 whenever $\re D_0>0$.
\end{itemize}
Then $i\R\subset \rho(A_e)$ and the closed-loop system is strongly stable.

  Assume in addition that $\mathcal{I}\subset \Z$ is finite, $(A,B,C,D)$ is exponentially stable,
  and there exists $R>0$ such that $\sup_{\abs{\omega}\geq R}\norm{R(i\omega,A_c )}<\infty$.
  If either
  $\limsup_{\abs{\omega}\to \infty}\norm{G(i\omega)P(i\omega)}<1$
  or
$\re P(i\omega)\geq \gg(\omega)\geq 0$ and $\re G(i\omega)\geq d_c(\omega)\geq 0$ such that 
  $\gg(\omega)+d_c(\omega)\geq \gg_0>0$ for some constant $\gg_0>0$ and for large $\abs{\omega}$,
  then the closed-loop system is exponentially stable.
\end{theorem}


\begin{proof}
  We begin by showing that $i\R\subset \rho(A_e)$. 
 Since the semigroup generated by $A_e$ is uniformly bounded by Lemma~\ref{lem:CLreg}, the strong stability of $T_e(t)$ then follows from the Arendt--Batty--Lyubich--V\~{u} Theorem~\cite{AreBat88,LyuVu88}.



Lemma~\ref{lem:Repostoinv}(d) implies that $u(t)=-D_c y(t)$ is an admissible output feedback for $(A,B,C,D)$, and by~\cite{Wei94} the resulting system  $(A^{cl},B^{cl},C^{cl},D^{cl})=(A-BD_c Q_1C_\Lambda,BQ_2,Q_1C_\Lambda,Q_1D)$ is regular. The assumption $i\R\subset \rho(A)$ and Lemma~\ref{lem:Afbreg} imply $i\R\subset \rho(A^{cl})$,
and by Lemma~\ref{lem:Repostoinv}(d) the transfer function $P_{cl}(\lambda)$ is given by $P_{cl}(i\omega)=P(i\omega)(I+D_c P(i\omega))^{-1}$ for all $\omega\in\R$. 
If $\omega\in\R$ and if we denote $R_{i\omega} = R(i\omega,A^{cl})$, then $i\omega-A_e$ has a bounded inverse given by
\eq{
  R(i\omega,A_e) 
  = \pmat{R_{i\omega}-R_{i\omega} B^{cl} C_{c\Lambda} S_A(i\omega)^{-1} B_c  C^{cl} R_{i\omega}&R_{i\omega} B^{cl} C_{c\Lambda} S_A(i\omega)^{-1}\\-S_A(i\omega)^{-1} B_c  C^{cl} R_{i\omega} &S_A(i\omega)^{-1}}
}
provided that the Schur complement 
  \eq{
    S_A(i\omega)
    &= i\omega-A_c  + B_c  D^{cl} C_{c\Lambda} + B_c  C^{cl} R(i\omega,A^{cl})B^{cl} C_{c\Lambda} \\
    &= i\omega-A_c  + B_c  P(i\omega)(I+D_c P(i\omega))^{-1} C_{c\Lambda}
  }
with domain $D(S_A(i\omega)) = \setm{z\in D(C_{c\Lambda})}{S_A(i\omega)z\in Z}$ 
has a bounded inverse. If $\omega=\omega_n$ for some $n\in\mathcal{I}$, then  $\re P(i\omega_n)>0$ and assumption~(3)
imply that $S_A(i\omega_n)$ is boundedly invertible. Thus $\set{i\omega_k}_{k\in\mathcal{I}}\subset \rho(A_e)$.


Now let $\omega\in \R\setminus \set{\omega_k}_{k\in\mathcal{I}}$. If $\re G(i\omega)=0$, then $I+G(i\omega)P(i\omega)$ is invertible by assumption. By Lemma~\ref{lem:Repostoinv}(a) the same is also true if $\re G(i\omega)>0$, since $I+G(i\omega)P(i\omega)=G(i\omega)(G(i\omega)^{-1} + P(i\omega))$.
Because
\eq{
I+D_c P(i\omega)+  C_{c\Lambda} R(i\omega,A_c ) B_c P(i\omega)
=  I+ G(i\omega)P(i\omega),
}
 the Woodbury formula implies that
$S_A(i\omega)$ has a bounded inverse 
  \eqn{
  \label{eq:SAinvWoodbury}
  S_A(i\omega)^{-1} 
  &= R(i\omega,A_c ) \bigl[I \hspace{-.2ex} - \hspace{-.2ex}  B_c P(i\omega)(I \hspace{-.2ex} + \hspace{-.2ex} G(i\omega) P(i\omega) )^{-1}
  C_{c\Lambda}  R(i\omega,A_c )\bigl]. \hspace{-1.6ex}
  }
  Thus $i\omega\in \rho(A_e)$ also for all $\omega\in\R \setminus\set{\omega_k}_{k\in\mathcal{I}}$. 
Since the semigroup $T_e(t)$ is contractive, the closed-loop system is strongly stable.

  Finally, assume that $\mathcal{I}\subset \Z$ is finite, $(A,B,C,D)$ is exponentially stable, and $\sup_{\abs{\omega}\geq R}\norm{R(i\omega,A_c )}<\infty$ for some $R>0$.
   The stability and regularity of $(A,B,C,D)$ imply that the norms $\norm{R(\cdot,A)}$, $\norm{R(\cdot,A)B}$, $\norm{C_\Lambda R(\cdot,A)}$, and $\norm{P(\cdot)}$ are uniformly bounded on $i\R$.
  Similarly the regularity of the controller implies that 
  $\norm{R(i\omega,A_c )}$, $\norm{R(i\omega,A_c )B_c}$, $\norm{C_{c\Lambda} R(i\omega,A_c )}$, and $\norm{C_{c\Lambda} R(i\omega,A_c ) B_c}$ are uniformly bounded with respect to $\omega\in\R$ with $\abs{\omega}\geq R$.
  If
  $\limsup_{\abs{\omega}\to \infty}\norm{G(i\omega)P(i\omega)}<1$
  the norms $\norm{P(i\omega)(I+G(i\omega)P(i\omega))^{-1}}$ are uniformly bounded for large $\abs{\omega}$.
  On the  other hand, if
  $\gg(\omega)+d_c(\omega)\geq \gg_0>0$, then Lemma~\ref{lem:Repostoinv}(b) implies $\norm{P(i\omega)(I+G(i\omega)P(i\omega))^{-1}}\lesssim \gg_0^{-1} $.
   Thus~\eqref{eq:SAinvWoodbury} implies that
   $\norm{R(i\omega,A_e)}$ is uniformly bounded for large $\abs{\omega}$. 
  Since $i\R\subset \rho(A_e)$ and $T_e(t)$ is contractive, the 
Gearhart--Pr\"{u}ss Theorem
  shows that the closed-loop system is exponentially stable.
\end{proof}

\begin{remark}
  \label{rem:Dc0}
  Condition~(2) is in particular satisfied if
$\re G(i\omega)>0$ for all 
 $\omega\in \R\setminus \set{\omega_k}_{k\in\mathcal{I}}$. If  
 $\re G(i\omega)\geq d_c>0$ for some constant $d_c>0$, then 
$\norm{P(i\omega)(I+G(i\omega) P(i\omega) )^{-1}}\leq d_c^{-1}$
 for all 
 $\omega\in \R\setminus \set{\omega_k