\section{Introduction}

We study output trajectory tracking for a class of nonlinear reaction diffusion equations such that a prescribed performance of the tracking error is achieved. To this end, we utilize the method of funnel control which was developed in~\cite{IlchRyan02b}, see also the survey~\cite{IlchRyan08}. The funnel controller is a model-free output-error feedback of high-gain type. Therefore, it is inherently robust and of striking simplicity. The funnel controller has been successfully applied e.g.\ in temperature control of chemical reactor models~\cite{IlchTren04}, control of industrial servo-systems~\cite{Hack17} and underactuated multibody systems~\cite{BergOtto19}, speed control of wind turbine systems~\cite{Hack14,Hack15b,Hack17}, current control for synchronous machines~\cite{Hack15a,Hack17}, DC-link power flow control~\cite{SenfPaug14}, voltage and current control of electrical circuits~\cite{BergReis14a}, oxygenation control during artificial ventilation therapy~\cite{PompAlfo14}, control of peak inspiratory pressure~\cite{PompWeye15} and adaptive cruise control~\cite{BergRaue18}.

A funnel controller for a large class of systems described by functional differential equations with arbitrary (well-defined) relative degree has been developed in~\cite{BergHoan18}. It is shown in~\cite{BergPuch19b} that this abstract class indeed allows for fairly general infinite-dimensional systems, where the internal dynamics are modeled by a (PDE). In particular, it was shown in~\cite{BergPuch19} that the linearized model of a moving water tank, where sloshing effects appear, belongs to the aforementioned system class. On the other hand, not even every linear, infinite-dimensional system has a well-defined relative degree, in which case the results as in~\cite{BergHoan18,IlchRyan02b} cannot be applied. Instead,  the feasibility of funnel control has to be investigated directly for the (nonlinear) closed-loop system, see~\cite{ReisSeli15b} for a boundary controlled heat equation and~\cite{PuchReis19} for a general class of boundary control systems.

The nonlinear reaction diffusion system that we consider in the present paper is known as the monodomain model and represents defibrillation processes of the human heart~\cite{Tung78}. The monodomain equations are a reasonable simplification of the well accepted bidomain equations, which arise in cardiac electrophysiology~\cite{SundLine07}. In the monodomain model the dynamics are governed by a parabolic reaction diffusion equation which is coupled with a linear ordinary differential equation that models the ionic current.

It is discussed in~\cite{KuniNagaWagn11} that, under certain initial conditions, reentry phenomena and spiral waves may occur. From a medical point of view, these situations can be interpreted as fibrillation processes of the heart that should be terminated by an external control, for instance by applying an external stimulus to the heart tissue, see~\cite{NagaKuniPlan13}.

The present paper is organized as follows: In Section~\ref{sec:mono_main} we introduce the mathematical framework, which strongly relies on preliminaries on Neumann elliptic operators. The control objective is presented in Section~\ref{sec:mono_controbj}, where we also state the main result on the feasibility of the proposed controller design in Theorem~\ref{thm:mono_funnel}. The proof of this result is given in Section~\ref{sec:mono_proof_mt} and it uses several auxiliary results derived in Appendices~\ref{sec:mono_prep_proof} and~\ref{sec:mono_prep_proof2}. We illustrate our result by a simulation in Section~\ref{sec:numerics}.

\textbf{Nomenclature}.
The set of bounded operators from $X$ to $Y$ is denoted by $\mathcal{L}(X,Y)$, $X'$ stands for the dual of a~Banach space $X$, and $B'$ is the dual of an operator $B$.\\
For a bounded and measurable set $\Omega\subset{\mathbb{R}}^d$, $p\in[1,\infty]$ and $k\in{\mathbb{N}}_0$, $W^{k,p}(\Omega;{\mathbb{R}}^n)$ denotes the Sobolev space of equivalence classes of $p$-integrable and $k$-times weakly differentiable functions $f:\Omega\to{\mathbb{R}}^n$, $W^{k,p}(\Omega;{\mathbb{R}}^n)\cong (W^{k,p}(\Omega))^n$, and the Lebesgue space of equivalence classes of $p$-integrable functions is $L^p(\Omega)=W^{0,p}(\Omega)$. For $r\in(0,1)$ we further set
\[
    W^{r,p}(\Omega) := \setdef{f\in L^p(\Omega)}{ \left( (x,y)\mapsto \frac{|f(x)-f(y)|}{|x-y|^{d/p+r}}\right) \in L^p(\Omega\times\Omega)}.
\]
For a domain $\Omega$ with smooth boundary, $W^{k,p}(\partial\Omega)$ denotes the Sobolev space at the boundary.\\
We identify functions with their restrictions, that is, for instance, if $f\in L^p(\Omega)$  $\Omega_0\subset \Omega$, then the restriction $f|_{\Omega_0}\in L^p(\Omega_0)$ is again dentoted by~$f$. For an interval $J\subset{\mathbb{R}}$, a Banach space $X$ and $p\in[1,\infty]$, we denote by $L^p(J;X)$ the vector space of equivalence classes of strongly measurable functions $f:J\to X$ such that $\|f(\cdot)\|_X\in L^p(J)$.
Note that if $J=(a,b)$ for $a,b\in{\mathbb{R}}$, the spaces $L^p((a,b);X)$, $L^p([a,b];X)$, $L^p([a,b);X)$ and $L^p((a,b];X)$ coincide, since the points at the boundary have measure zero. We will simply write $L^p(a,b;X)$, also for the case $a=-\infty$ or $b=\infty$. We refer to \cite{Adam75} for further details on Sobolev and Lebesgue spaces.\\
In the following, let $J\subset{\mathbb{R}}$ be an interval, $X$ be a Banach space and $k\in{\mathbb{N}}_0$. Then $C^k(J;X)$ is defined as the space of $k$-times continuously differentiable functions $f:J\to X$. The space of bounded $k$-times continuously differentiable functions with bounded first $k$ derivatives is denoted by $BC^k(J;X)$, and it is a Banach space endowed with the usual supremum norm. The space of bounded and uniformly continuous functions will be denoted by $BUC(J;X)$. The Banach space of H\"older continuous functions $C^{0,r}(J;X)$ with $r\in(0,1)$ is given by
\begin{align*}
C^{0,r}(J;X)&:=\setdef{f\in BC(J;X)}{[f]_{r}:=\sup_{t,s\in J,s<t}\frac{\|f(t)-f(s)\|}{(t-s)^r}<\infty},\\
\|f\|_r&:=\|f\|_\infty+[f]_{r},
\end{align*}
see \cite[Chap.~0]{Luna95}. We like to note that for all $0<r<q<1$ we have that
\[
    C^{0,q}(J;X) \subseteq C^{0,r}(J;X) \subseteq BUC(J;X).
\]
For $p\in[1,\infty]$, the symbol $W^{1,p}(J;X)$ stands for the Sobolev space of $X$-valued equivalance classes of weakly differentiable and $p$-integrable functions $f:J\to X$ with $p$-integrable weak derivative, i.e., $f,\dot{f}\in L^p(J;X)$. Thereby, integration (and thus weak differentiation) has to be understood in the Bochner sense, see~\cite[Sec.~5.9.2]{Evan10}.
The spaces $L^p_{\rm loc}(J;X)$ and $W^{1,p}_{\rm loc}(J;X)$ consist of all $f$ whose restriction to any compact interval $K\subset J$ are in $L^p(K;X)$ or $W^{1,p}(K;X)$, respectively.

\section{The FitzHugh-Nagumo model}
\label{sec:mono_main}

Throughout this paper we will frequently use the following assumption. For $d\in{\mathbb{N}}$ we denote the scalar product in $L^2(\Omega;{\mathbb{R}}^d)$ by $\scpr{\cdot}{\cdot}$ and the norm in $L^2(\Omega)$ by $\|\cdot\|$.

\begin{Ass}\label{Ass1}
Let $d\le 3$ and $\Omega\subset {\mathbb{R}}^d$ be a bounded domain with Lipschitz boundary $\partial\Omega$.
Further, let $D\in L^\infty(\Omega;{\mathbb{R}}^{d\times d})$ be symmetric-valued and satisfy the \emph{ellipticity condition}
\begin{equation}
\exists\,\delta>0:\ \ \text{for a.e.}\,\zeta\in\Omega\ \forall\, \xi\in{\mathbb{R}}^d:\ \xi^\top D(\zeta) \xi = \sum_{i,j=1}^d D_{ij}(\zeta)\xi_i\xi_j\geq\delta \|\xi\|_{{\mathbb{R}}^d}^2.\label{eq:ellcond}
\end{equation}
\end{Ass}

To formulate the model of interest, we consider the sesquilinear form
\begin{equation}
\mathfrak{a}:W^{1,2}(\Omega)\times W^{1,2}(\Omega)\to{\mathbb{R}},\ (z_1,z_2)\mapsto\scpr{\nabla z_1}{D\nabla z_2}.\label{eq:sesq}
\end{equation}
We can associate a linear operator to $\mathfrak{a}$.
\begin{Prop}\label{prop:Aop}
Let Assumption~\ref{Ass1} hold. Then there exists exactly one operator $\mathcal{A}:\mathcal{D}(\mathcal{A})\subset L^2(\Omega)\to L^2(\Omega)$ with
\[
    \mathcal{D}(\mathcal{A})=\setdef{\!\!z_2\in W^{1,2}(\Omega)\!\!}{\exists\, y_2\in L^2(\Omega)\ \forall\, z_1\in W^{1,2}(\Omega):\,\mathfrak{a}(z_1,z_2)=-\scpr{z_1}{y_2}\!\!},
\]
and
\[
    \forall\, z_1\in W^{1,2}(\Omega)\ \forall\, z_2\in \mathcal{D}(\mathcal{A}):\ \mathfrak{a}(z_1,z_2)=-\scpr{z_1}{\mathcal{A} z_2}.
\]
We call $\mathcal{A}$ the {\em Neumann elliptic operator on $\Omega$ associated to $D$}.  The operator $\mathcal{A}$ is closed, self-adjoint, and $\mathcal{D}(\mathcal{A})$ is dense in $W^{1,2}(\Omega)$.
\end{Prop}
\begin{proof}
Existence, uniqueness and closedness of $A$ as well as the density of $\mathcal{D}(\mathcal{A})$ in $W^{1,2}(\Omega)$
follow from Kato's First Representation Theorem~\cite[Sec.~VI.2, Thm~2.1]{Kato80}, whereas
self-adjointness is an immediate consequence of the property $\mathfrak{a}(z_1,z_2)={\mathfrak{a}(z_2,z_1)}$ for all $z_1,z_2\in W^{1,2}(\Omega)$.
\end{proof}

Note that the operator $\mathcal{A}$ in Proposition~\ref{prop:Aop} is well-defined, independent of any further smoothness requirements on $\partial\Omega$. In particular, the classical Neumann boundary trace, i.e., the derivative of a function in the direction of the outward normal unit vector $\nu:\partial\Omega\to{\mathbb{R}}^d$ does not need to exist.
However, if $\partial\Omega$ and the coefficient matrix $D$ are sufficiently smooth, then
\[
\mathcal{A} z={\rm div} D\nabla z,\quad z\in\mathcal{D}(\mathcal{A})=\setdef{z\in W^{2,2}(\Omega) }{ (\nu^\top\cdot D\nabla z)|_{\partial\Omega}=0},
\]
see \cite[Thm.~2.2.2.5]{Gris85}. This justifies to call $\mathcal{A}$ a Neumann elliptic operator. We collect some further properties of such operators in Appendix~\ref{sec:neum_lapl}.

Now we are in the position to introduce the model for the interaction of the electric current in a cell, namely
\begin{equation}\label{eq:FHN_model}
\begin{aligned}
\dot v(t)&=\mathcal{A} v(t)+p_3(v(t))-u(t)+I_{s,i}(t)+\mathcal{B} I_{s,e}(t),\quad&v(0)&=v_0,\\
\dot u(t)&=c_5v(t)-c_4u(t),&u(0)&=u_0,\\
y(t) &= \mathcal{B}'v(t),
\end{aligned}
\end{equation}
where
\[p_3(v):=-c_1v+c_2v^2-c_3v^3,\]
with constants $c_i>0$ for $i=1,\dots,5$, initial values $v_0,u_0\in L^2(\Omega)$, the  Neumann elliptic operator $\mathcal{A}:\mathcal{D}(\mathcal{A})\subseteq L^2(\Omega)\to L^2(\Omega)$ on $\Omega$ associated to $D\in L^\infty(\Omega;\mathbb R^{d\times d})$ and control operator $\mathcal{B}\in\mathcal{L}({\mathbb{R}}^m,W^{1,2}(\Omega)')$, where $W^{1,2}(\Omega)'$ is the dual of $W^{1,2}(\Omega)$ with respect to the pivot space $L^{2}(\Omega)$; consequently, $\mathcal{B}'\in\mathcal{L}(W^{1,2}(\Omega),{\mathbb{R}}^m)$.

System~\eqref{eq:FHN_model} is known as the FitzHugh-Nagumo model for the ionic current~\cite{Fitz61}, where
\[I_{ion}(u,v)=p_3(v)-u.\]
The functions  $I_{s,i}\in L^2_{{\rm loc}}(0,T;L^2(\Omega))$, $I_{s,e}\in L^2_{{\rm loc}}(0,T;{\mathbb{R}}^m)$
are the intracellular and extracellular stimulation currents, respectively. In particular, $I_{s,e}$ is the control input of the system, whereas $y$ is the output.\\
Next we introduce the solution concept.

\begin{Def}\label{def:solution}
Let Assumption~\ref{Ass1} hold and $\mathcal{A}$ be a Neumann elliptic operator on $\Omega$ associated to $D$ (see Proposition~\ref{prop:Aop}), let $\mathcal{B}\in \mathcal{L}({\mathbb{R}}^m,W^{1,2}(\Omega)')$, and
$u_0,v_0\in L^2(\Omega)$ be given. Further, let $T\in(0,\infty]$ and $I_{s,i}\in L^2_{{\rm loc}}(0,T;L^2(\Omega))$, $I_{s,e}\in L^2_{{\rm loc}}(0,T;{\mathbb{R}}^m)$. A triple of functions $(u,v,y)$ is called \emph{solution} of~\eqref{eq:FHN_model} on $[0,T)$, if
	\begin{enumerate}[(i)]
		\item $v\in L^2(0,T;W^{1,2}(\Omega))\cap C([0,T);L^{2}(\Omega))$ with $v(0)=v_0$;
        \item $u\in C([0,T);L^{2}(\Omega))$ with $u(0)=u_0$;
		\item for all $\chi\in L^2(\Omega)$, $\theta\in W^{1,2}(\Omega)$, the scalar functions $t\mapsto\scpr{u(t)}{\chi}$, $t\mapsto\scpr{v(t)}{\theta}$ are weakly differentiable on $[0,T)$, and for almost all $t\in (0,T)$ we have
		\begin{equation}\label{eq:solution}
		\begin{aligned}
		{\textstyle\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}}\scpr{v(t)}{\theta}&=-\mathfrak{a}(v(t),\theta)+\scpr{p_3(v(t))-u(t)+I_{s,i}(t)}{\theta}+\scpr{I_{s,e}(t)}{\mathcal{B}'\theta}_{{\mathbb{R}}^m},\\
		{\textstyle\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}} \scpr{u(t)}{\chi}&=\scpr{c_5v(t)-c_4u(t)}{\chi},\\
        y(t)&=\mathcal{B}'v(t),
		\end{aligned}
		\end{equation}
		where $\mathfrak{a}:W^{1,2}(\Omega)\times W^{1,2}(\Omega)\to{\mathbb{R}}$ is the sesquilinear defined as in \eqref{eq:sesq}.
	\end{enumerate}
\end{Def}
\begin{Rem}\label{rem:openloop}
	\hspace{1em}
\begin{enumerate}[a)]
\item\label{rem:openloop1} Weak differentiability of $t\mapsto\scpr{u(t)}{\chi}$, $t\mapsto\scpr{v(t)}{\theta}$ for all $\chi\in L^2(\Omega)$, $\theta\in W^{1,2}(\Omega)$ on $(0,T)$
further leads to $v\in W^{1,2}(0,T;W^{1,2}(\Omega)')$ and $u\in W^{1,2}(0,T;L^{2}(\Omega))$.
\item\label{rem:openloop2} The Sobolev Embedding Theorem \cite[Thm.~5.4]{Adam75} implies that the inclusion map $W^{1,2}(\Omega)\hookrightarrow L^6(\Omega)$ is bounded. This guarantees that $p_3(v)\in L^2(0,T;L^2(\Omega))$, whence the first equation in \eqref{eq:solution} is well-defined.
\item\label{rem:openloop3} Let $w\in L^2(\Omega)$. An input operator of the form $\mathcal{B} u=u\cdot w$ corresponds to distributed input, and we have $\mathcal{B}\in\mathcal{L}({\mathbb{R}},L^{2}(\Omega))$.
In this case, the output is given by
\[y(t)=\int_\Omega w(\xi)\cdot(v(t))(\xi)d\xi.\]
A typical situation is that $w$ is an indicator function on a subset of $\Omega$; such choices have been considered in~\cite{KuniSouz18} for instance.
\item\label{rem:openloop4} Let $w\in L^2(\partial\Omega)$.  An input operator with
\begin{equation}\mathcal{B}' z=\int_{\partial\Omega} w(\xi)\cdot z(\xi)d\sigma\label{eq:neumanncontr}\end{equation}
corresponds to a~Neumann boundary control
\[\nu(\xi)^\top\cdot (\nabla v(t))(\xi) =w(\xi)\cdot I_{s,e}(t),\quad \xi\in\partial\Omega.\]
In this case, the output is given by a~weighted integral of the Dirichlet boundary values. More precisely
\[y(t)=\int_{\partial\Omega} w(\xi)\cdot(v(t))(\xi)d\sigma.\]
Note that $\mathcal{B}'$ is the composition of the trace operator
\[\begin{aligned}
{\rm tr}:&& z&\,\mapsto \left. z\right|_{\partial\Omega}
\end{aligned}\]
and the inner product in $L^2(\partial\Omega)$ with respect to $w$. The trace operator satisfies ${\rm tr}\in \mathcal{L}(W^{1/2+\varepsilon,2}(\Omega),L^2(\partial\Omega))$ for all $\varepsilon>0$ by the Trace Theorem \cite[Thm.~1.39]{Yagi10}. In particular, ${\rm tr}\in \mathcal{L}(W^{1,2}(\Omega),L^2(\partial\Omega))$, which implies that
$\mathcal{B}'\in \mathcal{L}(W^{1,2}(\Omega),{\mathbb{R}})$ and $\mathcal{B}\in \mathcal{L}({\mathbb{R}},W^{1,2}(\Omega)')$.
\end{enumerate}
\end{Rem}

\section{Control objective}\label{sec:mono_controbj}

The objective is that the output~$y$ of the system~\eqref{eq:FHN_model} tracks a given reference signal which is $y_{\rm ref}\in W^{1,\infty}(0,\infty;{\mathbb{R}}^m)$ with a prescribed performance of the tracking error $e:= y- y_{\rm ref}$, that is~$e$ evolves within the performance funnel
\[
\mathcal{F}_\varphi := \setdef{ (t,e)\in[0,\infty)\times{\mathbb{R}}^m}{ \varphi(t) \|e\|_{{\mathbb{R}}^m} < 1}
\]
defined by a function~$\varphi$ belonging to
\[
\Phi_\gamma:=\setdef{\varphi\in W^{1,\infty}(0,\infty;{\mathbb{R}}) }{\varphi|_{[0,\gamma]}\equiv0,\ \forall\delta>0, \inf_{t>\gamma+\delta}\varphi(t)>0
},
\]
for some $\gamma>0$.

\begin{figure}[h]
	\begin{minipage}{0.45\textwidth}
		\begin{tikzpicture}[domain=0.001:4,scale=1.2]

		\fill[color=blue!20,domain=0.47:4] (0,0)-- plot (\x,{min(2.2,1/\x+2*exp(-3))})--(4,0)-- (0,0);
		\fill[color=blue!20] (0,0) -- (0,2.2) -- (0.47,2.2) -- (0.47,0) -- (0,0);

		\draw[->] (-0.2,0) -- (4.3,0) node[right] {$t$};
		\draw[->] (0,-0.2) -- (0,2.5) node[above] {};

		\draw[color=blue,domain=0.47:4] plot (\x,{min(2.2,1/\x+2*exp(-3))}) node[above] {$1/\varphi(t)$};

		\draw[smooth,color=red,style=thick] (1,0) node[below] {$\|e(t)\|_{{\mathbb{R}}^m}$}
		plot coordinates{(0,0.8)(0.5,1.2)(1,0)}--
		plot coordinates{(1,0)(1.25,0.6)(2,0.2)(3,0.3)(4,0.05)} ;
		\end{tikzpicture}
		\caption{Error evolution in a funnel $\mathcal F_{\varphi}$ with boundary~$1/\varphi(t)$.}
		\label{Fig:monodomain_funnel}
	\end{minipage}
	\quad
	\begin{minipage}{0.5\textwidth}
		The situation is illustrated in Fig.~\ref{Fig:monodomain_funnel}. The funnel boundary given by~$1/\varphi$ is unbounded in a small interval $[0,\gamma]$ to allow for an arbitrary initial tracking error. Since $\varphi$ is bounded there exists $\lambda>0$ such that $1/\varphi(t) \ge \lambda$ for all $t>0$. Thus, we seek practical tracking with arbitrary small accuracy $\lambda>0$, but asymptotic tracking is not required in general.
	\end{minipage}
\end{figure}
The funnel boundary is not necessarily monotonically decreasing, while in most situations
it is convenient to choose a monotone funnel. Sometimes, widening the funnel over some later time interval might be beneficial, for instance in the presence of periodic disturbances or strongly varying reference signals. For typical choices of funnel boundaries see e.g.~\cite[Sec.~3.2]{Ilch13}.\\
A controller which achieves the above described control objective is the funnel controller. In the present paper, it suffices to restrict ourselves to the simple version developed in~\cite{IlchRyan02b}, which is the feedback law
\begin{equation}\label{eq:monodomain_funnel_controller}
I_{s,e}(t)=-\frac{k_0}{1-\varphi(t)^2\|\mathcal{B}' v(t)-y_{\rm ref}(t)\|^2_{{\mathbb{R}}^m}}(\mathcal{B}'v(t)-y_{\rm ref}(t)),
\end{equation}
where $k_0>0$ is some constant used for scaling and agreement of physical units. Note that, by $\varphi|_{[0,\gamma]}\equiv0$, the controller satisfies
\[\forall\,  t\in[0,\gamma]:\ I_{s,e}(t)=-k_0(\mathcal{B}'v(t)-y_{\rm ref}(t)).\]
We are interested in considering solutions of~(\ref{eq:FHN_feedback}), which leads to the following weak solution framework.

\begin{Def}\label{def:solution_feedback}
Use the assumptions from Definition~\ref{def:solution}. Furthermore, let $k_0>0$, $y_{\rm ref}\in W^{1,\infty}(0,\infty;{\mathbb{R}}^m)$, $\gamma>0$ and $\varphi\in\Phi_\gamma$. A triple of functions $(u,v,y)$ is called \emph{solution} of system~\eqref{eq:FHN_model} with feedback~\eqref{eq:monodomain_funnel_controller} on $[0,T)$, if $(u,v,y)$ satisfies the conditions~(i)--(iii) from Definition~\ref{def:solution} with $I_{s,e}$ as in~\eqref{eq:monodomain_funnel_controller}.
\end{Def}

\begin{Rem}\
	\begin{enumerate}[a)]
		\item Inserting the feedback law~\eqref{eq:monodomain_funnel_controller} into the system~\eqref{eq:FHN_model}, we obtain the closed-loop system
\begin{equation}\label{eq:FHN_feedback}
\begin{aligned}
\dot v(t)&=\mathcal{A} v(t)+p_3(v)(t)-u(t)+I_{s,i}(t) -\frac{k_0 \mathcal{B} (\mathcal{B}' v(t)-y_{\rm ref}(t))}{1-\varphi(t)^2\|\mathcal{B}'v(t)-y_{\rm ref}(t)\|^2_{{\mathbb{R}}^m}},\\
\dot u(t)&=c_5v(t)-c_4u(t).
\end{aligned}
\end{equation}
Consequently, $(u,v,y)$ is a {solution} of~\eqref{eq:FHN_model},~\eqref{eq:monodomain_funnel_controller} (resp.~\eqref{eq:FHN_feedback}) if, and only if,
	\begin{enumerate}[(i)]
		\item $v\in L^2(0,T;W^{1,2}(\Omega))\cap C([0,T);L^{2}(\Omega)))$ with $v(0)=v_0$;
        \item $u\in C([0,T);L^{2}(\Omega))$ with $u(0)=u_0$;
		\item for all $\chi\in L^2(\Omega)$, $\theta\in W^{1,2}(\Omega)$, the scalar functions $t\mapsto\scpr{u(t)}{\chi}$, $t\mapsto\scpr{v(t)}{\theta}$ are weakly differentiable on $[0,T)$, and it holds that, for almost all $t\in (0,T)$,
		\begin{equation}\label{eq:solution_cl}
		\begin{aligned}
		{\textstyle\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}}\scpr{v(t)}{\theta}&=-\mathfrak{a}(v(t),\theta)+\scpr{p_3(v(t))-u(t)+I_{s,i}(t)}{\theta}\\&\qquad-{\frac{k_0 \scpr{\mathcal{B}'v(t)-y_{\rm ref}(t)}{\mathcal{B}'\theta}_{{\mathbb{R}}^m}}{1-\varphi(t)^2\|\mathcal{B}' v(t)-y_{\rm ref}(t)\|^2_{{\mathbb{R}}^m}}},\\[2mm]
		{\textstyle\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}}\scpr{u(t)}{\chi}&=\scpr{c_5v(t)-c_4u(t)}{\chi},\\
        y(t)&=\mathcal{B}'v(t).
		\end{aligned}
		\end{equation}
	\end{enumerate}
The system~\eqref{eq:FHN_feedback} is a nonlinear and non-autonomous PDE and any solution needs to satisfy that the tracking error evolves in the prescribed performance funnel~$\mathcal{F}_\varphi$. Therefore, existence and uniqueness of solutions is a nontrivial problem and even if a solution exists on a finite time interval $[0,T)$, it is not clear that it can be extended to a global solution.
		\item For global solutions it is desirable that $I_{s,e}\in L^\infty(\delta,\infty;{\mathbb{R}}^m)$ for all $\delta>0$. Note that this is equivalent to
		\[\limsup_{t\to\infty}\varphi(t)^2\|\mathcal{B}'v(t)-y_{\rm ref}(t)\|^2_{{\mathbb{R}}^m}<1.\]
        It is as well desirable that $y$ and $I_{s,e}$ have a certain smoothness.
	\end{enumerate}
\end{Rem}

In the following we state the main result of the present paper. We will show that the closed-loop system~\eqref{eq:FHN_feedback}
has a unique global solution so that all signals remain bounded. Furthermore, the tracking error stays uniformly away from the funnel boundary.
We further show that we gain more regularity of the solution, if $\mathcal{B}\in\mathcal{L}({\mathbb{R}}^m,W^{r,2}(\Omega)')$ for some $r\in [0,1)$ or even
$\mathcal{B}\in\mathcal{L}({\mathbb{R}}^m,W^{1,2}(\Omega))$. Recall that $\mathcal{B}\in\mathcal{L}({\mathbb{R}}^m,W^{r,2}(\Omega)')$ if, and only if, $\mathcal{B}'\in\mathcal{L}(W^{r,2}(\Omega),{\mathbb{R}}^m)$. Furthermore, for any $r\in(0,1)$ we have the inclusions
\[
    \mathcal{L}({\mathbb{R}}^m,W^{1,2}(\Omega)) \subset \mathcal{L}({\mathbb{R}}^m,L^2(\Omega)) \subset \mathcal{L}({\mathbb{R}}^m,W^{r,2}(\Omega)') \subset \mathcal{L}({\mathbb{R}}^m,W^{1,2}(\Omega)').
\]

\begin{Thm}\label{thm:mono_funnel}
Use the assumptions from Definition~\ref{def:solution_feedback}. Furthermore, assume that $\ker\mathcal{B}=\{0\}$ and $I_{s,i}\in L^\infty(0,\infty;L^2(\Omega))$. Then there exists a unique solution of~\eqref{eq:FHN_feedback} on $[0,\infty)$ and we have
	\begin{enumerate}[(i)]
		\item $u,\dot{u},v\in BC([0,\infty);L^2(\Omega))$;
		\item for all $\delta>0$ we have
		\begin{align*}
			v&\in BUC([\delta,\infty);W^{1,2}(\Omega))\cap C^{0,1/2}([\delta,\infty);L^{2}(\Omega)),\\
			y,I_{s,e}&\in BUC([\delta,\infty);{\mathbb{R}}^m);
		\end{align*}
        \item $\exists\,\varepsilon_0>0\ \forall\,\delta>0\ \forall\, t\geq\delta:\ \varphi(t)^2\|\mathcal{B}'v(t)-y_{\rm ref}(t)\|^2_{{\mathbb{R}}^m}\leq1-\varepsilon_0.$
		\end{enumerate}
        Furthermore,
		\begin{enumerate}[a)]
			\item if additionally $\mathcal{B}\in\mathcal{L}({\mathbb{R}}^m,W^{r,2}(\Omega)')$ for some $r\in (0,1)$, then for all $\delta>0$ we have that
			\begin{align*}
			v\in C^{0,1-r/2}([\delta,\infty);L^{2}(\Omega)),\quad
			y,I_{s,e}\in C^{0,1-r}([\delta,\infty);{\mathbb{R}}^m).
			\end{align*}
			\item if additionally $\mathcal{B}\in\mathcal{L}({\mathbb{R}}^m,L^2(\Omega))$, then for all $\delta>0$ and all $\lambda\in(0,1)$ we have
			\begin{align*}
			v\in C^{0,\lambda}([\delta,\infty);L^{2}(\Omega)),\quad
			y,I_{s,e}\in C^{0,\lambda}([\delta,\infty);{\mathbb{R}}^m).
			\end{align*}
			\item if additionally $\mathcal{B}\in\mathcal{L}({\mathbb{R}}^m,W^{1,2}(\Omega))$, then for all $\delta>0$ we have $y,I_{s,e}\in W^{1,\infty}(\delta,\infty;{\mathbb{R}}^m)$.
	\end{enumerate}
\end{Thm}

\begin{Rem}\label{rem:main}
	\hspace{1em}
\begin{enumerate}[a)]
\item\label{rem:main1} The condition $\ker \mathcal{B}=\{0\}$ is equivalent to $\im \mathcal{B}'$ being dense in ${\mathbb{R}}^m$. The latter is equivalent to $\im \mathcal{B}'={\mathbb{R}}^m$ by the finite-dimensionality of ${\mathbb{R}}^m$.\\
    Note that surjectivity of $\mathcal{B}'$ is mandatory for tracking control, since it is necessary that any reference signal $y_{\rm ref}\in W^{1,\infty}(0,\infty;{\mathbb{R}}^m)$ can actually be generated by the output $y(t)=\mathcal{B}' v$. This property is sometimes called \emph{right-invertibility}, see e.g.~\cite[Sec.~8.2]{TrenStoo01}.
\item\label{rem:main2} If the input operator corresponds to Neumann boundary control, i.e., $\mathcal{B}$ is as in~\eqref{eq:neumanncontr} for some $w\in L^2(\partial\Omega)$, then $\mathcal{B}\in\mathcal{L}({\mathbb{R}},W^{r,2}(\Omega)')$ for some $r\in(1/2,1)$, cf.\ Remark~\ref{rem:openloop}\,\ref{rem:openloop4}), and the assertions of Theorem~\ref{thm:mono_funnel}\,a) hold.
\item\label{rem:main3} If the input operator corresponds to distributed control, that is $\mathcal{B} u=u\cdot w$ for some $w\in L^2(\Omega)$, then $\mathcal{B}\in\mathcal{L}({\mathbb{R}},L^2(\Omega))$, cf.\ Remark~\ref{rem:openloop}\,\ref{rem:openloop3}), and the assertions of Theorem~\ref{thm:mono_funnel}\,b) hold.
\end{enumerate}
\end{Rem}

\section{Proof of Theorem~\ref{thm:mono_funnel}}
\label{sec:mono_proof_mt}

The proof is inspired by~the results of \cite{Jack90} on existence and uniqueness of (non-controlled) FitzHugh-Nagamo equations, which is based on a spectral approximation and subsequent convergence proofs by using arguments from~\cite{Lion69}. We divide the proof in two major parts. First, we show that there exists a unique solution on the interval $[0,\gamma]$. After that we show that the solution also exists on $(\gamma,\infty)$, is continuous at $t=\gamma$ and has the desired properties.

\subsection{Solution on $[0,\gamma]$}
\label{ssec:mono_proof_tleqgamma}

Assuming that $t\in[0,\gamma]$, we have that $\varphi(t)\equiv0$ so that we need to show existence of a pair of functions $(v,u)$ with the properties as in Definition~\ref{def:solution}~(i)--(iii), where~\eqref{eq:solution} simplifies to
\begin{equation}\label{eq:weak_uv_delta}
\begin{aligned}
\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\scpr{v(t)}{\theta}&=-\mathfrak{a}(v(t),\theta)+\scpr{p_3(v(t))-u(t)+I_{s,i}(t)}{\theta}+\scpr{I_{s,e}(t)}{\mathcal{B}'\theta}_{{\mathbb{R}}^m},\\
\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\scpr{u(t)}{\chi}&=\scpr{c_5 v(t)-c_4u(t)}{\chi},\\
I_{s,e}(t)&=-k_0(\mathcal{B}' v(t)-y_{\rm ref}(t)),\\
y(t)&= \mathcal{B}' v(t).
\end{aligned}
\end{equation}
Recall that $\mathfrak{a}:W^{1,2}(\Omega)\times W^{1,2}(\Omega)\to{\mathbb{R}}$ is the sesquilinear form \eqref{eq:sesq}.

\emph{Step 1: We show existence and uniqueness of a solution.}\\
\emph{Step 1a: We show existence of a local solution on $[0,\gamma]$.} To this end, let $(\theta_i)_{i\in{\mathbb{N}}_0}$ be the eigenfunctions of $-\mathcal{A}$ and $\alpha_i$ be the corresponding eigenvalues, with $\alpha_i\geq0$ for all $i\in{\mathbb{N}}_0$. Recall that $(\theta_i)_{i\in{\mathbb{N}}_0}$ form an orthonormal basis of $L^2(\Omega)$ by Proposition~\ref{prop:Aop_n}\,\ref{item:Aop5}). Hence, with $a_i :=  \scpr{v_0}{\theta_i}$ and $b_i :=  \scpr{u_0}{\theta_i}$ for $i\in{\mathbb{N}}_0$ and
\[
v_0^n:= \sum_{i=0}^na_{i}\theta_i,\quad u_0^n:= \sum_{i=0}^nb_{i}\theta_i,\quad n\in{\mathbb{N}},
\]
we have that $v^n_0\to v_0$ and $u^n_0\to u_0$ strongly in $L^2(\Omega)$.\\
Fix $n\in{\mathbb{N}}_0$ and let $\gamma_i:=\mathcal{B}'\theta_i$ for $i=0,\dots,n$. Consider, for $j=0,\ldots,n$, the differential equations
\begin{equation}\label{eq:muj-nuj-0gamma}
\begin{aligned}
\dot{\mu}_j(t)&=-\alpha_j\mu_j(t)-\nu_j(t)-\scpr{k_0\left(\sum_{i=0}^n\gamma_i\mu_i(t)-y_{\rm ref}(t)\right)}{\gamma_j}_{{\mathbb{R}}^m}+\scpr{I_{s,i}(t)}{\theta_j} \\
&\quad +\scpr{p_3\left(\sum_{i=0}^n\mu_i(t)\theta_i\right)}{\theta_j},\\
\dot{\nu}_j(t)&=-c_4\nu_j(t)+c_5\mu_j(t),\qquad\qquad \text{with}\ \mu_j(0)=a_j,\ \nu_j(0)=b_j,
\end{aligned}
\end{equation}
defined on ${\mathbb{D}}:=[0,\infty)\times{\mathbb{R}}^{2(n+1)}$. Since the functions on the right hand side of~\eqref{eq:muj-nuj-0gamma} are continuous, it follows from ODE theory, see e.g.~\cite[\S~10, Thm.~XX]{Walt98}, that there exists a weakly differentiable solution $(\mu^n,\nu^n)=(\mu_0,\dots,\mu_n,\nu_0,\dots,\nu_n):[0,T_n)\to{\mathbb{R}}^{2(n+1)}$ of~\eqref{eq:muj-nuj-0gamma} such that $T_n\in(0,\infty]$ is maximal. Furthermore, the closure of the graph of~$(\mu^n,\nu^n)$ is not a compact subset of ${\mathbb{D}}$.\\
Now, set $v_n(t):=\sum_{i=0}^n{\mu}_i(t)\theta_i$ and $u_n(t):=\sum_{i=0}^n{\nu}_i(t)\theta_i$. Invoking~\eqref{eq:muj-nuj-0gamma} and using the functions $\theta_j$ we have that for $j=0,\dots,n$ the functions $(v_n,u_n)$ satisfy
	\begin{equation}\label{eq:weak_delta}
	\begin{aligned} \scpr{\dot{v}_n(t)}{\theta_j}&=-\mathfrak{a}(v_n(t),\theta_j)- \scpr{u_n(t)}{\theta_j}+\scpr{p_3(v_n(t))}{\theta_j}+\scpr{I_{s,i}(t)}{\theta_j}\\
	&\quad -\scpr{k_0(\mathcal{B}' v_n(t)-y_{\rm ref}(t))}{\mathcal{B}'\theta_j}_{{\mathbb{R}}^m},\\
    \scpr{\dot{u}_n(t)}{\theta_j} &= -c_4\scpr{u_n(t)}{\theta_j}+c_5\scpr{v_n(t)}{\theta_j}.
	\end{aligned}
	\end{equation}
\emph{Step 1b: We show boundedness of $(v_n,u_n)$.} Consider the Lyapunov function candidate
\begin{equation}\label{eq:Lyapunov}
V:L^2(\Omega)\times L^2(\Omega)\to{\mathbb{R}},\ (v,u)\mapsto\frac12(c_5\|v\|^2+\|u\|^2).
\end{equation}
Observe that, since $(\theta_i)_{i\in{\mathbb{N}}_0}$ are orthonormal, we have $\|v_n\|^2 = \sum_{j=0}^n \mu_j^2$ and $\|u_n\|^2 = \sum_{j=0}^n \nu_j^2$. Hence we find that, for all $t\in[0, T_n)$,
\begin{align*}
\tfrac{\text{\normalfont d}}{\text{\normalfont d}t} V(v_n(t),u_n(t)) &\stackrel{\eqref{eq:muj-nuj-0gamma}}{=} c_5\sum_{j=0}^n\mu_j(t)\dot{\mu}_j(t)+\sum_{j=0}^n\nu_j(t)\dot{\nu}_j(t)\\ &=-c_5\sum_{j=0}^{n}\alpha_j\mu_j(t)^2-c_4\sum_{j=0}^{n}\nu_j(t)^2\\
&\quad -c_5\scpr{k_0\left(\sum_{i=0}^n\gamma_i\mu_i(t)-y_{\rm ref}(t)\right)}{\sum_{j=0}^n\gamma_j\mu_j(t)}_{{\mathbb{R}}^m}\\
&\quad +c_5\scpr{p_3\left(v_n(t)\right)}{v_n(t)}+c_5\scpr{I_{s,i}(t)}{v_n(t)}
\end{align*}
hence, omitting the argument~$t$ for brevity in the following,
\begin{equation}\label{eq:Lyapunov_delta}
\begin{aligned}
\tfrac{\text{\normalfont d}}{\text{\normalfont d}t} V(v_n,u_n)=&-c_5\mathfrak{a}(v_n,v_n)-c_4\|u_n\|^2+c_5\scpr{I_{s,i}}{v_n}\\
&-c_5k_0\|\overline{e}_n\|_{{\mathbb{R}}^m}^2+c_5k_0\scpr{\overline{e}_n}{y_{\rm ref}}_{{\mathbb{R}}^m}+c_5\scpr{p_3(v_n)}{v_n},
\end{aligned}
\end{equation}
where
\[\overline{e}_n(t):=\sum_{i=0}^n\gamma_i\mu_i(t)-y_{\rm ref}(t)=\mathcal{B}' v_n(t)-y_{\rm ref}(t).\]
Before proceeding, recall Young's inequality for products, i.e., for $a,b\ge 0$ and $p,q\ge 1$ such that $1/p + 1/q = 1$ we have that
\[
a b \le \frac{a^p}{p} + \frac{b^q}{q},
\]
which will be frequently used in the following. Note that
\begin{align*}
\scpr{p_3(v_n)}{v_n}&=-c_1\|v_n\|^2+c_2\scpr{v_n^2}{v_n}-c_3\|v_n\|^4_{L^4},\\
c_2|\scpr{v_n^2}{v_n}|&=|\scpr{\epsilon v_n^3}{\epsilon^{-1}c_2}|\leq \frac{3\epsilon^{4/3}}{4}\|v_n\|^4_{L^4}+\frac{c_2^4}{4\epsilon^4}|\Omega|,
\end{align*}
where the latter follows from Young's inequality with $p=\tfrac43$ and $q=4$. Choosing $\epsilon=\left(\tfrac23 c_3\right)^{\tfrac34}$ we obtain
\[\scpr{p_3(v_n)}{v_n}\leq \frac{27 c_2^4}{32 c_3^3}|\Omega|-c_1\|v_n\|^2-\frac{c_3}{2}\|v_n\|_{L^4}^4.\]
Moreover,
\[\scpr{\overline{e}_n}{y_{\rm ref}}_{{\mathbb{R}}^m}\leq\frac{1}{2}\|\overline{e}_n\|_{{\mathbb{R}}^m}^2+\frac{1}{2}\|y_{\rm ref}\|_{{\mathbb{R}}^m}^2\]
and
\[\scpr{I_{s,i}}{v_n}\leq \frac{c_1}{2}\|v_n\|^2+\frac{1}{2c_1}\|I_{s,i}\|^2,\]
such that \eqref{eq:Lyapunov_delta} can be estimated by
\begin{align*}
	\tfrac{\text{\normalfont d}}{\text{\normalfont d}t} V(v_n,u_n) \leq&-c_5\mathfrak{a}(v_n,v_n)-\frac{c_1c_5}{2}\|v_n\|^2-\frac{c_5k_0}{2}\|\overline{e}_n\|_{{\mathbb{R}}^m}^2-\frac{c_3 c_5}{2}\|v_n\|_{L^4}^4\\
	&+\frac{k_0c_5}{2}\|y_{\rm ref}\|_{{\mathbb{R}}^m}^2+\frac{1}{2c_1}\|I_{s,i}\|^2+\frac{27 c_2^4}{32 c_3^3}|\Omega|\\
	\leq&\ -c_5\mathfrak{a}(v_n,v_n)-\frac{c_1c_5}{2}\|v_n\|^2-\frac{c_5k_0}{2}\|\overline{e}_n\|_{{\mathbb{R}}^m}^2-\frac{c_3 c_5}{2}\|v_n\|_{L^4}^4\\
	&+\frac{k_0c_5}{2}\|y_{\rm ref}\|_{\infty}^2+\frac{1}{2c_1}\|I_{s,i}\|^2_{2,\infty}+\frac{27 c_2^4}{32 c_3^3}|\Omega|,
\end{align*}
where $\|I_{s,i}\|_{2,\infty} = \esssup_{t\ge 0} \left( \int_\Omega |I_{s,i}(\zeta,t)|^2 \ds{\zeta}\right)^{1/2}$. Setting
\[C_\infty:=\frac{k_0c_5}{2}\|y_{\rm ref}\|_{\infty}^2+\frac{1}{2c_1}\|I_{s,i}\|^2_{2,\infty}+\frac{27 c_2^4}{32 c_3^3}|\Omega|,\]
we obtain that, for all $t\in[0, T_n)$,
\begin{align*}
 &V(v_n(t),u_n(t))+c_5\int_0^t\mathfrak{a}(v_n(s),v_n(s))\ds{s}+\frac{c_1 c_5}{2}\int_0^t\|v_n(s)\|^2\ds{s}\\
	&+\frac{c_5k_0}{2}\int_0^t\|\overline{e}_n(s)\|_{{\mathbb{R}}^m}^2\ds{s} +\frac{c_3 c_5}{2}\int_0^t\|v_n(s)\|_{L^4}^4\ds{s}\leq V(v_0^n,u_0^n)+C_\infty t.
\end{align*}
Since $(u_n^0,v_n^0)\to (u_0,v_0)$ strongly in $L^2(\Omega)$ and we have for all $p\in L^2(\Omega)$ that
\[
\left\|\sum_{i=0}^n \scpr{p}{\theta_i} \theta_i\right\|^2=\sum_{i=0}^n \scpr{p}{\theta_i}^2\leq\sum_{i=0}^\infty \scpr{p}{\theta_i}^2=\left\|\sum_{i=1}^\infty \scpr{p}{\theta_i}\theta_i\right\|^2=\|p\|^2,
\]
it follows that, for all $t\in[0, T_n)$,
\begin{equation}\label{eq:bound_1}
\begin{aligned}
&c_5\|v_n(t)\|^2+\|u_n(t)\|^2+ 2c_5\int_0^t\mathfrak{a}(v_n(s),v_n(s))\ds{s}+ {c_1 c_5}\int_0^t\|v_n(s)\|^2\ds{s} \\
&+ {c_5k_0}\int_0^t\|\overline{e}_n(s)\|_{{\mathbb{R}}^m}^2\ds{s} +{c_3 c_5}\int_0^t\|v_n(s)\|_{L^4}^4\ds{s}\leq 2C_\infty t+c_5\|u_0\|^2+\|v_0\|^2.
\end{aligned}
\end{equation}
\emph{Step 1c: We show that $T_n = \infty$.} Assume that $T_n<\infty$, then it follows from~\eqref{eq:bound_1} together with~\eqref{eq:sesq} that $(v_n,u_n)$ is bounded, thus the solution $(\mu^n,\nu^n)$ of~\eqref{eq:muj-nuj-0gamma} is bounded on $[0,T_n)$. But this implies that the closure of the graph of $(\mu^n,\nu^n)$ is a compact subset of ${\mathbb{D}}$, a contradiction. Therefore, $T_n=\infty$ and in particular the solution is defined for all $t\in[0,\gamma]$.\\
\emph{Step 1d: We show convergence of $(v_n,u_n)$ to  a solution of~\eqref{eq:weak_uv_delta} on $[0,\gamma]$.} First note that it follows from~\eqref{eq:bound_1} that
\begin{equation}\label{eq:uv_bound_delta}
\forall\, t\in[0,\gamma]:\ \|v_n(t)\|^2\leq C_v,\quad \|u_n(t)\|^2\leq C_u
\end{equation}
for some $C_v, C_u>0$. From \eqref{eq:bound_1} and condition~\eqref{eq:ellcond} in Assumption~\ref{Ass1}  it follows that there is a constant $C_\delta>0$ such that
\[\int_0^\gamma\|\nabla v_n(t)\|^2\ds{t}\leq\delta^{-1}\int_0^\gamma\mathfrak{a}(v_n(t),v_n(t))\ds{t}\leq C_\delta.\]
This together with~\eqref{eq:bound_1}  and~\eqref{eq:uv_bound_delta}  implies that there exist constants $C_1,C_2>0$ with
\begin{equation}\label{eq:extra_bounds_delta}
\|v_n\|^4_{L^4(0,\gamma;L^{4}(\Omega))}\leq C_1,\quad \|v_n\|_{L^2(0,\gamma;W^{1,2}(\Omega))}\leq C_2.
\end{equation}
Note that \eqref{eq:extra_bounds_delta} directly implies that
\begin{equation}\label{eq:vn2}
\begin{aligned}\|v_n^2\|^2_{L^2(0,\gamma;L^{2}(\Omega))}\leq&\, C_1,\\
\|v_n^3\|_{L^{4/3}(0,\gamma;L^{4/3}(\Omega))} =&\, \left( \|v_n^2\|^2_{L^2(0,\gamma;L^{2}(\Omega))}\right)^{3/4} \le C_1^{3/4}.
\end{aligned}\end{equation}
Multiplying the second equation in~\eqref{eq:weak_delta} by $\dot{\nu}_j$ and summing up over $j\in\{0,\ldots,n\}$ leads to
\begin{align*}
    \|\dot{u}_n\|^2 &=  -\frac{c_4}{2}  \tfrac{\text{\normalfont d}}{\text{\normalfont d}t} \|u_n\|^2 + c_5 \scpr{v_n}{\dot{u}_n}\\
    &\le -\frac{c_4}{2} \tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|u_n\|^2 + \frac{c_5^2}{2}\|v_n\|^2+\frac{1}{2}\|\dot{u}_n\|^2,
\end{align*}
thus
\[\|\dot{u}_n\|^2\leq-c_4 \tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|u_n\|^2+c_5^2\|v_n\|^2.\]
Upon integration over $[0,\gamma]$ and using \eqref{eq:uv_bound_delta} this yields that
\[\int_0^\gamma\|\dot{u}_n(t)\|^2\ds{t}\leq c_4 C_u + c_5^2\int_0^\gamma\|v_n(t)\|^2\ds{t} \le \hat{C}_3\]
for some $\hat{C}_3>0$, where the last inequality is a consequence of~\eqref{eq:bound_1}. This together with \eqref{eq:uv_bound_delta} implies that there is $C_3>0$ such that $\|u_n\|_{W^{1,2}(0,\gamma;L^2(\Omega))}\leq C_3$.

Now, let $P_n$ be the orthogonal projection of $L^2(\Omega)$ onto the subspace generated by the set $\setdef{\theta_i}{i=1,\dots,n}$. Consider the norm
\[\|v\|_{W^{1,2}}=\left(\sum_{i=0}^n(1+\alpha_i)|\scpr{v}{\theta_i}|^2\right)^{1/2}\]
on $W^{1,2}(\Omega)$ according to Proposition \ref{prop:Aop2_n} and Remark \ref{rem:X_alpha}. By duality we have that
\[\|\hat v\|_{(W^{1,2})'}=\left(\sum_{i=0}^n(1+\alpha_i)^{-1}|\scpr{\hat v}{\theta_i}|^2\right)^{1/2}\]
is a norm on $W^{1,2}(\Omega)'$, cf.~\cite[Prop.~3.4.8]{TucsWeis09}. Note that we can consider $P_n:W^{1,2}(\Omega)'\to W^{1,2}(\Omega)'$, which is a bounded linear operator with norm one, independent of $n$. Using this together with the fact that the injection from $L^2(\Omega)$ into $W^{1,2}(\Omega)'$ is continuous and $\mathcal{A}\in\mathcal{L}(W^{1,2}(\Omega),W^{1,2}(\Omega)')$, we can rewrite the weak formulation~\eqref{eq:weak_delta} as
\begin{equation}\label{eq:approx_dual}
\dot{v}_n=P_n\mathcal{A} v_n+P_np_3(v_n)-P_nu_n+P_nI_{s,i}-P_n\mathcal{B} k_0(\mathcal{B}' v_n-y_{\rm ref}).
\end{equation}
Since $v_n\in L^2(0,\gamma;W^{1,2}(\Omega))$ and hence, by the Sobolev Embedding Theorem, $v_n\in L^2(0,\gamma;L^p(\Omega))$ for all $2\leq p\leq6$, we find that $p_3(v_n)\in L^2(0,\gamma;L^2(\Omega))$. We also have $\mathcal{A} v_n\in L^2(0,\gamma;W^{1,2}(\Omega)')$ and $\mathcal{B} k_0(\mathcal{B}' v_n-y_{\rm ref})\in L^2(0,\gamma;W^{1,2}(\Omega)')$ so that by using the previously derived estimates and~\eqref{eq:approx_dual}, there exists $C_4>0$ independent of $n$ and $t$ with
\[\|\dot{v}_n\|_{L^2(0,\gamma;W^{1,2}(\Omega)')}\leq C_4.\]
Now, by Lemma~\ref{lem:weak_convergence} we have that there exist subsequences of $(u_n)$, $(v_n)$ and $(\dot v_n)$, resp., again denoted in the same way, for which
\begin{equation}\label{eq:convergence_subseq}
\begin{aligned}
u_n\to u&\in W^{1,2}(0,\gamma;L^2(\Omega))\mbox{ weakly},\\
u_n\to u&\in W^{1,\infty}(0,\gamma;L^{2}(\Omega))\mbox{ weak}^\star,\\
v_n\to v&\in L^2(0,\gamma;W^{1,2}(\Omega))\mbox{ weakly},\\
v_n\to v&\in L^\infty(0,\gamma;L^{2}(\Omega))\mbox{ weak}^\star,\\
v_n\to v&\in L^4(0,\gamma;L^{4}(\Omega))\mbox{ weakly},\\
\dot{v}_n\to \dot{v}&\in L^2(0,\gamma;W^{1,2}(\Omega)')\mbox{ weakly}.
\end{aligned}
\end{equation}
Moreover, let
$p_0=p_1=2$ and $X=W^{1,2}(\Omega)$, $Y=L^2(\Omega)$, $Z=W^{1,2}(\Omega)'$. Then, \cite[Chap.~1, Thm.~5.1]{Lion69} implies that
\[W:=\setdef{u\in L^{p_0}(0,\gamma;X) }{ \dot{u}\in L^{p_1}(0,\gamma;Z) }\]
with norm $\|u\|_{L^{p_0}(0,\gamma;X)}+\|\dot{u}\|_{L^{p_1}(0,\gamma;Y)}$ has a compact injection into $L^{p_0}(0,\gamma;Y)$, so that the weakly convergent sequence $v_n\to v\in W$ converges strongly in $L^2(0,\gamma;L^2(\Omega))$ by \cite[Lem.~1.6]{HinzPinn09}. Further, $(u(0),v(0))=(u_0,v_0)$ and by $v\in W^{1,2}(0,\gamma;L^2(\Omega))$, $v\in L^2(0,\gamma;W^{1,2}(\Omega))$ and $\dot{v}\in L^2(0,\gamma;W^{1,2}(\Omega)')$ it follows that $u,v\in C([0,\gamma];L^2(\Omega))$, see for instance \cite[Thm.~1.32]{HinzPinn09}. Moreover, note that $\mathcal{B}' v-y_{\rm ref}\in L^2(0,\gamma;{\mathbb{R}}^m)$. Hence, $(u,v)$ is a solution of \eqref{eq:FHN_feedback} in $[0,\gamma]$ and
\begin{equation}\label{eq:strong_delta}
\dot{v}(t)=\mathcal{A} v(t)+p_3(v(t))-u(t)+I_{s,i}(t)-\mathcal{B} k_0(\mathcal{B}' v(t)-y_{\rm ref}(t))
\end{equation}
is satisfied in $W^{1,2}(\Omega)'$. Moreover, by~\eqref{eq:vn2},~\cite[Chap.~1, Lem.~1.3]{Lion69} and $v_n\to v$ in $L^4(0,\gamma;L^{4}(\Omega))$ we have that $v_n^3\to v^3$ weakly in $L^{4/3}(0,\gamma;L^{4/3}(\Omega))$ and $v_n^2\to v^2$ weakly in $L^2(0,\gamma;L^{2}(\Omega))$.\\
\emph{Step 1e: We show uniqueness of the solution $(v,u)$.} To this end, we separate the linear part of~$p_3$ so that
\[p_3(v)=-c_1v-c_3\hat{p}_3(v),\quad \hat{p}_3(v)\coloneqq v^2\left(v-c\right),\ \quad c\coloneqq c_2/c_3.\]
Assume that $(v_1,u_1)$ and $(v_2,u_2)$ are two solutions of~\eqref{eq:FHN_feedback} on $[0,\gamma]$ with the same initial values, $v_1(0) = v_2(0) = v_0$ and $u_1(0) = u_2(0) = u_0$. Let $t_0\in(0,\gamma]$ be given. Let $Q_0\coloneqq(0,t_0)\times\Omega$. Define
\[\Sigma(t,\zeta):= |v_1(t,\zeta)|+|v_2(t,\zeta)|,\]
and let
\[Q^\Lambda:= \{(t,\zeta)\in Q_0\ |\ \Sigma(t,\zeta)\leq\Lambda\},\quad \Lambda>0.\]
Note that, by convexity of the map $x\mapsto x^p$ on $[0,\infty)$ for $p>1$, we have that
\[
 \forall\, a,b\ge 0:\ \big(\tfrac12 a+ \tfrac12 b)^p \le \tfrac12 a^p + \tfrac12 b^p.
\]
Therefore, since $v_1,v_2\in L^4(0,\gamma;L^{4}(\Omega))$, we find that $\Sigma\in L^4(0,\gamma;L^{4}(\Omega))$. Hence, by the monotone convergence theorem, for all $\epsilon>0$ we may choose $\Lambda$ large enough such that
\[\int_{Q_0\setminus Q^\Lambda}|\Sigma(\zeta,t)|^4\ds{(\zeta,t)}<\epsilon.\]
Note that without loss of generality we may assume that $\Lambda>\frac{c}{3}$. Let $V:= v_2-v_1$ and $U:= u_2-u_1$, then, by~\eqref{eq:FHN_feedback},
\begin{align*}
  \dot V &= (\mathcal{A}-c_1I) V -c_3(\hat{p}_3(v_2) - \hat{p}_3(v_1)) - U - k_0 \mathcal{B}\cB' V,\\
  \dot U &= c_5 V - c_4 U.
\end{align*}
By~\cite[Thm.~1.32]{HinzPinn09}, we have for all $t\in(0,\gamma)$ that
\[\tfrac{1}{2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|V(t)\|^2=\scpr{\dot{V}(t)}{V(t)},\quad \tfrac{1}{2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|U(t)\|^2=\scpr{\dot{U}(t)}{U(t)},\]
thus we may compute that
\begin{align*}
    \tfrac{c_5}{2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|V\|^2+\tfrac{1}{2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|U\|^2&= \scpr{(\mathcal{A}-c_1I) V - U - k_0 \mathcal{B}\cB' V}{c_5 V} - c_4 \|U\|^2 + c_5 \langle U,V \rangle\\
    &\quad-c_5 c_3\scpr{ \hat{p}_3(v_2) - \hat{p}_3(v_1)}{V}\\
    &= c_5 \scpr{(\mathcal{A}-c_1I) V}{V} - c_5 k_0 \scpr{\mathcal{B}' V}{\mathcal{B}' V} - c_4 \|U\|^2 \\
    &\quad - c_5c_3 \scpr{\hat{p}_3(v_2)-\hat{p}_3(v_1)}{V}\\
    &\le-c_5c_3 \scpr{\hat{p}_3(v_2)-\hat{p}_3(v_1)}{V}.
\end{align*}
Integration over $[0,t_0]$ and using $(U(0),V(0))=(0,0)$ leads to
\begin{align*}
\tfrac{c_5}{2}\|V(t_0)\|^2+\tfrac{1}{2}\|U(t_0)\|^2
&=-c_5c_3\int_0^{t_0}\int_\Omega (\hat{p}_3(v_2(\zeta,t))-\hat{p}_3(v_1(\zeta,t)))V(\zeta,t)\ds{\zeta}\ds{t}\\
&=-c_5c_3\int_{Q^\Lambda} (\hat{p}_3(v_2(\zeta,t))-\hat{p}_3(v_1(\zeta,t)))V(\zeta,t)\ds{\zeta}\ds{t}\\
&\quad-c_5c_3\int_{Q_0\setminus Q^\Lambda} (\hat{p}_3(v_2(\zeta,t))-\hat{p}_3(v_1(\zeta,t)))V(\zeta,t)\ds{\zeta}\ds{t}.
\end{align*}
Note that on $Q^\Lambda$ we have $-\Lambda\leq v_1\leq\Lambda$ and $-\Lambda\leq v_2\leq\Lambda$. Let $a,b\in[-\Lambda,\Lambda]$, then the mean value theorem implies
\begin{align*}
(\hat{p}_3(b)-\hat{p}_3(a))(b-a)=\hat{p}_3'(\xi)(b-a)^2
\end{align*}
for some $\xi\in(-\Lambda,\Lambda)$. Since $\hat{p}_3'(\xi)=3\xi^2-2c\xi$ has a minimum at
\[\xi^\ast=\frac{c}{3}\]
we have that
\[(\hat{p}_3(b)-\hat{p}_3(a))(b-a)=\hat{p}_3'(\xi)(b-a)^2\geq-\frac{c^2}{3}(b-a)^2.\]
Using that in the above inequality leads to
\begin{align*}
\tfrac{c_5}{2}\|V(t_0)\|^2+\tfrac{1}{2}\|U(t_0)\|^2&\leq c_5c_3\frac{c^2}{3}\int_{Q^\Lambda} V(\zeta,t)^2\ds{\zeta}\ds{t}\\
&\quad-c_5c_3\int_{Q_0\setminus Q^\Lambda} (\hat{p}_3(v_2(\zeta,t))-\hat{p}_3(v_1(\zeta,t)))V(\zeta,t)\ds{\zeta}\ds{t}\\
&\leq c_5c_3\frac{c^2}{3}\int_{Q_0} V(\zeta,t)^2\ds{\zeta}\ds{t}\\
&\quad+c_5c_3\int_{Q_0\setminus Q^\Lambda} |\hat{p}_3(v_2(\zeta,t))-\hat{p}_3(v_1(\zeta,t))||V(\zeta,t)|\ds{\zeta}\ds{t}\\
&\le c_5c_3\frac{c^2}{3}\int_0^{t_0}\|V(t)\|^2\ds{t}+2c_5c_3\int_{Q_0\setminus Q^\Lambda}|\Sigma(\zeta,t)|^4\ds{(\zeta,t)}\\
&\leq  c_3\frac{c^2}{3}\int_0^{t_0}c_5\|V(t)\|^2+\|U(t)\|^2\ds{t}+2c_5c_3\epsilon.
\end{align*}
Since $\epsilon>0$ was arbitrary we may infer that
\[\tfrac{c_5}{2}\|V(t_0)\|^2+\tfrac{1}{2}\|U(t_0)\|^2\leq \frac{2c_3c^2}{3}\int_0^{t_0}\tfrac{c_5}{2}\|V(t)\|^2+\tfrac12\|U(t)\|^2\ds{t}.\]
Hence, by Gronwall's lemma and $U(0)=0,V(0)=0$ it follows that $U(t_0)=0$ and $V(t_0)=0$. Since $t_0$ was arbitrary, this shows that $v_1 = v_2$ and $u_1 = u_2$ on $[0,\gamma]$.

\emph{Step 2: We show that for all $\epsilon\in(0,\gamma)$ and all $t\in[\epsilon,\gamma]$ we have $v(t)\in W^{1,2}(\Omega)$.}\\
Fix $\epsilon\in(0,\gamma)$. First we show that $v\in BUC([\epsilon,\gamma];W^{1,2}(\Omega))$. Multiplying the first equation in~\eqref{eq:weak_delta}  by $\dot{\mu}_j$ and summing up over $j\in\{0,\dots,n\}$ we obtain
\begin{align*}
\|\dot{v}_n\|^2&=-\tfrac{1}{2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t} \mathfrak{a}(v_n,v_n)-\scpr{u_n}{\dot{v}_n}+\scpr{p_3(v_n)}{\dot{v}_n}+\scpr{I_{s,i}}{\dot{v}_n}\\
&\quad -k_0\scpr{\mathcal{B}' v_n-y_{\rm ref}}{\mathcal{B}'\dot{v}_n}_{{\mathbb{R}}^m}\\
&=-\tfrac{1}{2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t} \mathfrak{a}(v_n,v_n)-\scpr{u_n}{\dot{v}_n}+\scpr{p_3(v_n)}{\dot{v}_n}+\scpr{I_{s,i}}{\dot{v}_n}\\
&\quad -k_0\scpr{\mathcal{B}' v_n-y_{\rm ref}}{\mathcal{B}'\dot{v}_n-\dot{y}_{\rm ref}}_{{\mathbb{R}}^m} - k_0\scpr{\mathcal{B}' v_n-y_{\rm ref}}{\dot{y}_{\rm ref}}_{{\mathbb{R}}^m}
\end{align*}
Furthermore, we may derive that
\begin{align*}
  \tfrac{\text{\normalfont d}}{\text{\normalfont d}t} v_n^4 &= 4 v_n^3 \dot v_n = -\frac{4}{c_3}\big( p_3(v_n) - c_2 v_n^2 + c_1 v_n\big) \dot v_n,\quad \text{thus}\\
  p_3(v_n) \dot v_n &= -\frac{c_3}{4} \tfrac{\text{\normalfont d}}{\text{\normalfont d}t} v_n^4 + c_2 v_n^2 \dot v_n - c_1 v_n \dot v_n,
\end{align*}
and this implies, for any $\delta>0$,
\begin{align*}
    \scpr{p_3(v_n)}{\dot v_n} &\le -\frac{c_3}{4} \tfrac{\text{\normalfont d}}{\text{\normalfont d}t} \|v_n\|_{L^4}^4 + c_2  \scpr{v_n^2}{\dot v_n} - c_1 \scpr{v_n}{\dot v_n}\\
    &\le -\frac{c_3}{4} \tfrac{\text{\normalfont d}}{\text{\normalfont d}t} \|v_n\|_{L^4}^4 + \frac{c_2}{2} \left( \delta \|v_n\|_{L^4}^4 + \frac{1}{\delta} \|\dot v_n\|^2\right) + \frac{c_1}{2} \left(\delta \|v_n\|^2 + \frac{1}{\delta} \|\dot v_n\|^2\right)\\
    &\stackrel{\eqref{eq:uv_bound_delta}}{\le} -\frac{c_3}{4} \tfrac{\text{\normalfont d}}{\text{\normalfont d}t} \|v_n\|_{L^4}^4 + \frac{c_2}{2} \left( \delta \|v_n\|_{L^4}^4 + \frac{1}{\delta} \|\dot v_n\|^2 \right) + \frac{c_1}{2} \left(\delta C_v + \frac{1}{\delta} \|\dot v_n\|^2\right).
\end{align*}
Moreover, we find that, recalling $\overline{e}_n = \mathcal{B}' v_n-y_{\rm ref}$
\begin{align*}
\scpr{u_n}{\dot{v}_n} &\le \frac{\delta}{2} \|u_n\|^2 + \frac{1}{2\delta} \|\dot v_n\|^2 \stackrel{\eqref{eq:uv_bound_delta}}{\le} \frac{\delta C_u}{2} + \frac{1}{2\delta} \|\dot v_n\|^2,\\
\scpr{I_{s,i}}{\dot{v}_n} &\le \frac{\delta}{2} \|I_{s,i}\|_{2,\infty}^2 + \frac{1}{2\delta} \|\dot v_n\|^2,\\
\scpr{\overline{e}_n}{\dot{y}_{\rm ref}}_{{\mathbb{R}}^m} &\le \frac{1}{2} \|\overline{e}_n\|^2_{{\mathbb{R}}^m} + \frac12 \|\dot{y}_{\rm ref}\|_\infty^2.
\end{align*}
Therefore, choosing $\delta$ large enough, we obtain that there exist constants $Q_1,Q_2>0$ independent of $n$ such that
\begin{align*}
\|\dot{v}_n\|^2\le&-\frac{1}{2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t} \mathfrak{a}(v_n,v_n)-\frac{c_3}{4}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|v_n\|_{L^4}^4-\frac{k_0}{2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|\overline{e}_n\|_{{\mathbb{R}}^m}^2+\frac{1}{2}\|\dot{v}_n\|^2\\
&+Q_1\|v_n\|^4_{L^4}+Q_2+\frac{k_0}{2}\|\overline{e}_n\|_{{\mathbb{R}}^m}^2,
\end{align*}
thus,
\begin{equation}\label{eq:est-norm-dotvn}
\begin{aligned}
\|\dot{v}_n\|^2&+\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\left(\mathfrak{a}(v_n,v_n)+\frac{c_3}{2}\|v_n\|_{L^4}^4+k_0\|\overline{e}_n\|_{{\mathbb{R}}^m}^2\right)\\
\le &\ 2Q_1\|v_n\|^4_{L^4}+2Q_2+k_0\|\overline{e}_n\|_{{\mathbb{R}}^m}^2.
\end{aligned}
\end{equation}
As a consequence, we find that for all $t\in[0,\gamma]$ we have
\begin{align*}
t\|\dot{v}_n(t)\|^2&+\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\left(t\mathfrak{a}(v_n(t),v_n(t))+\frac{c_3t}{2}\|v_n(t)\|_{L^4}^4+k_0t\|\overline{e}_n(t)\|_{{\mathbb{R}}^m}^2\right)\\
\stackrel{\eqref{eq:est-norm-dotvn}}{\le}&\ \left(2Q_1 t +\frac{c_3}{2}\right)\|v_n(t)\|^4_{L^4}+\mathfrak{a}(v_n(t),v_n(t)) +2Q_2 t+k_0(t+1)\|\overline{e}_n(t)\|_{{\mathbb{R}}^m}^2.
\end{align*}
Since $t\|\dot{v}_n(t)\|^2\geq0$ and $t\le \gamma$ for all $t\in[0,\gamma]$, it follows that
\begin{align*}
\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}&\left(t\mathfrak{a}(v_n(t),v_n(t))+\frac{c_3t}{2}\|v_n(t)\|_{L^4}^4+k_0t\|\overline{e}_n(t)\|_{{\mathbb{R}}^m}^2\right)\\
\le&\ \left(2Q_1\gamma+\frac{c_3}{2}\right)\|v_n(t)\|^4_{L^4}+\mathfrak{a}(v_n(t),v_n(t)) +2Q_2\gamma+k_0(\gamma+1)\|\overline{e}_n(t)\|_{{\mathbb{R}}^m}^2.
\end{align*}
Integrating the former and using \eqref{eq:bound_1}, there exist $P_1,P_2>0$ independent of $n$ such that for $t\in[0,\gamma]$ we have
\begin{align*}
t\mathfrak{a}(v_n(t),v_n(t))+\frac{c_3t}{2}\|v_n(t)\|_{L^4}^4+k_0t\|\overline{e}_n(t)\|_{{\mathbb{R}}^m}^2
\le P_1+P_2t.
\end{align*}
Thus, there exist constants $C_5,C_6>0$ independent of $n$ such that
\[\forall\, t\in[0,\gamma]:\ t\mathfrak{a}(v_n(t),v_n(t))\leq C_5\ \wedge\ t\|\overline{e}_n(t)\|_{{\mathbb{R}}^m}\leq C_6.\]
Hence, for all $\epsilon\in(0,\gamma)$, it follows from the above estimates together with~\eqref{eq:bound_1} that $v_n\in L^\infty(\epsilon,\gamma;W^{1,2}(\Omega))$ and $\overline{e}_n\in L^\infty(\epsilon,\gamma;{\mathbb{R}}^m)$, so that in addition to~\eqref{eq:convergence_subseq}, from Lemma~\ref{lem:weak_convergence} we further have that there exists a subsequence such that
\[v_n\to v\in L^\infty(\epsilon,\gamma;W^{1,2}(\Omega))\mbox{ weak}^\star\]
and $\mathcal{B}'v\in L^\infty(\epsilon,\gamma;{\mathbb{R}}^m)$ for all $\epsilon\in(0,\gamma)$, hence $I_{s,e}\in L^2(0,\gamma;{\mathbb{R}}^m)\cap L^\infty(\epsilon,\gamma;{\mathbb{R}}^m)$. By the Sobolev Embedding Theorem, $W^{1,2}(\Omega)\hookrightarrow L^p(\Omega)$ for $2\leq p\leq 6$ we have that $p_3(v)\in L^\infty(\epsilon,\gamma;L^2(\Omega))$. Moreover, since \eqref{eq:strong_delta} holds, we can rewrite it as
\[\dot{v}(t)=(\mathcal{A}-c_1 I) v(t)+I_r(t)+\mathcal{B} I_{s,e}(t),\]
where $I_r:=c_2v^2-c_3v^3-u+I_{s,i}\in L^2(0,\gamma;L^2(\Omega))\cap L^\infty(\epsilon,\gamma;L^2(\Omega))$ and Proposition~\ref{prop:hoelder} (recall that $W^{1,2}(\Omega)' = X_{-1/2}$ and hence $\mathcal{B}\in\mathcal{L}({\mathbb{R}}^m,X_{-1/2})$) with $c=c_1$ implies that $v\in BUC([\epsilon,\gamma];W^{1,2}(\Omega))$. Hence, for all $\epsilon\in(0,\gamma)$, $v(t)\in W^{1,2}(\Omega)$ for $t\in[\epsilon,\gamma]$, so that in particular $v(\gamma)\in W^{1,2}(\Omega)$.

\subsection{Solution on $(\gamma,\infty)$}
\label{ssec:mono_proof_tgeqgamma}

The crucial step in this part of the proof is to show that the error remains uniformly bounded away from the funnel boundary while $v\in L^\infty(\gamma,\infty;W^{1,2}(\Omega))$. The proof is divided into several steps.

\emph{Step 1: We show existence of an approximate solution by means of a time-varying state-space transformation.}\\
Again, let $(\theta_i)_{i\in{\mathbb{N}}_0}$ be the eigenfunctions of $-\mathcal{A}$ and let $\alpha_i$ be the corresponding eigenvalues, with $\alpha_i\geq0$ for all $i\in{\mathbb{N}}_0$. Recall that $(\theta_i)_{i\in{\mathbb{N}}_0}$ form an orthonormal basis of $L^2(\Omega)$ by Proposition~\ref{prop:Aop_n}\ref{item:Aop5}). Let $(u_\gamma,v_\gamma):=(u(\gamma),v(\gamma))$, $a_i :=  \scpr{v_\gamma}{\theta_i}$ and $b_i :=  \scpr{u_\gamma}{\theta_i}$ for $i\in{\mathbb{N}}_0$ and
\[
v_\gamma^n:= \sum_{i=0}^na_{i}\theta_i,\quad u_\gamma^n:= \sum_{i=0}^nb_{i}\theta_i,\quad n\in{\mathbb{N}}.
\]
Then we have that $v^n_\gamma\to v_\gamma$ strongly in $W^{1,2}(\Omega)$ and $u^n_\gamma\to u_\gamma$ strongly in $L^2(\Omega)$. As stated in Remark~\ref{rem:main}\,\ref{rem:main1}) we have that $\ker\mathcal{B}=\{0\}$ implies $\mathcal{B}' \mathcal{D}(\mathcal{A})={\mathbb{R}}^m$. As a~consequence, there exist $q_1,\ldots,q_m\in\mathcal{D}(\mathcal{A})$ such that $\mathcal{B}'q_k=e_k$ for $k=1,\dots,m$. By Proposition~\ref{prop:Aop_n}\,\ref{item:Aop3}), we further have  $q_k\in C^{0,\nu}(\Omega)$ for some $\nu\in(0,1)$.\\
Note that $U\coloneqq\bigcup_{n\in{\mathbb{N}}} U_n$, where $U_n = \mathrm{span}\{\theta_i\}_{i=0}^n$, satisfies $\overline{U}=W^{1,2}(\Omega)$ with the respective norm. Moreover, $\overline{\mathcal{B}' U}={\mathbb{R}}^m$. Since ${\mathbb{R}}^m$ is complete and finite dimensional and $\mathcal{B}'$ is linear and continuous it follows that $\mathcal{B}' U={\mathbb{R}}^m$. By the surjectivity of $\mathcal{B}'$ we have that for all $k\in\{1,\dots,m\}$ there exist $n_k\in{\mathbb{N}}$ and $q_k\in U_{n_k}$ such that $\mathcal{B}' q_k=e_k$. Thus, there exists $n_0\in{\mathbb{N}}$ with $q_k\in U_{n_0}$ for all $k=\{1,\dots,m\}$, hence the $q_k$ are a (finite) linear combination of the eigenfunctions $\theta_i$.\\
Define $q\in W^{1,2}(\Omega;{\mathbb{R}}^m)\cap C^{0,\nu} (\Omega;{\mathbb{R}}^m)$ by $q(\zeta)=\big(q_1(\zeta),\ldots,q_m(\zeta)\big)^\top$ and $q\cdoty_{\rm ref}$ by
\[
(q\cdoty_{\rm ref}) (t,\zeta) :=  \sum_{k=1}^mq_k(\zeta)y_{{\rm ref},k}(t),\quad \zeta\in \Omega,\, t\ge 0.
\]
We may define $q\cdot\dot{y}_{\rm ref}$ analogously. Note that we have $(q\cdoty_{\rm ref})\in BC([0,\infty)\times\Omega)$, because
\[
|(q\cdoty_{\rm ref}) (t,\zeta)| \le \sum_{k=1}^m \|q_k\|_\infty\, \|y_{{\rm ref},k}\|_\infty
\]
for all $\zeta\in \Omega$ and $t\ge 0$, where we write $\|\cdot\|_\infty$ for the supremum norm. We define $q_{k,j}:= \scpr{q_k}{\theta_j}$ for $k=1,\ldots,m$, $j\in{\mathbb{N}}_0$ and $q_k^n:=\sum_{j=0}^nq_{k,j}$ for $n\in{\mathbb{N}}_0$. Similarly, $q^n:=(q_1^n,\dots,q_m^n)^\top$ for $n\in{\mathbb{N}}$, so that $q^n\to q$ strongly in $W^{1,2}(\Omega)$. In fact, since $q_k\in U_{n_0}$ for all $k=1,\ldots,m$, it follows that $q^n = q$ for all $n\ge n_0$.\\
Since $\mathcal{B}':W^{r,2}(\Omega)\to{\mathbb{R}}^m$ is continuous for some $r\in[0,1]$, it follows that for all $\theta\in W^{r,2}(\Omega)$ there exists $\Gamma_r>0$ such that
\[\|\mathcal{B}'\theta\|_{{\mathbb{R}}^m}\leq\Gamma_r\|\theta\|_{W^{r,2}}.\]
For $n\in{\mathbb{N}}_0$, let
\[\kappa_n\coloneqq \big((n+1) \Gamma_r (1+\|v_\gamma^n-q^n\cdoty_{\rm ref}(\gamma)\|^2_{W^{r,2}})\big)^{-1}.\]
Note that for $v_\gamma\in W^{1,2}(\Omega)$ it holds that $\kappa_n>0$ for all $n\in{\mathbb{N}}_0$, $(\kappa_n)_{n\in{\mathbb{N}}_0}$ is bounded by $\Gamma_r^{-1}$ (and monotonically decreasing) and $\kappa_n\to0$ as $n\to\infty$ and by construction
\[\forall\,n\in{\mathbb{N}}_0:\ \kappa_n\|\mathcal{B}' (v_\gamma^n-q^n\cdoty_{\rm ref}(\gamma))\|_{{\mathbb{R}}^m}<1.\]
Consider a modification of $\varphi$ induced by $\kappa_n$, namely
\[\varphi_n\coloneqq\varphi+\kappa_n,\quad n\in{\mathbb{N}}_0.\]
It is clear that for each $n\in{\mathbb{N}}_0$ we have $\varphi_n\in W^{1,\infty}([\gamma,\infty);{\mathbb{R}})$, the estimates $\|\varphi_n\|_\infty\leq\|\varphi\|_\infty+\Gamma_r^{-1}$ and $\|\dot{\varphi}_n\|_\infty=\|\dot{\varphi}\|_\infty$ are independent of $n$, and
$\varphi_n\to\varphi\in\Phi_\gamma$ uniformly. Moreover, $\inf_{t>\gamma}\varphi_n(t)>0$.\\
Now, fix $n\in{\mathbb{N}}_0$. For $t\ge \gamma$, define
\begin{align*}
\phi(e)&:= \frac{k_0}{1-\|e\|_{{\mathbb{R}}^m}^2}e,\quad e\in{\mathbb{R}}^m,\ \|e\|_{{\mathbb{R}}^m}<1,\\
\omega_0(t)&:= \dot{\varphi}_n(t)\varphi_n(t)^{-1},\\
F(t,z)&:= \varphi_n(t)f_{-1}(t)+\varphi_n(t)f_0(t)+f_1(t)z+\varphi_n(t)^{-1}f_2(t)z^2\\
&\quad \ -c_3\varphi_n(t)^{-2}z^3,\quad z\in{\mathbb{R}},\\
f_{-1}(t)&:= I_{s,i}(t)+\sum_{k=1}^my_{{\rm ref},k}(t)\mathcal{A} q_k,\\
f_0(t)&:= -q\cdot(\dot{y}_{\rm ref}(t)+c_1y_{\rm ref}(t))+c_2(q\cdoty_{\rm ref}(t))^2-c_3(q\cdoty_{\rm ref}(t))^3,\\
f_1(t)&:= (q\cdoty_{\rm ref}(t))(2c_2-3c_3(q\cdoty_{\rm ref}(t))),\\
f_2(t)&:= c_2-3c_3(q\cdoty_{\rm ref}(t)),\\
g(t)&:= c_5(q\cdoty_{\rm ref}(t)).
\end{align*}
We have that $f_{-1}\in L^\infty(\gamma,\infty;L^2(\Omega))$, since
\begin{align*}
\|f_{-1}\|_{2,\infty}  &:=  \esssup_{t\ge \gamma} \left(\int_\Omega f_{-1}(\zeta,t)^2 \ds{\lambda} \right)^{1/2}\\
&\le \|I_{s,i}\|_{2,\infty} + \sum_{k=1}^m \|y_{{\rm ref},k}\|_\infty\, \|A q_k\|_{L^2} < \infty.
\end{align*}
Furthermore, we have that $f_0\in L^\infty((\gamma,\infty)\times\Omega)$, because
\begin{align*}
|f_0(\zeta,t)|&\leq\ (\|\dot{y}_{\rm ref}\|_\infty+c_1\|y_{\rm ref}\|_\infty)\sum_{k=1}^m\|q_k\|_\infty+c_2\|y_{\rm ref}\|_\infty^2\left(\sum_{k=1}^m\|q_k\|_\infty\right)^2\\
&\quad +c_3\|y_{\rm ref}\|^3_\infty\left(\sum_{k=1}^m\|q_k\|_\infty\right)^3\text{ for a.a.\ $(\zeta,t)\in \Omega\times [\gamma,\infty)$},
\end{align*}
whence
\[\|f_0\|_{\infty,\infty}:= \esssup_{t\geq\gamma,\zeta\in\Omega}|f_0(\zeta,t)|<\infty.\]
Similarly $\|f_1\|_{\infty,\infty}<\infty$, $\|f_2\|_{\infty,\infty}<\infty$ and $\|g\|_{\infty,\infty}<\infty$.\\
Consider the system of $2(n+1)$ ODEs
\begin{equation}\label{eq:appr_ODE}
\begin{aligned}
\dot{\mu}_j(t)&=-\alpha_j\mu_j(t)-(c_1-\omega_0(t))\mu_j(t)-\nu_j(t)- \scpr{\phi\left(\sum_{i=0}^n \mathcal{B}'\theta_i \mu_i(t)\right)}{\mathcal{B}'\theta_j}_{{\mathbb{R}}^m} \\
&\quad +\scpr{F\left(t,\sum_{i=0}^n\mu_i(t)\theta_i\right)}{\theta_j},\\
\dot{\nu}_j(t)&=-(c_4-\omega_0(t))\nu_j(t)+c_5\mu_j(t)+\varphi_n(t)\scpr{g(t)}{\theta_j}
\end{aligned}
\end{equation}
defined on
\[
{\mathbb{D}}:= \setdef{(t,\mu_0,\dots,\mu_n,\nu_0,\dots,\nu_n)\in[\gamma,\infty)\times{\mathbb{R}}^{2(n+1)} }{ \left\|\sum_{i=0}^n\gamma_i\mu_i\right\|_{{\mathbb{R}}^m}<1 },
\]
with initial value
\[\mu_j(\gamma)=\kappa_n\left(a_j-\sum_{k=1}^mq_{k,j}y_{{\rm ref},k}(\gamma)\right),\quad \nu_j(\gamma)=\kappa_n b_j,\quad j\in{\mathbb{N}}_0.\]
Since the functions on the right hand side of~\eqref{eq:appr_ODE} are continuous, the set~${\mathbb{D}}$ is relatively open in $[\gamma,\infty)\times{\mathbb{R}}^{2(n+1)}$ and by construction the initial condition satisfies $(\gamma,\mu_0(\gamma),\dots,\mu_n(\gamma),\nu_0(\gamma),\dots,\nu_n(\gamma))\in{\mathbb{D}}$ it follows from ODE theory, see e.g.~\cite[\S~10, Thm.~XX]{Walt98}, that there exists a weakly differentiable solution
\[(\mu^n,\nu^n)=(\mu_0,\dots,\mu_n,\nu_0,\dots,\nu_n):[\gamma,T_n)\to{\mathbb{R}}^{2(n+1)}\] such that $T_n\in(\gamma,\infty]$ is maximal. Furthermore, the closure of the graph of~$(\mu^n,\nu^n)$ is not a compact subset of ${\mathbb{D}}$.\\
With that, we may define
\[z_n(t):=\sum_{i=0}^n\mu_i(t)\theta_i,\quad w_n(t):=\sum_{i=0}^n\nu_i(t)\theta_i,\quad e_n(t):= \sum_{i=0}^n\mathcal{B}'\theta_i\mu_i(t),\quad t\in[\gamma,T_n)\]
and note that
\[z_\gamma^n:=z_n(\gamma)=\kappa_n(v_\gamma^n-q^n\cdoty_{\rm ref}(\gamma)),\quad w_\gamma^n:=w_n(\gamma)=\kappa_n u_\gamma^n.\]
From the orthonormality of the $\theta_i$ we have that
\begin{equation}\label{eq:weak}
	\begin{aligned} \scpr{\dot{z}_n(t)}{\theta_j}&=-\mathfrak{a}(z_n(t),\theta_j)-(c_1-\omega_0(t))\scpr{z_n(t)}{\theta_j} - \scpr{w_n(t)}{\theta_j}\\
	&\quad -\scpr{\phi\left(\mathcal{B}' z_n(t)\right)}{\mathcal{B}' \theta_j}_{{\mathbb{R}}^m} +\scpr{F\left(t,z_n(t)\right)}{\theta_j},\\ \scpr{\dot{w}_n(t)}{\theta_j} &= -(c_4-\omega_0(t))\scpr{w_n(t)}{\theta_j}+c_5\scpr{z_n(t)}{\theta_j}+\varphi_n\scpr{g(t)}{\theta_j}.
\end{aligned}
	\end{equation}
Define now
\begin{equation}\label{eq:transformation}
\begin{aligned}
v_n(t)&\coloneqq \varphi_n(t)^{-1}z_n(t)+q^n\cdoty_{\rm ref}(t),\\
u_n(t)&\coloneqq \varphi_n(t)^{-1}w_n(t),\\
\tilde{\mu}_i(t) &\coloneqq \varphi_n(t)^{-1}\mu_i(t) +\sum_{k=1}^m q_{k,i}y_{{\rm ref},k}(t),\\
\tilde{\nu}_i(t) &\coloneqq \varphi_n(t)^{-1}\nu_i(t),
\end{aligned}
\end{equation}
then $v_n(t)=\sum_{i=0}^n\tilde{\mu}_i(t)\theta_i$ and $u_n(t)=\sum_{i=0}^n\tilde{\nu}_i(t)\theta_i$. With this transformation we obtain that $(v_n, u_n)$ satisfies, for all $\theta\in W^{1,2}(\Omega)$, $\chi\in L^2(\Omega)$ and all $t\in[\gamma,T_n)$ that
\begin{equation*}
\begin{aligned}
\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\scpr{v_n(t)}{\theta}=&-\mathfrak{a}(v_n(t),\theta)+\scpr{p_3(v_n(t)+(q-q^n)\cdoty_{\rm ref}(t))-u_n(t)}{\theta}\\
&+\scpr{I_{s,i}(t)-(q-q^n)\cdot\dot{y}_{\rm ref}(t)+\sum_{k=1}^my_{{\rm ref},k}(t)\mathcal{A}(q_k-q_k^n) }{\theta}\\
&+\scpr{I_{s,e}^n(t)}{\mathcal{B}'\theta}_{{\mathbb{R}}^m},\\
\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\scpr{u_n(t)}{\chi}=&\scpr{c_5(v_n(t)+(q-q^n)\cdoty_{\rm ref}(t))-c_4u_n(t)}{\chi},\\
I_{s,e}^n(t)=&-\frac{k_0}{1-\varphi_n(t)^2\|\mathcal{B}' (v_n(t)-q^n\cdoty_{\rm ref}(t))\|^2_{{\mathbb{R}}^m}}(\mathcal{B}'( v_n(t)-q^n\cdoty_{\rm ref}(t))),
\end{aligned}
\end{equation*}
with $(u_n(\gamma),v_n(\gamma))=(u_\gamma,v_\gamma)$. Since there exists some $n_0\in{\mathbb{N}}$ with $q^{n}=q$ for all $n\geq n_0$, we have for all $n\geq n_0$, $\theta\in W^{1,2}(\Omega)$ and $\chi\in L^2(\Omega)$ that
\begin{equation}\label{eq:weak_uv}
\begin{aligned}
\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\scpr{v_n(t)}{\theta}=&-\mathfrak{a}(v_n(t),\theta)+\scpr{p_3(v_n(t))-u_n(t)}{\theta}\\
&+\scpr{I_{s,i}(t)}{\theta}+\scpr{I_{s,e}^n(t)}{\mathcal{B}'\theta}_{{\mathbb{R}}^m},\\
\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\scpr{u_n(t)}{\chi}=&\scpr{c_5v_n(t)-c_4u_n(t)}{\chi},\\
I_{s,e}^n(t)=&-\frac{k_0}{1-\varphi_n(t)^2\|\mathcal{B}' v_n(t)-y_{\rm ref}(t)\|^2_{{\mathbb{R}}^m}}(\mathcal{B}' v_n(t)-y_{\rm ref}(t)),
\end{aligned}
\end{equation}

\emph{Step 2: We show boundedness of $(z_n,w_n)$ in terms of~$\varphi_n$.}\\
Consider again the Lyapunov function \eqref{eq:Lyapunov} and observe that $\|z_n(t)\|^2 = \sum_{j=0}^n \mu_j(t)^2$ and $\|w_n(t)\|^2 = \sum_{j=0}^n \nu_j(t)^2$. We find that, for all $t\in[\gamma, T_n)$,
\begin{align*}
\tfrac{\text{\normalfont d}}{\text{\normalfont d}t} V(z_n(t),w_n(t)) &= c_5\sum_{j=0}^n\mu_j(t)\dot{\mu}_j(t)+\sum_{j=0}^n\nu_j(t)\dot{\nu}_j(t)\\ &=-c_5\sum_{j=0}^{n}\alpha_j\mu_j(t)^2- c_5 (c_1-\omega_0(t))\sum_{j=0}^{n}\mu_j(t)^2\\
&\quad-(c_4-\omega_0(t))\sum_{j=0}^{n}\nu_j(t)^2 -c_5\scpr{\phi(e_n(t))}{e_n(t)}_{{\mathbb{R}}^m} \\
&\quad +\varphi_n (t) \scpr{g(t)}{\sum_{i=0}^n\nu_i(t)\theta_i}\\
&\quad+c_5\scpr{F\left(t,\sum_{i=0}^n\mu_i(t)\theta_i\right)}{\sum_{i=0}^n\mu_i(t)\theta_i},
\end{align*}
hence, omitting the argument~$t$ for brevity in the following,
\begin{equation}\label{eq:Lyapunov_boundary_1}
\begin{aligned}
\tfrac{\text{\normalfont d}}{\text{\normalfont d}t} V(z_n,w_n)=&-c_5\mathfrak{a}(z_n,z_n)-c_5(c_1-\omega_0)\|z_n\|^2-(c_4-\omega_0)\|w_n\|^2\\
&-c_5\frac{k_0\|e_n\|_{{\mathbb{R}}^m}^2}{1-\|e_n\|_{{\mathbb{R}}^m}^2}+c_5\scpr{F(t,z_n)}{z_n}+\varphi_n \scpr{g}{w_n}.
\end{aligned}
\end{equation}
Next we use some Young and Hölder inequalities to estimate the term
\begin{align*} \scpr{F(t,z_n)}{z_n}&=\underbrace{\varphi_n(t)\scpr{f_{-1}(t)}{z_n}}_{I_{-1}} +\underbrace{\varphi_n(t)\scpr{f_0(t)}{z_n}}_{I_0}+\underbrace{\scpr{f_1(t)z_n}{z_n}}_{I_1}\\	&\quad+\underbrace{\varphi_n(t)^{-1}\scpr{f_2(t)z_n^2}{z_n}}_{I_2}-c_3\varphi_n(t)^{-2}\underbrace{\scpr{z_n^3}{z_n}}_{=\|z_n\|_{L^4}^4}.
\end{align*}
For the first term we derive, using Young's inequality for products with $p=4/3$ and $q=4$, that
\begin{align*}
I_{-1}&\leq \scpr{\frac{2^{1/2}\varphi_n^{3/2} |I_{s,i}|}{c_3^{1/4}}}{\frac{c_3^{1/4}|z_n|}{2^{1/2}\varphi_n^{1/2}}}+ \sum_{k=1}^m\scpr{\frac{(4m)^{1/4}\varphi_n^{3/2}\|y_{\rm ref}\|_\infty |Aq_k|}{c_3^{1/4}}}{\frac{c_3^{1/4}|z_n|}{(4m)^{1/4}\varphi_n^{1/2}}}\\
&\leq\frac{2^{2/3} 3\varphi_n^2\|I_{s,i}\|_{2,\infty}^{4/3}|\Omega|^{1/3}}{4c_3^{1/3}}+ \sum_{k=1}^m\frac{3(4m)^{1/3}\varphi_n^2\|y_{\rm ref}\|_\infty^{4/3}\|Aq_k\|^{4/3}|\Omega|^{1/3}}{4c_3^{1/3}}+ \frac{c_3\|z_n\|^4_{L^4}}{8\varphi_n^2}
\end{align*}
and with the same choice we obtain for the second term
\[I_0\leq\scpr{\frac{2^{1/4}\varphi_n^{3/2}\|f_0\|_{\infty,\infty}}{c_3^{1/4}}}{\frac{c_3^{1/4}|z_n|}{2^{1/4}\varphi_n^{1/2}}}\leq \frac{2^{1/3}3\varphi_n^2\|f_0\|_{\infty,\infty}^{4/3}|\Omega|}{4c_3^{1/3}}+\frac{c_3\|z_n\|^4_{L^4}}{8\varphi_n^2}.\]
Using $p=q=2$ we find that the third term satisfies
\[ I_1\leq\scpr{\frac{2\varphi_n\|f_1\|_{\infty,\infty}}{\sqrt{c_3}}}{\frac{\sqrt{c_3}|z_n|^2}{2\varphi_n}}\leq\frac{2\varphi_n^2\|f_1\|_{\infty,\infty}^2|\Omega|}{c_3}+ \frac{c_3\|z_n\|^4_{L^4}}{8\varphi_n^2},
\]
and finally, with $p=4$ and $q=4/3$,
\begin{align*} I_2&\leq\scpr{\varphi_n^{-1}\|f_2\|_{\infty,\infty}}{|z_n|^3}= \scpr{\frac{3^{3/2}\varphi_n^{1/2}\|f_2\|_{\infty,\infty}}{c_3^{3/4}}}{\left|\frac{c_3^{1/4}z_n}{\varphi_n^{1/2}\sqrt{3}}\right|^3}\\
&\leq\frac{9^3\varphi_n^2\|f_2\|^4_{\infty,\infty}|\Omega|}{4c_3^3}+\frac{c_3}{12\varphi_n^2}\|z_n\|^4_{L^4}.
\end{align*}
Summarizing, we have shown that
\[\scpr{F(t,z_n)}{z_n}\leq K_0\varphi_n^2-\frac{13c_3}{24\varphi_n^2}\|z_n\|^4_{L^4}\leq K_0\varphi_n^2-\frac{c_3}{2\varphi_n^2}\|z_n\|^4_{L^4},\]
where
\begin{align*}
K_0\coloneqq  & \ \frac{2^{2/3} 3\|I_{s,i}\|_{2,\infty}^{4/3}|\Omega|^{1/3}}{4c_3^{1/3}}+ \sum_{k=1}^m\frac{3(4m)^{1/3}\|y_{\rm ref}\|_\infty^{4/3}\|Aq_k\|^{4/3}|\Omega|^{1/3}}{4c_3^{1/3}}\\
&+ \frac{2^{1/3}3\|f_0\|_{\infty,\infty}^{4/3}|\Omega|}{4c_3^{1/3}}+\frac{2\|f_1\|_{\infty,\infty}^2|\Omega|}{c_3}+\frac{9^3\|f_2\|^4_{\infty,\infty}|\Omega|}{4c_3^3}.
\end{align*}
Finally, using Young's inequality with $p=q=2$, we estimate the last term in~\eqref{eq:Lyapunov_boundary_1} as follows
\[\varphi_n \scpr{g}{w_n}\leq\frac{\varphi_n^2\|g\|_{\infty,\infty}^2|\Omega|}{2c_4}+\frac{c_4}{2}\|w_n\|^2.\]
We have thus obtained the estimate
\begin{equation}\label{eq:Lyapunov_boundary_2}
\begin{aligned}
\tfrac{\text{\normalfont d}}{\text{\normalfont d}t} V(z_n,w_n)\leq&-(\sigma-2\omega_0) V(z_n,w_n)\\
&-c_5\mathfrak{a}(z_n,z_n)-c_5\frac{k_0\|e_n\|_{{\mathbb{R}}^m}^2}{1-\|e_n\|_{{\mathbb{R}}^m}^2}-\frac{c_3c_5}{2\varphi_n^{2}}\|z_n\|_{L^4}^4+\varphi_n^2K_1,
\end{aligned}
\end{equation}
where
\begin{align*}
\sigma\coloneqq  2\min\{c_1,c_4\},\quad  K_1\coloneqq  c_5K_0+\frac{\|g\|_{\infty,\infty}^2|\Omega|}{2c_4}.
\end{align*}
In particular, we have the conservative estimate
\[
\tfrac{\text{\normalfont d}}{\text{\normalfont d}t} V(z_n,w_n) \le -(\sigma-2\omega_0) V(z_n,w_n) +\varphi_n^2K_1
\]
on $[\gamma,T_n)$, which implies that
\[
V(z_n(t),w_n(t)) \le \mathrm{e}^{-K(t,\gamma)} V(z_n(\gamma),w_n(\gamma)) + \int_\gamma^t \mathrm{e}^{-K(t,s)} \varphi_n(s)^2 K_1 \ds{s},
\]
where
\[
K(t,s) = \int_s^t \sigma -2\omega_0(\tau) \ds{\tau} = \sigma(t-s) - 2\ln \varphi_n(t) + 2 \ln \varphi_n(s),\quad \gamma\le s\le t < T_n.
\]
Therefore, invoking $\varphi_n(\gamma)=\kappa_n$, for all $t\in[\gamma, T_n)$ we have
\begin{align*}
&c_5\|z_n(t)\|^2+\|w_n(t)\|^2 = 2V(z_n(t),w_n(t))\\
&\leq 2\mathrm{e}^{-\sigma (t-\gamma)}\frac{\varphi_n(t)^2}{\kappa_n^2}V(z_n(\gamma),w_n(\gamma))+\frac{2K_1}{\sigma}\varphi_n(t)^2\\
&= \varphi_n(t)^2\left((c_5\|v_\gamma^n-q^n\cdoty_{\rm ref}(\gamma)\|^2+\|u_\gamma^n\|^2)\mathrm{e}^{-\sigma (t-\gamma)}+2K_1\sigma^{-1}\right)\\
&\le \varphi_n(t)^2\left(c_5\|v_\gamma-q\cdoty_{\rm ref}(\gamma)\|^2+\|u_\gamma\|^2+2K_1\sigma^{-1}\right).
\end{align*}
Thus there exist $M,N>0$ which are independent of $n$ and $t$ such that
\begin{equation}\label{eq:L2_bound}
\forall\, t\in[\gamma,T_n):\
\|z_n(t)\|^2\leq M\varphi_n(t)^2\ \ \text{and}\ \
\|w_n(t)\|^2\leq N\varphi_n(t)^2,
\end{equation}
and, as a consequence,
\begin{equation}\label{eq:L2_bound_uv}
\forall\, t\in[\gamma,T_n):\ \|v_n(t)-q^n\cdoty_{\rm ref}(t)\|^2\leq M\ \ \text{and}\ \ \|u_n(t)\|^2\leq N.
\end{equation}

\emph{Step 3: We show $T_n=\infty$ and that $e_n$ is uniformly bounded away from~1 on~$[\gamma,\infty)$.}\\
\emph{Step 3a: We derive some estimates for $\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^2$ and for an integral involving $\|z_n\|^4_{L^4}$.} In a similar way in which we have derived~\eqref{eq:Lyapunov_boundary_2} we can obtain the estimate
\begin{equation}\label{eq:energy_boundary_z}
\begin{aligned}
\tfrac{1}{2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^2\leq&-\mathfrak{a}(z_n,z_n)-(c_1-\omega_0)\|z_n\|^2+\|z_n\|\|w_n\|\\
&-\frac{k_0\|e_n\|^2_{{\mathbb{R}}^m}}{1-\|e_n\|^2_{{\mathbb{R}}^m}}-\frac{c_3}{2\varphi_n^2}\|z_n\|^4_{L^4}+K_0\varphi_n^2.
\end{aligned}
\end{equation}
Using \eqref{eq:L2_bound} and $-c_1\|z_n\|^2\le 0$ leads to
\begin{align*}
\tfrac{1}{2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^2\leq&-\mathfrak{a}(z_n,z_n)-\frac{k_0\|e_n\|^2_{{\mathbb{R}}^m}}{1-\|e_n\|^2_{{\mathbb{R}}^m}}-\frac{c_3}{2\varphi_n^2}\|z_n\|^4_{L^4}\\
&+\|\dot{\varphi}\|_\infty M\varphi_n+(K_0+\sqrt{MN})\varphi_n^2.
\end{align*}
Hence,
\begin{equation}\label{eq:energy_boundary_2}
\begin{aligned}
\tfrac{1}{2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^2\leq&-\mathfrak{a}(z_n,z_n)-\frac{k_0\|e_n\|^2_{{\mathbb{R}}^m}}{1-\|e_n\|^2_{{\mathbb{R}}^m}}-\frac{c_3}{2\varphi_n^2}\|z_n\|^4_{L^4}+K_1\varphi_n+K_2\varphi_n^2
\end{aligned}
\end{equation}
on $[\gamma,T_n)$, where $K_1\coloneqq  M\|\dot{\varphi}\|_\infty$ and $K_2\coloneqq  K_0+\sqrt{MN}$.
Observe that
\[
\frac{c_3}{2}\varphi_n^{-3}\|z_n\|^4_{L^4}\leq-\frac{\varphi_n^{-1}}{2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^2+K_3,
\]
where $K_3\coloneqq  K_1+K_2\|\varphi\|_\infty$. Therefore,
\begin{align*}
&\frac{c_3}{2}\int_\gamma^t\mathrm{e}^s\varphi_n(s)^{-3}\|z_n(s)\|^4_{L^4}\ds{s}\\
&\le K_3(\mathrm{e}^t-\mathrm{e}^\gamma)-\frac{1}{2}\int_\gamma^t\mathrm{e}^s\varphi_n(s)^{-1}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n(s)\|^2\ds{s}\\
&=  K_3(\mathrm{e}^t-\mathrm{e}^\gamma)-\frac{1}{2}\left(\mathrm{e}^t\varphi_n(t)^{-1}\|z_n(t)\|^2-\frac{\|z_\gamma^n\|^2}{\kappa_n}\mathrm{e}^\gamma\right)\\
&\quad +\frac{1}{2}\int_\gamma^t\mathrm{e}^s\varphi_n(s)^{-2}(\varphi_n(s)-\dot{\varphi}_n(s))\|z_n(s)\|^2\ds{s}\\
&\le \frac{\mathrm{e}^t}{2}(2K_3+(\|\varphi\|_\infty+\Gamma_r^{-1}+\|\dot{\varphi}\|_\infty)M)+\kappa_n\mathrm{e}^\gamma(\|v_\gamma\|^2+\|q\cdoty_{\rm ref}(\gamma)\|^2),
\end{align*}
and hence there exist $D_0,D_1>0$ independent of $n$ and $t$ such that
\begin{equation}
\label{eq:expz4_boundary}
\forall\, t\in[\gamma,T_n):\ \int_\gamma^t\mathrm{e}^s\varphi_n(s)^{-3}\|z_n(s)\|^4_{L^4}\ds{s}\leq D_1\mathrm{e}^t+\kappa_n D_0.
\end{equation}
\emph{Step 3b: We derive an estimate for $\|\dot z_n\|^2$.} Multiplying the first equation in~\eqref{eq:weak} by $\dot{\mu}_j$ and summing up over $j\in\{0,\ldots,n\}$ we obtain
\begin{align*}
\|\dot{z}_n\|^2=&-\frac{1}{2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\mathfrak{a}(z_n,z_n)-\frac{c_1}{2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^2+\frac{k_0}{2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\ln(1-\|e_n\|^2_{{\mathbb{R}}^m})\\
&+ \scpr{\omega_0z_n+F\left(t,z_n\right)-w_n}{\dot{z}_n}.
\end{align*}
We can estimate the last term above by
\begin{align*}
\scpr{\omega_0z_n}{\dot{z}_n}\leq&\ \frac{7}{2}\|\dot{\varphi}\|_\infty^2\varphi_n^{-2}\|z_n\|^2+\frac{1}{14}\|\dot{z}_n\|^2 \stackrel{\eqref{eq:L2_bound}}{\leq} \frac{7}{2}\|\dot{\varphi}\|_\infty^2M+\frac{1}{14}\|\dot{z}_n\|^2,\\
\scpr{-w_n}{\dot{z}_n}\leq&\ \frac{7}{2}\|w_n\|^2+\frac{1}{14}\|\dot{z}_n\|^2,\\
\scpr{F\left(t,z_n\right)}{\dot{z}_n}\leq&\ \frac{7}{2}\varphi_n^2\left(m\sum_{k=1}^m \|y_{{\rm ref},k}\|_\infty^2 \|\mathcal{A} q_k\|^2+\|I_{s,i}\|^2_{2,\infty}+\|f_0\|_{\infty,\infty}^2|\Omega|\right)\\
&+\frac{7}{2}\|f_1\|^2_{\infty,\infty}\|z_n\|^2+\frac{7}{2}\varphi_n^{-2}\|f_2\|_{\infty,\infty}^2\|z_n\|^4_{L^4}\\
&+\frac{5}{14}\|\dot{z}_n\|^2-\frac{c_3}{4\varphi_n^2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^4_{L^4}.
\end{align*}
Inserting these inequalities, substracting $\tfrac12 \|\dot{z}_n\|^2$ and then multiplying by~$2$ gives
\begin{equation*}\label{eq:energy_boundary_1}
\begin{aligned}
\|\dot{z}_n\|^2=&-\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\mathfrak{a}(z_n,z_n)-c_1\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^2+k_0\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\ln(1-\|e_n\|^2_{{\mathbb{R}}^m})-\frac{c_3}{2\varphi_n^2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^4_{L^4}\\
&+7\varphi_n^2\left(m\sum_{k=1}^m\|y_{{\rm ref},k}\|_\infty^2 \|\mathcal{A} q_k\|^2+\|I_{s,i}\|^2_{2,\infty}\!+\|f_0\|_{\infty,\infty}^2|\Omega|+\|f_1\|^2_{\infty,\infty}M\!+\!N\!\right)\\
&+7\|\dot{\varphi}\|_\infty^2M+7\varphi_n^{-2}\|f_2\|_{\infty,\infty}^2\|z_n\|^4_{L^4}.
\end{aligned}
\end{equation*}
Now we add and subtract $\frac{1}{2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^2$, thus we obtain
\begin{align*}
\|\dot{z}_n\|^2\leq&-\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\mathfrak{a}(z_n,z_n)-\left(c_1+\frac{1}{2}\right)\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^2+k_0\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\ln(1-\|e_n\|^2_{{\mathbb{R}}^m}) -\frac{c_3}{2\varphi_n^2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^4_{L^4}\\
&+7(\|\varphi\|_\infty+\Gamma_r^{-1})^2\left(m\sum_{k=1}^m\|y_{{\rm ref},k}\|_\infty^2\|\mathcal{A} q_k\|^2+\|I_{s,i}\|^2_{2,\infty}+\|f_0\|_{\infty,\infty}^2|\Omega|\right.\\
&\left.\phantom{\sum_{i=0}^n}\hspace*{-6mm}+\|f_1\|^2_{\infty,\infty}M\right)+7(N(\|\varphi\|_\infty+\Gamma_r^{-1})^2 +\|\dot{\varphi}\|_\infty^2M)+7\varphi_n^{-2}\|f_2\|_{\infty,\infty}^2\|z_n\|^4_{L^4}\\
&+\frac{1}{2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^2.
\end{align*}
By the product rule we have
\[-\frac{c_3}{2\varphi_n^2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^4_{L^4}=-\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\left(\frac{c_3}{2\varphi_n^2}\|z_n\|^4_{L^4}\right)- c_3\varphi_n^{-3}\dot{\varphi_n}\|z_n\|^4_{L^4},\]
thus we find that
\begin{equation}\label{eq:energy_boundary_3}
\begin{aligned}
\|\dot{z}_n\|^2&+\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\mathfrak{a}(z_n,z_n)-k_0\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\ln(1-\|e_n\|^2_{{\mathbb{R}}^m})+\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\left(\frac{c_3}{2\varphi_n^2}\|z_n\|^4_{L^4}\right)\\
\leq&-\left(c_1+\frac{1}{2}\right)\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^2+E_1+E_2\varphi_n^{-3}\|z_n\|^4_{L^4}+\frac{1}{2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^2,
\end{aligned}
\end{equation}
where
\begin{align*}
E_1&\coloneqq  \ 7(\|\varphi\|_\infty+\Gamma_r^{-1})^2\left(m\sum_{k=1}^m\|y_{{\rm ref},k}\|_\infty^2\|Aq_k\|^2+ \|I_{s,i}\|^2_{2,\infty}+\|f_0\|_{\infty,\infty}^2|\Omega|\right.\\
&\quad\ \ \left.\phantom{\sum_{i=0}^n}\hspace*{-6mm}+\|f_1\|^2_{\infty,\infty}M\right) +7\big(N(\|\varphi\|_\infty+\Gamma_r^{-1})^2+\|\dot{\varphi}\|_\infty^2M\big),\\
E_2&\coloneqq  7\|f_2\|_{\infty,\infty}^2(\|\varphi\|_\infty+\Gamma_r^{-1})+c_3\|\dot{\varphi}\|_\infty
\end{align*}
are independent of $n$ and $t$.\\
\emph{Step 3c: We show uniform boundedness of~$e_n$.} Using~\eqref{eq:energy_boundary_2} in~\eqref{eq:energy_boundary_3} we obtain
\begin{align*}
\|\dot{z}_n\|^2+\dot\rho_n\leq&-\left(c_1+\frac{1}{2}\right)\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^2+E_1+E_2\varphi_n^{-3}\|z_n\|^4_{L^4}\\
&-\mathfrak{a}(z_n,z_n)-\frac{k_0\|e_n\|^2_{{\mathbb{R}}^m}}{1-\|e_n\|^2_{{\mathbb{R}}^m}}-\frac{c_3}{2\varphi_n^2}\|z_n\|^4_{L^4}+K_1\varphi_n+K_2\varphi_n^2\\
=&-\left(c_1+\frac{1}{2}\right)\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^2+E_2\varphi_n^{-3}\|z_n\|^4_{L^4}\\
&-\mathfrak{a}(z_n,z_n)-\frac{k_0}{1-\|e_n\|^2_{{\mathbb{R}}^m}}-\frac{c_3}{2\varphi_n^2}\|z_n\|^4_{L^4}+\Lambda,
\end{align*}
where
\begin{align*}
\rho_n&\coloneqq  \mathfrak{a}(z_n,z_n)-k_0\ln(1-\|e_n\|^2_{{\mathbb{R}}^m})+\frac{c_3}{2\varphi_n^2}\|z_n\|^4_{L^4},\\
\Lambda&\coloneqq  E_1+K_1(\|\varphi\|_\infty+\Gamma_r^{-1})+K_2(\|\varphi\|_\infty+\Gamma_r^{-1})^2+k_0,
\end{align*}
and we have used the equality
$$\frac{\|e_n\|^2_{{\mathbb{R}}^m}}{1-\|e_n\|^2_{{\mathbb{R}}^m}}=-1+\frac{1}{1-\|e_n\|^2_{{\mathbb{R}}^m}}.$$
Adding and subtracting $k_0\ln(1-\|e_n\|^2_{{\mathbb{R}}^m})$ leads to
\begin{align}
\|\dot{z}_n\|^2+\dot\rho_n\leq&-\rho_n-\left(c_1+\frac{1}{2}\right)\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^2+E_2\varphi_n^{-3}\|z_n\|^4_{L^4}\notag\\
&-k_0\left(\frac{1}{1-\|e_n\|^2_{{\mathbb{R}}^m}}+\ln(1-\|e_n\|^2_{{\mathbb{R}}^m})\right)+\Lambda\notag\\
\leq&-\rho_n-\left(c_1+\frac{1}{2}\right)\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^2+E_2\varphi_n^{-3}\|z_n\|^4_{L^4}+\Lambda, \label{eq:L2zdot_boundary}
\end{align}
where for the last inequality we have used that
\[\forall\, p\in(-1,1):\ \frac{1}{1-p^2}\geq\ln\left(\frac{1}{1-p^2}\right) = -\ln(1-p^2).\]
We may now use the integrating factor $\mathrm{e}^t$ to obtain
\[
\tfrac{\text{\normalfont d}}{\text{\normalfont d}t} \left(\mathrm{e}^t\rho_n\right) = \mathrm{e}^t(\rho_n + \dot \rho_n) \leq -\mathrm{e}^t\left(c_1+\frac{1}{2}\right)\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^2+E_2\mathrm{e}^t\varphi_n^{-3}\|z_n\|^4_{L^4}+\Lambda \mathrm{e}^t \underset{\le 0}{\underbrace{- \mathrm{e}^t \|\dot z_n\|^2}}.
\]
Integrating and using \eqref{eq:expz4_boundary} yields that for all $t\in[\gamma,T_n)$ we have
\begin{align*}
\mathrm{e}^t\rho_n(t)-\rho_n(\gamma)\mathrm{e}^\gamma\leq&\ (E_2D_1+\Lambda)\mathrm{e}^t+\kappa_n E_2D_0-\int_\gamma^t\mathrm{e}^s\left(c_1+\frac{1}{2}\right)\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n(s)\|^2\ds{s}\\
\leq&\ (E_2D_1+\Lambda)\mathrm{e}^t+\kappa_n E_2D_0+\left(c_1+\frac{1}{2}\right)\|z_\gamma^n\|^2\mathrm{e}^\gamma\\
&+\left(c_1+\frac{1}{2}\right)\int_\gamma^t\mathrm{e}^s\|z_n(s)\|^2\ds{s}\\
\stackrel{\eqref{eq:L2_bound}}{\leq}&\ (E_2D_1+\Lambda)\mathrm{e}^t+\kappa_n E_2D_0+\left(c_1+\frac12\right)\kappa_n^2\mathrm{e}^\gamma(\|v_\gamma-q\cdoty_{\rm ref}(\gamma)\|^2)\\
&+\left(c_1+\frac{1}{2}\right)(\|\varphi\|_\infty+\Gamma_r^{-1})^2M\mathrm{e}^t.
\end{align*}
Thus, there exit $\Xi_1,\Xi_2,\Xi_3>0$ independent of $n$ and $t$, such that
\[\rho_n(t)\leq\rho_n(\gamma)\mathrm{e}^{-(t-\gamma)}+\Xi_1+\kappa_n(\Xi_2+\kappa_n\Xi_3)\mathrm{e}^{-(t-\gamma)}.\]
Invoking the definition of $\rho_n$ and that $\mathrm{e}^{-(t-\gamma)}\leq1$ for $t\geq\gamma$ we find that
\begin{equation}\label{eq:rho}
\forall\, t\in[\gamma,T_n):\  \rho_n(t)\leq \rho_{n}^0+\Xi_1+\kappa_n\Xi_2+\kappa_n^2\Xi_3,
\end{equation}
where
\begin{align*}
\rho_{n}^0\coloneqq  &\ \kappa_n^2\mathfrak{a}(v_\gamma^n\!-\!q^n\cdoty_{\rm ref}(\gamma),v_\gamma^n\!-\!q^n\cdoty_{\rm ref}(\gamma))\!-\!k_0\ln(1\!-\!\kappa_n^2\|\mathcal{B}' (v_\gamma^n\!-\!q^n\cdoty_{\rm ref}(\gamma))\|^2_{{\mathbb{R}}^m})\\
&+\kappa_n^2\|v_\gamma^n-q^n\cdoty_{\rm ref}(\gamma)\|_{L^4}^4 = \rho_n(\gamma).
\end{align*}
Note that by construction of $\kappa_n$ and the Sobolev Embedding Theorem, $(\rho_n^0)_{n\in{\mathbb{N}}}$ is bounded, $\rho_n^0\to0$ as $n\to\infty$, so that $\rho_n^0$ can be bounded independently of $n$.\\
Again using the definition of $\rho_n$ and~\eqref{eq:rho} we find that
\begin{align*}
k_0 \ln\left(\frac{1}{1-\|e_n\|^2_{{\mathbb{R}}^m}}\right)  = \rho_n - \mathfrak{a}(z_n,z_n)- \frac{c_3}{2\varphi_n^2}\|z_n\|^4_{L^4}
\le \rho_{n}^0+\Xi_1+\kappa_n\Xi_2+\kappa_n^2\Xi_3,
\end{align*}
and hence
\[\frac{1}{1-\|e_n\|^2_{{\mathbb{R}}^m}}\leq\exp\left(\frac{1}{k_0}\left(\rho_{n}^0+\Xi_1+\kappa_n\Xi_2+\kappa_n^2\Xi_3\right)\right)=:\varepsilon(n).\]
We may thus conclude that
\begin{equation}\label{eq:err_bounded_away}
\forall\, t\in[\gamma,T_n):\ \|e_n(t)\|^2_{{\mathbb{R}}^m}\leq1-\varepsilon(n),
\end{equation}
or, equivalently,
\begin{equation}\label{eq:err_bdd}
\forall\, t\in[\gamma,T_n):\ \varphi_n(t)^2\|\mathcal{B}'( v_n(t)-q^n\cdoty_{\rm ref}(t))\|^2_{{\mathbb{R}}^m}\leq1-\varepsilon(n).
\end{equation}
Moreover, from~\eqref{eq:rho}, the definition of $\rho$, $k_0\ln(1-\|e_n\|^2_{{\mathbb{R}}^m})\leq0$ and Assumption~\ref{Ass1} we have that
\begin{align*}
\delta\|\nabla z_n\|^2+\frac{c_3}{2\varphi_n^2}\|z_n\|^4_{L^4}
\leq\rho_{n}^0+\Xi_1+\kappa_n\Xi_2+\kappa_n^2\Xi_3.
\end{align*}
Reversing the change of variables leads to
\begin{equation}\label{eq:potential}
\begin{aligned}
\forall\, t\in[\gamma,T_n):\ \delta\varphi_n(t)^2\|\nabla(v_n(t)-q^n\cdoty_{\rm ref}(t))\|^2&+\varphi_n(t)^2\|v_n(t)-q^n\cdoty_{\rm ref}(t)\|_{L^4}^4\\
&\leq\rho_{n}^0+\Xi_1+\kappa_n\Xi_2+\kappa_n^2\Xi_3,
\end{aligned}
\end{equation}
which implies that for all $t\in[\gamma,T_n)$ we have $v_n(t)\in W^{1,2}(\Omega)$.\\
\emph{Step 3d: We show that~$T_n=\infty$.} Assuming $T_n<\infty$ it follows from~\eqref{eq:err_bounded_away} that the graph of the solution~$(\mu^n,\nu^n)$ from Step~2 would be a compact subset of~${\mathbb{D}}$, a contradiction. Therefore, we have $T_n=\infty$.

\emph{Step 4: We show convergence of the approximate solution, uniqueness and regularity of the solution in $[\gamma,\infty)\times \Omega$.}\\
\emph{Step~4a: we prove some inequalities for later use.} From~\eqref{eq:rho} we have that, on $[\gamma,\infty)$,
\[\varphi_n^{-2}\|z_n\|_{L^4}^4\leq\rho_{n}^0+\Xi_1+\kappa_n\Xi_2 + \kappa_n^2\Xi_3.\]
Using a similar procedure as for the derivation of~\eqref{eq:expz4_boundary} we may obtain the estimate
\begin{equation}\label{eq:noexpz4_boundary}
\forall\, t\ge 0:\ \int_\gamma^t\varphi_n(s)^{-3}\|z_n(s)\|^4_{L^4}\ds{s}\leq\kappa_n d_0+d_1t
\end{equation}
for $d_0,d_1>0$ independent of $n$ and $t$.
Further, we can integrate \eqref{eq:L2zdot_boundary} on the interval $[\gamma,t]$ to obtain, invoking $\rho_n(t)\ge 0$ and~\eqref{eq:noexpz4_boundary},
\[
\int_\gamma^t\|\dot{z}_n(s)\|^2\ds{s}\leq\rho_{n}^0+ \left(c_1+\frac12\right)\kappa_n^2(\|v_\gamma-q\cdoty_{\rm ref}(\gamma)\|^2)+E_2(\kappa_n d_0+d_1t)+\Lambda t
\]
for all $t\ge \gamma$. Hence, there exist $S_0,S_1,S_2>0$ independent of $n$ and $t$ such that
\begin{equation}\label{eq:intzdot}
\forall\, t\ge \gamma:\ \int_\gamma^t\|\dot{z}_n(s)\|^2\ds{s}\leq\rho_{n}^0+S_0\kappa_n+S_1\kappa_n^2+S_2t.
\end{equation}
This implies existence of $S_3,S_4>0$ such that
\begin{equation}\label{eq:dot_var_v}
\forall\, t\ge \gamma:\ \int_\gamma^t\left\|\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}(\varphi_nv_n)\right\|^2\ds{s}\leq\rho_{n}^0+S_0\kappa_n+S_1\kappa_n^2+S_3t+S_4.
\end{equation}
In order to improve~\eqref{eq:noexpz4_boundary}, we observe that from~\eqref{eq:energy_boundary_z} it follows
\begin{align*}
\tfrac{1}{2}\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\|z_n\|^2\leq&-\mathfrak{a}(z_n,z_n)-(c_1-\omega_0)\|z_n\|^2+\|z_n\|\|w_n\|\\
&-\frac{k_0\|e_n\|^2_{{\mathbb{R}}^m}}{1-\|e_n\|^2_{{\mathbb{R}}^m}}-\frac{c_3}{2\varphi_n^2}\|z_n\|^4_{L^4}+K_0\varphi_n^2\\
\leq&\ \omega_0\|z_n\|^2-\frac{c_3}{2\varphi_n^2}\|z_n\|^4_{L^4}+K_2\varphi_n^2 -\mathfrak{a}(z_n,z_n)-\frac{k_0\|e_n\|^2_{{\mathbb{R}}^m}}{1-\|e_n\|^2_{{\mathbb{R}}^m}},
\end{align*}
which gives
\[\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\varphi_n^{-2}\|z_n\|^2\leq 2K_2-c_3\varphi_n^{-4}\|z_n\|^4_{L^4}-2\varphi_n^{-2}\mathfrak{a}(z_n,z_n)-\frac{2k_0\varphi_n^{-2}\|e_n\|^2_{{\mathbb{R}}^m}}{1-\|e_n\|^2_{{\mathbb{R}}^m}}.\]
This implies that for all $t\ge \gamma$ we have
\begin{equation}\label{eq:vphi4z4}
\begin{aligned}
\int_\gamma^t c_3\varphi_n(s)^{-4}\|z_n(s)\|^4_{L^4}
&+2\varphi_n(s)^{-2}\mathfrak{a}(z_n(s),z_n(s))+\frac{2k_0\varphi_n(s)^{-2}\|e_n(s)\|^2_{{\mathbb{R}}^m}}{1-\|e_n(s)\|^2_{{\mathbb{R}}^m}}\ds{s}\\
&\leq2K_2t+\|v_\gamma-q\cdoty_{\rm ref}(\gamma)\|^2,
\end{aligned}
\end{equation}
which is bounded independently of $n$. This shows that for all $t\ge \gamma$ we have
\begin{equation}\label{eq:L4}
\begin{aligned}
c_3\int_\gamma^t\|v_n(s)-q^n\cdoty_{\rm ref}(s)\|^4_{L^4}\ds{s}+\int_\gamma^t2\mathfrak{a}(v_n(s)-q^n\cdot y_{\rm ref}(s),v_n(s)-q^n\cdot y_{\rm ref}(s))\ds{s}\\
+\int_\gamma^t\frac{2k_0\|\mathcal{B}' (v_n(s)-q^n\cdoty_{\rm ref}(s))\|^2_{{\mathbb{R}}^m}}{1-\varphi_n(s)^{2}\|\mathcal{B}' (v_n(s)-q^n\cdoty_{\rm ref}(s))\|^2_{{\mathbb{R}}^m}}\ds{s}\leq2K_2t+\|v_\gamma-q\cdoty_{\rm ref}(\gamma)\|^2.
\end{aligned}
\end{equation}
In order to prove that $\|\dot{w}_n\|^2$ is bounded independently of $n$ and $t$, a last calculation is required. Multiply the second equation in~\eqref{eq:weak} by $\dot{\nu}_j$ and sum over $j$ to obtain
\[\|\dot{w}_n\|^2=-(c_4-\omega_0)\scpr{w_n}{\dot{w}_n}+c_5\scpr{z_n}{\dot{w}_n}+\varphi_n \scpr{g}{\dot{w}_n}.\]
Using $(\omega_0 - c_4) w_n = (\dot\varphi_n - c_4\varphi_n) \varphi_n^{-1} w_n$ and the inequalities
\begin{align*} -(c_4-\omega_0)\scpr{w_n}{\dot{w}_n}&\leq\frac{3}{2} \| \dot\varphi - c_4\varphi\|_\infty^2 \varphi_n^{-2}\|w_n\|^2+\frac{\|\dot{w}_n\|^2}{6}\\
&\leq\frac{3}{2}(\| \dot{\varphi}\|_\infty + c_4(\|\varphi\|_\infty+\Gamma_r^{-1}))^2 N+\frac{\|\dot{w}_n\|^2}{6},\\
c_5\scpr{z_n}{\dot{w}_n}&\leq\frac{3c_5^2}{2}\|z_n\|^2+\frac{1}{6}\|\dot{w}_n\|^2\\
&\leq\frac{3c_5^2M}{2}(\|\varphi\|_\infty+\Gamma_r^{-1})^2+\frac{1}{6}\|\dot{w}_n\|^2,\\
\varphi_n \scpr{g}{\dot{w}_n}& \leq\frac{3}{2}(\|\varphi\|_\infty+\Gamma_r^{-1})^2\|g\|^2_{\infty,\infty}|\Omega|+\frac{1}{6}\|\dot{w}_n\|^2,
\end{align*}
it follows that for all $t\ge\gamma$ we have
\begin{equation}\label{eq:dotw}
\begin{aligned}
\|\dot{w}_n(t)\|^2\leq&3\|(\| \dot{\varphi}\|_\infty + c_4(\|\varphi\|_\infty+\Gamma_r^{-1}))^2 N\\&+3c_5^2M(\|\varphi\|_\infty+\Gamma_r^{-1})^2+3(\|\varphi\|_\infty+\Gamma_r^{-1})^2\|g\|^2_{\infty,\infty}|\Omega|,\end{aligned}
\end{equation}
which is bounded independently of $n$ and $t$. Multiplying the second equation in~\eqref{eq:weak} by $\varphi_n^{-1}$ and $\theta_i$ and summing up over $i\in\{0,\dots,n\}$ leads to
\[\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}(\varphi_n^{-1}w_n)=-\varphi^{-2}\dot{\varphi}_nw_n+\varphi_n^{-1}\dot{w}_n=-c_4\varphi_n^{-1}w_n+c_5\varphi_n^{-1}z_n+g_n,\]
where
\[g_n\coloneqq  \sum_{i=0}^n\scpr{g}{\theta_i}\theta_i.\]
Taking the norm of the latter gives
\begin{align*}
\left\|\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}(\varphi_n^{-1}w_n)\right\|&\leq c_4\varphi_n^{-1}\|w_n\|+c_5\varphi_n^{-1}\|z_n\|+\|g_n\|\\
&\leq c_4N+c_5M+\|g\|_{\infty,\infty},
\end{align*}
thus
\begin{equation}\label{eq:dot_u}
\forall\, t\ge \gamma:\ \|\dot{u}_n(t)\|\leq c_4N+c_5M+\|g\|_{\infty,\infty}.
\end{equation}
\emph{Step 4b: We show that $(v_n,u_n)$ converges weakly.} Let $T>\gamma$ be given. Using a similar argument as in Section~\ref{ssec:mono_proof_tleqgamma}, we have that $v_n\in L^2(\gamma,T;W^{1,2}(\Omega))$ and $\dot{v}_n\in L^2(\gamma,T;W^{1,2}(\Omega)')$, since~\eqref{eq:L4} together with~\eqref{eq:err_bdd} implies that $I_{s,e}^n\in L^2(\gamma,T;{\mathbb{R}}^m)$ and $v_n\in L^2(\gamma,T;W^{1,2}(\Omega))$.\\
Furthermore, analogously to Section~\ref{ssec:mono_proof_tleqgamma}, we have that there exist subsequences such that
\begin{align*}
u_n\to u&\in W^{1,2}(\gamma,T;L^{2}(\Omega))\mbox{ weakly},\\
v_n\to v&\in L^2(\gamma,T;W^{1,2}(\Omega))\mbox{ weakly},\\
\dot{v}_n\to\dot{v}&\in L^2(\gamma,T;(W^{1,2}(\Omega))')\mbox{ weakly},
\end{align*}
so that $u,v\in C([\gamma,T];L^2(\Omega))$. Also $v_n^2\to v^2$ weakly in $L^2((\gamma,T)\times\Omega)$ and $v_n^3\to v^3$ weakly in $L^{4/3}((\gamma,T)\times\Omega)$.\\
We may infer further properties of $u$ and $v$. By \eqref{eq:L2_bound_uv}, \eqref{eq:potential}, \eqref{eq:dot_var_v} \& \eqref{eq:dot_u} we have that $u_n,\dot{u}_n$ lie in a bounded subset of $L^\infty(\gamma,\infty;L^2(\Omega))$ and that $v_n$ lie in a bounded subset of $L^\infty(\gamma,\infty;L^2(\Omega))$. Moreover, $\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}(\varphi_nv_n)\in L^2_{\rm loc}(\gamma,\infty;L^2(\Omega))$. Then, using Lemma~\ref{lem:weak_convergence}, we find a subsequence such that
\begin{align*}
u_n\to u&\in L^\infty(\gamma,T;L^2(\Omega))\mbox{ weak}^\star,\\
\dot{u}_n\to \dot{u}&\in L^\infty(\gamma,T;L^2(\Omega))\mbox{ weak}^\star,\\
v_n\to v&\in L^\infty(\gamma,T;L^{2}(\Omega))\mbox{ weak}^\star,\\
\varphi_nv_n\to \varphi v&\in L^\infty(\gamma,T;W^{1,2}(\Omega))\mbox{ weak}^\star,\\
\dot{v}_n\to \dot{v}&\in L^2(\gamma,T;W^{1,2}(\Omega)')\mbox{ weakly},\\
\varphi_n\dot{v}_n\to \varphi\dot{v}&\in L^2(\gamma,T;L^2(\Omega))\mbox{ weakly},
\end{align*}
since $\varphi_n\to\varphi$ in $BC([\gamma,T];{\mathbb{R}})$. Moreover, by $\inf_{t>\gamma+\delta}\varphi(t)>0$, we also have that $v\in L^\infty(\gamma+\delta,T;W^{1,2}(\Omega))$ and $\dot{v}\in L^2(\gamma+\delta,T;L^2(\Omega))$ for all~$\delta>0$.\\
Further, $\kappa_n,\rho_n^0\to0$ and
\[\varepsilon(n)\underset{n\to\infty}{\to}\varepsilon_0\coloneqq  \exp\left(-k_0^{-1}\Xi_1\right).\]
Thus, by \eqref{eq:L2_bound_uv}, \eqref{eq:err_bdd}, \eqref{eq:potential} \& \eqref{eq:L4} we have $v\in L^4((\gamma,T)\times\Omega)$ and for almost all $t\in[\gamma,T)$ the following estimates hold:
\begin{equation}\label{eq:potential_limit}
\begin{aligned}
&\|v(t)-q\cdoty_{\rm ref}(t)\|\leq \sqrt{M},\\
&\|u(t)\|\leq \sqrt{N},\\
&\varphi(t)^2\|\mathcal{B}' v(t)-y_{\rm ref}(t)\|^2_{{\mathbb{R}}^m}\leq1-\varepsilon_0,\\
&\delta\varphi(t)^2\|\nabla(v(t)-q\cdoty_{\rm ref}(t))\|^2+\varphi(t)^2\|v(t)-q\cdoty_{\rm ref}(t)\|_{L^4}^4\leq\Xi_1,\\
&\int_\gamma^t\|v(s)-q\cdoty_{\rm ref}(s)\|^4_{L^4}\ds{s}\leq2K_2t+\|v_\gamma-q\cdoty_{\rm ref}(\gamma)\|^2.
\end{aligned}
\end{equation}
Moreover, as in Section~\ref{ssec:mono_proof_tleqgamma}, $v_n\to v$ strongly in $L^2(\gamma,T;L^2(\Omega))$ and $u,v\in C([\gamma,T);L^2(\Omega))$ with $(u(\gamma),v(\gamma))=(u_\gamma,v_\gamma)$.\\
Hence, for $\chi\in L^2(\Omega)$ and $\theta\in W^{1,2}(\Omega)$ we have that $(u_n,v_n)$ satisfy the integrated version of \eqref{eq:weak_uv}, thus we obtain that for $t\in(\gamma,T)$
\begin{align*}
\scpr{v(t)}{\theta}=&\ \scpr{v_\gamma}{\theta}+\int_\gamma^t-\mathfrak{a}(v(s),\theta)+\scpr{p_3(v(s))-u(s)+I_{s,i}(s)}{\theta}\ds{s},\\
&+\int_\gamma^T\scpr{I_{s,e}(s)}{\mathcal{B}'\theta}_{{\mathbb{R}}^m}\ds{s},\\
\scpr{u(t)}{\chi}=&\ \scpr{u_\gamma}{\chi}+\int_\gamma^t\scpr{c_5v(s)-c_4u(s)}{\chi}\ds{s},\\
I_{s,e}(t)=&\ -\frac{k_0}{1-\varphi(t)^2\|\mathcal{B}' v(t)-y_{\rm ref}(t)\|^2_{{\mathbb{R}}^m}}(\mathcal{B}' v(t)-y_{\rm ref}(t))
\end{align*}
by bounded convergence \cite[Thm.~II.4.1]{Dies77}.
Hence, $(u,v)$ is a solution of~\eqref{eq:FHN_feedback} in $(\gamma,T)$. Moreover,~\eqref{eq:strong_delta} also holds in $W^{1,2}(\Omega)'$ for $t\geq\gamma$, that is
\begin{equation}\label{eq:Xminus}
\dot{v}(t)=\mathcal{A} v(t)+p_3(v(t))+\mathcal{B} I_{s,e}(t)-u(t)+I_{s,i}(t).
\end{equation}

\emph{Step 5: We show uniqueness of the solution on $[0,\infty)$.}\\
Using the same arguments as in Step~1e of Section~\ref{ssec:mono_proof_tleqgamma} together with $v,u\in L^4((\gamma,T)\times\Omega)$, it can be shown that the solution $(v,u)$ of~\eqref{eq:FHN_feedback} is unique on $(\gamma,T)$ for any $T>0$. Combining this with uniqueness on $[0,\gamma]$ we obtain a unique solution on $[0,\infty)$.

\emph{Step 6: We show the regularity properties of the solution.}\\
To this end, note that for all $\delta>0$ we have that
\[v\in L^2_{\rm loc}(\gamma,\infty;W^{1,2}(\Omega))\cap L^\infty(\gamma+\delta,\infty;W^{1,2}(\Omega)),\]
so that $I_r\coloneqq I_{s,i}+c_2v^2-c_3v^3-u\in L^2_{\rm loc}(\gamma,\infty;L^{2}(\Omega))\cap L^\infty(\gamma+\delta,\infty;L^2(\Omega))$, and the application of Proposition \ref{prop:hoelder} yields that $v\in BC([\gamma,\infty);L^2(\Omega))\cap BUC((\gamma,\infty);W^{1,2}(\Omega))$. By the uniform continuity of $v$ and the completeness of $W^{1,2}(\Omega)$, $v$ has a limit at $t=\gamma$, see for instance \cite[Thm.~II.13.D]{Simm63}. Thus, $v\in L^\infty(\gamma,\infty;W^{1,2}(\Omega))$. From Section \ref{ssec:mono_proof_tleqgamma} and the latter we have that $v\in L^2_{\rm loc}(0,\infty;W^{1,2}(\Omega))\cap L^\infty(\delta,\infty;W^{1,2}(\Omega))$ for all $\delta>0$, so we have
\begin{align*}
I_{s,e}&\in L^2_{\rm loc}(0,\infty;{\mathbb{R}}^m)\cap L^\infty(\delta,\infty;{\mathbb{R}}^m),\\
v&\in L^2_{\rm loc}(0,\infty;W^{1,2}(\Omega))\cap L^\infty(\delta,\infty;W^{1,2}(\Omega))\\
&\quad \ \ \cap BC([0,\infty);L^2(\Omega)) \cap BUC([\delta,\infty);W^{1,2}(\Omega)),
\end{align*}
so that $I_r\coloneqq  I_{s,i}+c_2v^2-c_3v^3-u\in L^2_{\rm loc}(0,\infty;L^{2}(\Omega))\cap L^\infty(\delta,\infty;L^2(\Omega))$.\\
Recall that by assumption we have $\mathcal{B}\in\mathcal{L}({\mathbb{R}}^m,W^{r,2}(\Omega)')$ for some $r\in [0,1]$. Applying Proposition \ref{prop:hoelder} we have that  for all $\delta>0$ the unique solution of~\eqref{eq:Xminus} satisfies
\begin{equation}\label{eq:sol_reg}
\begin{aligned}
\text{if $r=0$:} &\quad \forall\,\lambda\in(0,1):\ 	v\in C^{0,\lambda}([\delta,\infty);L^2(\Omega)); \\
\text{if $r\in(0,1)$:} &\quad
	v\in C^{0,1-r/2}([\delta,\infty);L^2(\Omega));\\
\text{if $r=1$:} &\quad	v\in C^{0,1/2}([\delta,\infty);L^2(\Omega)).
\end{aligned}
\end{equation}
Since $u,v\in BC([0,\infty);L^2(\Omega))$ and $\dot{u}=c_4v-c_5u$, we also have $\dot{u}\in BC([0,\infty);L^2(\Omega))$.

Now, from~\eqref{eq:sol_reg} and $\mathcal{B}' \in\mathcal{L}(W^{r,2}(\Omega),{\mathbb{R}}^m)$ for $r\in[0,1]$ we obtain that
\begin{itemize}
	\item for $r=0$ and $\lambda\in(0,1)$:\ $y=  \mathcal{B}' v\in C^{0,\lambda}([\delta,\infty);{\mathbb{R}}^m)$;
	\item for $r\in(0,1)$:\ $y=  \mathcal{B}' v\in C^{0,1-r}([\delta,\infty);{\mathbb{R}}^m)$;
	\item for $r=1$:\ $y=  \mathcal{B}' v\in BUC([\delta,\infty);{\mathbb{R}}^m)$.
\end{itemize}
Further, from \eqref{eq:potential_limit} we have
\[\forall\,t\geq\delta:\ \varphi(t)^{2}\|\mathcal{B}' v(t)-y_{\rm ref}(t)\|^2_{{\mathbb{R}}^m}\leq1-\varepsilon_0,\]
hence $I_{s,e}\in L^\infty(\delta,\infty;{\mathbb{R}}^m)$ and $I_{s,e}$ has the same regularity properties as $y$, since we have that $\varphi\in\Phi_\gamma$ and $y_{\rm ref}\in W^{1,\infty}(0,\infty;{\mathbb{R}}^m)$. Therefore, we have proved statements (i)--(iii) in Theorem~\ref{thm:mono_funnel} as well as~a) and~b).

It remains to show~c), for which we additionally require that $\mathcal{B}\in\mathcal{L}({\mathbb{R}}^m,W^{1,2}(\Omega))$. Then there exist $b_1,\ldots,b_m\in W^{1,2}(\Omega)$ such that $(\mathcal{B}' x)_i=\scpr{x}{b_i}$ for all $i=1,\dots,m$ and $x\in L^2(\Omega)$. Using the $b_i$ in the weak formulation for $i=1,\dots,m$, we have
\[\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}\scpr{v(t)}{b_i}=-\mathfrak{a}(v(t),b_i)+\scpr{p_3(v(t))-u(t)+I_{s,i}(t)}{b_i}+\scpr{I_{s,e}(t)}{\mathcal{B}' b_i}_{{\mathbb{R}}^m}.\]
Since $(\mathcal{B}' v(t))_i=\scpr{v(t)}{b_i}$, this leads to
\[\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}(\mathcal{B}' v(t)_i)=-\mathfrak{a}(v(t),b_i)+\scpr{p_3(v(t))-u(t)+I_{s,i}(t)}{b_i}+\scpr{I_{s,e}(t)}{\mathcal{B}' b_i}_{{\mathbb{R}}^m}.\]
Taking the absolute value and using the Cauchy-Schwarz inequality yields
\begin{align*}
\left|\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}(\mathcal{B}' v(t))_i\right|\leq&\ \|D\|_{L^\infty}\|v(t)\|_{W^{1,2}}\|b_i\|_{W^{1,2}}+\|p_3(v(t))-u(t)+I_{s,i}(t)\|_{L^2}\|b_i\|_{L^2}\\
&+\|I_{s,e}(t)\|_{{\mathbb{R}}^m}\|\mathcal{B}' b_i\|_{{\mathbb{R}}^m},
\end{align*}
and therefore
\begin{align*}
\forall\, i=1,\ldots,m\ \forall\,\delta>:\ \left\|\tfrac{\text{\normalfont d}}{\text{\normalfont d}t}(\mathcal{B}' v)_i\right\|_{L^\infty(\delta,\infty;{\mathbb{R}}^m)}<\infty,
\end{align*}
by which $y = \mathcal{B}'v \in W^{1,\infty}(\delta,\infty;{\mathbb{R}}^m)$ as well as $I_{s,e} \in W^{1,\infty}(\delta,\infty;{\mathbb{R}}^m)$. This completes the proof of the theorem.
\ensuremath{\hfill\ensuremath{\square}}

\newlength\fheight
\newlength\fwidth
\setlength\fheight{0.3\linewidth}
\setlength\fwidth{0.9\linewidth}

\section{A numerical example}
\label{sec:numerics}

In this section, we illustrate the practical applicability of the funnel controller by means of a numerical example. The setup chosen here is a standard test example for termination of reentry waves and has been considered similarly e.g.\ in~\cite{BreiKuni17,KuniNagaWagn11}. All simulations are generated on an AMD Ryzen 7 1800X @ 3.68 GHz x 16,
64 GB RAM,  MATLAB\textsuperscript{\textregistered} \;Version 9.2.0.538062 (R2017a). The solutions of the ODE systems are obtained by the MATLAB\textsuperscript{\textregistered}\;routine \texttt{ode23}. The parameters for the FitzHugh-Nagumo model \eqref{eq:FHN_model} used here are as follows:
\begin{align*}
\Omega&=(0,1)^2,\ \ D=\begin{bmatrix} 0.015 & 0 \\ 0 & 0.015 \end{bmatrix}, \ \
\begin{pmatrix}
c_1 \\ c_2 \\ c_3 \\ c_4 \\ c_5
\end{pmatrix}
\approx
\begin{pmatrix}
1.614\\ 0.1403 \\ 0.012\\ 0.00015\\ 0.015
\end{pmatrix}.
\end{align*}
 The spatially discrete system of ODEs corresponds to a finite element discretization with piecewise linear finite elements on a uniform $64\times 64$ mesh.

  For the control action, we assume that $\mathcal{B}\in \mathcal{L}(\mathbb R^4,W^{1,2}(\Omega)')$, where the Neumann control operator is defined by
 \begin{align*}
 \mathcal{B}'z &= \begin{pmatrix} \int_{\Gamma_1} z(\xi)\, \mathrm{d}\sigma,\int_{\Gamma_2} z(\xi)\, \mathrm{d}\sigma,\int_{\Gamma_3} z(\xi)\, \mathrm{d}\sigma ,\int_{\Gamma_4} z(\xi) \,\mathrm{d}\sigma\end{pmatrix}^\top,

     \\ \Gamma_1 &= \{1\}\times [0,1],  \ \ \Gamma_2= [0,1]\times \{1\}, \ \ \Gamma_3 = \{0\}\times [0,1], \ \ \Gamma=[0,1]\times \{0\}.
 \end{align*}
 The purpose of the numerical example is to model a typical defibrillation process as a tracking problem as discussed above. In this context, system \eqref{eq:FHN_model} is initialized with $(v(0),u(0))=(v_0^*,u_0^*)$ and $I_{s,i}=0=I_{s,e}$, where $(v_0^*,u_0^*)$ is an arbitrary snapshot of a reentry wave. The resulting reentry phenomena are shown in Fig.~\ref{fig:reentry_waves} and resemble a dysfunctional heart rhythm which impedes  the intracellular stimulation current~$I_{s,i}$. The objective is to design a stimulation current $I_{s,e}$ such that the dynamics return to a natural heart rhythm modeled by a reference trajectory $y_{\text{ref}}$. The trajectory $y_{\text{ref}} = \mathcal{B}' v_{\text{ref}}$ corresponds to a solution $(v_{\text{ref}},u_{\text{ref}})$ of~\eqref{eq:FHN_model} with $(v_{\text{ref}}(0),u_{\text{ref}}(0))=(0,0)$, $I_{s,e}=0$ and
 \begin{align*}
  I_{s,i}(t) = 101\cdot w(\xi) (\chi_{[49,51]}(t) + \chi_{[299,301]}(t)),
  \end{align*}
   where the excitation domain of the intracellular stimulation current~$I_{s,i}$ is described by
 \begin{align*}
  w(\xi) = \begin{cases} 1 , & \text{if } (\xi_1-\frac{1}{2})^2+(\xi_2-\frac{1}{2})^2 \le 0.0225, \\
  0 , & \text{otherwise}. \end{cases}
 \end{align*}
  The smoothness of the signal is guaranteed by convoluting the original signal with a triangular function.
The function $\varphi$ characterizing the performance funnel (see Fig.~\ref{fig:funnel_error}) is chosen as
\begin{align*}
\varphi(t)=
\begin{cases}
0, & t \in [0,0.05] ,\\
\mathrm{tanh}(\frac{t}{100}), & t > 0.05 .
\end{cases}
\end{align*}

\begin{figure}[tb]
\begin{subfigure}{.5\linewidth}
\begin{center}
\includegraphics[scale=0.4]{reentry_1.pdf}

  \end{center}
\end{subfigure}
\begin{subfigure}{.5\linewidth}
\begin{center}
  \includegraphics[scale=0.4]{reentry_2.pdf}

  \end{center}
\end{subfigure}
\caption{Snapshots of reentry waves for $t=100$ (left) and $t=200$ (right).}
\label{fig:reentry_waves}
\end{figure}

\begin{figure}[tb]
\begin{center}
  \input{funnel_error.tikz}
  \end{center}
    \caption{Error dynamics and funnel boundary.}
  \label{fig:funnel_error}
\end{figure}
 Fig.~\ref{fig:y_funnel} shows the results of the closed-loop system for $(v(0),u(0))=(v_0^*,u_0^*)$ and the control law
\begin{align*}
 I_{s,e}(t)=-\frac{0.75}{1-\varphi(t)^2\|\mathcal{B}' v(t)-y_{\rm ref}(t)\|^2_{{\mathbb{R}}^m}}(\mathcal{B}'v(t)-y_{\rm ref}(t)),
\end{align*}
which is visualized in Fig.~\ref{fig:u_funnel}. Let us note that the sudden changes in the feedback law are due to the jump discontinuities of the intracellular stimulation current $I_{s,i}$ used for simulating a regular heart beat.

\setlength\fheight{0.3\linewidth}
\setlength\fwidth{0.4\linewidth}

\begin{figure}[tb]
\begin{subfigure}{.5\linewidth}
\begin{center}
  \input{y_funnel_1.tikz}

  \end{center}
\end{subfigure}\quad
\begin{subfigure}{.5\linewidth}
\begin{center}
  \input{y_funnel_2.tikz}

  \end{center}
\end{subfigure}
\begin{subfigure}{.5\linewidth}
\begin{center}
  \input{y_funnel_3.tikz}

  \end{center}
\end{subfigure}\quad
\begin{subfigure}{.5\linewidth}
\begin{center}
  \input{y_funnel_4.tikz}

  \end{center}
\end{subfigure}
\caption{Reference signals and outputs of the funnel controlled system.}
\label{fig:y_funnel}
\end{figure}
We see from Fig.~\ref{fig:y_funnel} that the controlled system tracks the desired reference signal with the prescribed performance. Also note that the performance constraints are not active on the interval $[0,0.05]$. Fig.~\ref{fig:u_funnel} further shows that the tracking is achieved with a comparably small control effort.
\begin{figure}[tb]
\begin{subfigure}{.5\linewidth}
\begin{center}
  \input{u_funnel_1.tikz}

  \end{center}
\end{subfigure}\quad
\begin{subfigure}{.5\linewidth}
\begin{center}
  \input{u_funnel_2.tikz}

  \end{center}
\end{subfigure}
\begin{subfigure}{.5\linewidth}
\begin{center}
  \input{u_funnel_3.tikz}

  \end{center}
\end{subfigure}\quad
\begin{subfigure}{.5\linewidth}
\begin{center}
  \input{u_funnel_4.tikz}

  \end{center}
\end{subfigure}
\caption{Funnel control laws.}
\label{fig:u_funnel}
\end{figure}

\begin{appendices}

\section{Neumann elliptic operators}\label{sec:neum_lapl}

We collect some further facts on Neumann elliptic operators as introduced in Proposition~\ref{prop:Aop}.

\begin{Prop}\label{prop:Aop_n}
If Assumption~\ref{Ass1} holds, then the Neumann elliptic operator $\mathcal{A}$ on $\Omega$ associated to $D$ has the following properties:
\begin{enumerate}[a)]
\item\label{item:Aop3} there exists $\nu\in(0,1)$ such that $\mathcal{D}(\mathcal{A})\subset C^{0,\nu}(\Omega)$;
\item\label{item:Aop4} $\mathcal{A}$ has compact resolvent;
\item\label{item:Aop5} there exists a real-valued and monotonically increasing sequence $(\alpha_j)_{j\in{\mathbb{N}}_0}$ such that
	\begin{enumerate}[(i)]
		\item $\alpha_0=0$, $\alpha_1>0$ and $\lim_{j\to\infty}\alpha_j=\infty$, and
		\item the spectrum of $\mathcal{A}$ reads $\sigma(\mathcal{A})=\setdef{-\alpha_j}{j\in{\mathbb{N}}_0}$
	\end{enumerate}
	and an orthonormal basis $(\theta_j)_{j\in{\mathbb{N}}_0}$ of $L^2(\Omega)$, such that
	\begin{equation}\forall\,x\in\mathcal{D}(\mathcal{A}):\ \mathcal{A} x=-\sum_{j=0}^\infty\alpha_j\scpr{x}{\theta_j}\theta_j,\label{eq:spectr}\end{equation}
and the domain of $\mathcal{A}$ reads
\begin{equation}\mathcal{D}(\mathcal{A})=\setdef{\sum_{j=0}^\infty \lambda_j \theta_j}{(\lambda_j)_{j\in{\mathbb{N}}_0}\text{ with }\sum_{j=1}^\infty \alpha_j^2 |\lambda_j|^2<\infty}.\label{eq:spectrda}\end{equation}
\end{enumerate}
\end{Prop}
\begin{proof}
Statement \ref{item:Aop3}) follows from \cite[Prop.~3.6]{Nitt11}.\\
To prove \ref{item:Aop4}), we first use that the ellipticity condition \eqref{eq:ellcond} implies
\begin{equation}\delta\|z\|+\|\mathcal{A} z\|\geq  \|z\|_{W^{1,2}}.\label{eq:Acoer}\end{equation}
Since $\partial\Omega$ is Lipschitz, $\Omega$ has the cone property \cite[p.~66]{Adam75}, and we can apply the Rellich-Kondrachov Theorem~\cite[Thm.~6.3]{Adam75}, which states that
$W^{1,2}(\Omega)$ is compactly embedded in $L^2(\Omega)$. Combining this with \eqref{eq:Acoer}, we obtain that $\mathcal{A}$ has compact resolvent.\\
We show~\ref{item:Aop5}). Since $\mathcal{A}$ has compact resolvent and is self-adjoint by Proposition~\ref{prop:Aop}, we obtain from \cite[Props.~3.2.9 \&~3.2.12]{TucsWeis09} that there exists a~real valued sequence $(\alpha_j)_{j\in{\mathbb{N}}_0}$ with $\lim_{j\to\infty}|\alpha_j|=\infty$ and \eqref{eq:spectr}, and the domain of $\mathcal{A}$ has the representation \eqref{eq:spectrda}. Further taking into account that
\[\forall\, z\in \mathcal{D}(\mathcal{A}):\ \scpr{z}{\mathcal{A} z}=-\mathfrak{a}(z,z)\leq0,\]
we obtain that $\alpha_j\geq0$ for all $j\in{\mathbb{N}}_0$. Consequently, it is no loss of generality to assume that $(\alpha_j)_{j\in{\mathbb{N}}_0}$ is monotonically increasing. It remains to prove that $\alpha_0=0$ and $\alpha_1>0$: On the one hand, we have that the constant function $1_\Omega\in L^2(\Omega)$ satisfies $\mathcal{A} 1_\Omega=0$, since
\[\forall\, z\in W^{1,2}(\Omega):\ \scpr{z}{ \cA1_\Omega}=-\mathfrak{a}(z,1_\Omega)=-\scpr{\nabla z_1}{D\nabla 1_\Omega}=0.\]
On the other hand, if $z\in\ker \mathcal{A}$, then
\[0=\scpr{z}{\mathcal{A} z}=-\mathfrak{a}(z,z)=-\scpr{\nabla z}{D\nabla z},\]
and the pointwise positive definiteness of $D$ implies $\nabla z=0$, whence $z$ is a constant function. This
gives $\dim \ker \mathcal{A}=1$, by which $\alpha_0=0$ and $\alpha_1>0$.
\end{proof}

\section{Interpolation spaces}
\label{sec:mono_prep_proof}

We collect some results on interpolation spaces, which are necessary for the proof of Theorem~\ref{thm:mono_funnel}. For a (more) general interpolation theory, we refer to \cite{Luna18}.

\begin{Def}
Let $X,Y$ be Hilbert spaces and let $\alpha\in[0,1]$. Consider the function
\[K:(0,\infty)\times (X+Y)\to{\mathbb{R}},\ (t,x)\mapsto \underset{x=a+b}{\inf_{a\in X,\, b\in Y,}}\, \|a\|_X+t\|b\|_Y.\]
The {\em interpolation space} $(X,Y)_{\alpha}$ is defined by
\[(X,Y)_{\alpha}:=\setdef{x\in X+Y}{\Big(t\mapsto t^{-\alpha} K(t,x)\Big)\in L^2(0,\infty)},\]
and it is a Hilbert space with the norm
\[\|x\|_{(X,Y)_\alpha}=\|t\mapsto t^{-\alpha} K(t,x)\|_{L^2}.\]
\end{Def}

Note that interpolation can be performed in a more general fashion for Banach spaces $X$, $Y$. More precise, we may utilize the $L^p$-norm of the map $t\mapsto t^{-\alpha} K(t,x)$ for some $p\in[1,\infty)$ instead of the $L^2$-norm in the above definition. However, this does not lead to Hilbert spaces $(X,Y)_\alpha$, not even when~$X$ and~$Y$ are Hilbert spaces.

For a~self-adjoint operator $A:\mathcal{D}(A)\subset X\to X$, $X$ a Hilbert space and $n\in{\mathbb{N}}$, we may define the space $X_n:=\mathcal{D}(A^n)$ by $X_0=X$ and $X_{n+1}:=\setdef{x\in X_n}{Ax\in X_n}$. This is a Hilbert space with norm $\|z\|_{X_{n+1}}=\|-\lambda z+Az\|_{X_n}$, where $\lambda\in{\mathbb{C}}$ is in the resolvent set of $A$.
Likewise, we introduce $X_{-n}$ as the completion of $X$ with respect to the norm $\|z\|_{X_{-n}}=\|(-\lambda I+A)^{-n}z\|$. Note that $X_{-n}$ is the dual of $X_n$ with respect to the pivot space $X$, cf.~\cite[Sec.~2.10]{TucsWeis09}.
Using interpolation theory, we may further introduce the spaces~$X_\alpha$ for any $\alpha\in{\mathbb{R}}$ as follows.

\begin{Def}\label{Def:int-space-A}
Let $\alpha\in{\mathbb{R}}$, $X$ a Hilbert space and $A:\mathcal{D}(A)\subset X\to X$ be self-adjoint. Further, let $n\in{\mathbb{Z}}$ be such that $\alpha\in[n,n+1)$.
The space $X_\alpha$ is defined as the interpolation space
\[X_\alpha=(X_{n},X_{n+1})_{\alpha-n}.\]
\end{Def}

The Reiteration Theorem, see~\cite[Cor.~1.24]{Luna18}, together with~\cite[Prop.~3.8]{Luna18} yields that for all $\alpha\in[0,1]$ and $\alpha_1,\alpha_2\in{\mathbb{R}}$ with $\alpha_1\leq\alpha_2$ we have that
\begin{equation}\label{eq:Xalpha-reit}
    (X_{\alpha_1},X_{\alpha_2})_{\alpha}=X_{\alpha_1+\alpha (\alpha_2-\alpha_1)}.
\end{equation}
Next we characterize interpolation spaces associated with the Neumann elliptic operator $\mathcal{A}$.

\begin{Prop}\label{prop:Aop2_n}
Let Assumption~\ref{Ass1} hold and $\mathcal{A}$ be the Neumann elliptic operator on $\Omega$ associated to $D$. Further let $X_\alpha$, $\alpha \in{\mathbb{R}}$, be the corresponding interpolation spaces with, in particular, $X=X_0=L^2(\Omega)$. Then
\[X_{r/2}=W^{r,2}(\Omega)\;\text{  for all $r\in[0,1]$}.\]
\end{Prop}
\begin{proof}
The equation $X_{1/2}=W^{1,2}(\Omega)$ is an immediate consequence of Kato's Second Representation Theorem~\cite[Sec.~VI.2, Thm.~2.23]{Kato80}. For general $r\in[0,1]$ equation~\eqref{eq:Xalpha-reit} implies
\[X_{r/2}=(X_{0},X_{1/2})_{r}.\]
Now using that $X_0=L^2(\Omega)$ by definition and, as already stated, $X_{1/2}=W^{1,2}(\Omega)$, it follows from~\cite[Thm.~1.35]{Yagi10} that
\[(L^2(\Omega),W^{1,2}(\Omega))_{r}=W^{r,2}(\Omega),\]
and thus $X_{r/2}=W^{r,2}(\Omega)$.
\end{proof}

\begin{Rem}
\label{rem:X_alpha}
In terms of the spectral decomposition \eqref{eq:spectr}, the space $X_\alpha$ has the representation
\begin{equation}X_\alpha=\setdef{\sum_{j=0}^\infty \lambda_j \theta_j}{(\lambda_j)_{j\in{\mathbb{N}}_0}\text{ with }\sum_{j=1}^\infty \alpha_j^{2\alpha} |\lambda_j|^2<\infty}.\label{eq:Xrspec}\end{equation}
This follows from a~combination of~\cite[Thm.~4.33]{Luna18} with~\cite[Thm.~4.36]{Luna18}.
\end{Rem}

\section{Abstract Cauchy problems and regularity}
\label{sec:mono_prep_proof2}

We consider mild solutions of certain abstract Cauchy problems and the concept of admissible control operators. This notion is well-known in infinite-dimensional linear systems theory with unbounded control and observation operators and we refer to~\cite{TucsWeis09} for further details.

Let $X$ be a real Hilbert space and recall that a semigroup $(\T_t)_{t\geq0}$ on $X$ is a $\mathcal{L}(X,X)$-valued map satisfying ${\mathbb{T}}_0=I_{X}$ and ${\mathbb{T}}_{t+s}={\mathbb{T}}_t {\mathbb{T}}_s$, $s,t\geq0$, where $I_{X}$ denotes the identity operator, and $t\mapsto {\mathbb{T}}_t x$ is continuous for every $x\in X$. Semigroups are characterized by their generator~$A$, which is a, not necessarily bounded, operator on~$X$. If $A:\mathcal{D}(A)\subset X\to X$ is self-adjoint with $\scpr{x}{Ax}\leq0$ for all $x\in\mathcal{D}(A)$, then
it generates a~contractive, analytic semigroup $(\T_t)_{t\geq0}$ on $X$, cf.~\cite[Thm.~4.2]{ArenElst12}. Furthermore, if additionally there exists $\omega_0>0$ such that $\scpr{x}{Ax}\leq-\omega_0 \|x\|^2$ for all $x\in\mathcal{D}(A)$, then the semigroup $(\T_t)_{t\geq0}$  generated by $A$ satisfies $\|{\mathbb{T}}_t\|\leq \mathrm{e}^{-\omega_0 t}$ for all $t\geq0$; the smallest number $\omega_0$ for which this is true is called \emph{growth bound} of $(\T_t)_{t\geq0}$. We can further conclude from~\cite[Thm.~6.13\,(b)]{Pazy83} that, for all $\alpha\in{\mathbb{R}}$, $(\T_t)_{t\geq0}$ restricts (resp.\ extends) to an analytic semigroup $(({\mathbb{T}}|_{\alpha})_t)_{t\ge0}$ on $X_\alpha$ with same growth bound as $(\T_t)_{t\geq0}$. Furthermore, we have $\im {\mathbb{T}}_t\subset X_r$ for all $t>0$ and $r\in{\mathbb{R}}$, see~\cite[Thm.~6.13(a)]{Pazy83}. In the following we present an estimate for the corresponding operator norm.

\begin{Lem}\label{lem:A_alpha}
	Assume that $A:\mathcal{D}(A)\subset X\to X$, $X$ a Hilbert space, is self-adjoint and there exists $\omega_0>0$ with $\scpr{x}{Ax}\leq-\omega_0 \|x\|^2$ for all $x\in\mathcal{D}(A)$. Then there exist $M,\omega>0$ such that  the semigroup $(\T_t)_{t\geq0}$ generated by $A$ satisfies
	\[
    \forall\, \alpha\in[0,2]\ \forall\, t>0:\ \|{\mathbb{T}}_t\|_{\mathcal{L}(X,X_\alpha)}\leq M(1+t^{-\alpha})\mathrm{e}^{-\omega t}.\]
	Thus, for each $\alpha\in[0,2]$ there exists $K>0$ such that
	\[\sup_{t\in[0,\infty)}t^\alpha\|{\mathbb{T}}_t\|_{\mathcal{L}(X,X_\alpha)}<K.\]
\end{Lem}

\begin{proof}
	Since $A$ with the above properties generates an exponentially stable analytic semigroup $(\T_t)_{t\geq0}$, the cases $\alpha\in[0,1]$ and $\alpha=2$ follow from~\cite[Cor.~3.10.8~\&~Lem.~3.10.9]{Staf05}. The result for $\alpha\in[1,2]$ is a consequence of~\cite[Lem~3.9.8]{Staf05} and interpolation between $X_1$ and $X_2$, cf.\ Appendix~\ref{sec:mono_prep_proof}.
\end{proof}

Next we consider the abstract Cauchy problem with source term.

\begin{Def}\label{Def:Cauchy}
	Let $X$ be a Hilbert space, $A:\mathcal{D}(A)\subset X\to X$ be self-adjoint with $\scpr{x}{Ax}\leq0$ for all $x\in\mathcal{D}(A)$, $T\in(0,\infty]$, and $\alpha\in[0,1]$. Let $(\T_t)_{t\geq0}$ be the semigroup on $X$ generated by $A$, and let $B\in\mathcal{L}({\mathbb{R}}^m,X_{-\alpha})$. For $x_0\in X$, $p\in[1,\infty]$, $f\in L^p_{\loc}(0,T;X)$ and $u\in L^p_{\loc}(0,T;{\mathbb{R}}^m)$, we call~$x:[0,T)\to X$ a \emph{mild solution} of
	\begin{equation}\label{eq:abstract_cauchy}
	\begin{aligned}
	\dot{x}(t)=Ax(t)+f(t)+Bu(t),\quad
	x(0)=x_0
	\end{aligned}
	\end{equation}
	on $[0,T)$, if it satisfies
	\begin{equation}\label{eq:mild_solution}
	\forall\, t\in[0,T):\ x(t)={\mathbb{T}}_tx_0+\int_0^t{\mathbb{T}}_{t-s}f(s)\ds{s}+\int_0^t({\mathbb{T}}|_{-\alpha})_{t-s}Bu(s)\ds{s}.
	\end{equation}
We further call $x:[0,T)\to X$ a \emph{strong solution} of \eqref{eq:abstract_cauchy} on $[0,T)$, if $x$ in~\eqref{eq:mild_solution}  satisfies $x\in C([0,T);X)\cap W^{1,p}_{\rm loc}(0,T;X_{-1})$.
\end{Def}

Definition~\ref{Def:Cauchy} requires that the integral $\int_0^t({\mathbb{T}}|_{-\alpha})_{t-s}Bu(s)\ds{s}$ is in $X$, whilst the integrand is not necessarily in $X$. This motivates the definition of admissibility, which is now introduced for self-adjoint $A$. Note that admissibility can  also be defined for arbitrary generators of semigroups, see~\cite{TucsWeis09}.

\begin{Def}\label{Def:Adm}
Let $X$ be a Hilbert space, $A:\mathcal{D}(A)\subset X\to X$ be self-adjoint with $\scpr{x}{Ax}\leq0$ for all $x\in\mathcal{D}(A)$, $T\in(0,\infty]$, $\alpha\in[0,1]$ and $p\in[1,\infty]$. Let $(\T_t)_{t\geq0}$ be the semigroup on $X$ generated by $A$, and let $B\in\mathcal{L}({\mathbb{R}}^m,X_{-\alpha})$. Then $B$ is called an {\em $L^p$-admissible (control operator) for $(\T_t)_{t\geq0}$}, if for some (and hence any) $t> 0$ we have
$$\forall\,  u\in L^p(0,t;{\mathbb{R}}^m):\ \Phi_{t}u :=\int_0^t({\mathbb{T}}|_{-\alpha})_{t-s}Bu(s)\ds{s} \in X.$$
By a closed graph theorem argument this implies that $\Phi_t\in \mathcal{L}(L^p(0,t;{\mathbb{R}}^m),X)$ for all $t> 0$. We call $B$ an  {\em infinite-time $L^p$-admissible (control operator) for $(\T_t)_{t\geq0}$}, if
\[
    \sup_{t>0} \|\Phi_t\| < \infty.
\]
\end{Def}

In the following we show that for $p\ge 2$ and $\alpha\le 1/2$ any~$B$ is admissible and the mild solution of the abstract Cauchy problem is indeed a strong solution.

\begin{Lem}\label{lem:abstract_solution}
	Let $X$ be a Hilbert space, $A:\mathcal{D}(A)\subset X\to X$ be self-adjoint with $\scpr{x}{Ax}\leq0$ for all $x\in\mathcal{D}(A)$, $B\in\mathcal{L}({\mathbb{R}}^m,X_{-\alpha})$ for some $\alpha\in[0,1/2]$, and $(\T_t)_{t\geq0}$ be the analytic  semigroup generated by $A$. Then for all $p\in[2,\infty]$ we have that~$B$ is $L^p$-admissible for $(\T_t)_{t\geq0}$.

	Furthermore, for all $x_0\in X$, $T\in(0,\infty]$, $f\in L^p_{\loc}(0,T;X)$ and $u\in L^p_{\loc}(0,T;{\mathbb{R}}^m)$, the function~$x$ in~\eqref{eq:mild_solution} is a strong solution of~\eqref{eq:abstract_cauchy} on $[0,T)$.
\end{Lem}
\begin{proof}
For the case $p=2$, there exists a unique strong solution in $X_{-1}$ (that is, we replace $X$ by $X_{-1}$ and $X_{-1}$ by $X_{-2}$ in the definition) given by~\eqref{eq:mild_solution} and at most one strong solution in $X$, see for instance~\cite[Thm.~3.8.2~(i)~\&~(ii)]{Staf05}, so we only need to check that all the elements are in the correct spaces. Since $A$ is self-adjoint, the semigroup generated by $A$ is self-adjoint as well. Further, by combining~\cite[Prop.~5.1.3]{TucsWeis09} with~\cite[Thm.~4.4.3]{TucsWeis09}, we find that~$B$ is an $L^2$-admissible control operator for~$(\T_t)_{t\geq0}$. Moreover, by~\cite[Prop.~4.2.5]{TucsWeis09} we have that
	\[\left(t\mapsto{\mathbb{T}}_t x_0+\int_0^t({\mathbb{T}}|_{-\alpha})_{t-s}Bu(s)\ds{s}\right)\in C([0,T);X)\cap W^{1,2}_{\rm loc}(0,T;X_{-1})\]
and from \cite[Thm.~3.8.2~(iv)]{Staf05},
	\[\left(t\mapsto\int_0^t{\mathbb{T}}_{t-s}f(s)\ds{s}\right)\in C([0,T);X)\cap W^{1,2}_{\rm loc}(0,T;X_{-1}),\]
	whence $x\in C([0,T);X)\cap W^{1,2}_{\rm loc}(0,T;X_{-1})$, which proves that~$x$ is a strong solution of~\eqref{eq:abstract_cauchy} on $[0,T)$.

Since $B$ is $L^2$-admissible, it follows from the nesting property of $L^p$ on finite intervals that~$B$ is an $L^p$-admissible control operator for~$(\T_t)_{t\geq0}$ for all $p\in[2,\infty]$. Furthermore, for $p>2$, set $\tilde{f}\coloneqq f+Bu$ and apply~\cite[Thm.~3.10.10]{Staf05} with $\tilde{f}\in L^\infty_{\rm loc}(0,T;X_{-\alpha})$ to conclude that~$x$ is a strong solution.
\end{proof}

Next we show the regularity properties of the solution of~\eqref{eq:abstract_cauchy}, if $A = \mathcal{A}$ and $B = \mathcal{B}$ are as in the model~\eqref{eq:FHN_model}. Note that this result also holds when considering some $t_0\ge 0$, $T\in(t_0,\infty]$, and the initial condition $x(t_0)=x_0$ (instead of $x(0)=x^0$) by some straightforward modifications, cf.~\cite[Sec.~3.8]{Staf05}.

\begin{Prop}\label{prop:hoelder}
Let Assumption~\ref{Ass1} hold, $\mathcal{A}$ be the Neumann elliptic operator on $\Omega$ associated to $D$, $T\in(0,\infty]$ and $c>0$. Further let $X=X_0=L^2(\Omega)$ and $X_r$, $r\in{\mathbb{R}}$, be the interpolation spaces corresponding to~$\mathcal{A}$ according to Definition~\ref{Def:int-space-A}.
	Define $\mathcal{A}_0\coloneqq  \mathcal{A}-cI$ with $\mathcal{D}(\mathcal{A}_0)=\mathcal{D}(\mathcal{A})$ and consider $\mathcal{B}\in\mathcal{L}({\mathbb{R}}^m,X_{-\alpha})$ for $\alpha\in[0,1/2]$, $u\in L_{\rm loc}^2(0,T;{\mathbb{R}}^m)\cap L^\infty(\delta,T;{\mathbb{R}}^m)$ and $f\in L_{\rm loc}^2(0,T;X)\cap L^\infty(\delta,T;X)$ for all $\delta>0$. Then for all $x_0\in X$ and all $\delta>0$ the mild solution of~\eqref{eq:abstract_cauchy} (with $A=\mathcal{A}_0$ and $B=\mathcal{B}$) on~$[0,T)$, given by~$x$ as in~\eqref{eq:mild_solution}, satisfies
	\begin{enumerate}[(i)]
		\item if $\alpha=0$, then
		\[\forall\, \lambda\in(0,1):\ x\in BC([0,T);X)\cap C^{0,\lambda}([\delta,T);X);\]
		\item if $\alpha\in(0,1/2)$, then
		\[x\in BC([0,T);X)\cap C^{0,1-\alpha}([\delta,T);X)\cap C^{0,1-2\alpha}([\delta,T);X_{\alpha});\]
		\item if $\alpha=1/2$, then
		\[x\in BC([0,T);X)\cap C^{0,1/2}([\delta,T);X)\cap BUC([\delta,T);X_{1/2}).\]
	\end{enumerate}
\end{Prop}
\begin{proof}
First observe that by Proposition~\ref{prop:Aop} the assumptions of Lemma~\ref{lem:abstract_solution} are satisfied with $p=2$, hence~$x$ as in~\eqref{eq:mild_solution} is a strong solution of~\eqref{eq:abstract_cauchy} on $[0,T)$ in the sense of Definition~\ref{Def:Cauchy}. In the following we restrict ourselves to the case $T=\infty$, and the assertions for $T<\infty$ follow from these arguments by considering the restrictions to $[0,T)$.
	Define, for $t\ge 0$, the functions
	\begin{align}
	x_h(t)\coloneqq  {\mathbb{T}}_tx_0,\quad	x_f(t)\coloneqq  \int_0^t{\mathbb{T}}_{t-s}f(s)\ds{s},\quad
	x_u(t)\coloneqq  \int_0^t({\mathbb{T}}|_{-\alpha})_{t-s}\mathcal{B} u(s)\ds{s},\label{eq:xhxfxu}
	\end{align}
	so that $x=x_h+x_f+x_u$.

\emph{Step 1}: We show that $x\in BC([0,\infty);X)$. The definition of $\mathcal{A}$ in Proposition~\ref{prop:Aop} implies that for all $z\in \mathcal{D}(\mathcal{A})$ we have $\langle z,\mathcal{A} z\rangle\leq -c\|z\|^2$. The self-adjointness of $\mathcal{A}$ moreover implies that $\mathcal{A}_0$ is self-adjoint, whence~\cite[Thm.~4.2]{ArenElst12} gives that $\mathcal{A}_0$ generates an analytic, contractive semigroup $(\T_t)_{t\geq0}$ on $X$, which satisfies
\begin{equation}\label{eq:est-sg-exp}
    \forall\, t\ge 0\ \forall\, x\in X:\ \|{\mathbb{T}}_t x\|\leq \mathrm{e}^{-ct}\|x\|.
\end{equation}
Since, by Lemma~\ref{lem:abstract_solution}, $x$ is a strong solution, we have $x\in C([0,\infty);X)\cap W^{1,2}_{\rm loc}(0,\infty;X_{-1})$. Further observe that $\mathcal{B}$ is $L^\infty$-admissible by Lemma~\ref{lem:abstract_solution}. Then it follows from~\eqref{eq:est-sg-exp} and~\cite[Lem.~2.9\,(i)]{JacoNabi18} that~$\mathcal{B}$ is infinite-time $L^\infty$-admissible, which implies that for $x_u$ as in~\eqref{eq:xhxfxu} we have
\[
    \|x_u\|_\infty \le \left( \sup_{t>0} \|\Phi_t\|\right) \|u\|_\infty < \infty,
\]
thus  $x_u\in BC([0,\infty);X)$. A direct calculation using~\eqref{eq:est-sg-exp} further shows that $x_h,x_f\in BC([0,\infty);X)$, whence $x\in BC([0,\infty);X)$.

\emph{Step 2}: We show~(i). Let $\delta>0$ and set $\tilde{f}:=f+Bu\in L^2_{\rm loc}(0,\infty;X)\cap L^\infty(\delta,\infty;X)$, then we may infer  from~\cite[Props.~4.2.3~\&~4.4.1\,(i)]{Luna95} that
	\[\forall\,\lambda\in(0,1):\ x\in C^{0,\lambda}([\delta,\infty);X).\]
From this together with Step~1 we may infer~(i).

\emph{Step 3}: We show~(ii). Let $\delta>0$, then it follows from~\cite[Props.~4.2.3~\&~4.4.1\,(i)]{Luna95} together with $x_0\in X$ and $f\in L^\infty(\delta,\infty;X)$, that
	\[\begin{aligned}
x_h+x_f&\in C^{0,1-\alpha}([\delta,\infty);X_{\alpha})\cap C^{1}([\delta,\infty);X)\\
&= C^{0,1-2\alpha}([\delta,\infty);X_{\alpha})\cap C^{0,1-\alpha}([\delta,\infty);X).\end{aligned}\]
Since we have shown in Step~1 that $x\in BC([0,\infty),X)$, it remains to show that
	$x_u\in C^{0,1-2\alpha}([\delta,\infty);X_{\alpha})\cap C^{0,1-\alpha}([\delta,\infty);X)$.\\
To this end, consider the space $Y:=X_{-\alpha}$. Then $(\T_t)_{t\geq0}$ extends to a~semigroup~$\big(({\mathbb{T}}|_{-\alpha})_{t}\big)_{t\ge 0}$ on $Y$ with generator $\mathcal{A}_{0,\alpha}:\mathcal{D}(\mathcal{A}_{0,\alpha})=X_{-\alpha+1}\subset X_{-\alpha}=Y$, cf.~\cite[pp.~50]{Luna95}. Now, for $r\in{\mathbb{R}}$, consider the interpolation spaces $Y_r$ as in Definition~\ref{Def:int-space-A} by means of the operator $\mathcal{A}_{0,\alpha}$. Then it is straightforward to show that $Y_n = \mathcal{D}(\mathcal{A}_{0,\alpha}^n) = X_{n-\alpha}$ for all $n\in{\mathbb{N}}$ using the representation~\eqref{eq:Xrspec}. Similarly, we may show that $Y_n = X_{n-\alpha}$ for all $n\in{\mathbb{Z}}$. Then the Reiteration Theorem, see~\cite[Cor.~1.24]{Luna18} and also \eqref{eq:Xalpha-reit}, gives
\[\forall\, r\in{\mathbb{R}}\,:\quad Y_r=X_{r-\alpha}.\]
Since $\mathcal{B}\in\mathcal{L}({\mathbb{R}}^m,Y)$, \cite[Props.~4.2.3~\&~4.4.1\,(i)]{Luna95} now imply
\[\begin{aligned}
x_u &\in C^{0,1-2\alpha}([\delta,\infty);Y_{2\alpha})\cap C^{0,1-\alpha}([\delta,\infty);Y_{\alpha})\\
&= C^{0,1-2\alpha}([\delta,\infty);X_{\alpha})\cap C^{0,1-\alpha}([\delta,\infty);X),
\end{aligned}\]
which completes the proof of (ii).

\emph{Step 4}: We show~(iii). The proof of $x\in C^{0,1/2}([\delta,\infty);X)$ is analogous to that of $x\in C^{0,1-\alpha}([\delta,\infty);X)$ in Step~3. Boundedness and continuity of $x$ on $[0,\infty)$ was proved in Step~1. Hence, it remains to show that $x$ is uniformly continuous: Again consider the additive decomposition of $x$ into $x_h$, $x_f$ and $x_u$ as in~\eqref{eq:xhxfxu}. Similar to Step~3 it can be shown that $x_h,x_f\in C^{0,1/2}([\delta,\infty);X_{1/2})$, whence $x_h,x_f\in BUC([\delta,\infty);X_{1/2})$. It remains to show that $x_u\in BUC([\delta,\infty);X_{1/2})$.

Note that Lemma \ref{lem:abstract_solution} gives that $x_\delta\coloneqq x(\delta)\in X$. Then $x_u$ solves $\dot z(t) = \mathcal{A}_0 z(t) + \mathcal{B} u(t)$ with $z(\delta) = x_u(\delta)$ and hence, for all $t\geq\delta$ we have
\begin{equation}\label{eq:x-T-delta}
\begin{aligned}
    x_u(t)=&\,{\mathbb{T}}_{t-\delta}x_u(\delta)+\underbrace{\int_\delta^t({\mathbb{T}}|_{-\alpha})_{t-s}\mathcal{B} u(s)\ds{s}}_{=:x_u^\delta(t)}\\
    \end{aligned}
\end{equation}
Since $x_u(\delta)\in X$ by Lemma~\ref{lem:abstract_solution}, it remains to show that $x_u^\delta\in BUC([\delta,\infty);X_{1/2})$.
We obtain from Proposition~\ref{prop:Aop_n}\,\ref{item:Aop5}) that $\mathcal{A}_0$ has an eigendecomposition of type~\eqref{eq:spectr}
	with eigenvalues $(-\beta_j)_{j\in{\mathbb{N}}_0}$, $\beta_j\coloneqq  \alpha_j+c$, and eigenfunctions $(\theta_j)_{j\in{\mathbb{N}}_0}$. Moreover, there exist $b_i\in X_{-1/2}$ for $i=1,\dots,m$ such that $\mathcal{B} \xi = \sum_{i=1}^m b_i \cdot \xi_i$ for all $\xi\in{\mathbb{R}}^m$. Therefore,
	\begin{align*}
	x^\delta_u(t)&=\int_\delta^t\sum_{j=0}^\infty \mathrm{e}^{-\beta_j(t-\tau)}\theta_j \sum_{i=1}^m\scpr{b_i \cdot u_i(\tau)}{\theta_j}\ds{\tau}\\
	&=\int_\delta^t\sum_{j=0}^\infty \mathrm{e}^{-\beta_j(t-\tau)}\theta_j \sum_{i=1}^m u_i(\tau) \scpr{b_i}{\theta_j}\ds{\tau},
	\end{align*}
	where the last equality holds since $u_i(\tau)\in{\mathbb{R}}$ and can be treated as a constant in~$X$. By considering each of the factors in the sum over $i=1,\dots,m$, we can assume without loss of generality that $m=1$ and $b\coloneqq b_1$, so that
	\[x_u^\delta(t)=\int_\delta^t\sum_{j=0}^\infty \mathrm{e}^{-\beta_j(t-\tau)} u(\tau) \scpr{b}{\theta_j} \theta_j \ds{\tau}.\]
	Define $b^j \coloneqq  \scpr{b}{\theta_j}$ for $j\in{\mathbb{N}}_0$. Since $b\in X_{-1/2}$ we have that $\sum_{j=0}^\infty b_j^2/\beta_j$ converges, which implies
	\begin{equation}S\coloneqq  \sum_{j=0}^\infty\frac{(b^j)^2}{\beta_j}<\infty.\label{eq:Sdef}\end{equation}
	Recall that the spaces $X_\alpha$, $\alpha\in{\mathbb{R}}$, are defined by using $\lambda\in{\mathbb{C}}$ belonging to the resolvent set of $\mathcal{A}$, and they are independent of the choice of $\lambda$. Since $c>0$ in the statement of the proposition is in the resolvent set of $\mathcal{A}$, the spaces $X_\alpha$ coincide for $\mathcal{A}$ and $\mathcal{A}_0=\mathcal{A}-cI$.\\
	Using the diagonal representation from Remark~\ref{rem:X_alpha} and \cite[Prop.~3.4.8]{TucsWeis09}, we may infer that $x_u^\delta(t)\in X_{1/2}$ for a.e. $t\geq\delta$, namely,
	\begin{align*}
	\|x_u^\delta(t)\|_{X_{1/2}}^2&\leq\sum_{j=0}^\infty\beta_j(b^j)^2\|u\|^2_{L^\infty(\delta,\infty)}\left(\int_\delta^t\mathrm{e}^{-\beta_j(t-s)}\ds{s}\right)^2\\
	&=\|u\|^2_{L^\infty(\delta,\infty)}\sum_{j=0}^\infty\frac{(b^j)^2}{\beta_j}\left(1-\mathrm{e}^{-\beta_j(t-\delta)}\right)^2\\
	&\leq\|u\|^2_{L^\infty(\delta,\infty)}\sum_{j=0}^\infty\frac{(b^j)^2}{\beta_j}<\infty.
	\end{align*}
	Hence,
	\begin{equation}\label{eq:xudelta_X12}
	\|x_u^\delta(t)\|_{X_{1/2}}\leq \|u\|_{L^\infty(\delta,\infty)}\sqrt{S}.
	\end{equation}
	Now let $t>s>\delta$ and $\sigma>0$ such that $t-s<\sigma$. By dominated convergence \cite[Thm.~II.2.3]{Dies77}, summation and integration can be interchanged, so that
	\begin{align*}
	\|x_u^\delta&(t)-x_u^\delta(s)\|_{X_{1/2}}^2\\
	\leq&\,\|u\|_{L^\infty(\delta,\infty)}^2\sum_{j=0}^\infty \beta_j (b^j)^2 \left(\int_\delta^s\mathrm{e}^{-\beta_j(s-\tau)}-\mathrm{e}^{-\beta_j(t-\tau)}\ds{\tau}+\int_s^t\mathrm{e}^{-\beta_j(t-\tau)}\ds{\tau}\right)^2\\
	\leq&\,4\|u\|_{L^\infty(\delta,\infty)}^2\sum_{j=0}^\infty  \frac{(b^j)^2}{\beta_j} \left(1-\mathrm{e}^{-\beta_j (t-s)}\right)^2\\
	\le&\, 4\|u\|_{L^\infty(\delta,\infty)}^2\sum_{j=0}^\infty \frac{(b^j)^2}{\beta_j} \left(1-\mathrm{e}^{-\beta_j \sigma}\right)^2.
	\end{align*}
	We can conclude from \eqref{eq:Sdef} that the series $F:(0,\infty)\to(0,S)$ with
	\[F(\sigma)\coloneqq  \sum_{j=0}^\infty\frac{(b^j)^2}{\beta_j}(1-\mathrm{e}^{-\beta_j\sigma})^2\]
	converges uniformly to a strictly monotone, continuous and surjective function. Therefore, $F$ has an inverse. The function $x_u^\delta$ is thus uniformly continuous on~$[\delta,\infty)$ and by~\eqref{eq:est-sg-exp} we obtain boundedness, i.e., $x_u^\delta\in BUC([\delta,\infty);X_{1/2})$.
\end{proof}

Finally we present a consequence of the Banach-Alaoglu Theorem, see e.g.~\cite[Thm.~3.15]{Rudi91}.

\begin{Lem}\label{lem:weak_convergence}
	Let $T>0$ and $Z$ be a reflexive and separable Banach space. Then
	\begin{enumerate}[(i)]
		\item every bounded sequence $(w_n)_{n\in{\mathbb{N}}}$ in $L^\infty(0,T;Z)$ has a weak$^\star$ convergent subsequence in $L^\infty(0,T;Z)$;\label{it:weak_star}
		\item every bounded sequence $(w_n)_{n\in{\mathbb{N}}}$ in $L^p(0,T;Z)$ with $p\in(1,\infty)$ has a weakly convergent subsequence in $L^p(0,T;Z)$.\label{it:weak}
	\end{enumerate}
\end{Lem}

\begin{proof}
	Let $p\in[1,\infty)$. Then $W:=L^p(0,T;Z')$ is a separable Banach space, see~\cite[Sec.~IV.1]{Dies77}. Since $Z$ is reflexive, by \cite[Cor.~III.4]{Dies77} it has the Radon-Nikodým property. Then it follows from~\cite[Thm.~IV.1]{Dies77} that $W'=L^q(0,T;Z)$ is the dual of $W$, where $q\in(1,\infty]$ such that $p^{-1}+q^{-1}=1$. Assertion~\eqref{it:weak_star} now follows from~\cite[Thm.~3.17]{Rudi91} with $p=1$ and $q=\infty$. On the other hand, statement~\eqref{it:weak} follows from~\cite[Thm.~V.2.1]{Yosi80} by further using that~$W$ is reflexive for $p\in(1,\infty)$.
\end{proof}

\end{appendices}

\section*{Acknowledgments}
The authors would like to thank Felix L.\ Schwenninger (U Twente) and Mark R.\ Opmeer (U Bath) for helpful comments on maximal regularity.

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