The slaughtering of these more highly evolved animals produces strong emotions such as fear and anger which remain as vibrations in the meat itself and increase people's naturally aggressive and/or fearful tendencies.
According to a study of more than 120,000 people carried out over 28 years it is scientifically proven that regularly eating red meat increases significantly risk of death from heart disease and cancer.
The food one eats has an effect on the clarity and upliftment of the mind; it plays an important role not only in bodily health but also in one's spiritual life.
In the pictures below look at the eyes of the cow they say it all.
Cow is Devine. if a person only has cow and nothing else he can still survive without eating her meat ofcourse. it can provixe you with everything that a mother does to its child.milk, shelter, security to name a few. if you have a cow at home there certain peace at home a feeling of content.
Over 50,000 cows are killed everyday in india. I dont know how to save them, in gujrat cow slaughter is banned. It should be banned everywhere. I used to buy leather products in past without checking if it was cow leather. Even hindus are sell cow leather. No wonder this country is going down in a negative direction. I really dont know how this is happening in this nation where cow is considered as mother. Shame on this modern country called india.
may Lord krishna save all the cows!!! And let every cow killer and eater rot in hell!!!
Good initiative brother.. feeling happy after reading your thoughts and research on indian culture..
The most important thing about Desi cow is that it is a storehouse of "Prana Tattva".. Its the prana tattva which can give healing powers.. Human body is not made just from gross material elements element but also contains Prana.. When thsi prana is disturbed, it leads to various disorders... Many diseases can be healed by fixing the prana tattva.. This is where cow becomes so important... the producst from a happy cow like milk, ghee, gaumutra have great healing properties as they are rich in Prana tattva..
It is years and years old practice to make leather from dead bodies and using it. Nothing is wrong in this, as now only body is left. But to kill just for this is wrong. But ones after natural death you can use the body. Even we are killing plants for food that is not considered a sin. Because life is this you need to depend on something for survival. But slaugthering of cow , where god designed cow to provide milk and because of religious reasons is wrong. But using leather as a byproduct cannot be wrong . As cow is not primerely or only slauthered for leather. Rest is on beliefe.


Une analyse de mon travail cinématographique par Jihoon Kim dans son livre consacré aux images hybrides dans le cinéma contemporain.
An analysis of my films work by Jihoon Kim in his book about Hybrid Moving Images.
Grounding its study in interdisciplinary framework of film studies, media studies, and contemporary art criticism,Between Film, Video, and the Digital offers a fresh insight on the post-media conditions of film and video under the pervasive influences of digital technologies, as well as on the crucial roles of media hybridity in the creative processes of giving birth to the emerging forms of the moving image. Incorporating in-depth readings of recent works by more than thirty artists and filmmakers, including Jim Campbell, Bill Viola, Sam Taylor-Johnson, David Claerbout, Fiona Tan, Takeshi Murata, Jennifer West, Ken Jacobs, Christoph Girardet and Matthias Müller, Hito Steyerl, Lynne Sachs, Harun Farocki, Doug Aitken, Douglas Gordon, Stan Douglas, Candice Breitz, among others, the book is the essential scholarly monograph for understanding how digital technologies simultaneously depend on and differ film previous time-based media, and how this juncture of similarities and differences signals a new regime of the art of the moving image.
Jihoon Kim is an assistant professor of cinema and media studies in the Department of Film Studies at Chung-ang University. He is the author of Between Film, Video, and the Digital: Hybrid Moving Images in the Post-media Age (New York: Bloomsbury Academic, 2016). His essays on film theory, experimental film and video, art of the moving image, cinema and contemporary art, digital cinema, and experimental documentary have appeared in Screen, Film Quarterly, Animation: An Interdisciplinary Journal, Millennium Film Journal, Leonardo Electronic Almanac, and the anthologies Global Art Cinema: New Histories and Theories (Oxford University Press, 2010), and Taking Place: Location and the Moving Image (University of Minnesota Press, 2011), among others. Currently He is writing his second book manuscript entitled Documentary's Expanded Fields: New Media, New Platforms, and the Documentary.


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Patented design provides two layers of protection. Exterior polycarbonate layer disperses impact while interior custom engineered soft acrylic layer absorbs shock.
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text:
Could Jeb Bush Still Enter the GOP Presidential Primary?
Kyle Munzenrieder | December 7, 2011 | 2:11pm
The Confederacy of Dunces that emerged to vie for the GOP presidential primary certainly must have some could-have-been candidates rethinking their decision. After the swift, and often surreal, rises and eventual falls of Michele Bachmann, Rick Perry, and Herman Cain, the race is pretty much down to Newt Ginrich (at least until America remembers why exactly it was he was so hated in the '90s) and Mitt Romney (a man whose boring factors rivals that of a single slice of cheap ham on Wonder Bread).
Now, the Wall Street Journal reports there's murmurs among the conservative elite of the possibility of drafting a new candidate to enter race. Among the names apparently being whispered: Jeb Bush.
This isn't much of a surprise considering that ever since leaving the Florida Governor's Office Jebbers has pretty much made a career out of not running for president. Anytime he needs some quick publicity he just lets someone know he's not running for Prez. Obviously, he ruled out a 2012 bid long ago (with the possibility of a 2016 or beyond bid open), but could he be convinced to go back on his word?
The actual WSJ Political Diary entry is behind a paywall, but RedState excerpts some of it:
Efforts are underway by some wealthy Republican donors and a group of conservative leaders to investigate whether a new Republican candidate could still get into the presidential race. The talk is still preliminary and somewhat wishful, but it reflects dissatisfaction with the two leading candidates, Newt Gingrich and Mitt Romney.
Conservative leaders are looking into whether it is feasible for a dark horse to get on the ballot in select states. The deadline to qualifying for the ballot has passed in Florida, South Carolina, Missouri, and New Hampshire. But a candidate could still get on the ballot in states like Tennessee, Oklahoma, Louisiana, Michigan and Texas. At the first-in-the-nation Iowa caucuses, voters write in their choice, so there is no formal filing deadline.
In addition to New York Mayor Rudy Giuliani, Indiana Gov. Mitch Daniels, businessman Donald Trump, Rep. Paul Ryan of Wisconsin and South Carolina Sen. Jim DeMint, RedState adds that Jeb Bush's name is also in the mix.
Alan Steinberg at PolitickerNJ says that Bush would be the perfect candidate:
He was a superb governor of Florida, and he retains a remarkably high degree of popularity in the Sunshine State. Jeb has first rate communication skills and an ability to forge solutions on quality of life issues, particularly education and the environment. Although he is from the center-right, he is able to please both movement conservatives and moderates. In most presidential election years, Jeb Bush would be considered to be an ideal Republican presidential candidate.
There's still two glaring problems with this however (aside from, you know, the small fact that he doesn't want to run): 1. He still has the last name Bush. 2. It's too late to get on the Florida ballot.
There's also some other potential problems. Namely his past connections to Lehman Brothers and questions over weather his legacy set up Florida's economy to fail during the recession. His children also have some interesting arrest records.
Considering Bush's popularity in the state he still could have likely swept into the race just before the deadline to get on the ballot in Florida and found an easy path to victory. Even if he didn't win Iowa, New Hampshire, or South Carolina, he almost certainly would have won Florida and that could have set him up with enough momentum through the rest of the primary season.
But with Florida now out of the picture, a last-minute Jebbers run makes considerably less sense. Though the fact that conservatives aren't happy with either Gingrich or Romney is good news for Bush's future presidential ambitions. 2012 may have been too early for Bush to escape the stigma of his last name, but by 2020 he could be long forgotten. 2016 seems perfect. Assuming Marco Rubio isn't running, that is.
Follow Miami New Times on Facebook and Twitter @MiamiNewTimes.

redpajama_set_name: RedPajamaCommonCrawl


text:
Joseph Rainey
Bishop Garrison Named Interim Executive Director of Truman National Security Project
By Rainey Center
Washington, D.C. – The Joseph Rainey Center for Public Policy is proud to announce that our co-founder and President Bishop Garrison has been chosen to serve as the Interim Executive Director of the Truman National Security Project.
The Truman National Security Project is a nationwide organization whose diverse leaders came together in the aftermath of 9/11 to shape and advocate for tough, smart national security solutions. The Truman membership includes more than 1,700 post-9/11 veterans, frontline civilians, policy experts, and political professionals from 47 different states.
Along with its sister organization, the Truman Center for National Policy, the Truman National Security Project identifies, trains, and positions rising leaders across America who share their worldview and values, working to shape the debate, fight for policy change, and deliver concrete solutions at the local, state, and national levels.
"I am humbled by the board's trust and confidence," said Garrison. "The Truman community means a great deal to me personally, and I'm honored to have the opportunity to help lead it at such a pivotal moment. I thank Michael Breen for his dedication and years of leadership to this organization. He has created an amazing foundation to build upon and a standard to follow that will not soon be forgotten."
In addition to his new role at the Truman National Security Project, Garrison will continue to serve as the President of the Board of Directors for the Rainey Center.
The Joseph Rainey Center for Public Policy is a post-partisan, 501(c)3 public policy research organization and incubator for diverse, non-traditional emerging policy leaders, named after Joseph Rainey, a former slave who was the first African-American to serve in the United States House of Representatives.
Sarah Rumpf
info@raineycenter.org
@RaineyCenter
Tags: bishop garrison, National Security, national security, press release


text:
Marta Amorós Romagosa (born 9 August 1970) is a former synchronized swimmer from Spain. She competed in the women's solo competition at both the 1988 and 1992 Summer Olympics.

References

1970 births
Living people
Spanish synchronized swimmers
Olympic synchronized swimmers of Spain
Synchronized swimmers at the 1988 Summer Olympics
Synchronized swimmers at the 1992 Summer Olympics

redpajama_set_name: RedPajamaWikipedia


text:
Q: jQuery Datatable serverside reload with new post data I have common function for all the server-side data-tables which accepts the table object, url and the post data. The post data consists of the values from the select boxes that are in the filter area.
UTIL.serverDatatable = function (table, api, data) {
        var token = "Bearer " + JSON.parse(UTIL.getItemLocalStorage('token'));
        return table.DataTable({
            "processing": true,
            "serverSide": true,
            "paging": true,
            "ajax": {
                url: api,
                type: "post",
                beforeSend: function (request) {
                    request.setRequestHeader("Authorization", token);
                },
                "data": data
            }
        });
    };

And another function to reload the datatable
UTIL.datatableReload = function(table) {
    table.ajax.reload();
};

This worked fine before these functions were made common. The post data was  changing when the filters changes and the reload function was called after that. But now when the data is changed, the post is not getting updated. Is there any solution for this? How to post the changed data via the ajax call?

A: Change the code as below
UTIL.serverDatatable = function (table, api, callback) {
    var token = "Bearer " + JSON.parse(UTIL.getItemLocalStorage('token'));
    return table.DataTable({
        "processing": true,
        "serverSide": true,
        "paging": true,
        "ajax": {
            url: api,
            type: "post",
            beforeSend: function (request) {
                request.setRequestHeader("Authorization", token);
            },
            "data": callback
        }
    });
};

Then call the util function as below
UTIL.serverDatatable(tablename, apiname, function(data){
   data.key= $('#input1').val();
});

And then call the function as
UTIL.datatableReload = function(table) {
    table.ajax.reload();
};

redpajama_set_name: RedPajamaStackExchange


Here you'll find travel reports about Limáni Sámou. Discover travel destinations in Greece of travelers writing a travel blog on FindPenguins.
Awesome traditional meal at "Epikairo"
Tonight for dinner we went to a "mezodopoleio" or a Greek tapas restaurant, Epikairo, in Vathi, the capital of Samos Island. We had eggplants Imam, domades, chicken with dates and fruit, octopus, and of course, ouzo.
Today, we visited the fascinating museum of archeological treasures in Vathy, Samos. The artifacts were all found at the Temple of Hera, another archeological site on the southern part of the island where the goddess Hera was said to have been born.
My hope is that the Tibetan people will get sometime his freedom again! The painting reminds me of the strenght and the confidence they have and the faith in a better future.


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Tottenham kept up momentum in the Premier League with today's 1-0 win away to Crystal Palace.
With 88 minutes on the clock, Harry Kane scored his 24th goal of the season, heading in Christian Eriksen's corner kick, to wrap up all three points at Selhurst Park.
The results sees Spurs climb up to fourth place in the Premier League standings on 55 points – two above Chelsea (53), who lost 2-1 today to Manchester United.


text:
\section{Introduction}\label{section-introduction}
A salient issue in biological regulatory networks is to properly understand the structure and temporal behaviour, which requires to integrate regulatory data into a formal dynamical model \cite{de2002modelling}. Although this problem has been recurrently solved by standard mathematical methods, such as differential or stochastic equations, it is still complicated owing to the diversity and sophistication of regulatory mechanisms, as well as the chronic lack of credible quantitative information \cite{faure2006dynamical}. Motivated by such defect, the intrinsically qualitative methods were developed to learn on Boolean algebra or generalisation thereof \cite{kauffman1969metabolic}. Boolean network (BN), as an effective approach for exploring the evolution patterns and structure, has increasingly attracted much interest. In a Boolean model, each node is valued as a binary logical variable, and its state is updated according to a certain specified logical rule composed of basic logical operators and its neighbors' states \cite{kauffman1969metabolic,azuma1,azuma2}.

Subsequently, probabilistic BNs (PBNs) were introduced in \cite{shmulevich2002probabilistic} to characterize the switch-like behavior of gene regulation networks. Such switch-like behavior is reflected when cells move from one state to another in a normal growth process and when cells respond to external signals. Particularly, switching in probability occurs in the discrete decision-making processes of the cell. To be more detailed, PBN is an effective tool to characterize the signal pathway of the mammalian cell-cycle with mutation phenotype \cite{faryabi2008regulatory}, which is further demonstrated in the simulation of this paper.

In recent years, Cheng and his cooperators \cite{chengdz2011springer} proposed an algebraic technique, called semi-tensor product (STP), for the analysis of Boolean (control) networks. The STP of matrices is defined as a novel matrix product for two arbitrary-dimensional matrices, thereby it breaks the traditional dimension-matching condition and is more flexible to utilize \cite{cheng2010analysis}. Based on the STP technique, several fundamental problems in control theory have been investigated for BNs and PBNs, including but not limited to, stability and stabilization \cite{zhusy2018tac,liht2017siam,liht2019siam}, controllability \cite{lu2018stabilization,Margaliot2018Auto}, observability \cite{margaliot2012aut1218,yyy2019observability,guoyq,guoyq2019auto}, synchronization \cite{chenhw2019}, optimal control \cite{wu2017policy,wuyhtac}, and other related problems \cite{lirui2018siam,yyyTAC,liuyang2017TAC}. In essence, the STP of matrices linearizes the algebraic function by enumerating the state space. It leads to a high computational complexity of many developed methods, and these methods are difficultly applied on large-dimensional BNs \cite{cheng2010analysis}.

Among the control-related issues, stabilization is a fundamental and essential problem in the therapeutic intervention and safety verification \cite{li2013state}. More precisely, a recent research discovers that gene activity emerges spontaneous and orderly collective behaviour \cite{huang1999gene}, and coincidentally, it can be properly demonstrated by the BN being stabilized to a certain state. In addition, in the long time evolution of genes in gene regulatory networks, steady states usually represent cell states, including cell death or unregulated growth. Hence, there is abundant justification in the assertion that one needs to design an efficient control strategy, under which the gene regulatory network is guided to a desirable state and remains at this state afterward.

In the last decades, the stabilization of BNs and PBNs has been studied mostly by means of traditional discrete control, including state feedback control \cite{lu2018stabilization,li2013state}, sampled-data control \cite{lu2018stabilization}, as well as event-triggered control \cite{zhusy2019}. These controllers are applied to either all the nodes or some randomly selected nodes of a BN; it may result in the greater control cost or redundant control inputs for some nodes.

Recently, one significative method, called pinning control strategy, has been introduced in \cite{lu2020tac} and has received considerable attentions. The main conception of pinning control is that only a fraction of nodes are determined to be imposed state feedback controllers, while the remaining nodes can be propagated through the coupling among nodes \cite{wu2013synchronization}. In the existing works, control design is almost from the point of state transition matrix \cite{liff2015pinning,liff2019set}. Unfortunately, such design approach will lead to a high-dimensional form of workable controllers associated with high computational complexity. For a BN with $n$ nodes, its state transition matrix is $2^{2n}$ dimension, and then the time complexity of designing pinning controllers is $O(2^{2n})$. In order to overcome this aporia, Zhong et al. \cite{zhong2020arxiv} developed a novel pinning control design strategy, which only utilized the local neighbors' information in the network structure, rather than the traditional global state transition space. Subsequently, Zhu et al. \cite{zhusy2020arxiv} further utilized such design method to address the observability problem for BNs. It is worth mentioning that this method successfully reduces the time complexity from $O(2^{2n})$ to $O(n2^{\alpha})$, where $\alpha$ is the largest in-degree of nodes in wring digraph. Hence, it provides a potential applications on some large-scale networks.

Inspired by the above discussions, we study the pinning control design for PBNs in consideration of the stochastic switching signal. The main contributions of this paper are concluded as follows:
\begin{enumerate}
  \item A more efficient pinning control is designed to stabilize PBNs with respect to the feature of each pinning node. More precisely, some pinning nodes can be imposed uniform state feedback controllers to reduce control cost; while other pinning nodes need to utilize different controllers for different possibilities. Particularly, a criterion is proposed to identify pinning nodes that can be applied on uniform controllers.
  \item The first group of pinning nodes (to achieve stabilization) is easily determined by finding a feasible feedback arc set (FAS) in the wiring digraph of the PBN. Besides, the second group of pinning nodes (to achieve stabilization at a certain state) can be selected by solving an optimization problem.
  \item The time complexity of our approach is dramatically reduced from $O(2^{2n})$ as in \cite{wanglq2019pinning,liff2019set} to $O(n2^{\max_{i\in[1,n]}|\mathcal{N}_i|})$, where $\mathcal{N}_i$ is the in-neighbor set of node $x_i$. Thereby, it is more implementable than conventional pinning control design methods when applied in large-scale networks, especially sparse networks.
\end{enumerate}

The reminder of this paper is organized as follows: Section \ref{section-preliminaries} introduces some basic definitions and model formulation. Sections \ref{section-further stability} and \ref{section-graphicaltheory} respectively design pinning control to achieve stabilization and stabilizing at a certain state. An biological example is established in Section \ref{section-conclusion}, followed by a brief conclusion in Section \ref{section-stabilization}.

\section{Preliminaries and model formulation}\label{section-preliminaries}

\subsection{Notations}
For the better expression, we list some basic notations, which are used throughout this study.
\begin{itemize}
  \item $\mathbb{N}_+$ is the set of positive integers;

  \item $[a,b]:=\{a,a+1,\cdots,b-1,b\}\subseteq \mathbb{N_+}$;

  \item $\mathscr{D}~:=\{0,1\}$ and
  $\mathscr{D}^n:=\underbrace{\mathscr{D}\times \mathscr{D} \times \cdots \times \mathscr{D}}_n$;

  \item $\text{Col}_j(A)$ (resp., $\text{Col}(A)$) is the $j$-th column (resp., column set) of matrix $A$;

  \item $\Delta_n:=\{\text{Col}_i(I_n)~|~i\in[1,n]\}$, where $I_n$ is the $n\times n$ identity matrix;

  \item $\mathscr{R}^{m\times n}$ (resp., $\mathscr{L}^{m\times n}$) is the set of $m\times n$ real matrices (resp., logical matrices). Moreover, matrix $A\in\mathscr{R}^{m\times n}$ is called a logical matrix, if $\text{Col}(A)\subseteq \Delta_{m}$;

  \item $\mathbf{r}:=(r_1, \cdots, r_p)^\top$ is a $p$-dimensional probability vector satisfying $r_i\geq 0,i\in[1,p]$ and $\Sigma_{i=1}^{p}r_i=1$;

  \item $\mathscr{P}^{m \times n}$ is the set of $m \times n$ probability matrices, whose columns are $m$-dimensional probability vectors;

  \item $\mathbb{B}_2$ is the set of $2$-dimensional probability vectors, where each element is larger than 0. Besides, $\mathbb{B}$ is the set of real numbers larger than 0 and less than 1;

  \item $[a_i]_{i\in I}:=a_{i_1},\cdots,a_{i_k}\in \{a_1,\cdots,a_n\}$ is a series of variables, where index set $I=\{i_1,\cdots,i_k\}\subseteq [1,n]$;

  \item $A^\top$ is the transposition of matrix $A$;

  \item $|\Omega|$ is the cardinal number of set $\Omega$.
\end{itemize}

\subsection{Problem Description}
PBN is a kind of logical network composed by certain number of nodes, in which each node is interacted by logical operators, including conjunction, disjunction, negation, and so on. Accordingly, a PBN with $n$ nodes considered in this paper can be described as:
\begin{equation}\label{equation-PBN}\left\{\begin{array}{l}
x_1(t+1)=f_1([x_j(t)]_{j\in \mathcal{N}_1}),\\
x_2(t+1)=f_2([x_j(t)]_{j\in \mathcal{N}_2}),\\
~~~~~~~~~~~~\vdots\\
x_n(t+1)=f_n([x_j(t)]_{j\in \mathcal{N}_n}),
\end{array}\right.
\end{equation}
where $x_i(t)\in \mathscr{D}, i\in [1,n]$ denotes the state of the $i$-th node at time instance $t$, set $\mathcal{N}_i\subseteq [1,n]$ contains the subscript indices of the in-neighbors of node $x_i$, $f_i:\mathscr{D}^{|\mathcal{N}_i|}\rightarrow \mathscr{D}, i\in[1,n]$ is the logical function for node $x_i$. Precisely, each logical function $f_i,i\in [1,n]$ for node $x_i$ has $l_i$ possibilities, and is chosen from a specific finite logical function set $\mathscr{F}_i=\{f_i^1,f_i^2,\cdots,f_i^{l_i}\}$. The probability of $f_i$ being $f_i^\kappa,\kappa\in[1,l_i]$ is assumed to be $p_i^\kappa$ with $\sum_{\kappa=1}^{l_i}p_i^\kappa=1$.
There are $\Pi_{i=1}^{n}l_{i}$ possibilities, and the $\lambda$-th model is denoted by $\Sigma_\lambda=\{f_1^{\lambda_1},f_2^{\lambda_2},\cdots,f_n^{\lambda_n}\}$, where $\lambda=\Sigma_{i=1}^{n-1}(\lambda_i-1)\Pi_{j=1}^{n-1}l_{j+1}+\lambda_n,\lambda_i \in [1,l_i]$. In this paper, the probability of each logical function is assumed to be mutually independent, then the probability of $\Sigma_\lambda$ being active is $P_\lambda=\Pi_{i=1}^{n}p_i^{\lambda_i}$. At each time step, $x(t)=(x_1(t),x_2(t),\cdots,x_n(t))\in \mathscr{D}^n$ is defined as the state of PBN (\ref{equation-PBN}).

\begin{definition}[Functional Variable]\label{functional-variable}
Consider logical function $f(x_1,\cdots,x_n):\mathscr{D}^{n}\rightarrow \mathscr{D}$, if $f(x_1,\cdots,x_i,\cdots,x_n)\neq f(x_1,\cdots,\neg x_i,\cdots,x_n)$, variable $x_i$ is called functional variable, otherwise it is called non-functional variable.
\end{definition}

Accordingly, denote $[x_j]_{j\in \mathcal{N}_{i,\kappa}}$ by functional variables for each possibility $f_i^\kappa\in \mathscr{F}_i,\kappa\in [1,l_i]$.
Thereupon, PBN (\ref{equation-PBN}) can be simplified by removing non-functional variables, and its network structure can be depicted by a wiring digraph defined as follows.

\begin{definition}[Wiring Digraph]\label{wiring-digraph}
The wiring digraph of a BN (or a PBN) with $n$ state components is denoted by $\mathbf{G}=(\mathbf{V},\mathbf{E})$. Thereinto, node set $\mathbf{V}$ is equivalent to $\{x_1,\cdots,x_n\}$, and edge set $\mathbf{E}$ consists of all edges $x_j\rightarrow x_i, i\in [1,n]$, where variable $x_j$ is a functional variable of logical function $f_i$ with positive probability.
\end{definition}

\begin{definition}[Global Stability]\label{global-stability}
The state $x(t)\in \mathscr{D}^n$ of PBN (\ref{equation-PBN}) is called a steady state if $P\{x(t+1)=x(t)\}=1$. Besides, $\{x(t),x(t+1),\cdots,x(t+s-1)\}\subseteq \mathscr{D}^n$ of PBN (\ref{equation-PBN}) is called an $s$-length cyclic attractor if $P\{x(t+s)=x(t)~\mathrm{and}~ x(t+\tau)\neq x(t), \tau\in [1,s-1]\}=1$. Then, PBN (\ref{equation-PBN}) is globally stable, if there exists a unique steady state as the attractor without cyclic attractor.
\end{definition}

\subsection{Model Transformation}
To convert the logical equation (\ref{equation-PBN}) into an algebraic representation, we introduce the STP tool. Moreover, its detailed properties and applications can be acquired in \cite{chengdz2011springer,cheng2010analysis}.

\begin{definition}[\cite{chengdz2011springer}]\label{stp}
Given two matrices $A\in \mathscr{R}^{m \times n}$ and $B\in \mathscr{R}^{p\times q}$, the STP of $A$ and $B$, termed as $A \ltimes B$, is defined as
\[A \ltimes B=(A\otimes I_{\frac{\alpha}{n}})(B\otimes I_{\frac{\alpha}{p}}).\]
Here, `$~\otimes$' is the Kronecker product, and $\alpha$ is the least common multiple of $n$ and $p$.
In addition, the symbol `$\ltimes$' is hereafter omitted if no confusion occurs.
\end{definition}

\begin{proposition}[\cite{chengdz2011springer}]\label{proposition-properties}
Some operation properties:
\begin{itemize}
  \item[1)] Let $A\in \mathscr{R}^{m \times n}$, $B\in \mathscr{R}^{p \times q}$ and $C\in \mathscr{R}^{r\times d}$, then $(A\ltimes B)\ltimes C = A\ltimes (B \ltimes C)$;

  \item[2)] Let $A\in \mathscr{R}^{p\times 1}$ and $B\in \mathscr{R}^{m\times n}$, then $A \ltimes B=(I_{p} \otimes B)\ltimes A$;

  \item[3)] Let $A\in \mathscr{R}^{p\times 1}$ and $B\in \mathscr{R}^{d\times 1}$, then $B \ltimes A= W_{[p,d]} \ltimes A \ltimes B$, where $W_{[p,d]}=[I_{d}\otimes \delta_{p}^{1}, \cdots, I_{d} \otimes \delta_{p}^{p}]\in \mathscr{L}^{pd\times pd}$ is called a swap matrix.
\end{itemize}
\end{proposition}

\begin{lemma}[\cite{chengdz2011springer}]\label{lemma-reducing matrix}
Let $\mathbf{x}=\ltimes_{i=1}^n \mathbf{x}_i=\mathbf{x}_1\ltimes \mathbf{x}_2\ltimes\ldots\ltimes \mathbf{x}_n$ with $\mathbf{x}_i\in\Delta_2, i\in[1,n]$, then it holds that $\mathbf{x}^2=\Phi_n \mathbf{x}$ with $\Phi_n=\Pi_{i=1}^n I_{2^{i-1}}\otimes [(I_2 \otimes W_{[2,2^{n-i}]})M_r]$, where $M_r$ denotes the power-reducing matrix satisfying $\mathbf{x}_i^2=M_r \mathbf{x}_i$.
\end{lemma}

\begin{definition}[\cite{RaoSolutions}]
Given two matrices $A\in \mathscr{R}^{m \times n}$ and $B\in \mathscr{R}^{w\times n}$, the Khatri-Rao product of $A$ and $B$, termed as $A\ast B$, is denoted by
\begin{equation*}
A\ast B=[\mathrm{Col}_1(A)\ltimes \mathrm{Col}_1(B),\cdots,\mathrm{Col}_n(A)\ltimes \mathrm{Col}_n(B)]\in \mathscr{R}^{mw \times n}.
\end{equation*}
\end{definition}

Denote a bijection $\sigma : \mathscr{D}\rightarrow \Delta_2$ as $\sigma(x)=\delta_2^{2-x}=\mathbf{x}$. Then, based on the STP of matrices, the algebraic representation for logical function $f:\mathscr{D}^n\rightarrow\mathscr{D}$ can be also derived.

\begin{lemma}[\cite{chengdz2011springer}]\label{lemma-structure matrix}
For a logical function $f(x_1, x_2,\ldots,x_n):\mathscr{D}^n\rightarrow \mathscr{D}$, its algebraic form $\mathbf{f}(\mathbf{x}_1, \mathbf{x}_2,\ldots,\mathbf{x}_n): (\Delta_2)^n\rightarrow \Delta_2$ satisfying
\begin{equation}
\mathbf{f}(\mathbf{x}_1, \mathbf{x}_2,\ldots,\mathbf{x}_n)=H_f \ltimes_{i=1}^n \mathbf{x}_i,
\end{equation}
where $H_f\in\mathscr{L}^{2\times 2^n}$ is unique and is called the structure matrix of logical function $f$.
\end{lemma}

According to Lemma \ref{lemma-structure matrix}, the structure matrix of logical function $f^\kappa_i,i\in [1,n]$ can be expressed as $F^\kappa_i\in \mathscr{L}^{2\times 2^{|\mathcal{N}_{i,\kappa}|}}$. Let $\widehat{F}_i^\kappa\in \mathscr{L}^{2\times 2^{|\mathcal{N}_{i}|}}$ be its extended structure matrix such that $\widehat{F}_i^\kappa \ltimes_{j\in \mathcal{N}_i}\mathbf{x}_j(t)=F_i^\kappa \ltimes_{j\in \mathcal{N}_{i,\kappa}}\mathbf{x}_j(t)$.  Then on the basis of the STP method, the algebraic representation of PBN (\ref{equation-PBN}) reads:
\begin{equation}\left\{\begin{array}{l}\label{equation-PBN-a1}
\mathbf{x}_1(t+1)=F_1\ltimes_{j\in \mathcal{N}_1}\mathbf{x}_j(t),\\
\mathbf{x}_2(t+1)=F_2\ltimes_{j\in \mathcal{N}_2}\mathbf{x}_j(t),\\
~~~~~~~~~~~~\vdots\\
\mathbf{x}_n(t+1)=F_n\ltimes_{j\in \mathcal{N}_n}\mathbf{x}_j(t),
\end{array}\right.
\end{equation}
where $F_i, i\in[1,n]$ is chosen from matrix set $\{\widehat{F}_i^\kappa\in \mathscr{L}^{2\times 2^{|\mathcal{N}_i|}}~|~\kappa\in [1,l_i]\}$ subject to the probability distribution $\{0\leq p_i^\kappa\leq1~|~\kappa\in [1,l_i]\}$. Furthermore, the mathematical expectation of $\mathbf{x}_i(t+1),i\in[1,n]$ is defined as
\begin{equation}\label{equation-PBN-algebraic}
\mathbb{E}\{\mathbf{x}_i(t+1)\}=\mathbb{E}\{F_i\ltimes_{j\in \mathcal{N}_i}\mathbf{x}_j(t)\}=\widehat{F}_i\mathbb{E}\{\ltimes_{j\in \mathcal{N}_i}\mathbf{x}_j(t)\},
\end{equation}
where $\widehat{F}_i=\sum_{\kappa=1}^{l_i}p_i^\kappa \widehat{F}_i^\kappa$, and $\mathbb{E}\{\cdot\}$ represents the mathematical expectation throughout this paper.

\begin{remark}
Reviewing the existing literatures \cite{liht2017siam,lu2018stabilization} and \cite{guoyq}, many fundamental results on stability, controllability and observability are generally based on state transition matrix $L\in \mathscr{P}^{2^n\times 2^n}$, which can be obtained by the following process:
\begin{itemize}
  \item Construct a set of matrices $\Upsilon_i \in \mathscr{P}^{2^{|\mathcal{N}_i|} \times {2^n}}, i\in [1,n]$, such that $\mathbb{E}\{\ltimes_{j\in \mathcal{N}_i}\mathbf{x}_j(t)\}=\Upsilon_i\mathbb{E}\{\ltimes_{j=1}^{n}\mathbf{x}_j(t)\}$.
  \item Obtain the corresponding augmented system:
\begin{equation}\begin{aligned}\label{equation-PBN-a2}
\mathbb{E}\{\mathbf{x}(t+1)\}=&\ltimes_{i=1}^n \widehat{F}_i\mathbb{E}\{\ltimes_{j\in \mathcal{N}_i}\mathbf{x}_j(t)\}\\
=&\ltimes_{i=1}^n \widehat{F}_i\Upsilon_i\mathbb{E}\{\ltimes_{j=1}^{n}\mathbf{x}_j(t)\}\\
\triangleq&L\mathbb{E}\{\mathbf{x}(t)\},
\end{aligned}
\end{equation}
where $L=(\widehat{F}_1\Upsilon_1)\ast \cdots \ast (\widehat{F}_n\Upsilon_n)\in \mathscr{P}^{2^n\times 2^n}$ is called the state transition matrix.
\end{itemize}
On the one hand, some control-related problems can be easily handled in virtue of the converted form (\ref{equation-PBN-a2}). On the other hand, the scale of matrix $L$ expands rapidly with the increase of network nodes. It reveals that designing controllers based on state transition matrix causes high computational complexity.
\end{remark}

\section{Pinning Control Design for Stabilizing PBNs}\label{section-further stability}
In this section, we investigate the stability criterion for PBN (\ref{equation-PBN}) based on network structure rather than state transition matrix.
The relationship between global stability of a prespecified PBN and the acyclic structure of its wiring digraph is derived based on the following Lemma \ref{lemma-robert}.
\begin{lemma} [\cite{discrete2012robert}]\label{lemma-robert}
A BN is globally stable, if its wiring digraph has no cycle.
\end{lemma}

It implies that in order to recover the global stability of BNs, we need to reconstruct an acyclic wiring digraph by deleting certain edges of original wiring diagraph.
This action can be achieved by imposing feasible control strategy on certain nodes as indicated in \cite{zhong2020arxiv} and \cite{zhusy2020arxiv}. Hereafter, we prove that Lemma \ref{lemma-robert} also holds for the case of PBNs.

\begin{proposition}\label{proposition-pbn}
If the wiring digraph of PBN (\ref{equation-PBN}) is acyclic, the wiring digraph of each possible model is also acyclic.
\end{proposition}

\begin{proof}
The probability of the $\lambda$-th model $\Sigma_\lambda=\{f_1^{\lambda_1},f_2^{\lambda_2},\cdots,f_n^{\lambda_n}\}$ being active is $P_\lambda=\Pi_{i=1}^{n}p_i^{\lambda_i}$. Then, for node $x_i,i\in[1,n]$, if $x_j$ is the functional variable of $f_i^{\lambda_i}$ with probability $p_i^{\lambda_i}$, there is an edge $x_j\rightarrow x_i$ in the wiring digraph of the $\lambda$-th model, and this edge inevitably exists in the wiring digraph of PBN (\ref{equation-PBN}). Thereby, if there is a cycle in one of $\Pi_{i=1}^{n}l_{i}$ possible models, this cycle must exist in PBN  (\ref{equation-PBN}), which conflicts with the hypothesis.
\end{proof}

\begin{theorem}
PBN (\ref{equation-PBN}) is globally stable, if its wiring digraph is acyclic and all possible models are stabilized at the same steady state.
\end{theorem}
\begin{proof}
If the wiring digraph of PBN (\ref{equation-PBN}) is acyclic, then according to Proposition \ref{proposition-pbn}, each possible model has no cycle. Moreover, in the light of Lemma \ref{lemma-robert}, each possible model is globally stable. Furthermore, in order to avoid multiple attractors, it should guarantee that all possible models are stable at the same steady state.
\end{proof}

In what follows, we design two kinds of state feedback controllers, that is, uniform one (independent of switching signal) and non-uniform one (related to each possible model), to stabilize PBN (\ref{equation-PBN}), before which the method of selecting pinning nodes is presented.
\subsection{Selecting Pinning Nodes}
Initially, we introduce the concept of feedback arc set.
\begin{definition}[Feedback Arc Set \cite{bang2008algorithems}]\label{feedback-arc-set}
Feedback arc set (FAS) is a subset of edges, deleting which the wiring diagraph possesses acyclic structure. In other words, an FAS contains all fixed points and at least one edge of each cycle.
\end{definition}

Fortunately, the fixed points and cycles existing in wiring diagraph $\mathbf{G}=(\mathbf{V},\mathbf{E})$ can be directly determined by the depth-first search algorithm. Accordingly, denote a possible FAS by $\{e_1,\cdots,e_\varpi\}\subseteq \mathbf{E}$. For edges $e_1,\cdots,e_\varpi \in \mathbf{E}$, their corresponding starting nodes and ending nodes are respectively denoted by $v_{-}(e_1),\cdots,v_{-}(e_\varpi)\in \mathbf{V}$ and $v_{+}(e_1),\cdots,v_{+}(e_\varpi)\in \mathbf{V}$. With respect to the subgraph
$\mathbf{G}_i=(\mathbf{V}_i,\mathbf{E}_i)\subseteq\mathbf{G},i\in [1,n]$ induced by $x_i$, it reveals that $\bigcup_{e\in\mathbf{E}_i}v_{-}(e)=\bigcup_{\kappa=1}^{l_i}\bigcup_{j\in\mathcal{N}_{i,\kappa}}x_j(t)$ and $\bigcup_{e\in\mathbf{E}_i}v_{+}(e)=x_i(t)$. Moreover, subgraph $\mathbf{G}_i$ has the property that each starting node is connected by a certain edge, but each ending node may be connected by several different edges.

To proceed, we suppose that $\bigcup_{j=1}^{\varpi}v_{+}(e_j)=\{x_{\gamma_1},x_{\gamma_2},\cdots,x_{\gamma_\tau}\}\subseteq \{x_1,\cdots,x_n\}$, then it has
\begin{equation*}
\bigcup_{j=\omega_1^1}^{\omega_1^{\alpha_1}}v_{+}(e_j)=x_{\gamma_1},
\bigcup_{j=\omega_2^1}^{\omega_2^{\alpha_2}}v_{+}(e_j)=x_{\gamma_2},\cdots,
\bigcup_{j=\omega_\tau^1}^{\omega_\tau^{\alpha_\tau}}v_{+}(e_j)=x_{\gamma_\tau},
\end{equation*}
where $\{\omega_1^1,\cdots,\omega_1^{\alpha_1},\cdots,\omega_\tau^1,\cdots,\omega_\tau^{\alpha_\tau}\}=[1,\varpi]$ with $\omega_1^1<\cdots<\omega_1^{\alpha_1}<\cdots<\omega_\tau^1<\cdots<\omega_\tau^{\alpha_\tau}$. Correspondingly, the starting nodes are classified as
\[\bigcup_{j=\omega_1^1}^{\omega_1^{\alpha_1}}v_{-}(e_j)=\{x_{\pi_1^1},\cdots,x_{\pi_1^{\alpha_1}}\},
\bigcup_{j=\omega_2^1}^{\omega_2^{\alpha_2}}v_{-}(e_j)=\{x_{\pi_2^1},\cdots,x_{\pi_2^{\alpha_2}}\}, \]
\[\cdots,\bigcup_{j=\omega_\tau^1}^{\omega_\tau^{\alpha_\tau}}v_{-}(e_j)=\{x_{\pi_\tau^1},\cdots,x_{\pi_\tau^{\alpha_\tau}}\},\]
where $\pi_1^1,\cdots,\pi_1^{\alpha_1},\cdots,\pi_\tau^1,\cdots,\pi_\tau^{\alpha_\tau}$ with $\sum_{j=1}^\tau \alpha_j=\varpi$.

Afterwards, we reconstruct the acyclic wiring digraph based on the obtained FAS $\{e_1,\cdots,e_\varpi\}$, edges in which connect starting nodes $x_{\pi^1_1},\cdots,x_{\pi_1^{\alpha_1}}$ $,\cdots,x_{\pi^1_\tau},\cdots,x_{\pi_\tau^{\alpha_\tau}}\in \mathbf{V}$ and ending nodes $x_{\gamma_1},\cdots,x_{\gamma_\tau}\in \mathbf{V}$. It should be pointed out that all ending nodes are chosen to be pinning nodes, then the first group of pinning nodes (to achieve stabilization) are $x_{\gamma_1},\cdots,x_{\gamma_\tau}$, whose subscript indices are collected into set $\Lambda$.
For each pinning node $x_{\gamma_\sigma},\sigma\in[1,\tau]$, the subscript indices of its starting nodes are collected into set $\mathcal{N}^\star_{\gamma_\sigma}=\{\pi_\sigma^1,\cdots,\pi_\sigma^{\alpha_\sigma}\}$, and denote $\mathcal{N}^\circ_{\gamma_\sigma}=\mathcal{N}_{\gamma_\sigma}\backslash\mathcal{N}^\star_{\gamma_\sigma}$. Then, there exists a pair of probability matrices $\mathbf{T}_{\gamma_\sigma} \in \mathscr{P}^{2\times 2^{|\mathcal{N}_{\gamma_\sigma}|}}$ and $G_{\gamma_\sigma} \in \mathscr{P}^{2\times 2^{|\mathcal{N}^\circ_{\gamma_\sigma}|}}$ such that
\begin{equation}\label{stable-struture-matrix-0}
\mathbf{T}_{\gamma_\sigma}=G_{\gamma_\sigma}(I_{2^{|\mathcal{N}^\circ_{\gamma_\sigma}|}}\otimes \mathbf{1}^\top_{2^{|\mathcal{N}^\star_{\gamma_\sigma}|}}).
\end{equation}
\begin{remark}
Notice that there are many available matrix pairs ($\mathbf{T}_{\gamma_\sigma}$ and $G_{\gamma_\sigma}$) satisfying equations (\ref{stable-struture-matrix-0}), and they correspond to different control inputs. However, we can choose arbitrary one set of effective solutions for further analysis.
\end{remark}

Hereby, we transform the evolution equation of pinning node $x_{\gamma_\sigma},\sigma\in[1,\tau]$ from (\ref{equation-PBN-algebraic}) to the following form:
\begin{equation}\begin{aligned}\label{Ex(t+1)}
\mathbb{E}\{\mathbf{x}_{\gamma_\sigma}(t+1)\}&=\widehat{F}_{\gamma_\sigma}\mathbb{E}\{\ltimes_{j\in \mathcal{N}_{\gamma_\sigma}}\mathbf{x}_j(t)\}\\
&=\widehat{F}_{\gamma_\sigma}\mathbb{E}\{\mathbf{W}_{\gamma_\sigma}\ltimes_{j\in \mathcal{N}^\circ_{\gamma_\sigma}}\mathbf{x}_j(t)\ltimes_{j\in \mathcal{N}^\star_{\gamma_\sigma}}\mathbf{x}_j(t)\}\\
&\triangleq \mathbf{L}_{\gamma_\sigma}\mathbb{E}\{\ltimes_{j\in \mathcal{N}^\circ_{\gamma_\sigma}}\mathbf{x}_j(t)\ltimes_{j\in \mathcal{N}^\star_{\gamma_\sigma}}\mathbf{x}_j(t)\},
\end{aligned}
\end{equation}
where $\mathbf{W}_{\gamma_\sigma}=[\ltimes_{\sigma=1}^{\alpha_j}W_{[2,2^{|\pi_j^\sigma|\mathcal{N}_{\gamma_\sigma}}]}W_{[2^{|\mathcal{N}^\star_{\gamma_\sigma}|},2^{|\mathcal{N}^\circ_{\gamma_\sigma}|}]}]\otimes I_{2^{|\mathcal{N}_{\gamma_\sigma}|-|\mathcal{N}^\star_{\gamma_\sigma}|}}\in \mathscr{L}^{2^{|\mathcal{N}_{\gamma_\sigma}|}\times2^{|\mathcal{N}_{\gamma_\sigma}|}}$, and $\mathbf{L}_{\gamma_\sigma}\triangleq \widehat{F}_{\gamma_\sigma}\mathbf{W}_{\gamma_\sigma}\in \mathscr{P}^{2\times2^{|\mathcal{N}_{\gamma_\sigma}|}}$.

\subsection{Design Uniform State Feedback Controllers}
To achieve global stabilization for PBN (\ref{Ex(t+1)}), the state feedback controller imposed on pinning node $x_{\gamma_\sigma},\sigma\in [1,\tau]$ is designed as follows:
\begin{equation}\label{state-feedback-pinning-controllers}
u_{\gamma_\sigma}(t)=\varphi_{\gamma_\sigma}([x_j(t)]_{j\in \mathcal{N}_{\gamma_\sigma}}),
\end{equation}
where $u_{\gamma_\sigma}(t)\in \mathscr{D}$ denotes control input, and $\varphi_{\gamma_\sigma}:\mathscr{D}^{{|\mathcal{N}_{\gamma_\sigma}|}}\rightarrow\mathscr{D}$ is the logical function determined by node $x_{\gamma_\sigma}$'s in-neighbors. Subsequently, the pinning controlled PBN is described as:
\begin{equation}x_i(t+1)=\left\{\begin{array}{ll}\label{equation-BCN}
f_i([x_j(t)]_{j\in \mathcal{N}_i}),&i\in [1,n]\backslash \Lambda,\\
u_i(t) \odot_i f_i([x_j(t)]_{j\in \mathcal{N}_i}),&i\in\Lambda,
\end{array}\right.
\end{equation}
where $\odot_i:\mathscr{D}^2\rightarrow \mathscr{D}, i\in \Lambda$ is logical function connecting state feedback controller $u_i(t)$ and original dynamic equation $f_i([x_j(t)]_{j\in \mathcal{N}_i})$.
Furthermore, by resorting to the STP tool, the structure matrices of $\varphi_{\gamma_\sigma}$ and $\odot_{\gamma_\sigma}$ are respectively denoted by $\Psi_{\gamma_\sigma}\in \mathscr{L}^{2\times2^{|\mathcal{N}_{\gamma_\sigma}|}}$ and $M_{\odot_{\gamma_\sigma}}\in \mathscr{L}^{2\times 4}$, where $\gamma_\sigma\in\Lambda$. Then one derives that
\begin{equation}\begin{aligned}
\mathbf{u}_{\gamma_\sigma}(t)=&\Psi_{\gamma_\sigma}\ltimes_{j\in \mathcal{N}_{\gamma_\sigma}}\mathbf{x}_j(t)\\
=&\Psi_{\gamma_\sigma}\mathbf{W}_{\gamma_\sigma}\ltimes_{j\in \mathcal{N}^\circ_{\gamma_\sigma}}\mathbf{x}_j(t)\ltimes_{j\in \mathcal{N}^\star_{\gamma_\sigma}}\mathbf{x}_j(t)\\
\triangleq& \widehat{\Psi}_{\gamma_\sigma}\ltimes_{j\in \mathcal{N}^\circ_{\gamma_\sigma}}\mathbf{x}_j(t)\ltimes_{j\in \mathcal{N}^\star_{\gamma_\sigma}}\mathbf{x}_j(t),
\end{aligned}
\end{equation}
where $\widehat{\Psi}_{\gamma_\sigma}\triangleq \Psi_{\gamma_\sigma}\mathbf{W}_{\gamma_\sigma}\in \mathscr{L}^{2\times2^{|\mathcal{N}_{\gamma_\sigma}|}}$.
Consequently, the state updating of each pinning controlled node $x_{\gamma_\sigma},\sigma\in [1,\tau]$ is described in the algebraic form:
\begin{equation}\begin{aligned}
\mathbb{E}&\{\mathbf{x}_{\gamma_\sigma}(t+1)\}=M_{\odot_{\gamma_\sigma}}\mathbf{u}_{\gamma_\sigma}(t)\widehat{F}_i\mathbb{E}\{\ltimes_{j\in \mathcal{N}_i}\mathbf{x}_j(t)\}\\
=&M_{\odot_{\gamma_\sigma}}\widehat{\Psi}_{\gamma_\sigma}(I_{2^{|\mathcal{N}_{\gamma_\sigma}|}}\otimes \mathbf{L}_{\gamma_\sigma}) \Phi_{|\mathcal{N}_{\gamma_\sigma}|}\mathbb{E}\{\ltimes_{j\in\mathcal{N}^\circ_{\gamma_\sigma}}\mathbf{x}_j(t)\ltimes_{j\in \mathcal{N}^\star_{\gamma_\sigma}}\mathbf{x}_j(t)\}.
\end{aligned}
\end{equation}
If pinning controlled PBN (\ref{equation-BCN}) achieves global stabilization, it satisfies that
\begin{equation}\label{stable-structure-matrix}\begin{aligned}
M_{\odot_{\gamma_\sigma}}\widehat{\Psi}_{\gamma_\sigma}(I_{2^{|\mathcal{N}_{\gamma_\sigma}|}}\otimes \mathbf{L}_{\gamma_\sigma}) \Phi_{|\mathcal{N}_{\gamma_\sigma}|}&=G_{\gamma_\sigma}(I_{2^{|\mathcal{N}^\circ_{\gamma_\sigma}|}}\otimes \mathbf{1}^\top_{2^{|\mathcal{N}^\star_{\gamma_\sigma}|}})\\
&:=\mathbf{T}_{\gamma_\sigma}.
\end{aligned}
\end{equation}
Therefore, the effective state feedback controller (or structure matrices $M_{\odot_{\gamma_\sigma}}\in \mathscr{L}^{2\times 4}$ and $\widehat{\Psi}_{\gamma_\sigma}\in \mathscr{L}^{2\times 2^{|\mathcal{N}_{\gamma_\sigma}|}}$) for pinning node $x_{\gamma_\sigma}$ can be obtained by solving the above equations. Particularly, if equation (\ref{stable-structure-matrix}) is solvable,
logical functions $\odot_{\gamma_\sigma}$ and $\varphi_{\gamma_\sigma}$ can be uniquely derived according to Lemma \ref{lemma-structure matrix}.
Subsequently, we study the solvability criterion of equation (\ref{stable-structure-matrix}), before which several subsets are constructed as follows:
\begin{equation}\begin{aligned}\label{theorem-condition}
\Omega_1&=\{\mathrm{Col}_i(\mathbf{T}_{\gamma_\sigma})~|~\mathrm{Col}_i(\mathbf{T}_{\gamma_\sigma})=\mathrm{Col}_i(\mathbf{L}_{\gamma_\sigma})\},\\
\Omega_2&=\{\mathrm{Col}_i(\mathbf{T}_{\gamma_\sigma})~|~\mathrm{Col}_i(\mathbf{T}_{\gamma_\sigma})=\mathrm{Col}_i(\mathbf{L}_{\gamma_\sigma})=(1,0)^\top;~\mathrm{or}~\\
&~~~~\mathrm{Col}_i(\mathbf{T}_{\gamma_\sigma})=(1,0)^\top,\mathrm{Col}_i(\mathbf{L}_{\gamma_\sigma})=(0,1)^\top;~\mathrm{or}~\\
&~~~~\mathrm{Col}_i(\mathbf{T}_{\gamma_\sigma})=(1,0)^\top,\mathrm{Col}_i(\mathbf{L}_{\gamma_\sigma})\in \mathbb{B}_2\},\\
\Omega_3&=\{\mathrm{Col}_i(\mathbf{T}_{\gamma_\sigma})~|~\mathrm{Col}_i(\mathbf{T}_{\gamma_\sigma})=(1,1)^\top-\mathrm{Col}_i(\mathbf{L}_{\gamma_\sigma})\},\\
\Omega_4&=\{\mathrm{Col}_i(\mathbf{T}_{\gamma_\sigma})~|~\mathrm{Col}_i(\mathbf{T}_{\gamma_\sigma})=\mathrm{Col}_i(\mathbf{L}_{\gamma_\sigma})=(0,1)^\top;~\mathrm{or}~\\
&~~~~~\mathrm{Col}_i(\mathbf{T}_{\gamma_\sigma})=(0,1)^\top,\mathrm{Col}_i(\mathbf{L}_{\gamma_\sigma})=(1,0)^\top;~\mathrm{or}~\\
&~~~~~\mathrm{Col}_i(\mathbf{T}_{\gamma_\sigma})=(0,1)^\top,\mathrm{Col}_i(\mathbf{L}_{\gamma_\sigma})\in \mathbb{B}_2\}.
\end{aligned}
\end{equation}

\begin{theorem} \label{theorem-2-condition}
Equations (\ref{stable-structure-matrix}) is solvable for pinning node $x_{\gamma_\sigma},\sigma\in [1,\tau]$ if and only if $\mathrm{Col}(\mathbf{T}_{\gamma_\sigma})$ can be completely covered by at most two of $\Omega_1,\Omega_2,\Omega_3$ and $\Omega_4$.
\end{theorem}

\begin{proof}
(Sufficiency.) With respect to equation (\ref{stable-structure-matrix}), we presume its parameters as
\begin{equation*}M_{\odot_{\gamma_\sigma}}=\left(\begin{array}{cccc}
\alpha_1 & \alpha_2 & \alpha_3 & \alpha_4 \\
1-\alpha_1 & 1-\alpha_2 & 1-\alpha_3 & 1-\alpha_4
\end{array}\right),
\end{equation*}

\begin{equation*}\widehat{\Psi}_{\gamma_\sigma}=\left(\begin{array}{cccc}
\beta_1 & \beta_2 & \cdots & \beta_{2^{|\mathcal{N}_{\gamma_\sigma}|}} \\
1-\beta_1 & 1-\beta_2 & \cdots & 1-\beta_{2^{|\mathcal{N}_{\gamma_\sigma}|}}
\end{array}\right),
\end{equation*}

\begin{equation*}\mathbf{L}_{\gamma_\sigma}=\left(\begin{array}{cccc}
\xi_1 & \xi_2 & \cdots & \xi_{2^{|\mathcal{N}_{\gamma_\sigma}|}} \\
1-\xi_1 & 1-\xi_2 & \cdots & 1-\xi_{2^{|\mathcal{N}_{\gamma_\sigma}|}}
\end{array}\right),
\end{equation*}
\begin{equation*}\mathrm{and}~\mathbf{T}_{\gamma_\sigma}=\left(\begin{array}{cccc}
\eta_1 & \eta_2 & \cdots & \eta_{2^{|\mathcal{N}_{\gamma_\sigma}|}} \\
1-\eta_1 & 1-\eta_2 & \cdots & 1-\eta_{2^{|\mathcal{N}_{\gamma_\sigma}|}}
\end{array}\right),
\end{equation*}
where $\alpha_1,\alpha_2,\alpha_3,\alpha_4,\beta_1,\cdots,\beta_{2^{|\mathcal{N}_{\gamma_\sigma}|}}\in \mathscr{D}$ and $\xi_1,\cdots,\xi_{2^{|\mathcal{N}_{\gamma_\sigma}|}},$ $\eta_1,\cdots,\eta_{2^{|\mathcal{N}_{\gamma_\sigma}|}}\in \mathbb{B}\bigcup\{0,1\}$. After that, equations (\ref{stable-structure-matrix}) is converted into the following equations:
\begin{equation}\left\{\small{\begin{aligned}\label{equations-BCN-proof}
[\alpha_1\beta_1+\alpha_3(1-\beta_1)]\xi_1+&[\alpha_2\beta_1+\alpha_4(1-\beta_1)](1-\xi_1)=\eta_1,\\
[\alpha_1\beta_2+\alpha_3(1-\beta_2)]\xi_2+&[\alpha_2\beta_2+\alpha_4(1-\beta_2)](1-\xi_2)=\eta_2,\\
&\vdots \\
[\alpha_1\beta_i+\alpha_3(1-\beta_i)]\xi_i+&[\alpha_2\beta_i+\alpha_4(1-\beta_i)](1-\xi_i)=\eta_i,\\
&\vdots \\
[\alpha_1\beta_{2^{|\mathcal{N}_{\gamma_\sigma}|}}+\alpha_3(1-&\beta_{2^{|\mathcal{N}_{\gamma_\sigma}|}})]
\xi_{2^{|\mathcal{N}_{\gamma_\sigma}|}}+\\
[\alpha_2\beta_{2^{|\mathcal{N}_{\gamma_\sigma}|}}+\alpha_4(1-&\beta_{2^{|\mathcal{N}_{\gamma_\sigma}|}})](1-\xi_{2^{|\mathcal{N}_{\gamma_\sigma}|}})=\eta_{2^{|\mathcal{N}_{\gamma_\sigma}|}}.
\end{aligned}}\right.
\end{equation}
In what follows, we discuss the solution of the $i$-th ($i\in[1,2^{|\mathcal{N}_{\gamma_\sigma}|}]$) equation from the following three cases:
$$(a)~\eta_i=\xi_i;~~(b)~\eta_i=1-\xi_i;~~(c)~\eta_i\in \{0,1\},\xi\in \mathbb{B}.$$
First, we prove that cases $(a)\sim(c)$ have covered all situations. Since $\beta_i\in \mathscr{D}$, it derives $\alpha_3\xi_i+\alpha_4(1-\xi_i)=\eta_i$ when $\beta_i=0$; and $\alpha_1\xi_i+\alpha_2(1-\xi_i)=\eta_i$ when $\beta_i=1$. Owing to $\alpha_1,\alpha_2,\alpha_3,\alpha_4\in \mathscr{D}$, we can classify it into four situations:
(i) $\alpha_3=\alpha_4=0$; or $\alpha_1=\alpha_2=0$,
(ii) $\alpha_3=0,\alpha_4=1$; or $\alpha_1=0,\alpha_2=1$,
(iii) $\alpha_3=1,\alpha_4=0$; or $\alpha_1=1,\alpha_2=0$, and
(iv) $\alpha_3=\alpha_4=1$; or $\alpha_1=\alpha_2=1$, which respectively derive
(i) $\eta_i=0$,
(ii) $\eta_i=1-\xi_i$,
(iii) $\eta_i=\xi_i$,
and (iv) $\eta_i=1$. Moreover, case $(c)$ means $\eta_i\neq\xi_i$ and $\eta_i\neq 1-\xi_i$, then it holds $0<\xi_i<1$ since $\eta_i$ only equals to 0 or 1.

For case $(a)$, if $\beta_i=0$, one has $\alpha_3\xi_i+\alpha_4(1-\xi_i)=\xi_i$, that is, $(1-\alpha_3+\alpha_4)\xi_i=\alpha_4$. There are two solutions:
\begin{equation*}
(a1)~1-\alpha_3+\alpha_4=\alpha_4=0;~~~
(a2)~\xi_i=\frac{\alpha_4}{1-\alpha_3+\alpha_4},
\end{equation*}
which derives
\begin{equation*}\begin{array}{l}
(a1')~\xi_i=\eta_i, \alpha_3=1,\beta_i=\alpha_4=0; \\
(a2')~\xi_i=\eta_i=0,\beta_i=\alpha_4=0,\alpha_3=0;\\
(a3')~\xi_i=\eta_i=1,\beta_i=0,\alpha_3=\alpha_4=1;\\
(a4')~\xi_i=\eta_i=\frac{1}{2},\beta_i=\alpha_3=0,\alpha_4=1.
\end{array}
\end{equation*}
If $\beta_i=1$, it has $\alpha_1\xi_i+\alpha_2(1-\xi_i)=\xi_i$ and derives
\begin{equation*}\begin{array}{l}
(a1'')~\xi_i=\eta_i,\beta_i=\alpha_1=1,\alpha_2=0; \\ (a2'')~\xi_i=\eta_i=0,\beta_i=1,\alpha_2=0,\alpha_1=0;\\
(a3'')~\xi_i=\eta_i=1,\beta_i=\alpha_1=\alpha_2=1;\\
(a4'')~\xi_i=\eta_i=\frac{1}{2},\alpha_1=0,\beta_i=\alpha_2=1.
\end{array}
\end{equation*}

For case $(b)$, if $\beta_i=0$, one acquires $\alpha_3\xi_i+\alpha_4(1-\xi_i)=1-\xi_i$. It also has two solutions:
\begin{equation*}
(b1)~\alpha_3-\alpha_4+1=1-\alpha_4=0;~~~
(b2)~\xi_i=\frac{1-\alpha_4}{\alpha_3-\alpha_4+1},
\end{equation*}
which obtains
\begin{equation*}\begin{array}{ll}
(b1')~\eta_i=1-\xi_i,\beta_i=\alpha_3=0,\alpha_4=1;\\ (b2')~\eta_i=1-\xi_i=\frac{1}{2},\alpha_3=1,\beta_i=\alpha_4=0;\\
(b3')~\eta_i=1-\xi_i=1,\beta_i=0,\alpha_3=\alpha_4=1;\\
(b4')~\eta_i=1-\xi_i=0,\beta_i=\alpha_3=\alpha_4=0.
\end{array}
\end{equation*}
Else if $\beta_i=1$, it acquires $\alpha_1\xi_i+\alpha_2(1-\xi_i)=1-\xi_i$, which derives
\begin{equation*}\begin{array}{ll}
(b1'')~\eta_i=1-\xi_i,\alpha_1=0,\beta_i=\alpha_2=1; \\ (b2'')~\eta_i=1-\xi_i=\frac{1}{2},\beta_i=\alpha_1=1,\alpha_2=0;\\
(b3'')~\eta_i=1-\xi_i=1,\beta_i=\alpha_1=\alpha_2=1;\\
(b4'')~\eta_i=1-\xi_i=0,\beta_i=1,\alpha_1=\alpha_2=0.
\end{array}
\end{equation*}

Similarly, for case $(c)$, if $\beta_i=0$, it gains $\alpha_3\xi_i+\alpha_4(1-\xi_i)=0~\mathrm{or}~1$, that is,
\begin{equation*}\begin{array}{ll}
(c1')~\eta_i=0,\xi_i\in \mathbb{B}, \beta_i=\alpha_3=\alpha_4=0;~\mathrm{or}\\ (c2')~\eta_i=1,\xi_i\in\mathbb{B},\beta_i=0,\alpha_3=\alpha_4=1.
\end{array}
\end{equation*}
Else if $\beta_i=1$, it gains $\alpha_1\xi_i+\alpha_2(1-\xi_i)=0~\mathrm{or}~1$, that is,
\begin{equation*}\begin{array}{ll}
(c1'')~\eta_i=0,\xi_i\in \mathbb{B}, \beta_i=1,\alpha_1=\alpha_2=0; ~\mathrm{or}\\ (c2'')~\eta_i=1,\xi_i\in\mathbb{B},\beta_i=1,\alpha_1=\alpha_2=1.
\end{array}
\end{equation*}

The above solutions can be classified into four groups:
\begin{description}
  \item[\textbf{Group 1:}]~~~~$\eta_i=\xi_i$; $\eta_i=\xi_i=0$; $\eta_i=1-\xi_i=\frac{1}{2}$.

  \item[Note 1:]~$\eta_i=\xi_i\Leftrightarrow$ $(a1')$ or $(a1'')$; $\eta_i=\xi_i=0\Leftrightarrow$ $(a2')$ or $(a2'')$;
  $\eta_i=1-\xi_i=\frac{1}{2}\Leftrightarrow$ $(b2')$ or $(b2'')$.

  \item[\textbf{Group 2:}]~~~~$\eta_i=\xi_i=1$; $\eta_i=1-\xi_i=1$; $\eta_i=1,\xi_i\in \mathbb{B}$.
  \item[Note 2:]~$\eta_i=\xi_i=1\Leftrightarrow$ $(a3')$ or $(a3'')$;
  $\eta_i=1-\xi_i=1\Leftrightarrow$ $(b3')$ or $(b3'')$;
  $\eta_i=1,\xi_i\in \mathbb{B}\Leftrightarrow$ $(c2')$ or $(c2'')$.

  \item[\textbf{Group 3:}]~~~~$\eta_i=\xi_i=\frac{1}{2}$; $\eta_i=1-\xi_i$; $\eta_i=1-\xi_i=1$ (occurred in Group 2).
  \item[Note 3:]~$\eta_i=\xi_i=\frac{1}{2}\Leftrightarrow$ $(a4')$ or $(a4'')$;
  $\eta_i=1-\xi_i\Leftrightarrow$ $(b1')$ or $(b1'')$.

  \item[\textbf{Group 4:}]~~~~$\eta_i=\xi_i=0$ (occurred in Group 1), $\eta_i=1-\xi_i=0$, and $\eta_i=0,\xi_i\in \mathbb{B}$.
  \item[Note 4:]~~$\eta_i=1-\xi_i=0\Leftrightarrow$ $(b4')$ or $(b4'')$;
  $\eta_i=0,\xi_i\in \mathbb{B}\Leftrightarrow$ $(c1')$ or $(c1'')$.
\end{description}
Particularly, $\eta_i=\xi_i\in \{0,1,\frac{1}{2}\}$ and $\eta_i=1-\xi_i=\frac{1}{2}$ can be viewed as $\eta_i=\xi_i$ belonging to \textbf{Group 1}, while $\eta_i=1-\xi_i\in \{0,1,\frac{1}{2}\}$ and $\eta_i=\xi_i=\frac{1}{2}$ can be regarded as $\eta_i=1-\xi_i$ belonging to \textbf{Group 3}. Then, the above four groups respectively correspond to $\Omega_1,\Omega_2,\Omega_3$ and $\Omega_4$ defined in (\ref{theorem-condition}).
Besides, the cases in the same group can be handled by the same pinning controllers. More precisely, only when the solution of equations (\ref{equations-BCN-proof}) can be covered by the cases within at most two groups, can we obtain effective solutions.

(Necessity.) Suppose that matrix $\mathbf{T}_{\gamma_\sigma}$ is covered by at most two sets, then we prove that equations (\ref{equations-BCN-proof}) is solvable.

Case 1: $\mathrm{Col}(\mathbf{T}_{\gamma_\sigma})$ is contained by one of sets $\Omega_1,\cdots,\Omega_4$. If part of $\mathrm{Col}_i(\mathbf{T}_{\gamma_\sigma}), i\in [1,2^{|\mathcal{N}_{\gamma_\sigma}|}]$ is equal to $\mathrm{Col}_i(\mathbf{L}_{\gamma_\sigma})$, the other part is equal to $1-\mathrm{Col}_i(\mathbf{L}_{\gamma_\sigma})=\frac{1}{2}$, it derives
$M_{\odot_{\gamma_\sigma}}=\delta_2[\ast,\ast,1,2],\widehat{\Psi}_{\gamma_\sigma}=\delta_2[\underbrace{2,\cdots,2}_{2^{|\mathcal{N}_{\gamma_\sigma}|}}]$; or $M_{\odot_{\gamma_\sigma}}=\delta_2[1,2,\ast,\ast],\widehat{\Psi}_{\gamma_\sigma}=\delta_2[\underbrace{1,\cdots,1}_{2^{|\mathcal{N}_{\gamma_\sigma}|}}]$,
where the elements in the positions of `$\ast$' can be arbitrary chosen from $\{0,1\}$. Besides, other cases can similarly derive effective pinning controllers ($M_{\odot_{\gamma_\sigma}}$ and $\widehat{\Psi}_{\gamma_\sigma}$).

Case 2: $\mathrm{Col}(\mathbf{T}_{\gamma_\sigma})$ is contained by arbitrary two of sets $\Omega_1,\cdots,\Omega_4$. Next, we verify one situation and the others can be similarly proved.
Let $\Theta\subset [1,2^{|\mathcal{N}_{\gamma_\sigma}|}]$ be the index set of columns $\mathrm{Col}_i(\mathbf{T}_{\gamma_\sigma})$ belonging to $\Omega_1$, then the other columns satisfy $\mathrm{Col}_i(\mathbf{T}_{\gamma_\sigma})\in \Omega_2,i\in [1,2^{|\mathcal{N}_{\gamma_\sigma}|}]\backslash \Theta$. In this case, the structure matrices can be determined as
\begin{equation}\left\{\small{\begin{array}{l}\label{eg1}
M_{\odot_{\gamma_\sigma}}=\delta_2[1,1,1,2],\\
\widehat{\Psi}_{\gamma_\sigma}=\delta_2[\alpha_1,\alpha_2,\cdots,\alpha_{2^{|\mathcal{N}_{\gamma_\sigma}|}}],\\
\alpha_i=2~\mathrm{if}~i\in \Theta,\\
\alpha_i=1~\mathrm{if}~i\in [1,2^{|\mathcal{N}_{\gamma_\sigma}|}]\backslash\Theta,\\
\end{array}}\right.
\end{equation}
or
\begin{equation}\left\{\small{\begin{array}{l}\label{eg2}
M_{\odot_{\gamma_\sigma}}=\delta_2[1,2,1,1],\\
\widehat{\Psi}_{\gamma_\sigma}=\delta_2[\alpha_1,\alpha_2,\cdots,\alpha_{2^{|\mathcal{N}_{\gamma_\sigma}|}}],\\
\alpha_i=1~\mathrm{if}~i\in \Theta,\\
\alpha_i=2~\mathrm{if}~i\in [1,2^{|\mathcal{N}_{\gamma_\sigma}|}]\backslash\Theta.\\
\end{array}}\right.
\end{equation}
Taking (\ref{eg1}) or (\ref{eg2}) into equations (\ref{equations-BCN-proof}), it obviously holds. Therefore, the proof is completed.
\end{proof}

\subsection{Design Non-Uniform State Feedback Controllers}
For each pinning node $x_{\gamma_\sigma},\sigma\in [1,\tau]$, designing specific state feedback controller for each possibility is apparently feasible, which has been proved in \cite{liff2015pinning} and \cite{liff2019set}. However, for the sake of saving control cost, we preferentially choose uniform state feedback controllers if equations (\ref{stable-structure-matrix}) is solvable.

Sequentially, examining the solvability criterion in Theorem \ref{theorem-2-condition}, and collecting the subscript indices of pinning nodes, whose equations (\ref{stable-structure-matrix}) is unsolvable, into set $\Lambda_2\subseteq \Lambda$. Besides, denote $\Lambda_1=\Lambda\backslash \Lambda_2$ as the solvable index set. The non-uniform state feedback controller imposed on $x_{\gamma_\varrho}\in\Lambda_2$ depends on each possibility $f_{\gamma_\varrho}^\kappa,\kappa\in[1,l_{\gamma_\varrho}]$, and is given as that:
\begin{equation}\label{state-feedback-pinning-controller-1}
u^\kappa_{\gamma_\varrho}(t)=\varphi^\kappa_{\gamma_\varrho}([x_j(t)]_{j\in \mathcal{N}_{\gamma_\varrho}}).
\end{equation}
After that, the state updating of each node $x_i,i\in[1,n]$ obeys the following rule:
\begin{equation*}
x_i(t+1)=g_i([x_j(t)]_{j\in \mathcal{N}^\circ_i}),
\end{equation*}
where $g_i\in \{g^\kappa_i([x_j(t)]_{j\in \mathcal{N}^\circ_i})|\kappa\in[1,l_i]\}$, and
\begin{equation}g^\kappa_i=\left\{\begin{array}{ll}\label{g}
f^\kappa_i([x_j(t)]_{j\in \mathcal{N}_i}), &i\in[1,n]\backslash \Lambda,\\
u_i(t) \odot_i f^\kappa_i([x_j(t)]_{j\in \mathcal{N}_i}), &i\in \Lambda_1,\\
u^\kappa_i(t) \odot^\kappa_i f^\kappa_i([x_j(t)]_{j\in \mathcal{N}_i}), &i\in \Lambda_2.
\end{array}\right.
\end{equation}
Thereinto, $\odot^\kappa_i:\mathscr{D}^2\rightarrow \mathscr{D}$ is logical function connecting control input $u^\kappa_i(t)$ and original dynamic equation $f^\kappa_i([x_j(t)]_{j\in \mathcal{N}_i})$.
Moreover, denote $\widehat{\Psi}^\kappa_i\in \mathscr{L}^{2\times 2^{|\mathcal{N}_{i,\kappa}|}}$ and $M^\kappa_{\odot_i}\in \mathscr{L}^{2\times 4}$ as the structure matrix of $u^\kappa_i$ and $\odot^\kappa_i$.

Similarly, for each pinning node $x_{\gamma_\varrho}\in\Lambda_2$, the state feedback pinning controllers can be designed by solving structure matrices $\widehat{\Psi}^\kappa_{\gamma_\varrho}$ and $M^\kappa_{\odot_{\gamma_\varrho}},\kappa\in [1,l_{\gamma_\varrho}]$ from the below equations:
\begin{equation}\label{stable-structure-matrices}\begin{aligned}
M^\kappa_{\odot_{\gamma_\varrho}}\widehat{\Psi}^\kappa_{\gamma_\varrho}(I_{2^{|\mathcal{N}_{\gamma_\varrho,\kappa}|}}\otimes \mathbf{L}^\kappa_{\gamma_\varrho}) \Phi_{|\mathcal{N}_{\gamma_\varrho,\kappa}|}&=G^\kappa_{\gamma_\varrho}(I_{2^{|\mathcal{N}^\circ_{\gamma_\varrho,\kappa}|}}\otimes \mathbf{1}^\top_{2^{|\mathcal{N}^\star_{\gamma_\varrho,\kappa}|}})\\
&:=\mathbf{T}^\kappa_{\gamma_\varrho},
\end{aligned}
\end{equation}
where $\mathbf{L}^\kappa_{\gamma_\varrho}=F_{\gamma_\varrho}^\kappa\mathbf{W}_{\gamma_\varrho}$, $G^\kappa_{\gamma_\varrho} \in \mathscr{L}^{2\times 2^{|\mathcal{N}^\circ_{\gamma_\varrho,\kappa}|}}$, and $\mathbf{T}^\kappa_{\gamma_\varrho}\in \mathscr{L}^{2\times 2^{|\mathcal{N}_{\gamma_\varrho,\kappa}|}}$. Fortunately, equations (\ref{stable-structure-matrices}) is always solvable, whose detailed proof can be found in \cite{liff2015pinning,liff2019set}.
Subsequently, denote the extended structure matrix of $G^\kappa_{\gamma_\rho}\in \mathscr{L}^{2\times 2^{|\mathcal{N}^\circ_{\gamma_\varrho,\kappa}|}}$ by $\widehat{G}^\kappa_{\gamma_\rho}\in \mathscr{L}^{2\times 2^{|\mathcal{N}^\circ_{\gamma_\rho}|}}$ satisfying $\widehat{G}^\kappa_{\gamma_\rho}\ltimes_{j\in \mathcal{N}^\circ_{\gamma_\rho}}\mathbf{x}_j(t)=G^\kappa_{\gamma_\rho}\ltimes_{j\in \mathcal{N}^\circ_{\gamma_\rho,\kappa}}\mathbf{x}_j(t)$.

Eventually, the mathematical expectation of pinning controlled PBN is derived as that:
\begin{equation}
\mathbb{E}\{\mathbf{x}_i(t+1)\}=G_i\mathbb{E}\{\ltimes_{j\in \mathcal{N}^\circ_i}\mathbf{x}_j(t)\},
\end{equation}
where $G_i=\widehat{F}_i$ if $i\in [1,n]\backslash\Lambda$; $G_i=\sum_{\kappa=1}^{l_i}p^\kappa_iG^\kappa_i$ if $i\in \Lambda_1$; and $G_i=\sum_{\kappa=1}^{l_i}p^\kappa_i\widehat{G}^\kappa_i$ if $i\in \Lambda_2$.

\section{Pinning Control for Stabilizing PBNs to A certain State}\label{section-graphicaltheory}
Notice that if pinning controlled PBN (\ref{equation-BCN}) has achieved stabilization, there may occur several different steady states. Hence, in this section, we further impose another pinning control on PBN (\ref{equation-BCN}) to achieve stabilized at a unique and specific steady state.
In the following sequel, let steady state be $\{\epsilon_1,\epsilon_2,\cdots,\epsilon_n\},\epsilon_i\in \mathscr{D},i\in [1,n]$ and its algebraic form be $\delta_{2^n}^\epsilon=\ltimes_{i=1}^n \delta_2^{2-\epsilon_i}$.

Thereupon, we find matrices $\mathbf{Q}_i,\in\mathscr{L}^{2\times 2^{|\mathcal{N}^\circ_i|}},i\in[1,n]$, and binary variables $\lambda_1,\cdots,\lambda_n\in \mathscr{D}$ such that
\begin{equation}\label{objective-function}
\mathrm{min}~~~~ \Xi\triangleq\sum_{i=1}^n \lambda_i,
\end{equation}
subjects to
\begin{equation}\left\{\begin{array}{c}\label{constraint}
\delta_2^{2-\epsilon_1}=[(1-\lambda_1)G_1+\lambda_1\mathbf{Q}_1]\ltimes_{j\in \mathcal{N}^\circ_1}\delta_2^{2-\epsilon_j},\\
\vdots\\
\delta_2^{2-\epsilon_n}=[(1-\lambda_n)G_n+\lambda_n\mathbf{Q}_n]\ltimes_{j\in \mathcal{N}^\circ_n}\delta_2^{2-\epsilon_j}.
\end{array}\right.
\end{equation}
\begin{remark}
Here, $\Xi$ is termed as control cost function, and objective function (\ref{objective-function}) determines
the number of pinning nodes as minimal as possible. This optimization problem can be deemed as a linear programming problem and can be solved by existing methods.
\end{remark}

Assuming that the solution of the above optimization problem is $\Xi=\pi$ with $\lambda_{\zeta_1}=\lambda_{\zeta_2}=\cdots=\lambda_{\zeta_\pi}=1$, then the second group of pinning nodes (to achieve stabilizing at a certain state) is $x_{\zeta_1},\cdots,x_{\zeta_\pi}$, whose subscript indices are collected into set $\Gamma$.
The state feedback controller imposed on node $x_{\zeta_\iota},\iota\in[1,\pi]$ is described as:
\begin{equation}\left\{\begin{array}{l}\label{equation-BCN-2}
x_{\zeta_\iota}(t+1)=v_{\zeta_\iota}(t) \oplus_{\zeta_\iota} g_{\zeta_\iota}([x_j(t)]_{j\in \mathcal{N}^\circ_{\zeta_\iota}}),\\
v_{\zeta_\iota}(t)=\phi_{\zeta_\iota}([x_j(t)]_{j\in \mathcal{N}^\circ_{\zeta_\iota}}),
\end{array}\right.
\end{equation}
where $\phi_{\zeta_\iota}([x_j(t)]_{j\in \mathcal{N}^\circ_{\zeta_\iota}})$ is logical function decided by node $x_{\zeta_\iota}$'s in-neighbors, and $\oplus_{\zeta_\iota}: \mathscr{D}^2\rightarrow \mathscr{D}$ is logical function connecting state feedback controller $v_{\zeta_\iota}$ and $g_{\zeta_\iota}([x_j(t)]_{j\in \mathcal{N}^\circ_{\zeta_\iota}})$.

Likewise, in resorting to STP, the structure matrices of $\phi_{\zeta_\iota}$ and
$\oplus_{\zeta_\iota},\iota\in [1,\pi]$ are respectively denoted by $\Upsilon_\iota\in \mathscr{L}^{2\times2^{|\mathcal{N}^\circ_\iota|}}$ and $M_{\oplus_\iota}\in \mathscr{L}^{2\times 4}$. Then we can obtain the algebraic form of (\ref{equation-BCN-2}) as below:
\begin{equation}\left\{\small{\begin{array}{l}
\mathbb{E}\{\mathbf{x}_{\zeta_\iota}(t+1)\}=M_{\oplus_{\zeta_\iota}}\Upsilon_{\zeta_\iota}(I_{2^{|\mathcal{N}^\circ_{\zeta_\iota}|}}\otimes G_{\zeta_\iota})\Phi_{|\mathcal{N}^\circ_{\zeta_\iota}|}\mathbb{E}\{\ltimes_{j\in\mathcal{N}^\circ_{\zeta_\iota}}\mathbf{x}_j(t)\},\\
v_{\zeta_\iota}(t)=\Upsilon_{\zeta_\iota}\ltimes_{j\in\mathcal{N}^\circ_{\zeta_\iota}}\mathbf{x}_j(t),
\end{array}}\right.
\end{equation}
where $\Upsilon_{\zeta_\iota}$ and $M_{\oplus_{\zeta_\iota}}$ can be solved from equations:
\begin{equation}\label{stable-structure-matrix-2}
M_{\oplus_{\zeta_\iota}}\Upsilon_{\zeta_\iota}(I_{2^{|\mathcal{N}^\circ_{\zeta_\iota}|}}\otimes G_{\zeta_\iota})\Phi_{|\mathcal{N}^\circ_{\zeta_\iota}|}=\mathbf{Q}_{\zeta_\iota}.
\end{equation}

\begin{remark}
The solvability of equations (\ref{stable-structure-matrix-2}) is guaranteed. Since each $\delta_2^{2-\epsilon_i}, i\in [1,n]$ is equal to $\delta_2^1$ or $\delta_2^2$, its corresponding matrix $\mathbf{Q}_i, i\in [1,n]$ can be directly valued as
$\delta_2[\underbrace{1,\cdots,1}_{2^{|\mathcal{N}^\circ_i|}}]$ or $\delta_2[\underbrace{2,\cdots,2}_{2^{|\mathcal{N}^\circ_i|}}]$. It implies that the column set of each $\mathbf{Q}_i, i\in [1,n]$ can be completely covered by $\Omega_2$ or $\Omega_4$. Therefore, equation (\ref{stable-structure-matrix-2}) must be solvable according to Theorem \ref{theorem-2-condition}.
\end{remark}

Here, the pinning controlled PBN (\ref{equation-PBN}) is represented as that:
\begin{equation}x_i(t+1)=\left\{\begin{array}{l}\label{equation-BCN-final}
v_i(t) \oplus_i [u_i(t)\odot_i f_i([x_j(t)]_{j\in \mathcal{N}_i})], ~i\in\Lambda\cap \Gamma, \\
u_i(t)\odot_i f_i([x_j(t)]_{j\in \mathcal{N}_i}), ~~~~~~~~~i\in\Lambda\backslash\Gamma, \\
v_i(t) \oplus_i f_i([x_j(t)]_{j\in \mathcal{N}_i}),~~~~~~~~~i\in\Gamma\backslash\Lambda,\\
f_i([x_j(t)]_{j\in \mathcal{N}_i}),~~~~~~~~~~~~~i\in[1,n]\backslash(\Lambda\cap\Gamma).
\end{array}\right.
\end{equation}
Thereinto, $u_i,\odot_i, i\in \Lambda$ are obtained by solving equations (\ref{stable-structure-matrix}) or (\ref{stable-structure-matrices}),
and $v_i,\oplus_i, i\in \Gamma$ are derived by solving (\ref{stable-structure-matrix-2}).
To conclude, the procedure of designing pinning control for stabilizing PBN (\ref{equation-PBN}) to a prescribed state $\delta_{2^n}^\epsilon$ is established by the following steps:
\begin{enumerate}
  \item Deduce wiring digraph $\mathbf{G}=(\mathbf{V},\mathbf{E})$ of PBN (\ref{equation-PBN}). If $\mathbf{G}$ is acyclic, then goes to step 6, otherwise goes to next step.
  \item Find the fixed points and cycles existing in $\mathbf{G}$ by the depth-first search algorithm, thereupon determine a possible FAS $\{e_1,\cdots,e_\kappa\}$ and collect ending nodes $x_{\gamma_1},\cdots,x_{\gamma_\tau}$ into $\Lambda$.
  \item For each pinning node $x_{\gamma_\sigma},\sigma\in [1,\tau]$, if equations (\ref{stable-structure-matrix}) is solvable, then do the first operation; otherwise let $\Lambda_2\leftarrow\Lambda_2 \cup \{\gamma_\sigma\}$ and do the second operation:
      \begin{itemize}
        \item Solve structure matrices $M_{\odot_{\gamma_\sigma}}$ and $\widehat{\Psi}_{\gamma_\sigma}$ from (\ref{stable-structure-matrix}). (*Impose uniform controller (\ref{state-feedback-pinning-controllers}) to $x_{\gamma_\sigma}\in \Lambda_1$*)
        \item Solve a series of structure matrices $M^\kappa_{\odot_{\gamma_\sigma}}$ and $\widehat{\Psi}^\kappa_{\gamma_\sigma},\kappa\in[1,l_{\gamma_\sigma}]$ from (\ref{stable-structure-matrices}). (*Impose non-uniform controller (\ref{state-feedback-pinning-controller-1}) to each $f^\kappa_{\gamma_\sigma},\kappa\in[1,l_{\gamma_\sigma}]$ of $x_{\gamma_\sigma}\in \Lambda_2$*)
      \end{itemize}
  \item Solve optimization objective (\ref{objective-function}) subjecting to (\ref{constraint}), and derive $\Xi=\pi$ and matrices $\mathbf{Q}_{\zeta_1},\cdots,\mathbf{Q}_{\zeta_\pi}$ with $\lambda_{\zeta_1}=\cdots=\lambda_{\zeta_\pi}=1$. Then, collect pinning nodes $x_{\zeta_1},\cdots,x_{\zeta_\pi}$ into $\Gamma$.
  \item For node $x_{\zeta_\iota},\iota\in[1,\pi]$, solve structure matrices $M_{\oplus_{\zeta_\iota}}$ and $\Upsilon_{\zeta_\iota}$ from (\ref{stable-structure-matrix-2}).
(*Impose uniform controllers (\ref{equation-BCN-2}) to pinning nodes $x_{\zeta_1}\in \Gamma$*)
  \item The solved state feedback controllers are respectively imposed on the different kinds of pinning nodes as (\ref{equation-BCN-final}).
\end{enumerate}

\begin{remark}
Compared with traditional pinning control design in \cite{wanglq2019pinning} and \cite{liff2019set}, our control design strategy is more practical and can be implemented to sparse large-scale networks. Besides, the time complexity is reduced from $O(2^{2n})$ to $O(n2^{\max_{i\in[1,n]}|\mathcal{N}_i|})$.
\end{remark}

\section{Simulation}\label{section-stabilization}
In this section, a reduced mammalian cell cycle network \cite{faure2006dynamical} is performed to demonstrate the design process of our pinning control. In this setup, the mammalian cell-cycle encountering a mutated phenotype is postulated as a PBN, which has nine variables: $x_1,x_2,\cdots$ and $x_9$ (respectively representing genes Rb, E2F, CycE, CycA, p27, Cdc 20, Cdh 1, UbcH 10, and CycB). Particularly, this mammalian cell cycle may encounter mutation with probability 0.01, and the mutated mammalian cell cycle network has two possible constituent forms with the same probability 0.005. The form of the mutation depends on the expression status of external input CycD \cite{faryabi2008regulatory}.

To be specific, the logical dynamics of normal mammalian cell cycle network is described as:
\begin{equation}\left\{\begin{aligned}\label{ex-1}
f_1^1(\ast)=&(\neg x_3\wedge \neg x_4 \wedge \neg x_9)\vee (x_5 \wedge \neg x_9),\\
f_2^1(\ast)=&(\neg x_1\wedge \neg x_4 \wedge \neg x_9)\vee (x_5 \wedge \neg x_1\wedge \neg x_9),\\
f_3^1(\ast)=&x_2\wedge \neg x_1,\\
f_4^1(\ast)=&[x_2 \wedge\neg x_1\wedge \neg x_6 \wedge \neg (x_7\wedge x_8)]\vee \\
&~~[\neg x_1 \wedge \neg x_6 \wedge x_4 \wedge \neg (x_7\wedge x_8)],\\
f_5^1(\ast)=&(\neg x_3\wedge \neg x_4 \wedge \neg x_9)\vee [x_5 \wedge \neg (x_3\wedge x_4)\wedge \neg x_9],\\
f_6^1(\ast)=&x_9,\\
f_7^1(\ast)=&(\neg x_4\wedge \neg x_9)\vee x_6 \vee (x_5 \wedge \neg x_9),\\
f_8^1(\ast)=&\neg x_7 \vee [x_7 \wedge x_8 \wedge (x_6\vee x_4 \vee x_9)], \\
f_9^1(\ast)=&\neg x_6 \wedge \neg x_7.
\end{aligned}\right.
\end{equation}
Moreover, when the value of CycD is 0, the mutated mammalian cell cycle network is performed as form A:
\begin{equation}\left\{\begin{array}{l}\label{ex-2}
f^{2}_1(\ast)=\neg x_3\wedge \neg x_4 \wedge \neg x_9,~~f^{2}_2(\ast)=\neg x_1\wedge \neg x_4 \wedge \neg x_9,\\
f^{2}_3(\ast)=f_3(\ast),~~f^{2}_4(\ast)=f_4(\ast),~~f_5^{2}(\ast)=f_5(\ast),\\
f_6^{2}(\ast)=f_6(\ast),~~f_7^{2}(\ast)=(\neg x_4\wedge \neg x_9)\vee x_6,\\
f_8^{2}(\ast)=f_8(\ast),~~f_9^{2}(\ast)=f_9(\ast);
\end{array}\right.
\end{equation}
and when the value of CycD is 1, it is performed as form B:
\begin{equation}\left\{\begin{array}{l}\label{ex-3}
f^{3}_1(\ast)=0,~~~~~~~f^{3}_2(\ast)=f^{2}_2(\ast),~~f^{3}_3(\ast)=f^{2}_3(\ast),\\
f^{3}_4(\ast)=f^{2}_4(\ast),~~f_5^{3}(\ast)=f^{2}_5(\ast),~~f_6^{3}(\ast)=f^{2}_6(\ast),\\
f_7^{3}(\ast)=f^{2}_7(\ast),~~f_8^{3}(\ast)=f^{2}_8(\ast),~~f_9^{3}(\ast)=f^{2}_9(\ast).
\end{array}\right.
\end{equation}
As indicated in \cite{faryabi2008regulatory}, gene-activity profile would be forced to the desirable state $(1,1,1,0,0,0,1,0,0)^\top$ equivalent to $\delta_{512}^{44}$. Thereby, in the following, we execute pinning control to stabilize this PBN to $\delta_{512}^{44}$.

\textbf{Step 1:} the wiring digraph $\mathbf{G}$ of PBN (\ref{equation-PBN}) defined by (\ref{ex-1}), (\ref{ex-2}) and (\ref{ex-3}) is established in Fig. \ref{fig1}, and it has several fixed points and cycles. Then proceed to \textbf{Step 2}, one obtains a feasible FAS as $\{x_3\rightarrow x_1,x_1\rightarrow x_4,x_2\rightarrow x_4,x_4\rightarrow x_4,x_7\rightarrow x_4,x_8\rightarrow x_4,x_3\rightarrow x_5, x_5\rightarrow x_5,x_9\rightarrow x_6,x_8\rightarrow x_8,x_7\rightarrow x_9\}$, then the first group of pinning nodes are $x_1,x_4,x_5,x_6,x_8$ and $x_9$.

\begin{figure}[h!]
\centering
\includegraphics[width=0.27\textwidth=0.2]{cyjzd.jpg}
\caption{The wiring diagrams of PBN (\ref{equation-PBN}) defined in the section \ref{section-stabilization}. In this figure, each directed edge $i\rightarrow j,i,j\in[1,9]$ expresses $x_i$ being the functional variable of $f^1_j$ or $f^2_j$ or $f^3_j$. Besides, red edges are the edges in the FAS; and blue nodes are the selected pinning nodes.}\label{fig1}
\end{figure}
Before operating \textbf{Step 3}, we denote the updating matrix of gene $x_i,i\in[1,9]$ by $\textbf{L}_i\in \mathscr{P}^{2\times 2^{|\mathcal{N}_i|}}$, which is easily derived by (\ref{Ex(t+1)}).
Besides, the structure matrices after deleting edges in FAS are obtained as follows:
\begin{equation*}G_1=\left(\begin{array}{cccccccc}
0&0.99&0&0&0.995&0.995&0&0.995 \\
1&0.01&1&1&1&0.005&1&0.005
\end{array}\right),
\end{equation*}
\begin{equation*}G_2=\textbf{L}_2=\left(\begin{array}{c}
0~\cdots~0~0.99~0~0~0~1~0~1 \\
1~\cdots~1~0.01~1~1~1~0~1~0
\end{array}\right)\in \mathscr{P}^{2\times 16},
\end{equation*}
$G_3=\textbf{L}_3=\delta_2[2,1,2,2]$, $G_4=\delta_2[2,1]$, $G_5=\delta_2[2,2,2,1]$, $G_6=\delta_2^2$, \begin{equation*}G_7=\textbf{L}_7=\left(\begin{array}{c}
1~1~0~0.99~1~1~0~0~1~1~0~1~1~1~0~1 \\
0~0~1~0.01~0~0~1~1~0~0~1~0~0~0~1~0
\end{array}\right),
\end{equation*}
$G_8=\delta_2[1,\cdots,1,2,1,1]\in \mathscr{P}^{2\times 16}$ and $G_9=\delta_2[2,1]$.
Meanwhile, according to equation (\ref{stable-struture-matrix-0}), it derives that:
\begin{equation*}\mathbf{T}_1=\left(\small{\begin{array}{c}
0~0~0.99~0.99~\underbrace{0~\cdots~0}_6~0.995~0.995~0~0~0.995~0.995 \\
1~1~0.01~0.01~\underbrace{1~\cdots~1}_6~0.005~0.005~1~1~0.005~0.005
\end{array}}\right),
\end{equation*}
$\mathbf{T}_4=\delta_2[\underbrace{2,\cdots,2}_{32},\underbrace{1,\cdots,1}_{32}]$, $\mathbf{T}_5=\delta_2[\underbrace{2,\cdots,2}_{12},\underbrace{1,\cdots,1}_4]$, $\mathbf{T}_6=\delta_2[2,2]$,
$\mathbf{T}_8=\delta_2[\underbrace{1,\cdots,1}_{26},2,2,\underbrace{1,\cdots,1}_4]$, and $\mathbf{T}_9=\delta_2[2,2,1,1]$.
Then, we can examine the solvability of equations (\ref{stable-structure-matrix}) for pinning nodes $x_i,i\in \{1,4,5,6,8,9\}$ according to Theorem \ref{theorem-2-condition}, it reveals that $\mathbf{T}_1=M_{\odot_1}\widehat{\Psi}_1(I_{16}\otimes \mathbf{L}_1) \Phi_4\in\mathscr{P}^{2\times 16}$ is unsolvable, while $\mathbf{T}_4=M_{\odot_4}\widehat{\Psi}_4(I_{64}\otimes \mathbf{L}_4) \Phi_6$,
$\mathbf{T}_5=M_{\odot_5}\widehat{\Psi}_5(I_{16}\otimes \mathbf{L}_5)\Phi_4$,
$\mathbf{T}_6=M_{\odot_6}\widehat{\Psi}_6(I_{2}\otimes \mathbf{L}_6) \Phi_1$,
$\mathbf{T}_8=M_{\odot_8}\widehat{\Psi}_8(I_{32}\otimes \mathbf{L}_8)\Phi_5$ and
$\mathbf{T}_9=M_{\odot_9}\widehat{\Psi}_9(I_{4}\otimes \mathbf{L}_9) \Phi_2$ are solvable.
Thereby, with respect to pinning node $x_1$, we obtain structure matrices (\ref{stable-structure-matrices}) as $\mathbf{T}^1_1=\delta_2[2,2,1,1,2,\cdots,2,1,1,2,2,1,1]$, $\mathbf{T}^2_1=\delta_2[2,\cdots,2,1,1,2,2,1,1]$ and $\mathbf{T}^{3}_1=\delta_2[2,\cdots,2]$. Then, the state feedback pinning controllers imposed on $f^1_1$, $f^2_1$ and $f^3_1$ can be respectively solved from $\mathbf{T}^1_1=M^1_{\odot_1}\widehat{\Psi}^1_1(I_{16}\otimes \mathbf{L}^1_1) \Phi_4$, $\mathbf{T}^2_1=M^2_{\odot_1}\widehat{\Psi}^2_1(I_{16}\otimes \mathbf{L}^2_1) \Phi_4$, and $\mathbf{T}^{3}_1=M^{3}_{\odot_1}\widehat{\Psi}^{3}_1(I_{16}\otimes \mathbf{L}^{3}_1) \Phi_4$. It derives that
$$M^1_{\odot_1}=M^2_{\odot_1}=M^{3}_{\odot_1}=\delta_2[1,1,1,2],\widehat{\Psi}^1_1=\delta_2[\underbrace{2,\cdots,2}_{14},1,1],$$
$$\widehat{\Psi}^2_1=\delta_2[\underbrace{2,\cdots,2}_{10},1,2,2,2,1,2],~\mathrm{and}~\widehat{\Psi}^{3}_1=\delta_2[\underbrace{2,\cdots,2}_{16}].$$
Simultaneously, based on equation (\ref{stable-structure-matrix}),
the efficient state feedback controllers applied on pinning genes $x_4,x_5,x_6,x_8,x_9$ can be respectively designed as $$M_{\odot_4}=\delta_2[1,1,2,2],\widehat{\Psi}_4=\delta_2[\underbrace{2,\cdots,2}_{32},\underbrace{1,\cdots,1}_{32}];$$ $$M_{\odot_5}=\delta_2[1,1,2,2],\widehat{\Psi}_5=\delta_2[\underbrace{2,\cdots,2}_{12},\underbrace{1,\cdots,1}_{4}];$$
$$M_{\odot_6}=\delta_2[2,2,2,2],\widehat{\Psi}_6=\delta_2[1,1];$$ $$M_{\odot_8}=\delta_2[1,1,2,2],\widehat{\Psi}_8=\delta_2[\underbrace{1,\cdots,1}_{26},2,2,\underbrace{1,\cdots,1}_4];$$ $$\mathrm{and}~M_{\odot_9}=[1,1,2,2], \widehat{\Psi}_9=[2,2,1,1].$$
After that, the wiring digraph of this pinning controlled PBN is acyclic, and all possible models achieve globally stabilized.

To proceed, perform \textbf{Step 4} on the stabilized PBN. The solution of optimization problem (\ref{objective-function}) is $\Xi=8$ and $\lambda_1=\lambda_2=\lambda_3=\lambda_4=\lambda_5=\lambda_7=\lambda_8=\lambda_9=1$. Subsequently, we obtain $\mathbf{Q}_1=\delta_2[1,\cdots,1]\in\mathscr{L}^{2\times 8}$ $\mathbf{Q}_2=\mathbf{Q}_7=\delta_2[1,\cdots,1]\in\mathscr{L}^{2\times 16}$, $\mathbf{Q}_3=\mathbf{Q}_5=\delta_2[1,\cdots,1]\in\mathscr{L}^{2\times 4}$, $\mathbf{Q}_4=\mathbf{Q}_9=\delta_2[0,0]$, $\mathbf{Q}_8=\delta_2[2,\cdots,2]\in\mathscr{L}^{2\times 16}$.
Then, go to \textbf{Step 5} and obtain
$$M_{\oplus_1}=M_{\oplus_2}=M_{\oplus_3}=M_{\oplus_5}=M_{\oplus_7}=\delta_2[1,1,1,1],$$ $$M_{\oplus_4}=M_{\oplus_8}=M_{\oplus_9}=\delta_2[2,2,2,2],$$
$$\Upsilon_1=\delta_2[1,\cdots,1]\in\mathscr{L}^{2\times8},\Upsilon_2=\Upsilon_7=\Upsilon_8=\delta_2[1,\cdots,1]\in\mathscr{L}^{2\times16},$$ $$\Upsilon_3=\Upsilon_5=\delta_2[1,1,1,1],~\mathrm{and}~\Upsilon_4=\Upsilon_9=\delta_2[1,1].$$
Finally, the pinning node sets are designed as $\Lambda\cap\Gamma=\{1,4,5,8,9\}$, $\Lambda\backslash\Gamma=\{6\}$, $\Gamma\backslash\Lambda=\{2,3,7\}$ and $[1,n]\backslash(\Lambda\cup\Gamma)=\emptyset$, and then under the following designed pinning control, PBN (\ref{equation-PBN}) defined here can be stabilized to state $\delta_{512}^{44}$.

\small{\begin{equation*}\left\{\begin{array}{l}
x^1_1(\ast)=f^1_1(\ast)\vee u^1_1(\ast)\vee v_1(\ast),\\
u^1_1(\ast)=x_3\wedge \neg x_4\wedge \neg x_9, v^1_1(\ast)=\neg x_5\wedge x_9,\\
x^2_1(\ast)=f^2_1(\ast)\vee u^2_1(\ast)\vee v^2_1(\ast),\\
u^2_1(\ast)=x_3\vee x_4 \vee x_9,v^2_1(\ast)=x_4 \vee x_9,\\
x^3_1(\ast)=f^3_1(\ast)=0,\\
x_2(\ast)=f_2(\ast)\vee v_2(\ast),v_2(\ast)=\neg x_5\vee x_1\vee x_9,\\
x_3(\ast)=f_3(\ast)\vee v_3(\ast),v_3(\ast)=\neg x_2 \vee x_1,\\
x_4(\ast)=f_4(\ast)\vee u_4(\ast)\wedge v(\ast), u_4(\ast)=\neg x_6,v(\ast)=x_6, \\
x_5(\ast)=f_5(\ast)\vee u_5(\ast)\wedge v_5(\ast),\\
u_5(\ast)=\neg x_4 \wedge \neg x_9,v_5(\ast)=x_4\vee x_9,\\
x_6(\ast)=f_6(\ast)\wedge u_6(\ast),u_6(\ast)=\neg x_9,\\
x_7(\ast)=f_7(\ast)\vee v_7(\ast),v_7(\ast)=\neg x_6,\\
x_8(\ast)=f_8(\ast)\vee u_8(\ast)\wedge v_8(\ast),\\
u_8(\ast)=x_7\wedge \neg x_8\wedge(x_4\vee x_6 \vee x_9),v_8(\ast)=x_7\wedge \neg (x_4\vee x_6\vee x_9), \\
x_9(\ast)=f_9(\ast)\vee u_9(\ast)\wedge v_9(\ast), u_9(\ast)=\neg x_6\wedge x_7, v_9(\ast)=x_6.
\end{array}\right.
\end{equation*}}

\section{Conclusion}\label{section-conclusion}
This paper studied the stabilization of PBNs via pinning control based on network structure. First, the pinning nodes were selected by finding the FAS. Concerning the stochasticity of PBNs, uniform and non-uniform state feedback controllers were respectively designed to stabilize PBNs. Moreover, a criterion was obtained to determine whether each pinning node could utilize uniform controllers. Since a stable PBN may have multiple steady states, we designed another pinning control to further stabilize it at the desired state. This control design strategy was based on the network structure (local neighbors' information), rather than state transition matrix (global information). Thereby, the designed pinning control could be utilized to handle the networks with sparse connections, but not so sparse that the graph is in danger of becoming disconnected. Specially, we required $n\gg k \gg \ln(n)\gg 1$, where $k \gg \ln(n)$ guaranteed that the wiring digraph would be connected.
Eventually, the obtained results was demonstrated by a biological example about the mammalian cell-cycle encountering a mutated phenotype.

redpajama_set_name: RedPajamaArXiv


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Item 726 - Statement on the visit by President Mandela to Uganda, Germany and Switzerland: 25-30 January 1999
Statement on the visit by President Mandela to Uganda, Germany and Switzerland: 25-30 January 1999
Visit by President Mandela to Uganda, Germany and Switzerland
President Nelson Mandela, accompanied by Foreign Minister Alfred Nzo, will visit Uganda, Germany and Switzerland from 25-30 January 1999. President Mandela will visit Uganda on 25 and 26 January 1999.
The occasion will afford him the opportunity to further cement ties between South Africa and Uganda and discuss regional issues and other matters of common interest. South Africa is a major investor in Uganda and trade relations with this East African partner are expanding rapidly. President Mandela will also open a newly completed sport stadium outside Kampala, which will be named after him.
President Mandela will then pay a private visit to Baden-Baden in Germany on 28 January 1999 to receive the prestigious "Deutsche Medien Preis" 1998, presented by a German media research company, "Media Control". It is an internationally acknowledged prize honouring prominent newsmakers. Previous recipients have included Presidents Mitterand, Rabin, Arafat, Jeltsin, Chancellor Kohl and King Hussein.
President Mandela will also meet with the new German Chancellor, Gerhard Schroeder on the same day.
President Mandela will bestow the Order of Good Hope on Juergen Schrempp, Chairman of Daimler Chrysler and the South African Honorary Consul-General in the German Federal States of Baden Wuerttemberg, Rhineland-Palatinate and Saarland, since 1995, in recognition of his outstanding services to South Africa.
Mrs Graca Machel will accompany the President and travel to Strasbourg on 28 January 1999 to receive the North-South Prize from the North-South Centre, European Centre for Global Interdependence and Solidarity, Council of Europe.
President Mandela will also attend the annual World Economic Forum (WEF) meeting in Davos, Switzerland, from 29-30 January 1999. The World Economic Forum counts international business people among its members and the Davos meeting brings together these CEOs with heads of state and government and other senior politicians from all over the world.
The South African Government has, for the past few years, been an institutional member of the Geneva-based World Economic Forum and has been an active participant in its activities at senior political level, especially the Davos Annual Meeting. President Mandela has attended a number of these meetings since 1992 and will use this opportunity to bid farewell to these business leaders.
The South African business community will be widely represented at the WEF and will also attend the Davos meeting.
President Mandela will also meet with the President of the Swiss Confederation, Ruth Dreyfuss, the first woman to hold this position, during this visit.


text:
Teen Book Review: Seeker
Seeker by ARWEN ELYS DAYTON
Quin Kincaid has been trained all her life to become a Seeker: someone who protects the weak and less fortunate to keep peace in the universe. She has trained beside her two best friends, John and Shinobu, and the three of them plan to fight as Seekers together. The best part is that Quin will be with the one she loves, who also happens to be one of her best friends.
The night of her Oath to become a Seeker however, she finds that being a Seeker is not what she thought at all- and neither is her family, or the one she calls her true love.
Seeker is a fast paced novel filled with action, romance, and betrayal; it is the first of three books, highly recommended for fans of science fiction!
Review by,
Anoushka Sharma
Former TAB member and current Library Page
745.5 Square One!
The Teen Advisory Board helped out during the library's grand opening of Square One!
On Sunday, July 2, from 2-5 pm, the Woodland Public Library opened 745.5 Square One, a nearly 2000 square foot makerspace and community learning lab located on the ground floor of the Library.
A makerspace is an area dedicated to education through creation where users can imagine, explore, and collaborate in hands-on learning, design, and production. The Makerspace Movement has gained momentum in recent years worldwide as a creative, collaborative space outside home, school, and work to explore and learn. This movement aligns with the library's mission to provide education and literacy to the community. "Libraries have always been on the forefront of providing public access to emerging technology that isn't readily available to most consumers," stated Greta Galindo, Woodland Public Library Director.
The name of the space, 745.5 Square One, refers to the Dewey Decimal classification (how the library sorts nonfiction books) for 'handcrafts' (745.5) and to the place people return when they need to rethink an idea or try again (Square One).
Square One will provide the opportunity for patrons to learn important skills that can be used in the workforce—from drafting and design to the hands-on work of crafting circuit boards. Square One is designed to be a modular space to bring together the community to share knowledge and collaboratively problem solve. The central design hub, sponsored by Sunrise Rotary, consists of a dozen laptops and 8 work tables that can be rearranged for classes, large projects, or smaller group work. Surrounding the design hub are 5 work areas: a prototyping area, an art and textiles area, an enclosed wood shop, an electronics bench, and a digital arts station. These areas have all been created to be flexible spaces that can expand or contract as classes, presentations, and use require.
The Prototyping area, sponsored by the Dowling family, features a laser cutter for cutting and engraving plastics, wood, and some metals, as well as 2 3-D printers, large and small computer controlled cutting devices, and supplies to design and create projects. The Art and Textiles area, sponsored by the Thordsen Family and Woodland Sewing and Vacuum Center, consists of two portable self-contained sewing machine units, jewelry-making supplies, and other fiber arts. The wood shop contains a CNC-router, panel saw, drill press, band saw, and a full complement of hand tools for woodworking projects. The Electronics Bench, sponsored by Luna Vista Rotary, holds 3 vented soldering stations and supplies for projects ranging from introduction soldering to advanced robotics. The Digital Arts corner, also sponsored by Luna Vista Rotary, holds 2 computer workstations outfitted with myriad programs and applications for designing objects to be produced in the prototyping and wood working areas as well as for audio and video editing. This area also features video and still cameras for use.
This Open House marks the culmination of two years of work, planning, construction, and collaboration with numerous organizations. This project was funded through Measure E and J funds, and the generous financial support of the California State Library and the Library Services and Technology Act (LSTA), the Dowling Family, Dan Chervinskas from American Industrial Coatings, the Thordsen Family, Luna Vista Rotary, Sunrise Rotary, Meg and Tom Stallard, and Pavel and Alla Tkachuk at the Woodland Sewing and Vacuum Center. The Woodland Public Library thanks our generous community for their support of Square One.


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