{\top}.$

We then fix $\eta>0$ so that $\lambda_{2}+\eta\neq\lambda_{j},\ \forall
j\in\left\\{1,3,\dots,N_{0}\right\\}$.

###### Remark 6

Note that the other configurations, in terms of multiplicity of the different
eigenvalues, can be handled in a similar way by setting the number of scalar
measurements $M$ as the maximum of multiplicity for the eigenvalues
$\lambda_{1},\ldots,\lambda_{N_{0}}$.

###  3.1 Preliminary control design

For $\gamma>0$ large enough and for each $v\in L^{2}(\Gamma_{1})$, there
exists a unique solution, $D$, to the equation

(10)
$\displaystyle\mathcal{A}D-2\sum_{i=1}^{N_{0}}\lambda_{i}\left<D,\psi_{i}\right>\varphi_{i}-\eta\left<D,\psi_{2}\right>\varphi_{2}+\gamma
D=0\text{ in }\mathcal{O};$ $\displaystyle D=v\text{ on }\Gamma_{1},\
D=0\text{ on }\Gamma_{2}.$

Indeed, arguing as in [17], this is a direct consequence of the application of
the Lax-Milligram theorem. This allows us to introduce the following:

###### Definition 7

Let $D_{\gamma}:L^{2}(\Gamma_{1})\rightarrow L^{2}(\mathcal{O})$ be defined by
$D_{\gamma}v:=D$ where, for any given $v\in L^{2}(\Gamma_{1})$, $D\in
L^{2}(\mathcal{O})$ is the unique solution to (10).

###### Lemma 8

We have

(11) $\left(\begin{array}[]{c}\left<D_{\gamma}v,\psi_{1}\right>\\\
\left<D_{\gamma}v,\psi_{2}\right>\\\ \vdots\\\
\left<D_{\gamma}v,\psi_{N_{0}}\right>\end{array}\right)=-\Lambda_{\gamma}\left(\begin{array}[]{c}\left<v,\sum_{i=1}^{d}n_{i}\tilde{a}_{i}\partial_{i}\varphi_{1}\right>_{L^{2}(\Gamma_{1})}\\\
\left<v,\sum_{i=1}^{d}n_{i}\tilde{a}_{i}\partial_{i}\varphi_{2}\right>_{L^{2}(\Gamma_{1})}\\\
\vdots\\\
\left<v,\sum_{i=1}^{d}n_{i}\tilde{a}_{i}\partial_{i}\varphi_{N_{0}}\right>_{L^{2}(\Gamma_{1})}\end{array}\right),$

where

(12)
$\Lambda_{\gamma}=\mathrm{diag}\left(\frac{1}{\gamma-\lambda_{1}},\frac{1}{\gamma-\lambda_{2}-\eta},\frac{1}{\gamma-\lambda_{3}},\dots,\frac{1}{\gamma-\lambda_{N_{0}}}\right),$

while

(13)
$\left<D_{\gamma}v,\psi_{k}\right>=-\frac{1}{\gamma+\lambda_{k}}\left<v,\sum_{i=1}^{d}n_{i}\tilde{a}_{i}\partial_{i}\varphi_{k}\right>_{L^{2}(\Gamma_{1})},\
k\geq N_{0}+1.$

Proof. Scalarly multiplying equation (10) by $\psi_{k}=\mu\varphi_{k},\
k=1,2,\ldots,N_{0},$ while taking advantage of the bi-orthogonality of the
sequences $\left\\{\varphi_{i}\right\\}_{i\geq 1}$ and
$\left\\{\psi_{i}\right\\}_{i\geq 1}$, we get

(14)
$\displaystyle\left<\mu\mathcal{A}D_{\gamma}v,\varphi_{k}\right>-2\lambda_{k}\left<D_{\gamma}v,\psi_{k}\right>-\eta\left<D_{\gamma}v,\psi_{2}\right>\delta_{2,k}+\gamma\left<D_{\gamma}v,\psi_{k}\right>=0.$

Let us compute $\left<\mu\mathcal{A}D_{\gamma}v,\varphi_{k}\right>$. Using the
integration by parts formula and invoking the boundary conditions of
$\varphi_{j}$ and $D_{\gamma}v$, we obtain that

$\displaystyle\left<\mu\mathcal{A}D_{\gamma}v,\varphi_{k}\right>$
$\displaystyle=\left<-\sum_{i=1}^{d}\partial_{i}(\tilde{a}_{i}\partial_{i}D_{\gamma}v)+\tilde{c}D_{\gamma}v,\varphi_{k}\right>$
$\displaystyle=\sum_{i=1}^{d}\left<D_{\gamma}v,n_{i}\tilde{a}_{i}\partial_{i}\varphi_{k}\right>_{L^{2}(\Gamma_{1})}+\left<D_{\gamma}v,-\sum_{i=1}^{d}\partial_{i}(\tilde{a}_{i}\partial_{i}\varphi_{k})+\tilde{c}\varphi_{k}\right>$
$\displaystyle=\left<v,\sum_{i=1}^{d}n_{i}\tilde{a}_{i}\partial_{i}\varphi_{k}\right>_{L^{2}(\Gamma_{1})}+\lambda_{k}\left<D_{\gamma}v,\psi_{k}\right>.$

Substituting the right hand side of the latter identity into equation (14), we
infer that

$(-\lambda_{k}-\eta\delta_{2,k}+\gamma)\left<D_{\gamma}v,\psi_{k}\right>=-\left<v,\sum_{i=1}^{d}n_{i}\tilde{a}_{i}\partial_{i}\varphi_{k}\right>_{L^{2}(\Gamma_{1})},\;k\in\\{1,\ldots,N_{0}\\}.$

This gives (11). Proceeding similarly, we infer that (13) holds.

We next fix $N_{0}$ positive constants
$0<\gamma_{1}<\gamma_{2}<\ldots<\gamma_{N_{0}}$, selected sufficiently large
such that, for each $k\in\\{1,2,\ldots,N_{0}\\}$,

  1. 1.

equation (10) is well-posed for $\gamma=\gamma_{k}$;

  2. 2.

$\gamma_{k}\pm(\lambda_{i}+\eta\delta_{2,i})\neq 0$ for all $1\leq k\leq
N_{0}$ and $i\in\mathbb{N}^{*}$

Following Definition 7, we denote for $k\in\\{1,2,\ldots,N_{0}\\}$ by
$D_{\gamma_{k}}$ the corresponding operators as defined by (10).

We now introduce the Gram matrix $\mathbf{B}$, defined by:

(15)
$\textbf{B}:=\left(\left<\sum_{i=1}^{d}n_{i}\tilde{a}_{i}\partial_{i}\varphi_{k},\sum_{i=1}^{d}n_{i}\tilde{a}_{i}\partial_{i}\varphi_{l}\right>_{L^{2}(\Gamma_{1})}\right)_{1\leq
k,l\leq N_{0}}$

and we set

(16)
$B_{k}:=\Lambda_{\gamma_{k}}\textbf{B}\Lambda_{\gamma_{k}},\;k\in\\{1,2,\ldots,N_{0}\\}.$

We also define

(17)
$\mathcal{L}(x)=\left(\sum_{i=1}^{d}n_{i}(x)\tilde{a}_{i}(x)\partial_{i}\varphi_{1}(x),\ldots,\sum_{i=1}^{d}n_{i}(x)\tilde{a}_{i}(x)\partial_{i}\varphi_{N_{0}}(x)\right)^{\top}$

for all $x\in\Gamma_{1}$. Invoking iteratively the unique continuation
property of the eigenfunctions of the operator $\mathcal{A}$, stated by Lemma
5, we observe that $\mathcal{L}$ has non-zero entries for all $x$ in a non-
zero measure subset of $\Gamma_{1}$. Then, because
$-\lambda_{k}-\eta\delta_{2,k}$ for $k\in\\{1,2,\dots,N_{0}\\}$ are 2 by 2
distinct and owing to [22, Proposition 2.1], we have that
$B_{1}+B_{2}+\ldots+B_{N_{0}}$ is an invertible matrix. Therefore, we define:

(18) $A:=(B_{1}+B_{2}+\ldots+B_{N_{0}})^{-1}.$

Next, introducing an auxiliary command input
$U:[0,\infty)\rightarrow\mathbb{R}^{N_{0}},$ that will be specified later in
Subsection 3.3, we define

(19)
$u_{k}(x,t):=\left<\Lambda_{\gamma_{k}}AU(t),\mathcal{L}(x)\right>_{N_{0}},\;x\in\Gamma_{1},\;t\geq
0,$

for all $k=1,2,\ldots,N_{0}$. Then, we define the boundary control $u$
appearing in the plant (1) as

(20) $\displaystyle u(x,t):$
$\displaystyle=u_{1}(x,t)+u_{2}(x,t)+\ldots+u_{N_{0}}(x,t)$
$\displaystyle=\sum_{k=1}^{N_{0}}\left<\Lambda_{\gamma_{k}}AU(t),\mathcal{L}(x)\right>_{N_{0}}.$

###### Lemma 9

It holds, for all $k=1,\ldots,N_{0}$,

(21) $\left(\begin{array}[]{c}\left<D_{\gamma_{k}}u_{k},\psi_{1}\right>\vskip
3.0pt plus 1.0pt minus 1.0pt\\\ \left<D_{\gamma_{k}}u_{k},\psi_{2}\right>\\\
\vdots\\\
\left<D_{\gamma_{k}}u_{k},\psi_{N_{0}}\right>\end{array}\right)=-B_{k}AU,$

Proof.

Based on (11), we infer that

$\displaystyle\left(\begin{array}[]{c}\left<D_{\gamma_{k}}u_{k},\psi_{1}\right>\\\
\left<D_{\gamma_{k}}u_{k},\psi_{2}\right>\\\ \vdots\\\
\left<D_{\gamma_{k}}u_{k},\psi_{N_{0}}\right>\end{array}\right)=-\Lambda_{\gamma_{k}}\left(\begin{array}[]{c}\left<u_{k},\displaystyle\sum_{i=1}^{d}n_{i}\tilde{a}_{i}\partial_{i}\varphi_{1}\right>_{L^{2}(\Gamma_{1})}\\\
\left<u_{k},\displaystyle\sum_{i=1}^{d}n_{i}\tilde{a}_{i}\partial_{i}\varphi_{2}\right>_{L^{2}(\Gamma_{1})}\\\
\vdots\\\
\left<u_{k},\displaystyle\sum_{i=1}^{d}n_{i}\tilde{a}_{i}\partial_{i}\varphi_{N_{0}}\right>_{L^{2}(\Gamma_{1})}\end{array}\right).$

Taking into account (15) and (19), this yields

$\left(\begin{array}[]{c}\left<D_{\gamma_{k}}u_{k},\psi_{1}\right>\\\
\left<D_{\gamma_{k}}u_{k},\psi_{2}\right>\\\ \vdots\\\
\left<D_{\gamma_{k}}u_{k},\psi_{N_{0}}\right>\end{array}\right)=-\Lambda_{\gamma_{k}}\mathbf{B}\Lambda_{\gamma_{k}}AU$

and so, owing to the definition of $B_{k}$ given by (16), the claimed identity
(21) is proved.

###  3.2 Spectral reduction

Our objective is now to specify the auxiliary command input $U$ that appears
in (19). To do so, we first need to carry on a spectral reduction of the
system formed by the plant (1) along with the preliminary control input (20).
This is done in this subsection.

###### Lemma 10

Consider the following change of variable :

(22) $w:=z-\sum_{k=1}^{N_{0}}D_{\gamma_{k}}u_{k}$

and define the coefficients of projection
$w_{n}(t)=\left<w(t,\cdot),\psi_{n}\right>$ and
$z_{n}(t)=\left<z(t,\cdot),\psi_{n}\right>$. Then we have

$\displaystyle\frac{d}{dt}z_{n}$
$\displaystyle=-\lambda_{n}z_{n}+\sum_{k=1}^{N_{0}}(\lambda_{n}+\gamma_{k})\left<D_{\gamma_{k}}u_{k},\psi_{n}\right>$
(23)
$\displaystyle\phantom{=}\;-2\sum_{k,i=1}^{N_{0}}\lambda_{i}\left<D_{\gamma_{k}}u_{k},\psi_{i}\right>\delta_{i,n}-\eta\sum_{k=1}^{N_{0}}\left<D_{\gamma_{k}}u_{k},\psi_{2}\right>\delta_{2,n}$

for all $n\geq 1$. Moreover, we have

(24) $\displaystyle\frac{d}{dt}w_{n}$
$\displaystyle=-\lambda_{n}w_{n}+\sum_{k=1}^{N_{0}}\gamma_{k}\left<D_{\gamma_{k}}u_{k},\psi_{n}\right>-\sum_{k=1}^{N_{0}}\left<D_{\gamma_{k}}\frac{d}{dt}u_{k},\psi_{n}\right>$

for all $n\geq N_{0}+1$.

Proof. We first equivalently rewrite (1) as an internal-type control problem.
More precisely, invoking the change of variable (22), we have

$\displaystyle\frac{d}{dt}w$
$\displaystyle=\frac{d}{dt}z-\frac{d}{dt}\sum_{k=1}^{N_{0}}D_{\gamma_{k}}u_{k}$
$\displaystyle\overset{\eqref{e1}}{=}-\mathcal{A}z-\frac{d}{dt}\sum_{k=1}^{N_{0}}D_{\gamma_{k}}u_{k}$
$\displaystyle=-\mathcal{A}_{0}w-\sum_{k=1}^{N_{0}}\mathcal{A}D_{\gamma_{k}}u_{k}-\frac{d}{dt}\sum_{k=1}^{N_{0}}D_{\gamma_{k}}u_{k}$
$\displaystyle\overset{\eqref{e5}}{=}-\mathcal{A}_{0}w-2\sum_{k,i=1}^{N_{0}}\lambda_{i}\left<D_{\gamma_{k}}u_{k},\psi_{i}\right>\varphi_{i}-\eta\sum_{k=1}^{N_{0}}\left<D_{\gamma_{k}}u_{k},\psi_{2}\right>\varphi_{2}$
(25)
$\displaystyle\phantom{=}\;+\sum_{k=1}^{N_{0}}\gamma_{k}D_{\gamma_{k}}u_{k}-\frac{d}{dt}\sum_{k=1}^{N_{0}}D_{\gamma_{k}}u_{k},\
t>0.$

Then, recalling that $w_{n}(t)=\left<w(t,\cdot),\psi_{n}\right>$ and
$z_{n}(t)=\left<z(t,\cdot),\psi_{n}\right>$, the projection of (25) gives

$\displaystyle\frac{d}{dt}w_{n}$
$\displaystyle=-\lambda_{n}w_{n}-\sum_{k=1}^{N_{0}}\left<D_{\gamma_{k}}\frac{d}{dt}u_{k},\psi_{n}\right>-2\sum_{k,i=1}^{N_{0}}\lambda_{i}\left<D_{\gamma_{k}}u_{k},\psi_{i}\right>\delta_{i,n}$
(26)
$\displaystyle\phantom{=}\;-\eta\sum_{k=1}^{N_{0}}\left<D_{\gamma_{k}}u_{k},\psi_{2}\right>\delta_{2,n}+\sum_{k=1}^{N_{0}}\gamma_{k}\left<D_{\gamma_{k}}u_{k},\psi_{n}\right>$

for all $n\geq 1$. This gives (24) because $\delta_{i,n}=0$ for all $1\leq
i\leq N_{0}$ and $n\geq N_{0}+1$. Now, in view of the the change of variable
formula (22), we have

(27)
$w_{n}=z_{n}-\sum_{k=1}^{N_{0}}\left<D_{\gamma_{k}}u_{k},\psi_{n}\right>.$

Thus, combining (26)-(27), we infer that (23) holds.

Our objective is now to write an ODE describing the dynamics of a finite
number of modes of the system composed of the plant (1) and the control input
(20). To do so, we define

$Z^{N_{0}}:=(\left<z,\psi_{1}\right>,\left<z,\psi_{2}\right>,\ldots,\left<z,\psi_{N_{0}}\right>)^{\top}$

and

$\Xi:=\mathrm{diag}(0,\eta,0,\dots,0).$

In view of equation (23) and taking into account the relation (21), we deduce
that

(32) $\displaystyle\frac{d}{dt}Z^{N_{0}}(t)$
$\displaystyle=A_{0}Z^{N_{0}}(t)+\sum_{k=1}^{N_{0}}\left[-A_{0}+\gamma_{k}I\right]\left(\begin{array}[]{c}\left<D_{\gamma_{k}}u_{k},\psi_{1}\right>\\\
\left<D_{\gamma_{k}}u_{k},\psi_{2}\right>\\\ \ldots\\\
\left<D_{\gamma_{k}}u_{k},\psi_{N_{0}}\right>\end{array}\right)$ (41)
$\displaystyle\phantom{=}\;+\sum_{k=1}^{N_{0}}2A_{0}\left(\begin{array}[]{c}\left<D_{\gamma_{k}}u_{k},\psi_{1}\right>\\\
\left<D_{\gamma_{k}}u_{k},\psi_{2}\right>\\\ \ldots\\\
\left<D_{\gamma_{k}}u_{k},\psi_{N_{0}}\right>\end{array}\right)-\sum_{k=1}^{N_{0}}\Xi\left(\begin{array}[]{c}\left<D_{\gamma_{k}}u_{k},\psi_{1}\right>\\\
\left<D_{\gamma_{k}}u_{k},\psi_{2}\right>\\\ \ldots\\\
\left<D_{\gamma_{k}}u_{k},\psi_{N_{0}}\right>\end{array}\right)$
$\displaystyle=A_{0}Z^{N_{0}}(t)-\sum_{k=1}^{N_{0}}\left[A_{0}+\gamma_{k}I-\Xi\right]B_{k}AU(t)$
(42)
$\displaystyle\overset{\eqref{ie9}}{=}A_{0}Z^{N_{0}}(t)+\left(-A_{0}-\sum_{k=1}^{N_{0}}\gamma_{k}B_{k}A+\Xi\right)U(t).$

where we recall that $A_{0}=-\text{diag}(\lambda_{1},\ldots,\lambda_{N_{0}})$.
Introducing now a second integer $N\geq N_{0}+1$ to be specified later, we
define

$Z^{N-N_{0}}:=(\left<z,\psi_{N_{0}+1}\right>,\left<z,\psi_{N_{0}+2}\right>,\ldots,\left<z,\psi_{N}\right>)^{\top}$

along with $A_{1}=-\text{diag}(\lambda_{N_{0}+1},\ldots,\lambda_{N})$ and
$H^{N-N_{0}}:\mathbb{R}^{N_{0}}\rightarrow\mathbb{R}^{N-N_{0}}$ defined by

$\displaystyle H^{N-N_{0}}U$
$\displaystyle=\left(\sum_{k=1}^{N_{0}}(\lambda_{N_{0}+1}+\gamma_{k})\left<D_{\gamma_{k}}u_{k},\psi_{N_{0}+1}\right>,\ldots,\sum_{k=1}^{N_{0}}(\lambda_{N}+\gamma_{k})\left<D_{\gamma_{k}}u_{k},\psi_{N}\right>\right)^{\top}$
$\displaystyle\overset{\eqref{rq1}}{=}\sum_{k=1}^{N_{0}}\left(\begin{array}[]{c}\left<u_{k},\sum_{i=1}^{d}n_{i}\tilde{a}_{i}\partial_{i}\varphi_{N_{0}+1}\right>_{L^{2}(\Gamma_{1})}\\\
\left<u_{k},\sum_{i=1}^{d}n_{i}\tilde{a}_{i}\partial_{i}\varphi_{N_{0}+2}\right>_{L^{2}(\Gamma_{1})}\\\
\ldots\\\
\left<u_{k},\sum_{i=1}^{d}n_{i}\tilde{a}_{i}\partial_{i}\varphi_{N}\right>_{L^{2}(\Gamma_{1})}\end{array}\right)$

where $u_{k}$ is expressed in function of the auxiliary control input $U$
based on (19). Then, using again (23), we deduce that

(43) $\frac{d}{dt}Z^{N-N_{0}}(t)=A_{1}Z^{N-N_{0}}(t)+H^{N-N_{0}}U(t),\;t>0.$

###  3.3 Observer design and definition of the auxiliary control input $U$

We are now in position to properly define in this subsection the auxiliary
command input $U$ that appears in (19). We select the measurement locations
$\xi_{1},\xi_{2}\in\mathcal{O}$ from (9) such that

(44) $|\varphi_{i}(\xi_{1})|+|\varphi_{i}(\xi_{2})|\neq 0,\;\forall
i\in\\{1,4,5,\ldots,N_{0}\\};\qquad\det\left|\begin{array}[]{cc}\varphi_{2}(\xi_{1})&\varphi_{3}(\xi_{1})\\\
\varphi_{2}(\xi_{2})&\varphi_{3}(\xi_{2})\end{array}\right|\neq 0.$

Note that such a selection is always possible due to the fact that
$\varphi_{i}$, $i\in\\{1,2,\ldots,N_{0}\\}$, cannot vanish on any open ball
included in $\mathcal{O}$ and the fact that $\varphi_{2},\varphi_{3}$
correspond to the same eigenvalue and are linearly independent.

Second, introducing $\tilde{y}(t)=(w(\xi_{1},t),w(\xi_{2},t))^{\top}$, we
deduce from the change of variable formula (22) that

(45)
$y(t)=\tilde{y}(t)+\sum_{k=1}^{N_{0}}\begin{pmatrix}D_{\gamma_{k}}u_{k}(\xi_{1},t)\\\
D_{\gamma_{k}}u_{k}(\xi_{2},t)\end{pmatrix}=\sum_{i\geq
1}\begin{pmatrix}\varphi_{i}(\xi_{1})\\\
\varphi_{i}(\xi_{2})\end{pmatrix}w_{i}(t)+\sum_{k=1}^{N_{0}}\begin{pmatrix}D_{\gamma_{k}}u_{k}(\xi_{1},t)\\\
D_{\gamma_{k}}u_{k}(\xi_{2},t)\end{pmatrix}.$

Defining $C_{0}=\left(\begin{array}[]{c}\varphi_{1}(\xi_{1})\
\varphi_{2}(\xi_{1})\ \ldots\ \varphi_{N_{0}}(\xi_{1})\\\
\varphi_{1}(\xi_{2})\ \varphi_{2}(\xi_{2})\ \ldots\
\varphi_{N_{0}}(\xi_{2})\end{array}\right)$ and
$C_{1}=\left(\begin{array}[]{c}\varphi_{N_{0}+1}(\xi_{1})\
\varphi_{N_{0}+2}(\xi_{1})\ldots\ \varphi_{N}(\xi_{1})\\\
\varphi_{N_{0}+1}(\xi_{2})\ \varphi_{N_{0}+2}(\xi_{2})\ldots\
\varphi_{N}(\xi_{2})\end{array}\right)$, it immediately follows from (44) that
the pair $(A_{0},C_{0})$ satisfies the Kalman condition. Hence, we can fix
$L\in M_{N_{0}\times 2}(\mathbb{R})$ such that $A_{0}-LC_{0}$ is Hurwitz with
arbitrary spectral abscissa.

In view of (27), (42), and (43), we now define the following observer
dynamics:

(46a) $\displaystyle\hat{w}_{n}$
$\displaystyle=\hat{z}_{n}-\sum_{k=1}^{N_{0}}\left<D_{\gamma_{k}}u_{k},\psi_{n}\right>,\quad
1\leq n\leq N$ (46b) $\displaystyle\frac{d}{dt}\hat{Z}^{N_{0}}(t)$
$\displaystyle=A_{0}\hat{Z}^{N_{0}}(t)+\left(-A_{0}-\sum_{k=1}^{N_{0}}\gamma_{k}B_{k}A+\Xi\right)U(t)$
$\displaystyle\phantom{=}\;-L\left\\{\sum_{i=1}^{N}\begin{pmatrix}\varphi_{i}(\xi_{1})\\\
\varphi_{i}(\xi_{2})\end{pmatrix}\hat{w}_{i}(t)+\sum_{k=1}^{N_{0}}\begin{pmatrix}\left(D_{\gamma_{k}u_{k}}\right)(\xi_{1},t)\\\
\left(D_{\gamma_{k}u_{k}}\right)(\xi_{2},t)\end{pmatrix}-y(t)\right\\}$ (46c)
$\displaystyle\frac{d}{dt}\hat{Z}^{N-N_{0}}(t)$
$\displaystyle=A_{1}\hat{Z}^{N-N_{0}}(t)+H^{N-N_{0}}U(t)$

for $t>0$. This allows us to complete the definition of the control strategy
by setting the auxiliary control input $U$ as:

(47) $U=\hat{Z}^{N_{0}}.$

Overall, the control strategy is composed of (20), (46), and (47).

###  3.4 Main stabilization result

We can now state the main result of this work.

###### Theorem 11

Assume that Assumption 1 holds. Let $\delta>0$ and $N_{0}\geq 1$ be such that
$\lambda_{n}>\delta$ for all $n\geq N_{0}+1$. Assume that the first $N_{0}$
eigenvalues of the operator $\mathcal{A}$ are simple except of the second and
the third one which are equal. With corresponding measurement (9), pick
$\xi_{1},\xi_{2}\in\mathcal{O}$ so that (44) holds true. Let
$L\in\mathbb{R}^{N_{0}}$ be such that $A_{0}-LC_{0}$ is Hurwitz with
eigenvalues that have a real part strictly less than $-\delta$. Then,
$\gamma_{1}<\gamma_{2}<\ldots<\gamma_{N_{0}}$ can be selected large enough
such that the matrix $-\sum_{k=1}^{N_{0}}\gamma_{k}B_{k}A+\Xi$ is Hurwitz with
eigenvalues that have a real part strictly less than $-\delta$. Furthermore,
for $N\geq N_{0}+1$ selected to be large enough, there exists a constant $C>0$
such that, for any initial condition $z_{o}\in H^{2}(\mathcal{O})$, the
trajectory of the closed-loop system composed of the plant (1), the internal
measurement (2) and the controller (20) with $U$ given by (47), satisfies

(48) $\|z(\cdot,t)\|_{H^{1}(\mathcal{O})}+\sum_{k=1}^{N}|\hat{z}_{k}(t)|\leq
Ce^{-\delta
t}\left(\|z(0,t)\|_{H^{1}(\mathcal{O})}+\sum_{k=1}^{N}|\hat{z}_{k}(0)|\right).$

###### Example 12

Consider the plant (1) with $d=2$, $\mathcal{O}=(0,\pi)\times(0,\pi)$,
$\Gamma_{1}=\left\\{x_{1}\in(0,\pi),\ x_{2}=0\right\\}$, and
$\mathcal{A}f=-\Delta f-3\nabla\cdot f-10f$. Hence, introducing
$\mu=e^{3x_{1}+3x_{2}}$, which is positive and bounded, $1\leq\mu(x)\leq
e^{6\pi},$ Assumption 1 is fulfilled because:

$\mu\mathcal{A}f=-\sum_{i=1}^{2}\partial_{i}(e^{3x_{1}+3x_{2}}\partial_{i}f)-10e^{3x_{1}+3x_{2}}f.$

The associated eigenfunctions are described by
$\varphi_{i,j}(x)=\frac{2}{\pi}e^{\frac{-3x_{1}-3x_{2}}{2}}\sin(ix_{1})\sin(jx_{2})$
with the corresponding eigenvalues $\lambda_{i,j}=\frac{4i^{2}+4j^{2}-31}{4}$.
Then, it is seen that the bi-orthogonal system is given by
$\psi_{i,j}=\frac{2}{\pi}e^{\frac{3x_{1}+3x_{2}}{2}}\sin(ix_{1})\sin(jx_{2})$,
$i,j\in\mathbb{N}\setminus\left\\{0\right\\}$.

In this setting, the open-loop system is unstable with a total of three modes
that are not exponentially stable: $-\lambda_{1,1}=23/4$ and
$-\lambda_{1,2}=-\lambda_{2,1}=11/4$ with the corresponding three
eigenfunctions

$\left\\{\frac{2}{\pi}e^{\frac{-3x_{1}-3x_{2}}{2}}\sin x_{1}\sin x_{2},\
\frac{2}{\pi}e^{\frac{-3x_{1}-3x_{2}}{2}}\sin 2x_{1}\sin x_{2},\
\frac{2}{\pi}e^{\frac{-3x_{1}-3x_{2}}{2}}\sin x_{1}\sin 2x_{2}\right\\}.$

Hence, fixing $N_{0}=3$, the choice for the spectral multiplicity, of the
first three eigenvalues, is verified. Moreover, (44) holds true as soon as
$\xi_{i}=(\xi_{i1},\ \xi_{i2})\in(0,\pi)^{2},\ i=1,2,$ such that
$\cos\xi_{11}\cos\xi_{22}-\cos\xi_{12}\cos\xi_{21}\neq 0$. This allows the
application of Theorem 1.

Proof. We first need to rewrite the dynamics of the closed-loop system formed
by (20), (46), and (47) in a suitable format for the upcoming stability
analysis. To do so, we first define the errors of observation as

$E^{N_{0}}:=Z^{N_{0}}-\hat{Z}^{N_{0}},\quad\tilde{E}^{N-N_{0}}:=\Lambda^{N-N_{0}}(Z^{N-N_{0}}-\hat{Z}^{N-N_{0}})$

where $\Lambda^{N-N_{0}}=\text{diag}(\lambda_{Q_{0}+1},\ldots,\lambda_{N})$
while

$\zeta:=\sum_{n\geq N+1}\begin{pmatrix}\varphi_{n}(\xi_{1})\\\
\varphi_{n}(\xi_{2})\end{pmatrix}w_{n},\quad\tilde{C}_{1}:=C_{1}\left(\Lambda^{N-N_{0}}\right)^{-1}.$

Hence, we obtain from (20), (46), and (47) that

(49a) $\displaystyle\frac{d}{dt}\hat{Z}^{N_{0}}$
$\displaystyle=\left(-\sum_{k=1}^{N_{0}}\gamma_{k}B_{k}A+\Xi\right)\hat{Z}^{N_{0}}+LC_{0}E^{N_{0}}+L\tilde{C}_{1}\tilde{E}^{N-N_{0}}+L\zeta,$
(49b) $\displaystyle\frac{d}{dt}E^{N_{0}}$
$\displaystyle=(A_{0}-LC_{0})E^{N_{0}}-L\tilde{C}_{1}\tilde{E}^{N-N_{0}}-L\zeta,$
(49c) $\displaystyle\frac{d}{dt}\hat{Z}^{N-N_{0}}$
$\displaystyle=A_{1}\hat{Z}^{N-N_{0}}+H^{N-N_{0}}\hat{Z}^{N_{0}},$ (49d)
$\displaystyle\frac{d}{dt}\tilde{E}^{N-N_{0}}$
$\displaystyle=A_{1}\tilde{E}^{N-N_{0}}.$

Introducing the finite-dimensional state vector
$X=(\hat{Z}^{N_{0}},E^{N_{0}},\tilde{E}^{N-N_{0}})^{\top}$ along with the
matrices

(50)
$F:=\begin{pmatrix}-\sum_{k=1}^{N_{0}}\gamma_{k}B_{k}A+\Xi&LC_{0}&L\tilde{C}_{1}\\\
0&A_{0}-LC_{0}&-L\tilde{C}_{1}\\\
0&0&A_{1}\end{pmatrix},\quad\mathcal{L}:=\begin{pmatrix}L\\\ -L\\\
0\end{pmatrix},$

we infer that the closed-loop system dynamics is described by

(51a) $\displaystyle\frac{d}{dt}X$ $\displaystyle=FX+\mathcal{L}\zeta$ (51b)
$\displaystyle\frac{d}{dt}\hat{Z}^{N-N_{0}}$
$\displaystyle=A_{1}\hat{Z}^{N-N_{0}}+H^{N-N_{0}}\hat{Z}^{N_{0}}$ (51c)
$\displaystyle\frac{d}{dt}w_{n}$
$\displaystyle=-\lambda_{n}w_{n}+\sum_{k=1}^{N_{0}}\gamma_{k}\left<D_{\gamma_{k}}u_{k},\psi_{n}\right>-\sum_{k=1}^{N_{0}}\left<D_{\gamma_{k}}\frac{d}{dt}u_{k},\psi_{n}\right>,\quad
n\geq N+1$

Let $c>1$ be an arbitrarily given constant. Let us now show that we can fix
the real numbers $0<\gamma_{1}<\gamma_{2}<\ldots<\gamma_{N_{0}}<c\gamma_{1}$
large enough such that the matrix $-\sum_{k=1}^{N_{0}}\gamma_{k}B_{k}A+\Xi$ is
Hurwitz with eigenvalues that have a real part strictly less than $-\delta<0$.
To do so, let $\lambda\in\mathbb{C}$ and a non-zero vector
$Z\in\mathbb{C}^{N_{0}}$ be such that
$\left(-\sum_{k=1}^{N_{0}}\gamma_{k}B_{k}A+\Xi\right)Z=\lambda Z$. Recalling
that $A$ is defined by (18), $A$ is symmetric definite positive and
$\sum_{k=1}^{N_{0}}B_{k}A=I$. Hence, it follows that

$\displaystyle\lambda\|A^{\frac{1}{2}}Z\|^{2}_{N_{0}}$
$\displaystyle=\left<\lambda
Z,AZ\right>_{N_{0}}=-\sum_{k=1}^{N_{0}}\gamma_{k}\left<B_{k}AZ,AZ\right>_{N_{0}}+\left<\Xi
Z,AZ\right>_{N_{0}}$
$\displaystyle=-\gamma_{1}\left<Z,AZ\right>_{N_{0}}+\gamma_{1}\sum_{k=1}^{N_{0}}\left<B_{k}AZ,AZ\right>_{N_{0}}-\sum_{k=1}^{N_{0}}\gamma_{k}\left<B_{k}AZ,AZ\right>_{N_{0}}$
$\displaystyle\phantom{=}\;+\left<A^{\frac{1}{2}}\Xi
A^{-\frac{1}{2}}A^{\frac{1}{2}}Z,A^{\frac{1}{2}}Z\right>_{N_{0}}$
$\displaystyle=-\gamma_{1}\|A^{\frac{1}{2}}Z\|^{2}_{N_{0}}+\sum_{k=2}^{N_{0}}(\gamma_{1}-\gamma_{k})\left<B_{k}AZ,AZ\right>_{N_{0}}+\left<A^{\frac{1}{2}}\Xi
A^{-\frac{1}{2}}A^{\frac{1}{2}}Z,A^{\frac{1}{2}}Z\right>_{N_{0}}.$

We have for all $k\in\\{1,\ldots,N_{0}\\}$ that $\gamma_{1}\leq\gamma_{k}$ and
$B_{k}$ is positive semi-definite (in view of its definition (16)). Moreover,
by the definition of $A$, using the Landau notation, we have

$\|A^{\frac{1}{2}}\|=O(\gamma_{1}),\quad\|A^{-\frac{1}{2}}\|=O\left(\frac{1}{\gamma_{1}}\right)$

as $\gamma_{1}\rightarrow+\infty$. Hence $\|A^{\frac{1}{2}}\Xi
A^{-\frac{1}{2}}\|\leq C\eta,$ for some constant $C>0$ independent of
$0<\gamma_{1}<\gamma_{2}<\ldots<\gamma_{N_{0}}<c\gamma_{1}$. It then yields
from the above that

$\Re(\lambda)\|A^{\frac{1}{2}}Z\|^{2}_{N_{0}}\leq(-\gamma_{1}+C\eta)\|A^{\frac{1}{2}}Z\|_{N_{0}}^{2}.$

Since $\|A^{\frac{1}{2}}Z\|_{N_{0}}\neq 0$, we obtain for $\gamma_{1}$ large
enough that $\Re(\lambda)<-\delta$, which proves our claim. This, in
particular, implies that the matric $F$ defined by (50) is Hurwitz with
eigenvalues that have a real part strictly less than $-\delta$.

We now carry on a Lyapunov stability analysis. In view of (6), let us
introduce the Lyapunov function, for all $(X,w)\in\mathbb{R}^{N+N_{0}}\times
H^{1}(\mathcal{O})$

(52) $V(X,w)=X^{\top}PX+\sum_{n\geq N+1}(\lambda_{n}+\nu)w_{n}^{2}.$

The computation of the time derivative of $V$ along the system trajectories
(51) gives

$\displaystyle\dot{V}$ $\displaystyle=2X^{\top}P(FX+\mathcal{L}\zeta)$
$\displaystyle\phantom{=}\,+2\sum_{n\geq
N+1}(\lambda_{n}+\nu)\left\\{-\lambda_{n}w_{n}+\sum_{k=1}^{N_{0}}\gamma_{k}\left<D_{\gamma_{k}}u_{k},\psi_{n}\right>-\sum_{k=1}^{N_{0}}\left<D_{\gamma_{k}}\frac{d}{dt}u_{k},\psi_{n}\right>\right\\}w_{n}$
$\displaystyle=\tilde{X}^{\top}\begin{pmatrix}F^{\top}P+PF&P\mathcal{L}\\\
\mathcal{L}^{\top}P&0\end{pmatrix}\tilde{X}-2\sum_{n\geq
N+1}\lambda_{n}(\lambda_{n}+\nu)w_{n}^{2}$
$\displaystyle\phantom{=}\;+2\sum_{n\geq
N+1}(\lambda_{n}+\nu)\sum_{k=1}^{N_{0}}\gamma_{k}\left<D_{\gamma_{k}}u_{k},\psi_{n}\right>w_{n}-2\sum_{n\geq
N+1}(\lambda_{n}+\nu)\sum_{k=1}^{N_{0}}\left<D_{\gamma_{k}}\frac{d}{dt}u_{k},\psi_{n}\right>w_{n}$

where $\tilde{X}:=\mathrm{col}(X,\zeta)$. Let us estimate the before last
term. To do so, let us note that

$\displaystyle\left<D_{\gamma_{k}}u_{k},\psi_{n}\right>$
$\displaystyle\overset{\eqref{ie10}}{=}-\left<D_{\gamma_{k}}\left<\Lambda_{\gamma_{k}}AU,L\right>_{N_{0}},\psi_{n}\right>$
$\displaystyle=-\sum_{l=1}^{N_{0}}\tilde{\gamma}_{k,l}A_{L_{l}}U\left<D_{\gamma_{k}}L_{l},\psi_{n}\right>$

where $\tilde{\gamma}_{k,l}$ in the $l$th term on the diagonal of the matrix
$\Lambda_{\gamma_{k}}$ defined by (12), $A_{L_{l}}$ is the $l$th line of the
matrix $A$, and $L_{l}(x)$ is the $l$th component of $L(x)$ defined by (17).
Therefore, using Young’s inequality, we deduce for any $\epsilon>0$ that

$\displaystyle 2\sum_{n\geq
N+1}(\lambda_{n}+\nu)\sum_{k=1}^{N_{0}}\gamma_{k}\left<D_{\gamma_{k}}u_{k},\psi_{n}\right>w_{n}$
$\displaystyle=-2\sum_{n\geq
N+1}(\lambda_{n}+\nu)\sum_{k,l=1}^{N_{0}}\gamma_{k}\tilde{\gamma}_{k,l}A_{L_{l}}U\left<D_{\gamma_{k}}L_{l},\psi_{n}\right>w_{n}$
$\displaystyle\leq 2\sum_{k,l=1}^{N_{0}}\sum_{n\geq
N+1}|\gamma_{k}||\tilde{\gamma}_{k,l}|\|A_{L_{l}}\||\left<D_{\gamma_{k}}L_{l},\psi_{n}\right>|\|U\|\times(\lambda_{n}+\nu)|w_{n}|$
$\displaystyle\leq\epsilon\sum_{k,l=1}^{N_{0}}\gamma_{k}^{2}\tilde{\gamma}_{k,l}^{2}\|A_{L_{l}}\|^{2}\left\\{\sum_{n\geq
N+1}\left<D_{\gamma_{k}}L_{l},\psi_{n}\right>^{2}\right\\}\|U\|^{2}+\frac{1}{\epsilon}\sum_{k,l=1}^{N_{0}}\sum_{n\geq
N+1}(\lambda_{n}+\nu)^{2}w_{n}^{2}$
$\displaystyle\overset{\eqref{eq_riez_ineq}}{\leq}\epsilon\underbrace{c_{1}\sum_{k,l=1}^{N_{0}}\gamma_{k}^{2}\tilde{\gamma}_{k,l}^{2}\|A_{L_{l}}\|^{2}\|\mathcal{R}_{N}D_{\gamma_{k}}L_{l}\|^{2}}_{:=S_{1,N}}\|U\|^{2}+\frac{N_{0}^{2}}{\epsilon}\sum_{n\geq
N+1}(\lambda_{n}+\nu)^{2}w_{n}^{2}$

where $\mathcal{R}_{N}f:=\sum_{n\geq N+1}\left<f,\psi_{n}\right>\varphi_{n}$
and $U=\hat{Z}^{N_{0}}=E_{1}X$ with $E_{1}=\begin{bmatrix}I&0&0\end{bmatrix}$.
Similarly, we have

$\displaystyle 2\sum_{n\geq
N+1}(\lambda_{n}+\nu)\sum_{k=1}^{N_{0}}\left<D_{\gamma_{k}}\frac{d}{dt}u_{k},\psi_{n}\right>w_{n}$
$\displaystyle\overset{\eqref{eq_riez_ineq}}{\leq}\epsilon\underbrace{c_{1}\sum_{k,l=1}^{N_{0}}\tilde{\gamma}_{k,l}^{2}\|A_{L_{l}}\|^{2}\|\mathcal{R}_{N}D_{\gamma_{k}}L_{l}\|^{2}}_{:=S_{2,N}}\left\|\frac{d}{dt}U\right\|^{2}+\frac{N_{0}^{2}}{\epsilon}\sum_{n\geq
N+1}(\lambda_{n}+\nu)^{2}w_{n}^{2}$

where $\frac{d}{dt}U=\frac{d}{dt}\hat{Z}^{N_{0}}=E_{2}\tilde{X}$ with
$E_{2}=\begin{bmatrix}-\sum_{k=1}^{N_{0}}\gamma_{k}B_{k}A+\Xi&LC_{0}&L\tilde{C}_{1}&L\end{bmatrix}$.
Finally, we have by Cauchy-Schwarz inequality that

$\displaystyle\|\zeta\|^{2}$ $\displaystyle=\left(\sum_{n\geq
N+1}\varphi_{n}(\xi_{1})w_{n}\right)^{2}+\left(\sum_{n\geq
N+1}\varphi_{n}(\xi_{2})w_{n}\right)^{2}$
$\displaystyle\leq\underbrace{\sum_{n\geq
N+1}\frac{\varphi_{n}(\xi_{1})^{2}+\varphi_{n}(\xi_{2})^{2}}{(\lambda_{n}+\nu)^{2}}}_{:=S_{\varphi,N}}\times\sum_{n\geq
N+1}(\lambda_{n}+\nu)^{2}w_{n}^{2}.$

Gathering all the the above estimates, we deduce that

(53) $\dot{V}+2\delta V\leq\tilde{X}^{\top}\Theta_{1}\tilde{X}+\sum_{n\geq
N+1}(\lambda_{n}+\nu)\Psi_{n}w_{n}^{2}$

for $\delta>0$ fixed such that $F+\delta I$ is Hurwitz and where

$\displaystyle\Theta_{1}$ $\displaystyle=\begin{pmatrix}F^{\top}P+PF+2\delta
P+\epsilon S_{1,N}E_{1}^{\top}E_{1}&P\mathcal{L}\\\ \mathcal{L}^{\top}P&-\eta
I\end{pmatrix}+\epsilon S_{2,N}E_{2}^{\top}E_{2},$ $\displaystyle\Psi_{n}$
$\displaystyle=\left[-2\left(1-\frac{N_{0}^{2}}{\epsilon}\right)+\eta
S_{\varphi,N}\right]\lambda_{n}+\left[\frac{2N_{0}^{2}}{\epsilon}+\eta
S_{\varphi,N}\right]\nu+2\delta,\quad n\geq N+1$

for an arbitrary $\eta>0$.

Assume for the moment that $\Theta_{1}\preceq 0$ and $\Psi_{n}\leq 0$ for all
$n\geq N+1$. Then, in view of (53), we get that $\dot{V}+2\delta V\leq 0$.
Combining this estimate with the definition (52) of the Lyapunov function $V$,
the direct integration of the dynamics $\hat{Z}^{N-N_{0}}$ from (51), the use
of the estimates (6), and invoking the change of variable formula (22), we
directly infer the existence of a constant $C>0$, independent of the initial
condition, such that the claimed stability estimate (48) holds.

To conclude the proof, it thus remains to show that $N$ can be selected so
that $\Theta_{1}\preceq 0$ and $\Psi_{n}\leq 0$ for all $n\geq N+1$. To do so,
let us set $\epsilon=2N_{0}^{2}$ and $\eta=1/\sqrt{S_{\varphi,N}}$ if
$S_{\varphi,N}\neq 0$, $\eta=N$ otherwise. This implies that
$\eta\rightarrow+\infty$ while $\eta S_{\varphi,N}\rightarrow 0$ as
$N\rightarrow+\infty$. Hence, for $N$ large enough we have $\eta
S_{\varphi,N}\leq 1/2$ and $\lambda_{n}\geq\lambda_{N+1}>0$ for all $n\geq
N+1$, which implies that

$\displaystyle\Psi_{n}$
$\displaystyle\leq\Theta_{2}:=-\frac{1}{2}\lambda_{N+1}+\frac{3}{2}\nu+2\delta,\quad
n\geq N+1$

with $\Theta_{2}\rightarrow-\infty$ as $N\rightarrow+\infty$. Now, since $F$
defined by (50) is Hurwitz, we define $P\succ 0$ as the unique solution to the
Lyapunov equation $F^{\top}P+PF+2\delta P=-I$. Owing to Lemma 3, it can be
seen that $\|\tilde{C}_{1}\|=O(1)$, hence $\|L\tilde{C}_{1}\|=O(1)$, as
$N\rightarrow+\infty$. Therefore, a result similar to [19, Lemma in Appendix]
shows that $\|P\|=O(1)$ as $N\rightarrow+\infty$. Therefore, we have

(54) $\Theta_{1}=\underbrace{\begin{pmatrix}-I+\epsilon
S_{1,N}E_{1}^{\top}E_{1}&P\mathcal{L}\\\ \mathcal{L}^{\top}P&-\eta
I\end{pmatrix}}_{:=\Theta_{1,p}}+\epsilon S_{2,N}E_{2}^{\top}E_{2}.$

Using the Schur complement for $N$ sufficiently large so that $\eta>1/2$, we
see that $\Theta_{1,p}\preceq-\frac{1}{2}I$ if and only if
$-\frac{1}{2}I+\epsilon
S_{1,N}E_{1}^{\top}E_{1}+\frac{1}{\eta-\frac{1}{2}}P\mathcal{L}\mathcal{L}^{\top}P\preceq
0$. Noting that $\|P\|=O(1)$ and $S_{2,N}\rightarrow 0$ as
$N\rightarrow+\infty$ while $\|E_{1}\|$ and $\|\mathcal{L}\|$ are constants
independent of $N$, we deduce that $\Theta_{1,p}\preceq-\frac{1}{2}I$ for all
$N$ selected to be large enough. In that case,
$\Theta_{1}\preceq-\frac{1}{2}I+\epsilon S_{2,N}E_{2}^{\top}E_{2}$ for all $N$
selected to be large enough. Since $\|E_{2}\|=O(1)$ and $S_{2,N}\rightarrow 0$
as $N\rightarrow+\infty$, we deduce that $\Theta_{1}\preceq 0$ for $N$ large
enough.

##  4 Conclusions

This paper discussed the design of an observer-based feedback stabilizing
controller for multi-dimensional parabolic type equations governed by
diagonalizable second order differential operators. To fix the ideas and to
ease the presentation, we focused the developments on the case of three
unstable eigenvalues: one of multiplicity one and one of multiplicity two.
However, the approach reported in this paper easily extends to any other case
with a finite number of unstable modes with arbitrary finite multiplicity.

To conclude, it is worth to mention that, based on the technique presented in
this work, a natural perspective is the study of non-linear multi-dimensional
parabolic equations combining the present design method with [20], and the
study of multi-dimensional parabolic equations with delays as done in e.g.,
[21, 18].

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