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\TopicEN{Control theory, Partial differential equations}

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\title{Some fundamental properties of the approximate synchronization by groups for a coupled system of wave equations with internal controls}
\alttitle{Quelques propri\'et\'es fondamentales de la synchronisation approch\'ee par groupes pour un syst\`eme coupl\'e d'\'equations des ondes avec contr\^oles internes}

\author{\firstname{Tatsien} \lastname{Li}}
\address{School of Mathematical Sciences, Fudan University, Shanghai 200433, China}
\email{dqli@fudan.edu.cn}

\author{\firstname{Bopeng} \lastname{Rao}\CDRorcid{0000-0002-8991-5893}}
\address{Institut de Recherche Math\'ematique Avanc\'ee, Strasbourg University, 67000 Strasbourg, France}
\email{bopeng.rao@math.unistra.fr}

\thanks{This work was partially supported by the National Natural Science Foundation of China (11831011), and Fudan Scholar Program 2023}
\CDRGrant[National Natural Science Foundation of China]{11831011}

\subjclass{93B05, 93C20, 35L53}

\keywords{\kwd{Kalman's rank condition} \kwd{approximate synchronization by groups} \kwd{system of wave equations}}
\altkeywords{\kwd{Condition du rang de Kalman} \kwd{synchronisation approchée par groupes} \kwd{système d'équations d'ondes}}

\begin{abstract}
In this paper, we show that the independence of approximately synchronizable state $u$ by $p$-groups with respect to applied controls, the linear independence of the components, the non extensibility of the approximate synchronization by $p$-groups as well as the necessity of the condition of $C_p$-compatibility, all these properties are the consequence of the minimality of Kalman's rank condition and vice versa.
These results reveal the role of Kalman rank conditions on control problems from different aspects, and further develop the synchronization theory.
\end{abstract}

\begin{altabstract}
Dans cette note, nous montrons que l'indépendance de l'état de synchronisation approchée par rapport aux contrôles, la non extensibilité de la synchronisation approchée, l'indépendance linéaire des composants de l'état de synchronisation approchée ainsi que la condition de $C_p$-compatibilité, toutes ces propriétés sont la conséquence de la minimalité de la condition du rang de Kalman.
\end{altabstract}

\begin{document}

\maketitle

\section{Introduction}

The research on the synchronization for a coupled system of wave equations is a progressive field with broad application prospects. The related topic is closely connected with the control theory. We introduced the concepts of exact synchronization in~\cite{Rao2013} and approximate synchronization in~\cite{Siam2016} on partial differential equations. The relevant results about synchronization achieved only through boundary controls were collected in the monograph~\cite{book}, published by Birkh\"auser Publishing House in 2019. In~\cite{Interne, COCV2023}, we carried on the study of synchronization using locally distributed controls.

For the sake of a concise writing and the reader's convenience, we briefly review the main results. Let $A$ be a matrix of order $N$ and $D$ be a full column-rank matrix of order $N \times M $. Both $A$ and $D$ are composed of constant entries. Let $\Omega\subset\mathbb R^m$ be a bounded domain with smooth boundary $\Gamma$. We consider the following problem for the variable $U = (u^{(1)},\dotsc,u^{(N)})^T$:
\begin{equation}\label{1.1}
\begin{cases}
U''-\Delta U+AU=D\chi_\omega H 	& \text{in} \ (0, +\infty) \times \Omega, \\
U = 0							& \text{on} \ (0, +\infty) \times \Gamma
\end{cases}
\end{equation}
associated with the initial condition:
\begin{equation}\label{1.2}
t=0 \colon \quad U= \widehat U_0, \quad U'= \widehat U_1 \quad \text{in} \ \Omega, 
\end{equation}
where $H = (h^{(1)},\dotsc,h^{(M)})^T$ denotes the internal control and $\chi_\omega$ is the characteristic function of the subdomain $\omega\subset\Omega$.

The following well-posedness can be shown by the classic approach in~\cite{Balakrishna, Haraux, Pazy}.

\begin{prop}\label{th1.4}
Let $\Omega\subset \mathbb R^m$ be a bounded domain with smooth boundary $\Gamma$ and $ \omega$ be a subdomain of\/ $\Omega$. For any given initial data $(\widehat U_0, \widehat U_1) \in \parens[\big]{H_0^1(\Omega)\times L^2(\Omega)}^N$ and any given function $H\in \parens[\big]{L^1_{\loc} \parens[\big]{0, +\infty; L^2(\Omega)}}^M$,
system~\eqref{1.1} admits a unique weak solution:
\begin{equation}
U\in \parens[\Big]{C^0_{\loc}\parens[\big]{[0, +\infty); H_0^1(\Omega)} \cap C^1_{\loc}\parens[\big]{[0, +\infty); L^2(\Omega)}}^N
\end{equation}
with continuous dependence.
\end{prop}

System~\eqref{1.1} is approximately controllable at the time $T>0$ if for any given initial data $ (\widehat U_0, \widehat U_1)\in \parens[\big]{H^1_0(\Omega) \times L^2(\Omega)}^N$, there exists a sequence of internal controls $\{H_n\} _{n\in \mathbb N} \in \parens[\big]{L^2\parens[\big]{0, +\infty; L^2(\Omega)}}^M$ with compact support in $[0, T]$, such that the sequence $\{U_n\} _{n\in \mathbb N} $ of the corresponding solutions to system~\eqref{1.1} satisfies
\begin{equation}
U_n\to 0\quad \text{in} \ \parens[\Big]{C^0_{\loc} \parens[\big]{[T, +\infty); H^1_0(\Omega)}\cap C^1_{\loc} \parens[\big]{[T, +\infty); L^2(\Omega)}}^N
\end{equation}
as $n\to +\infty.$

\begin{theo}[{\cite[Theorem~13]{Interne}}]\label{th1.1}
Let\/ $\Omega\subset \mathbb R^m$ be a bounded domain with smooth boundary\/ $\Gamma$ and $\omega$ be a subdomain of\/ $\Omega$. If
system~\eqref{1.1} is approximately controllable at the time $T>0$, then we necessarily have the following Kalman's rank condition:
\begin{equation}\label{1.3}
\rank (D, AD,\dotsc, A^{N-1} D)= N.
\end{equation}
Conversely, assume that Kalman's rank condition~\eqref{1.3} holds, then system~\eqref{1.1} is approximately controllable at the time $T>0$, provided that $T>2d(\Omega)$, where $d(\Omega)$ denotes the geodesic diameter of\/~$\Omega$.
\end{theo}

Let $p\geqslant 1$ be an integer and
\begin{equation}
0=n_0< n_1<\dots<n_p=N
\end{equation}
be a partition with $n_{r}-n_{r-1} \geqslant 2$ for $1\leqslant r\leqslant p$. We re-arrange the components of the state variable~$U$ into $p$ groups
\begin{equation}
(u^{(1)},\dotsc, u^{(n_1)}), \ (u^{(n_1+1)},\dotsc, u^{(n_2)}), \ \dotsc, \ (u^{(n_{p-1} +1)},\dotsc, u^{(n_p)}).
\end{equation}

System~\eqref{1.1} is approximately synchronizable by $p$-groups in the consensus sense at the time $T>0$, if for any given initial data $(\widehat U_0, \widehat U_1)$ in $\parens[\big]{H^1_0(\Omega) \times L^2(\Omega)}^N$, there exists a sequence $\{H_n\} _{n\in \mathbb N} $ of controls in $\parens[\big]{L^2 \parens[\big]{0, +\infty; L^2(\Omega)}}^M$ with compact support in $[0, T]$, such that the sequence $\{U_n\} _{n\in \mathbb N} $ of the corresponding solutions to system~\eqref{1.1} satisfies
\begin{equation}\label{1.4}
u_n^{(k)}-u_n^{(l)} \to 0 \quad \text{in} \ C_{\loc} ^0\parens[\big]{[T, +\infty); H^1_0(\Omega)} \cap C_{\loc} ^1\parens[\big]{[T, +\infty); L^2(\Omega)}
\end{equation}
as $ n\to +\infty $ for all $n_{r-1} +1\leqslant k, l\leqslant n_r$ and $ 1\leqslant r\leqslant p$.

Let $S_r$ be a full row-rank matrix of order $(n_r-n_{r-1}-1)\times (n_r-n_{r-1})$:
\begin{equation}
S_r =
{\begin{pmatrix} 1&-1\\
&1&-1&\\&&\ddots&\ddots\\
&&&1&-1
\end{pmatrix}}, \quad 1\leqslant r\leqslant p.
\end{equation}
We define the $(N-p)\times N$ matrix $C_p$ of synchronization by $p$-groups as
\begin{equation}
C_p=
{\begin{pmatrix}
S_1\\ &S_2\\ &&\ddots\\ &&&S_p
\end{pmatrix}}.
\end{equation}
Then~\eqref{1.4} can be equivalently written as
\begin{equation}\label{1.5}
C_pU_n\to 0 \quad \text{in} \ \parens[\Big]{C_{\loc} ^0\parens[\big]{[T, +\infty); H^1_0(\Omega)} \cap C_{\loc} ^1\parens[\big]{[T, +\infty); L^2(\Omega)}}^{N-p}
\end{equation}
as $ n\to+\infty$.

Assume that $A$ satisfies the condition of $C_p$-compatibility:
\begin{equation}\label{1.6}
A\Ker(C_p)\subseteq \Ker(C_p).
\end{equation}
By~\cite[Proposition~2.15]{book}, there exists a matrix $A_p$ of order $(N-p)$, such that
\begin{equation}\label{1.7}
C_pA= A_pC_p.
\end{equation}
Applying $C_p$ to system~\eqref{1.1} and setting $Y_p= C_pU$ and $ D_p= C_pD$, we get the following reduced system:
\begin{equation}
\begin{cases}\label{1.8}
Y_p''-\Delta Y_p+ A_pY_p = D_p \chi_\omega H	& \text{in} \ (0, +\infty)\times\Omega, \\
Y_p=0										& \text{on} \ (0, +\infty)\times\Gamma.
\end{cases}
\end{equation}
Thus the approximate internal synchronization by $p$-groups of system~\eqref{1.1} is transformed into the approximate internal controllability of the reduced system~\eqref{1.8}. Moreover, we have the following result.
\begin{theo}[{\cite[Theorem~6.3]{COCV2023}}]\label{th1.2}
Let\/ $\Omega\subset \mathbb R^m$ be a bounded domain with smooth boundary\/ $\Gamma$ and $ \omega$ be a subdomain of\/ $\Omega$. Let $T>2d(\Omega)$, where $d(\Omega)$ denotes the geodesic diameter of\/ $ \Omega$. Assume that $A$ satisfies the condition of\/ $C_p$-compatibility~\eqref{1.6}, then system~\eqref{1.1} is approximately
synchronizable by $p$-groups if and only if
\begin{equation}\label{1.9}
{\rank} \parens[\big]{D_p, A_p D_p,\dotsc, A_p^{N-p-1} D_p} = N-p,
\end{equation}
or equivalently,
\begin{equation}\label{1.10}
{\rank} \parens[\big]{C_p(D, AD,\dotsc, A^{N-1} D)}=N-p. 
\end{equation}
\end{theo}
Furthermore, if there exist some functions $u_1,\dotsc, u_p$ such that
\begin{equation}\label{1.11}
u_n^{(k)} \to u_r \quad \text{in} \ C_{\loc} ^0\parens[\big]{[T, +\infty); H^1_0(\Omega)}\cap C_{\loc} ^1\parens[\big]{[T, +\infty); L^2(\Omega)}
\end{equation}
as $ n\to+\infty$ for all $n_{r-1} +1\leqslant k\leqslant n_r$ and $ 1\leqslant r\leqslant p$, system~\eqref{1.1} will be called approximately synchronizable by p-groups in the pinning sense, and $u=(u_1,\dotsc, u_p)^T$ will be called approximately synchronizable state by p-groups.

Let
\begin{equation}\label{1.12}
\Ker(C_p)={\Span}\braces[\big]{e_1,\dotsc,e_p},
\end{equation}
where
\begin{equation}\label{1.12b}
e_r= (0,\dotsc, 0, \overset{(n_{r-1} +1)} {1},\dotsc, \overset{(n_r)} {1}, 0,\dotsc, 0)^T, \quad 1\leqslant r\leqslant p.
\end{equation}
Then~\eqref{1.11} can be equivalently written as
\begin{equation}\label{1.13}
U_n\to \sum_{r=1} ^pu_re_r \quad \text{in} \ \parens[\Big]{C_{\loc} ^0\parens[\big]{[T, +\infty); H^1_0(\Omega)}\cap C_{\loc} ^1\parens[\big]{[T, +\infty); L^2(\Omega)}}^{N}
\end{equation}
as $ n\to+\infty.$

Clearly, the approximate synchronization in the pinning sense implies that in the consensus sense.

\begin{theo}[{\cite[Theorem~7.3]{COCV2023}}]\label{th1.3}
Assume that system~\eqref{1.1} is approximately
synchronizable by $p$-groups under the minimal rank condition
\begin{equation}\label{1.15}
\rank (D, AD,\dotsc, A^{N-1} D)= N-p.
\end{equation}
Then it is approximately synchronizable by $p$-groups in the pinning sense, and the approximately synchronizable state by $p$-groups $u = (u_1,\dotsc, u_p)^T$ is independent of applied controls. Furthermore the matrix $A$ satisfies the condition of $C_p$-compatibility~\eqref{1.6}.
\end{theo}

Through in-depth analysis, it can be found that due to the use of internal controls, more deep-going results on the approximate synchronization by groups can be obtained. In Section~\ref{section_2}, we will first characterize the feasibility of Kalman's rank condition~\eqref{1.15} via the diagonalization by blocks of $A$ on a suitable basis of type $\IM(C_p^T)\oplus V$. Then in Section~\ref{section_3}, we will show that the independence of approximately synchronizable state by groups with respect to applied controls, the linear independence of the components of the approximately synchronizable state by groups, and the possibility of the extensibility of approximate synchronization etc., all these important properties are the consequence of the minimality of Kalman's rank condition~\eqref{1.15}. Thus, we will give a complete answer to these fundamental questions, which have plagued us for a long time.

\section{Algebraic structure of synchronization}\label{section_2}

In this section we will describe the minimality of Kalman's rank condition~\eqref{1.15} by the algebraic structure of the matrix $A$.

We first recall the following basic property on Kalman's matrix.

\begin{lemm}[{\cite[Lemma~2.5]{Siam2016}}]\label{th2.1}
Let $d\geqslant 0$ be an integer. Kalman's rank condition
\begin{equation}
\rank (D, AD,\dotsc, A^{N-1} D)= N-d
\end{equation}
holds if and only if $d$ is the dimension of the largest subspace $V$, which is invariant for $A^T$ and contained in\/ $\Ker(D^T)$. Moreover, $V$ is given by
\begin{equation}
V = \Ker (D, AD,\dotsc, A^{N-1} D)^T.
\end{equation}
\end{lemm}

\begin{theo}\label{th2.2}
Let\/ $\Omega \subset \mathbb{R}^{m}$ be a bounded domain with smooth boundary $\Gamma$ and $ \omega$ be a subdomain of\/ $\Omega$. Assume that system~\eqref{1.1} is approximately synchronizable by $p$-groups. Then the following equivalences hold:
\begin{enumerate}\alphenumi
\item \label{th2.2_a} the matrix $A$ satisfies Kalman's rank condition~\eqref{1.15};
\item \label{th2.2_b} the subspace $\IM(C_p^T)$ is invariant for $A^T$ and admits a supplement $V$, on which the projection of system~\eqref{1.1} is independent of applied controls;
\item \label{th2.2_c} the subspace $\IM(C_p^T)$ is invariant for $A^T$ and admits a supplement $V$, which is contained in $\Ker(D^T)$ and invariant for $A^T$.
\end{enumerate}
\end{theo}

\begin{proof}
\begin{proof}[(\ref{th2.2_a})$\Longrightarrow$(\ref{th2.2_b})]
By Theorem~\ref{th1.3}, $A$ satisfies the condition of $C_p$-compatibility~\eqref{1.6}, namely, $\IM(C_p^T)$ is invariant for $A^T$. Moreover, by~\cite[Proposition~6.2]{COCV2023}, we have
\begin{equation}\label{2.1}
V\cap\IM(C_p^T)=\{0\},
\end{equation}
where $V$ denotes the largest subspace invariant for $A^T$ and contained in $\Ker(D^T).$ By Lemma~\ref{th2.1}, the rank condition~\eqref{1.15} implies that $V$ has the dimension $p$. Noting that $\dim\IM(C_p^T)= N-p$, $V$ is a supplement of $\IM(C_p^T)$.

Let $V=\Span\braces[\big]{\mathcal E_1,\dotsc, \mathcal E_{p}}$. There exists reals $ c_{rs} (1\leqslant r, s\leqslant d$) such that
\begin{equation}
A^T\mathcal E_{r} = \sum_{s=1}^p c_{rs}\mathcal E_s \quad \text{and} \quad D^T\mathcal E_r=0, \quad r=1,\dotsc, p.
\end{equation}
For $r= 1,\dotsc, p$, applying $\mathcal E_r^T$ to system~\eqref{1.1} and setting $\psi_r= \mathcal E_r^TU,$ we get a system of projection:
\begin{equation}\label{2.2}
\begin{dcases}
\psi_{r} ''- \Delta \psi_r + \sum_{s=1} ^p c_{rs} \psi_s=0	& \text{in} \ (0,+\infty)\times \Omega, \\
\psi_r = 0 													& \text{on} \ (0, +\infty)\times\Gamma,
\end{dcases}
\end{equation}
which is independent of applied controls.
\let\qed\relax
\end{proof}

\begin{proof}[(\ref{th2.2_b})$\Longrightarrow$(\ref{th2.2_c})]
By the decomposition $\mathbb R^N = V\oplus \IM(C_p^T)$, for any given $ \mathcal E\in V$ we have
\begin{align*}
A^T\mathcal E= \mathcal E_1 + \mathcal E_2, \quad \mathcal E_1\in V, \ \mathcal E_2\in \IM(C_p^T).
\end{align*}
Applying $\mathcal E^T$ to~\eqref{1.1}, we get
\begin{equation}
\begin{cases}
\mathcal E^T{U} '' - \Delta
\mathcal E^T {U} +
\mathcal E_1^T {U} + \mathcal E_2^T {U} =
\mathcal E^TD\chi_\omega H 					& \text{in} \ (0,+\infty)\times \Omega, \\
\mathcal E^T{U} = 0							& \text{on} \ (0, +\infty)\times\Gamma.
\end{cases}
\end{equation}
Since the projections $ \mathcal E^T{U} $ and $\mathcal E_1^T{U} $ are independent of applied controls $H$, and the mapping $H\to U$ is linear, it follows that
\begin{equation}\label{2.3}
\mathcal E_2^T {U} = \mathcal E^TD\chi_\omega H, \quad \forall H \in \parens[\Big]{L^2\parens[\big]{0, +\infty; L^2 (\Omega)}}^{M}.
\end{equation}
Let $ H = D^T\mathcal E h$ with $h\in L^2\parens[\big]{0, +\infty; L^2(\Omega)}.$ By Proposition~\ref{th1.4}, the mapping
\begin{equation}
\mathcal R\colon h\to U
\end{equation}
is continuous from $L_{\loc}^2\parens[\big]{0, +\infty; L^2(\Omega)}$ to $\parens[\big]{C_{\loc}^0\parens[\big]{0, +\infty; H_0^1(\Omega)}\cap C_{\loc}^1\parens[\big]{0, +\infty; L^2 (\Omega)}}^N$. By~\cite[Theorem~4.1]{Lionsbis}, or~\cite[Corollary~5, p.~86]{Simon}, the embedding
\begin{equation}
C^0\parens[\big]{[T, 2T]; H^1_0(\Omega)}\cap C^1\parens[\big]{[T, 2T]; L^2(\Omega)} \longhookrightarrow L^2\parens[\big]{T, 2T; L^2(\Omega)}
\end{equation}
is compact, so is the mapping $\mathcal R$ from $L^2\parens[\big]{0, T; L^2(\Omega)}$ to $(L^2\parens[\big]{0, T; L^2(\Omega)})^N$. It follows from~\eqref{2.3} with $H= D^T\mathcal E h$ that
\begin{equation}
\norm{D^T\mathcal E}^2\norm{h}_{L^2\parens{0, T; L^2(\Omega)}} \leqslant \norm{\mathcal E_2^T}\norm{\mathcal Rh}_{\parens{L^2\parens{0, T; L^2(\Omega)}}^N}.
\end{equation}
The compactness of $\mathcal R$ leads to $D^T\mathcal E =0$, namely, $V\subseteq \Ker(D^T)$.

Now, writing $\mathcal E_2= C_p^Tx$, it follows from~\eqref{2.3} that
\begin{equation}\label{2.4}
t\geqslant 0 \colon \quad \mathcal E_2^TU = x^TC_pU=0.
\end{equation}
Since the reduced system~\eqref{1.8} is approximately controllable, the state variable $ Y_p = C_pU$ can approximately touch any given target in $ \parens[\big]{H^1_0(\Omega)\times L^2(\Omega)}^{N-p}$. Then it follows from~\eqref{2.4} that $x=0$, then $\mathcal E_2=0$. Therefore, the subspace $V$ is invariant for $A^T$.
\let\qed\relax
\end{proof}

\begin{proof}[(\ref{th2.2_c})$\Longrightarrow$(\ref{th2.2_a})]
Noting that $\dim (V) = p$, by Lemma~\ref{th2.1}, we get
\begin{equation}
\rank (D, AD,\dotsc, A^{N-1} D) \leqslant N-p,
\end{equation}
which together with~\eqref{1.10} implies~\eqref{1.15}.
\end{proof}
\let\qed\relax
\end{proof}

The approximate synchronization by $p$-groups under Kalman's rank condition~\eqref{1.15} requires that $A^T$ should be diagonalizable by blocks on the basis: $\mathbb R^N = \IM(C_p^T)\bigoplus V$. The following result confirms the converse.

\begin{prop}
Let\/ $\Omega \subset \mathbb{R}^{m}$ be a bounded domain with smooth boundary\/ $\Gamma$ and $ \omega$ be a subdomain of\/ $\Omega$. Assume that $\IM(C_p^T)$ admits a supplement $V$, both $\IM(C_p^T)$ and $V$ being invariant for $A^T$. Define $D$ by $\Ker(D^T) = V.$ Then, system~\eqref{1.1} is approximately synchronizable by $p$-groups with Kalman's rank condition~\eqref{1.15}.
\end{prop}

\begin{proof}
Noting $\dim(V)= p$, Lemma~\ref{th2.1} implies~\eqref{1.15} and
\begin{equation}
\IM(C_p^T)\cap \Ker(D, AD,\dotsc, A^{N-1}D)^T = \IM(C_p^T)\cap V = \{0\}.
\end{equation}
It follows that
\begin{equation}
\rank C_p(D, AD,\dotsc, A^{N-1}D)= \rank(C_p) = N-p.
\end{equation}
Thus by Theorem~\ref{th1.2}, we get the approximate synchronization by $p$-groups of system~\eqref{1.1}.
\end{proof}

\section{Stability of approximate internal synchronization by groups}\label{section_3}

In this section, we will examine the minimality of Kalman's rank condition~\eqref{1.15} from the point of view of controllability. We show that the independence of the approximately synchronizable state by groups with respect to applied controls, the linear independence of the components of the approximately synchronizable state by groups, and the non-extensibility of approximate synchronization, these properties are all the consequences of the minimality of Kalman's rank condition~\eqref{1.15}, and vice versa.

We first construct the extension matrix. For $1\leqslant i\leqslant m$, let $\lambda_i$ be the eigenvalues of $A^T$ and denote by
\begin{equation}\label{3.1}
\mathcal E_{i0} =0,\quad A^T\mathcal E_{ij} = \lambda_i \mathcal E_{ij} + \mathcal E_{i, j-1}, \quad 1\leqslant j\leqslant d_{i},
\end{equation}
the corresponding Jordan chain (see~\cite{Smale}).

Assume that $A$ satisfies the condition of $C_p$-compatibility~\eqref{1.6}, namely, $\IM (C_p^T)$ is invariant for~$A^T$. Then\footnote{This property can be verified after a basis transformation, and $\IM(C_p^T)$ will be called strongly $A$-marked in~\cite{marked}.} the Jordan chain of $A^T$ contained in $\IM(C_p^T)$ can be extended by adding root vectors to a Jordan chain in $\mathbb C^N$. Then we can find a set of indices:
\begin{equation}\label{3.2}
I= \braces[\big]{i: \mathcal E_{i\alpha_i}\in \IM(C_p^T) \ \text{with} \ 1 \leqslant \alpha_i \leqslant d_i}
\end{equation}
such that
\begin{equation}\label{3.3a}
\IM (C_p^T) = \bigoplus_{i\in I} \Span\braces[\big]{\mathcal E_{i1},\dotsc, \mathcal E_{i \alpha_i}}.
\end{equation}
Define the extension matrix of order $(N-q)\times N$ by
\begin{equation}\label{3.3}
\IM (C_q^T) = \bigoplus_{i\in I} \Span\braces[\big]{\mathcal E_{i1},\dotsc, \mathcal E_{id_i}} \quad \text{with} \quad \Ker(C_q)= \Span\{\epsilon_1,\dotsc, \epsilon_q\}.
\end{equation}
where
\begin{equation}\label{3.5}
q= N-\sum_{i\in I} d_i.
\end{equation}
We say that system~\eqref{1.1} is approximately $C_q$-synchronizable if for any given initial data $ (\widehat U_0, \widehat U_1)\in \parens[\big]{H_0^1(\Omega)\times L^2(\Omega)}^N$, there exists a sequence $\{H_n\} _{n\in \mathbb N} $ of internal controls in $\parens[\big]{L ^2 \parens[\big]{0, +\infty; L^2 (\Omega)}}^M$ with compact support in $[0, T]$, such that the sequence $\{U_n\} _{n\in \mathbb N} $ of the corresponding solutions to system~\eqref{1.1} satisfies
\begin{equation}
C_qU_n \to 0 \quad \text{in} \ \parens[\Big]{C^0_{\loc} \parens[\big]{[T, +\infty); H^1_0(\Omega)}\cap C^1_{\loc} \parens[\big]{[T, +\infty); L^2(\Omega)}}^{N-q}
\end{equation}
as $ n\to+\infty $. Furthermore, if there exists some functions $v_1,\dotsc, v_q$ such that
\begin{equation}\label{4.3}
U_n\to\sum_{s=1} ^q v_s\epsilon_s \quad \text{in} \ \parens[\Big]{C^0_{\loc} \parens[\big]{[T, +\infty); H^1_0(\Omega)}\cap C^1_{\loc} \parens[\big]{[T, +\infty); L^2(\Omega)}}^{N}
\end{equation}
as $ n\to+\infty$, system~\eqref{1.1} will be called approximately $C_q$-synchronizable in the pining sense.

\begin{lemm}\label{th3.1}
Let $C_q$ be a full row-rank matrix of order $N\times (N-q)$, such that $\Ker(C_q)\subseteq \Ker(C_p)$ and $C_qC_q^T=I$.
We have
\begin{equation}
C_p = C_pC^T_qC_q.\label{3.7}
\end{equation}
Furthermore, $C_pC^T_q$ is a full row-rank matrix of order $(N - p)\times (N-q)$, such that
\begin{equation}
\Ker(C_pC^T_q) = C_q\Ker(C_p). \label{3.8}
\end{equation}
\end{lemm}

\begin{theo}\label{th3.2}
Let\/ $\Omega \subset \mathbb{R}^{m}$ be a bounded domain with smooth boundary\/ $\Gamma$ and $\omega$ be a subdomain of\/~$\Omega$. Let $A$ satisfy the condition of\/ $C_p$-compatibility~\eqref{1.6}.
The following assertions are equivalent:
\begin{enumerate}\alphenumi
\item \label{th3.2_a} system~\eqref{1.1} is approximately synchronizable by $p$-groups under Kalman's rank condition~\eqref{1.15};
\item \label{th3.2_b} system~\eqref{1.1} is approximately synchronizable by $p$-groups in the pinning sense, and the approximately synchronizable state by $p$-groups $u=(u_1,\dotsc, u_p)^T$ is independent of applied controls;
\item \label{th3.2_c} system~\eqref{1.1} is approximately synchronizable by $p$-groups in the pinning sense and the components $u_1,\dotsc, u_p$ of the approximately synchronizable state by $p$-groups are linearly independent;
\item \label{th3.2_d} system~\eqref{1.1} is approximately synchronizable by $p$-groups in the pinning sense, and this property cannot be extended to an approximate $C_q$-synchronization with $q<p$.
\end{enumerate}
\end{theo}

\begin{proof}
\begin{proof}[(\ref{th3.2_a})$\Longrightarrow$(\ref{th3.2_b})]
Let $V = \Span\{\mathcal E_1,\dotsc, \mathcal E_p\}$ denote the largest subspace invariant for $A^T$ and contained in $\Ker(D^T)$. Let $\IM(Q_p) =V^\perp$. Noting~\eqref{2.1}, by~\cite [Proposition~2.8]{book}, $\IM(C_p^T)$ and $V^\perp$
are bi-orthonormal and $\Ker(C_p)\oplus\IM(Q_p) = \mathbb R^N$. Without loss of generality, we may assume that $C_pQ_p = I_{N-p}$ and $E^T_re_s = \delta_{rs}$ for all $1 \leqslant r, s \leqslant p.$ A straightforward computation gives
\begin{equation}
U_n =\sum_{r=1}^p\psi_re_r + Q_pC_pU_n,
\end{equation}
where $U_n$ denotes the solution to system~\eqref{1.1} with $H$ replaced by $H_n$ and $U$ replaced by $U_n$ respectively. The projection $\psi_r= \mathcal E^T_rU_n$ for $r = 1,\dotsc, p$ are governed by the homogeneous system~\eqref{2.2}, therefore, independent of applied controls. Noting~\eqref{1.5}, we get~\eqref{1.11} with $u_r= \psi_r$ for $r = 1,\dotsc, p$.
\let\qed\relax
\end{proof}

\begin{proof}[(\ref{th3.2_b})$\Longrightarrow$(\ref{th3.2_c})]
Assume by absurd that there exist $p$ reals $c_1,\dotsc, c_p$ not all zero, such that
\begin{equation}\label{3.9}
t \geqslant T \colon \quad \sum_{r=1}^pc_r(n_r-n_{r-1})u_r=0
\end{equation}
for any given initial data $(\widehat U_0, \widehat U_1)$ in the space $\parens[\big]{H^1_0(\Omega)\times L^2(\Omega)}^N$. Define the row vector $c_{N-p+1}$ by
\begin{equation}
c_{N-p+1} = c_1e_1^T + \dotsb + c_pe_p^T.
\end{equation}
We construct the extended matrix of order $(N-p+1)\times N$ by
\begin{equation}
C_{p-1}=
{\begin{pmatrix} C_p\\
c_{N-p+1}
\end{pmatrix}}.
\end{equation}
By~\eqref{1.11} and~\eqref{3.9}, system~\eqref{1.1} possesses the approximate synchronization by $(p-1)$-groups in the pinning sense. Furthermore, by~\cite[Theorem~8.7]{COCV2023}, system~\eqref{1.1} possesses the approximate $C_q$-synchronization with $q<p$. On the other hand, $A$ satisfies the condition of $C_q$-compatibility~\eqref{1.6} with $C_p$ replaced by $C_q$, by~\cite[Proposition~2.15]{book}, there exists a matrix $A_q$ of order $(N-q)$, such that $C_qA = A_qC_q$. Setting $Y_q=C_qU$, the approximate $C_q$-synchronization implies the approximate controllability of the following reduced system:
\begin{equation}\label{3.10}
\begin{cases}
Y_q''-\Delta Y_q+ A_qY_q = C_qD\chi_\omega H	& \text{in} \ (0, +\infty)\times\Omega, \\
Y_q=0										& \text{on} \ (0, +\infty)\times\Gamma.
\end{cases}
\end{equation}
By Lemma~\ref{th3.1}, $ C_pC_q^T$ is a full row-rank matrix of order $(N-p)\times (N-q)$. Using~\eqref{1.7} and~\eqref{3.8}, we get
\begin{equation}
C_pC_q^T A_q\Ker (C_pC_q^T)
=C_pC_q^TA_qC_q\Ker (C_p) =C_pC_q^TC_qA\Ker (C_p).
\end{equation}
Next, by~\eqref{1.6} and~\eqref{3.7}, we get
\begin{equation}
C_pC_q^TC_qA\Ker (C_p)
= C_pA\Ker (C_p) \subseteq C_p\Ker (C_p) =\{0\}.
\end{equation}
Then, it follows that
\begin{equation}
A_q\Ker(C_pC_q^T)\subseteq \Ker(C_pC_q^T),
\end{equation}
namely, $A_q$ satisfy the condition of $ C_pC_q^T $-compatibility. By~\cite[Proposition~2.15]{book} with $C_p$ replaced by $C_pC_q^T$, there exists a matrix $A_{pq}$ of order $(N-p)$, such that
\begin{equation}
C_pC_q^T A_q= A_{pq} C_pC_q^T.
\end{equation}
Applying the matrix $ C_pC_q^T $ to system~\eqref{3.10}, we get
\begin{equation}\label{3.11}
\begin{cases}
C_pC_q^T Y_q''-\Delta C_pC_q^T Y_q+ A_{pq} C_pC_q^T Y_q = 0	& \text{in} \ (T, +\infty)\times\Omega, \\
C_pC_q^T Y_q=0												& \text{on} \ (T, +\infty)\times\Gamma.
\end{cases}
\end{equation}
With the initial data at $t=T$,
\begin{equation}\label{3.12}
t=T \colon \quad
C_pC_q^T Y_q = 0, \quad C_pC_q^T Y_q'= 0,
\end{equation}
in the homogeneous system~\eqref{3.11}, we get
\begin{equation}\label{3.13b}
t \geqslant T \colon \quad C_pC_q^T Y_q= C_pC_q^T Y_q'=0.
\end{equation}

Now we return to the reduced system~\eqref{3.10} with the initial data $(\widehat U_0, \widehat U_1)= (0, 0)$. By the approximate controllability, for any given functions $\parens[\big]{Y_q(T), Y_q'(T)}\in \parens[\big]{H^1_0(\Omega)\times L^2(\Omega)}^{N-q} $, there exists a sequence $\{H_n\}_{n\in \mathbb N} $ of internal controls, such that the sequence $\{U_n\} _{n\in \mathbb N} $ of the corresponding solutions satisfies
\begin{equation}\label{3.13}
t=T \colon \quad C_q (U_n, U_n')\to (Y_q, Y_q') \quad \text{as} \ n \rightarrow +\infty,
\end{equation}
Applying $C_pC_q^T$ to~\eqref{3.13} and noting~\eqref{3.7}, we get
\begin{equation}
t=T \colon \quad C_p(U_n, U_n')\to C_pC_q^T (Y_q, Y_q') \quad \text{as} \ n \rightarrow +\infty,
\end{equation}
which together with~\eqref{3.13b} gives
\begin{equation}\label{3.13c}
C_p(U_n, U_n') \to 0 \quad \text{in} \ \parens[\Big]{C^0_{\loc}\parens[\big]{[T, +\infty); H^1_0(\Omega))\times L^2(\Omega)}}^{N-p}
\end{equation}
as $n\to +\infty.$

On the other hand, by the independence of $(u_1,\dotsc, u_p)^T$ with respect to applied controls, we deduce that $(u_1,\dotsc, u_p)^T \equiv 0$, which together with~\eqref{3.13c} gives
\begin{equation}
(U_n, U_n') \to 0 \quad \text{in} \ \parens[\Big]{C^0_{\loc}\parens[\big]{[T, +\infty); H^1_0(\Omega)\times L^2(\Omega)}}^{N}
\end{equation}
as $n\to +\infty$. Passing to the limit in~\eqref {3.13} as $n\to +\infty$, we get
\begin{equation}
t= T \colon \quad Y_q= Y_q'=0.
\end{equation}
Thus, we get $\Ker(C_pC_q^T)=\{0\}$, which together with~\eqref{3.8} gives
\begin{equation}
C_q\Ker(C_{p}) = \Ker(C_pC_q^T)=\{0\},
\end{equation}
namely, $\Ker(C_p) \subseteq \Ker(C_{q})$, which contradicts the condition $q<p$.
\let\qed\relax
\end{proof}

\begin{proof}[(\ref{th3.2_c})$\Longrightarrow$(\ref{th3.2_d})]
Assume that system~\eqref{1.1} can be extended to an approximate $C_q$-synchronization. Applying $C_q$ to~\eqref{1.11}, we get
\begin{equation}
t \geqslant T \colon \quad \sum_{r=1}^p
C_qe_ru_r = 0
\end{equation}
for all the initial data $(\widehat U_0, \widehat U_1) \in \parens[\big]{H_0^1(\Omega)\times L^2(\Omega)}^N.$ Since $\Ker(C_q) \subset\Ker(C_p)$, there exists some $r (1\leqslant r \leqslant p)$ such that $C_qe_r\not = 0. $ This contradicts the linear independence of the function $u_1,\dotsc, u_p$.
\let\qed\relax
\end{proof}

\begin{proof}[(\ref{th3.2_d})$\Longrightarrow$(\ref{th3.2_a})]
Since system~\eqref{1.1} is not approximately $C_q$-synchronizable with $q<p$, we have $C_q=C_p$ and $p=q$ in~\eqref{3.3}. It follows that
\begin{equation}\label{3.14}
\IM (C_p^T) = \bigoplus_{i\in I} \Span\braces[\big]{\mathcal E_{i1},\dotsc, \mathcal E_{id_i}}.
\end{equation}
Noting~\eqref{1.10}, we can write
\begin{equation}\label{3.15}
\rank (D, AD,\dotsc, A^{N-1}D) = N-p'\quad \text{with} \ p' \leqslant p.
\end{equation}

Let $V =\Ker (D, AD,\dotsc, A^{N-1}D)^T$ denote the largest subspace invariant for $A^T$ and contained in $\Ker(D^T)$. Moreover, noting~\eqref{3.14} and~\eqref{2.1}, we have
\begin{equation}
V\cap \bigoplus_{i\in I} \Span\braces[\big]{\mathcal E_{i1},\dotsc, \mathcal E_{id_i}} =V\cap \IM(C_p^T) = \{0\}.
\end{equation}
Then, we have
\begin{equation}
V\subseteq \bigoplus_{i\in I^c} \Span\braces[\big]{\mathcal E_{i1},\dotsc, \mathcal E_{id_i}}.
\end{equation}
Noting~\eqref{3.14} and $\dim\IM(C_p^T)= N-p$, we have
\begin{equation}
\dim \bigoplus_{i\in I^c} \Span\braces[\big]{\mathcal E_{i1},\dotsc, \mathcal E_{id_i}} = p.
\end{equation}\label{3.16}
If $\dim (V) = p'<p$, there exists at least an integer $i_0\in I^c$ and $\alpha_{i_0}<d_{i_0}$, such that
\begin{equation}
\mathcal E_{i_01}, \mathcal E_{i_02},\dotsc, \mathcal E_{i_0 \alpha_{i_0}-1}\in V \qquad \text{and} \qquad \mathcal E_{i_0\alpha_{i_0}} \not\in V.
\end{equation}

For $j=1,\dotsc, m -1$, applying the vectors $\mathcal E_{i_0j}^T$ to system~\eqref{1.1} and setting $ \psi_j=\mathcal E_{i_0j}^TU$, we get a system on the sub-chain $\mathcal E_{i_01}, \mathcal E_{i_02},\dotsc, \mathcal E_{i_0\alpha_{i_0}-1}$:
\begin{equation}
\psi_0=0,\quad
\begin{cases}\psi_j''-\Delta \psi_j + \lambda_{i_0}\psi_j + \psi_{j-1}=0	& \text{in} \ (0, +\infty) \times \Omega,\\
\psi_j=0																	& \text{on} \ (0,+\infty)\times \Gamma.
\end{cases}
\end{equation}
Furthermore, applying $\mathcal E_{i_0 \alpha_{i_0}}^T$ to system~\eqref{1.1} and setting $\psi_{\alpha_{i_0}}= \mathcal E_{i_0 \alpha_{i_0}}^T\!\!U$, we get an inhomogeneous system:
\begin{equation}\label{3.17}
\begin{cases}
\psi_{\alpha_{i_0}}''-\Delta \psi_{\alpha_{i_0}} + \lambda_{i_0}\psi_{\alpha_{i_0}} + \psi_{\alpha_{i_0} -1}=\mathcal E_{i_0 \alpha_{i_0}}^T\!\!D\chi_\omega H 		& \text{in} \ (0,+\infty)\times \Omega,\\
\psi_{\alpha_{i_0}}=0	& \text{on} \ (0,+\infty)\times \Gamma.
\end{cases}
\end{equation}
Define a function $ \theta$ by
\begin{equation}
\begin{cases}
\theta''-\Delta \theta + \lambda_{i_0}\theta + \psi_{\alpha_{i_0}-1}=0	& \text{in} \ (0,+\infty)\times \Omega, \\
\theta=0 																& \text{on} \ (0,+\infty)\times \Gamma.
\end{cases}
\end{equation}
The function $\psi_{\alpha_{i_0} -1}$ is independent of applied controls, so is the function $\theta$. Then, inserting the new variable
\begin{equation}
\eta=\psi_{\alpha_{i_0}}-\theta
\end{equation}
into system~\eqref{3.17}, we get the following self-closed system:
\begin{equation}\label{3.18}
\begin{cases}
\eta''-\Delta \eta+ \lambda_{i_0}\eta =\mathcal E_{i_0 \alpha_{i_0}}^T\!\!D\chi_\omega H	& \text{in} \ (0,+\infty)\times \Omega,\\
\eta= 0																					& \text{on} \ (0,+\infty)\times \Gamma.
\end{cases}
\end{equation}
Thus, setting
\begin{equation}
\widetilde A_p =
{\begin{pmatrix}
A_p&0 \\ 0& \lambda_{i_0}
\end{pmatrix}},
\quad
\widetilde D_p=
{\begin{pmatrix}
C_pD \\ \mathcal E_{i_0 \alpha_{i_0}}^T\!\!D
\end{pmatrix}},
\quad
\widetilde Y_p=
{\begin{pmatrix}
Y_p \\\eta
\end{pmatrix}},
\end{equation}
and combining the two systems~\eqref{1.8} and~\eqref{3.18} together, we get an extended system:
\begin{equation}\label{3.19}
\begin{cases}
\widetilde Y_p'' -\Delta\widetilde Y_p + \widetilde A_p\widetilde Y_p=\widetilde D_p\chi_\omega H	& \text{in} \ (0, +\infty)\times\Omega,\\
\widetilde Y_p =0																				& \text{on} \ (0, +\infty)\times\Gamma.
\end{cases}
\end{equation}

Let us admit for the moment the following rank condition:
\begin{equation}\label{3.20}
\rank(\widetilde D_p, \widetilde A_p\widetilde D_p,\dotsc, \widetilde A_p^{N-p}\widetilde D) = N-p + 1.
\end{equation}
By Theorem~\ref{th1.1} with $p$ replaced by $(p-1)$, system~\eqref{3.19} is approximately controllable. Then, for any given initial data $(\widehat U_0, \widehat U_1)$, there exists a sequence of controls $\{H_n\}_{n\in \mathbb N}$, such that the sequence of corresponding solutions $\{\widetilde Y_{pn}\}_{n\in \mathbb N}$ to system~\eqref{3.19} satisfies
\begin{equation}
t=T \colon \quad \widetilde Y_{pn} =
\begin{pmatrix}
C_pU_n \\
\mathcal E_{i_0 \alpha_{i_0}}^T\!\!U_n- \eta
\end{pmatrix}
\to
\begin{pmatrix}
0 \\
- \eta
\end{pmatrix}
\end{equation}
as $n\to \infty$, namely,
\begin{equation}\label{3.21}
t=T \colon \quad C_pU_n \to 0 \quad \text{and} \quad \mathcal E_{i_0 \alpha_{i_0}}^T\!\!U_n \to 0
\end{equation}
as $n\to +\infty.$ Noting~\eqref{1.8} with $Y_{pn}=C_pU_n$ and that $H_n$ has the compact support in $[0, T]$, we deduce from the first condition in~\eqref{3.21} that
\begin{equation}
t\geqslant T \colon \quad C_pU_n \to 0 \quad \text{as} \ n \rightarrow +\infty.
\end{equation}
Then the approximate pinning synchronization by $p$-groups implies the existence of functions $u_1,\dotsc, u_p$, such that
\begin{equation}
U_n \to \sum_{r=1}^pe_ru_r\quad \text{in} \ \parens[\Big]{C^0_{\loc}\parens[\big]{[T, +\infty); H_0^1(\Omega)} \cap C^1_{\loc}\parens[\big]{[T, +\infty); L^2(\Omega)}}^N
\end{equation}
as $n\to +\infty$. Passing to the limit in the second condition in~\eqref{3.21}, we get
\begin{equation}\label{3.22}
t= T \colon \quad \sum_{r=1}^p\mathcal E_{i_0 \alpha_{i_0}}^T\!\!e_ru_r 
=0.
\end{equation}
By~\cite[Theorem~3.3]{Zu}, $u= (u_1,\dotsc, u_p)^T$ fills the subspace $\parens[\big]{H_0^1(\Omega)\times L^2(\Omega)}^p$, then it follows from~\eqref{3.22} that
\begin{equation}
\mathcal E_{i_0 \alpha_{i_0}}^T\!\!e_1 =0, \ \dotsc, \ \mathcal E_{i_0 \alpha_{i_0}}^T\!\!e_p=0,
\end{equation}
namely, $\mathcal E_{i_0 \alpha_{i_0}}\in \braces[\big]{\Ker(C_p)}^\perp= \IM(C_p^T)$. Noting~\eqref{3.14} and $i_0\in I^c$, this contradicts the condition $p'<p$.

Finally, we show the rank condition~\eqref{3.20}. Let
\[
\widetilde x_p =
\begin{pmatrix}x_p\\y
\end{pmatrix}\in \mathbb R^{N-p+1}\quad \text{with} \ x_p\in \mathbb R^{N-p} \ \text{and} \ y\in \mathbb R,
\]
such that $\widetilde A_p^T\widetilde x_p =
\lambda_{i_0} \widetilde x_p,$ namely,
\begin{equation}\label{3.23}
\begin{cases}
D^T(C_p^Tx_p + y\mathcal E_{i_0 \alpha_{i_0}})=0,\\
A_p^Tx_p = \lambda_{i_0}x_p.
\end{cases}
\end{equation}

We claim that $y =0$. Otherwise, without loss of generality, we may take $y=1$. Using~\eqref{1.7}, \eqref{3.1}, \eqref{3.16} and the first formula in~\eqref{3.23}, we successively get
\begin{equation}
\begin{cases}
A^T(\mathcal E_{i_0 \alpha_{i_0}} \!\! + C_p^Tx_p) =\mathcal E_{i_0 \alpha_{i_0}-1} \!\! + \lambda_{i_0} (\mathcal E_{i_0 \alpha_{i_0}} \!\! + C_p^Tx_p) \subseteq V + \lambda_{i_0} (\mathcal E_{i_0 \alpha_{i_0}} \!\! + C_p^Tx_p),\\
\mathcal E_{i_0 \alpha_{i_0}} + C_p^Tx_p \in \Ker(D^T).
\end{cases}
\end{equation}
Since $V$ is also invariant for $A^T$ and contained in $\Ker(D^T)$, so is the subspace
\begin{equation}
W = V \bigoplus \Span \braces[\big]{\mathcal E_{i_0 \alpha_{i_0}} + C_p^Tx_p}.
\end{equation}
Noting~\eqref{2.1} and~\eqref{3.16}, we get
\begin{equation}
V \cap \Span \braces[\big]{\mathcal E_{i_0 \alpha_{i_0}} + C_p^Tx_p} \subseteq V \cap \Span\braces[\big]{\mathcal E_{i_0 \alpha_{i_0}}} + V \cap \IM(C_p^T) = \{0\}.
\end{equation}
Thus, by Lemma~\ref{th2.1}, we get
\begin{equation}
\rank (D, AD,\dotsc, A^{N-1}D) \leqslant N-\dim (W)
= N-(p'+1),
\end{equation}
which contradicts~\eqref{3.15}. Thus, inserting $y=0$ in~\eqref{3.23}, $x_p$ is an eigenvector of $A_p^T$ contained in $\Ker(D_p^T)$. On the other hand, condition~\eqref{1.9} implies that the largest subspace invariant for $A_p^T$ and contained in $\Ker(D_p^T)$ is reduced to $\{0\}$, then $x_p=0$, namely, $ \widetilde x_p = 0$. Consequently, the largest subspace invariant for $\widetilde A_p^T$ and contained in $\Ker(\widetilde D_p^T)$ is reduced to $\{0\},$ which gives Kalman's rank condition~\eqref{3.20}.
\end{proof}
\let\qed\relax
\end{proof}

The present work mainly concentrates on the internal controllability and synchronization for the problem with Dirichlet boundary condition. However, the approaches can be easily adapted to the problem with Neumann boundary condition, and some other problems. In particular, the sufficiency of Kalman's rank condition~\eqref{1.15} for the approximate synchronization by $p$-groups is only used in the implication $\eqref{th3.2_d}\Longrightarrow\eqref{th3.2_a}$, and the other ones have an abstract character, then can be applicable to the approximate synchronization by groups for the problems with boundary controls considered in~\cite{book}.

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

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