In this section, we give the main results of this paper. Firstly, we convert the dynamics of a constrained Boolean network with pinning control into its equivalent constrained algebraic form, based on which we obtain a necessary and sufficient condition for the reachability of constrained BCNs with pinning control. Secondly, we present the pinning control design procedure for the feedback stabilization of constrained BCNs.
3.1 Constrained algebraic form
A Boolean network with n network nodes and r pinning controls can be described as
$$ \textstyle\begin{cases} x_{1}(t+1)=f_{1}(x_{1}(t),\ldots,x_{n}(t),u_{1}(t)), \\ \vdots \\ x_{r}(t+1)=f_{r}(x_{1}(t),\ldots,x_{n}(t),u_{r}(t)), \\ x_{r+1}(t+1)=f_{r+1}(x_{1}(t),\ldots,x_{n}(t)), \\ \vdots \\ x_{n}(t+1)=f_{n}(x_{1}(t),\ldots,x_{n}(t)), \\ y_{i}(t)=h_{i}(x_{1}(t),\ldots,x_{n}(t)),\quad i=1,\ldots,p, \end{cases} $$
(6)
where nodes \(i_{1},\ldots,i_{r}\) are selected to be pinning controlled, and \(1\leq r< n \). Without loss of generality, we assume that \(i_{s}=s\), \(s=1,\ldots, r\). \(x_{i}(t)\in\mathcal{D}\), \(i=1,\ldots ,n\), \(u_{i}(t)\in\mathcal{D}\), \(i=1,\ldots,r\), and \(y_{i}(t)\in\mathcal {D}\), \(i=1,\ldots,p\), are the states, the control inputs, and the outputs of system (6), respectively, and \(f_{i}: \mathcal {D}^{n+1}\mapsto\mathcal{D}\), \(i=1,\ldots,r\), \(f_{i}: \mathcal {D}^{n}\mapsto\mathcal{D}\), \(i=r+1,\ldots,n\), and \(h_{i}: \mathcal {D}^{n}\mapsto\mathcal{D}\), \(i=1,\ldots,p\), are logical functions.
In order to convert system (6) into an algebraic form, we define \(x(t)=\ltimes_{i=1}^{n}x_{i}(t)\in\Delta_{2^{n}}\), \(x^{1}(t)=\ltimes_{i=1}^{r}x_{i}(t)\in\Delta_{2^{r}}\), \(x^{2}(t)=\ltimes_{i=r+1}^{n}x_{i}(t)\in\Delta_{2^{nr}}\), \(u(t)=\ltimes_{i=1}^{r}u_{i}(t)\in\Delta_{2^{r}}\), and \(y(t)=\ltimes _{i=1}^{p}y_{i}(t)\in\Delta_{2^{p}}\). Assume that the structural matrix of \(f_{i}\) and \(h_{i}\) are \(F_{i}\), \(i=1,\ldots, n\), and \(H_{i}\), \(i=1,\ldots,p\), respectively. Using Lemma 1, system (6) can be expressed as
$$\begin{aligned} &\begin{aligned}[b] x^{1}(t+1)={}& F_{1}x(t)u_{1}(t)F_{2}x(t)u_{2}(t) \cdots F_{r}x(t)u_{r}(t) \\ ={}& F_{1}(I_{2^{n+1}}\otimes F_{2})x(t)u_{1}(t)x(t)u_{2}(t)F_{3}x(t)u_{3}(t) \cdots F_{r}x(t)u_{r}(t) \\ ={}& F_{1}(I_{2^{n+1}}\otimes F_{2})W_{[2^{n},2^{n+1}]}M_{r,2^{n}}x(t)u_{1}(t)u_{2}(t) \\ &{}\ltimes F_{3}x(t)u_{3}(t)\cdots F_{r}x(t)u_{r}(t) \\ ={}&F_{1}\prod_{i=2}^{r} \bigl[(I_{2^{n+i1}}\otimes F_{i}) W_{[2^{n},2^{n+i1}]}M_{r,2^{n}} \bigr]x(t)u(t) \\ :={}&L_{1}x(t)u(t), \end{aligned} \end{aligned}$$
(7)
$$\begin{aligned} &\begin{aligned}[b] x^{2}(t+1)={}&F_{r+1}x(t)F_{r+2}x(t) \cdots F_{n}x(t) \\ ={}&F_{r+1}\ast F_{r+2}\ast\cdots\ast F_{n}x(t) \\ :={}&L_{2}x(t), \end{aligned} \end{aligned}$$
(8)
and
$$ \begin{aligned}[b] y(t)={}&H_{1}x(t)H_{2}x(t)\cdots H_{p}x(t) \\ ={}&H_{1}\ast H_{2}\ast\cdots\ast H_{p}x(t) \\ :={}&Hx(t), \end{aligned} $$
(9)
where
$$\begin{aligned}& L_{1}=F_{1}\prod_{i=2}^{r} \bigl[(I_{2^{n+i1}}\otimes F_{i}) W_{[2^{n},2^{n+i1}]}M_{r,2^{n}} \bigr]\in\mathcal{L}_{2^{r}\times2^{n+r}}, \\& M_{r,2^{n}}=\operatorname{diag}\bigl\{ \delta_{2^{n}}^{1}, \delta_{2^{n}}^{2},\ldots ,\delta_{2^{n}}^{2^{n}} \bigr\} \in\mathcal{L}_{2^{2n}\times2^{n}}, \\& L_{2}=F_{r+1}\ast F_{r+2}\ast\cdots\ast F_{n}\in\mathcal {L}_{2^{nr}\times2^{n}}, \end{aligned}$$
and
$$H=H_{1}\ast H_{2}\ast\cdots\ast H_{p}\in \mathcal{L}_{2^{p}\times2^{n}}. $$
Summarizing, we obtain the following algebraic form of system (6):
$$ \textstyle\begin{cases} x(t+1)=Qu(t)x(t), \\ y(t)=Hx(t), \end{cases} $$
(10)
where \(Q=L_{1}(I_{2^{n+r}}\otimes L_{2})W_{[2^{n},2^{n+r}]}M_{r,2^{n}}W_{[2^{r},2^{n}]}\in\mathcal {L}_{2^{n}\times2^{n+r}}\).
Now, we consider system (10) with state and input constraints. For any \(t\in\mathbb{N}\), we assume that \(x(t)\in S_{x}\subseteq\bigtriangleup_{2^{n}}\) and \(u(t)\in S_{u}\subseteq \bigtriangleup_{2^{r}}\). Let \(S_{x}=n_{1}\leq2^{n}\) and \(S_{u}=r_{1}\leq2^{r}\), where \(S_{x}\) denotes the cardinality of the set \(S_{x}\). Then, \(S_{x}\) and \(S_{u}\) can be expressed as
$$\begin{aligned}& S_{x}=\bigl\{ \delta _{2^{n}}^{i_{k}}:k=1, \ldots,n_{1}; 1\leq i_{1}< \cdots< i_{n_{1}}\leq 2^{n}\bigr\} , \end{aligned}$$
(11)
$$\begin{aligned}& S_{u}=\bigl\{ \delta _{2^{r}}^{j_{k}}:k=1, \ldots,r_{1};1\leq j_{1}< \cdots< j_{r_{1}}\leq 2^{r}\bigr\} . \end{aligned}$$
(12)
Denote the trajectory of system (10) with a pinning control sequence \(\{(u_{1}(t), u_{2}(t),\ldots, u_{r}(t)):t \in\mathbb{N}\} \subseteq S_{u}\) and an initial state \(x_{0}\in S_{x}\) by \(x(t;x_{0},(u_{1}(t), u_{2}(t),\ldots, u_{r}(t)))\).
In the following, we convert system (10) with state and input constraints into an equivalent constrained algebraic form.
Define the following set of matrices:
$$ J_{i}^{(p,q)}:= [\underbrace{0_{q\times q}\quad \cdots\quad 0_{q\times q}\quad \underbrace{I_{q}}_{i\mathrm{th}}\quad 0_{q\times q}\quad \cdots\quad 0_{q\times q}}_{p} ], $$
(13)
where \(J_{i}^{(p,q)}\in\mathbb{R}^{q\times pq}\), \(i=1,2,\ldots,p\), \(0_{q\times q}\) denotes the \(q\times q\) zero matrix, and \(I_{q}\in \mathcal{L}_{q\times q}\) is the \(q\times q\) identity matrix.
Proposition 4
([35])
1. Given a matrix
\(A\in\mathbb{R}^{pq\times r}\), split
A
as
$$ A=\left [ \textstyle\begin{array}{c} A_{1}\\ \vdots\\ A_{p} \end{array}\displaystyle \right ], $$
(14)
where
\(A_{i}\in\mathbb{R}^{q\times r}\). Then,
$$ J_{i}^{(p,q)}A=A_{i}. $$
(15)
2. Given a matrix
\(B\in\mathbb{R}^{r\times pq}\), split
B
as
$$ B=[B_{1}\quad \cdots\quad B_{p}], $$
(16)
where
\(B_{i}\in\mathbb{R}^{r\times q}\). Then,
$$ B\bigl(J_{i}^{(p,q)}\bigr)^{T}=B_{i}. $$
(17)
Based on Proposition 4, set
$$\begin{aligned}& \Psi_{x}= \left [ \textstyle\begin{array}{c} J_{i_{1}}^{(2^{n},1)}\\ \vdots\\ J_{i_{n_{1}}}^{(2^{n},1)} \end{array}\displaystyle \right ], \end{aligned}$$
(18)
$$\begin{aligned}& \Psi_{u}= \left [ \textstyle\begin{array}{ccc} J_{j_{1}}^{(2^{r},1)}\\ \vdots\\ J_{j_{r_{1}}}^{(2^{r},1)} \end{array}\displaystyle \right ]. \end{aligned}$$
(19)
Denote \(\delta_{n_{1}}^{0}=0_{n_{1}\times1}\) and \(\delta _{r_{1}}^{0}=0_{r_{1}\times1}\). Then, we convert the state \(x(t)\in \Delta _{2^{n}}\) and input \(u(t)\in \Delta _{2^{r}}\) of the constrained system into the following form:
$$\begin{aligned}& \widehat{x}(t)=\Psi_{x}x(t)\in \widehat{S}_{x}, \end{aligned}$$
(20)
$$\begin{aligned}& \widehat{u}(t)=\Psi_{u}u(t)\in \widehat{S}_{u}, \end{aligned}$$
(21)
where \(\widehat{S}_{x}=\{\delta_{n_{1}}^{1},\delta _{n_{1}}^{2},\ldots,\delta_{n_{1}}^{n_{1}}\}\cup\{\delta _{n_{1}}^{0}\}\) and \(\widehat{S}_{u}=\{\delta_{r_{1}}^{1},\delta _{r_{1}}^{2},\ldots,\delta_{r_{1}}^{r_{1}}\}\cup\{\delta _{r_{1}}^{0}\}\).
For system (10), let \(Q=[Q_{1},\ldots,Q_{2^{r}}]\), \(Q_{l}\in \mathcal{L}_{2^{n}\times2^{n}}\), \(l\in\{1,\ldots,2^{r}\}\). Set
$$\begin{aligned}& \widehat{Q}= [\widehat{Q}_{1}\quad \cdots\quad \widehat{Q}_{2^{r}} ]\ltimes \bigl[\bigl(J_{j_{1}}^{(2^{r},n_{1})} \bigr)^{T}\quad \cdots \quad\bigl(J_{j_{r_{1}}}^{(2^{r},n_{1})} \bigr)^{T} \bigr]\in\mathcal{B}_{n_{1}\times n_{1}r_{1}}, \end{aligned}$$
(22)
$$\begin{aligned}& \widehat{H}=H\Psi_{x}^{T}\in\mathcal {L}_{2^{p}\times n_{1}}, \end{aligned}$$
(23)
where
$$ \widehat{Q}_{l}=\Psi_{x}Q_{l} \Psi _{x}^{T}\in\mathcal{B}_{n_{1}\times n_{1}},\quad l\in\bigl\{ 1, \ldots,2^{r}\bigr\} . $$
(24)
Remark 1
Since Q is a logical matrix, we can easily conclude that each column of Q̂ has at most one element ‘1’.
Based on the above transformation, we convert system (10) into the following form:
$$ \textstyle\begin{cases} \widehat{x}(t+1)=\widehat{Q}\widehat{u}(t)\widehat{x}(t), \\ \widehat{y}(t)=\widehat{H}\widehat{x}(t). \end{cases} $$
(25)
Proposition 5
The state trajectories of system (10) with
\(S_{x}\)
and
\(S_{u}\)
are equivalent to that of system (25) with
\(\widehat{S}_{x}\)
and
\(\widehat{S}_{u}\).
Proof
On one hand, \(\forall t\in\mathbb{N}\), \(\forall u(t)=\delta _{2^{r}}^{j_{s}}\in S_{u}\), and \(\forall x(t)=\delta_{2^{n}}^{i_{\alpha}}\in S_{x}\), if \(x(t+1)=Qu(t)x(t)\in S_{x}\), say, \(x(t+1)=\delta_{2^{n}}^{i_{\beta }}\), \(\beta\in\{1, \ldots, n_{1}\}\), a simple calculation shows that
$$\widehat{x}(t+1)=\widehat{Q}\widehat{u}(t)\widehat{x}(t)=\delta _{n_{1}}^{\beta}\in\widehat{S}_{x}\setminus\bigl\{ \delta_{n_{1}}^{0}\bigr\} , $$
where \(\widehat{x}(t)=\delta_{n_{1}}^{\alpha}\in\widehat{S}_{x}\) and \(\widehat{u}(t)=\delta_{r_{1}}^{s}\in\widehat{S}_{u}\). If \(x(t+1)=Qu(t)x(t)=\delta_{2^{n}}^{i}\notin S_{x}\), then \(\widehat {x}(t+1)=\widehat{Q}\widehat{u}(t)\widehat{x}(t)=\delta_{n_{1}}^{0}\). Hence, in both cases, \(\widehat{x}(t+1)=\Psi_{x}x(t+1)\).
On the other hand, \(\forall t\in\mathbb{N}\), \(\forall \widehat {u}(t)=\delta_{r_{1}}^{s}\in\widehat{S}_{u}\setminus\{\delta _{r_{1}}^{0}\}\), and \(\forall\widehat{x}(t)=\delta_{n_{1}}^{\alpha}\in\widehat {S}_{x}\setminus\{\delta_{n_{1}}^{0}\}\), if \(\widehat {x}(t+1)=\widehat{Q}\widehat{u}(t)\widehat{x}(t)\in\widehat {S}_{x}\setminus\{\delta_{n_{1}}^{0}\}\), say, \(\widehat {x}(t+1)=\delta_{n_{1}}^{\beta}\), \(\beta\in\{1, \ldots, n_{1}\}\), we can obtain that
$$x(t+1)=Qu(t)x(t)=\delta_{2^{n}}^{i_{\beta}}\in S_{x}. $$
If \(\widehat{x}(t+1)=\widehat{Q}\widehat{u}(t)\widehat{x}(t)=\delta _{n_{1}}^{0}\), then
$$x(t+1)=Qu(t)x(t)\notin S_{x}. $$
Hence, we have \(\widehat{x}(t+1)=\Psi_{x}x(t+1)\).
Therefore, the state trajectories of system (10) with \(S_{x}\) and \(S_{u}\) are equivalent to that of system (25) with \(\widehat{S}_{x}\) and \(\widehat{S}_{u}\). □
Remark 2
We call (25) the constrained algebraic form of the original system. Based on Proposition 5, we can convert the feedback stabilization of the original system to that of system (25).
3.2 Reachability analysis
In this subsection, we study the reachability of system (25), which is crucial to pinning control design for the feedback stabilization.
We give the definition of the reachability for system (25) as follows.
Definition 2
For system (25), given two states \(\widehat{x}_{0}, \widehat{x}_{d} \in\widehat {S}_{x}\setminus\{\delta_{n_{1}}^{0}\}\) and a integer \(k>0\), \(\widehat{x}_{d}\) is said to be reachable from \(\widehat{x}_{0}\) at time k if there is a pinning control sequence \((\widehat{u}_{1}(t), \widehat{u}_{2}(t),\ldots,\widehat{u}_{r}(t))\) with \(\widehat {u}(t)=\ltimes_{i=1}^{r}\widehat{u}_{i}(t)\in\widehat{S}_{u}\setminus \{\delta_{r_{1}}^{0}\}\), \(t\in\{1,2,\ldots,k1\}\), such that \(\widehat{x}(k; \widehat{x}_{0}, (\widehat{u}_{1}(t), \widehat {u}_{2}(t),\ldots,\widehat{u}_{r}(t)))=\widehat{x}_{d}\).
For system (25), consider two given states \(\widehat {x}_{0}=\delta_{n_{1}}^{\alpha}\in\widehat{S}_{x}\setminus\{\delta _{n_{1}}^{0}\}\), \(\widehat{x}_{d}=\delta_{n_{1}}^{\beta}\in\widehat {S}_{x}\setminus\{\delta_{n_{1}}^{0}\}\) and a given integer \(k>0\). Let \(P(k;\widehat{x}_{0}, \widehat{x}_{d})\) denote the number of different paths such that \(\widehat{x}_{d}\) is reachable from \(\widehat{x}_{0}\) at time k.
Lemma 2
Consider system (25) with two given states
\(\widehat{x}_{0}=\delta_{n_{1}}^{\alpha}\in\widehat {S}_{x}\setminus\{\delta_{n_{1}}^{0}\}\), \(\widehat{x}_{d}=\delta _{n_{1}}^{\beta}\in\widehat{S}_{x}\setminus\{\delta_{n_{1}}^{0}\}\)
and a given integer
\(k>0\). Then,
$$ P(k;\widehat{x}_{0},\widehat {x}_{d})= \bigl(\widehat{x}_{d}^{T}\bigr) (\overline{Q})^{k}( \widehat{x}_{0}), $$
(26)
where
\(\overline{Q}=\sum_{i=1}^{r_{1}}\widehat{Q}_{i}\), and
\(\widehat{Q}_{i}\)
is defined in (24).
Proof
We prove this lemma by induction.
Firstly, letting \(k=1\), assume that \(\widehat{u}^{1}=\delta _{r_{1}}^{\rho_{1}},\widehat{u}^{2}=\delta_{r_{1}}^{\rho _{2}},\ldots,\widehat{u}^{s}=\delta_{r_{1}}^{\rho_{s}}\in\widehat {S}_{u}\setminus\{\delta_{r_{1}}^{0}\}\) are different control sequences such that \(\widehat{x}_{d}\) is reachable from \(\widehat {x}_{0}\) at one step. Let \(\widehat{u}^{s+1}=\delta_{r_{1}}^{\rho _{s+1}},\widehat{u}^{s+2}=\delta_{r_{1}}^{\rho_{s+2}},\ldots ,\widehat{u}^{r_{1}}=\delta_{r_{1}}^{\rho_{r_{1}}}\in\widehat {S}_{u}\setminus\{\delta_{r_{1}}^{0}\}\) be different control sequences such that \(\widehat{x}_{0}\) cannot reach \(\widehat{x}_{d}\) in one step. Hence, it is easy to see that \((\widehat{Q}_{\rho_{l}})_{\beta ,\alpha}=1\), \(\forall l\in\{1,\ldots,s\}\), and \((\widehat{Q}_{\rho _{l}})_{\beta,\alpha}=0\), \(\forall l\in\{s+1,\ldots,r_{1}\}\), which implies that \((\sum_{i=1}^{r_{1}}\widehat{Q}_{i} )_{\beta ,\alpha}=s\). Since \((\widehat{x}_{d}^{T})(\overline{Q})(\widehat {x}_{0})= (\sum_{i=1}^{r_{1}}\widehat{Q}_{i} )_{\beta,\alpha }\), we have
$$P(1;\widehat{x}_{0},\widehat{x}_{d})=s=\bigl(\widehat {x}_{d}^{T}\bigr) (\overline{Q}) (\widehat{x}_{0}). $$
Thus, (26) holds for \(k=1\).
Suppose that (26) holds for an integer \(k\geq1\). Then, we consider the case of \(k+1\). It is easy to see that
$$\begin{aligned} \widehat{x}_{d}^{T}(\overline{Q})^{k+1}\widehat {x}_{0} =&\bigl(\delta_{n_{1}}^{\beta} \bigr)^{T}(\overline{Q})^{k+1}\delta _{n_{1}}^{\alpha}= \bigl(\overline{Q}^{k}\overline{Q}\bigr)_{(\beta,\alpha )} \\ =&\sum_{p=1}^{n_{1}}\bigl( \delta_{n_{1}}^{\beta}\bigr)^{T}\bigl(\overline {Q}^{k}\bigr) \bigl(\delta_{n_{1}}^{p}\bigr) \bigl( \delta_{n_{1}}^{p}\bigr)^{T}(\overline {Q}) \delta_{n_{1}}^{\alpha} \\ =&\sum_{p=1}^{n_{1}}P\bigl(k; \delta_{n_{1}}^{p},\widehat {x}_{d}\bigr)P\bigl(1; \widehat{x}_{0},\delta_{n_{1}}^{p}\bigr) \\ =&P(k+1;\widehat{x}_{0},\widehat{x}_{d}), \end{aligned}$$
(27)
which shows that (26) holds for \(k+1\).
By induction, (26) holds for any integer \(k>0\). This completes the proof. □
Based on Lemma 2, we give the following result on the reachability of system (25).
Theorem 1
For system (25), \(\widehat {x}_{d}=\delta_{n_{1}}^{\beta}\in\widehat{S}_{x}\setminus\{\delta _{n_{1}}^{0}\}\)
is reachable from
\(\widehat{x}_{0}=\delta_{n_{1}}^{\alpha}\in\widehat {S}_{x}\setminus\{\delta_{n_{1}}^{0}\}\)
at time
k
if and only if
$$\bigl(\overline{Q}^{k} \bigr)_{\beta,\alpha}>0. $$
Proof
By Lemma 2 we get \(P(k;\widehat{x}_{0},\widehat {x}_{d})=(\delta_{n_{1}}^{\beta})^{T}(\overline{Q})^{k}(\delta _{n_{1}}^{\alpha})= (\overline{Q}^{k} )_{\beta,\alpha}\). Therefore, \(P(k;\widehat{x}_{0},\widehat{x}_{d})\) denotes the number of different paths from \(\widehat{x}_{0}\) to \(\widehat{x}_{d}\) at time k. If \((\overline{Q}^{k})_{\beta,\alpha}>0\), then there are \((\overline{Q}^{k})_{\beta,\alpha}\) different paths from \(\widehat {x}_{0}\) to \(\widehat{x}_{d}\). Thus, \(\widehat{x}_{d}\) can be reached from \(\widehat{x}_{0}\) at time k. Conversely, if \(\widehat{x}_{d}\) is reachable from \(\widehat{x}_{0}\) at time k, then we can find at least one pinning control sequence \((\widehat{u}_{1}(t), \widehat {u}_{2}(t),\ldots,\widehat{u}_{r}(t))\) with \(\widehat{u}(t)=\ltimes _{i=1}^{r}\widehat{u}_{i}(t)\in\widehat{S}_{u}\setminus\{\delta _{r_{1}}^{0}\}\), \(t\in\{1,2,\ldots,k1\}\), such that \(\widehat{x}(k; \widehat{x}_{0}, (\widehat{u}_{1}(t), \widehat{u}_{2}(t),\ldots ,\widehat{u}_{r}(t)))=\widehat{x}_{d}\), which implies that \((\overline{Q}^{k} )_{\beta,\alpha}=P(k;\widehat{x}_{0},\widehat {x}_{d})>0\). □
Remark 3
From the proof of Theorem 1 we easily see that
$$P(k;\widehat{x}_{0},\widehat{x}_{d})=\bigl( \delta_{n_{1}}^{\beta }\bigr)^{T}(\overline{Q})^{k} \bigl(\delta_{n_{1}}^{\alpha}\bigr) = \bigl(\overline{Q}^{k} \bigr)_{\beta,\alpha}. $$
All of them denote the number of different paths such that \(\widehat {x}_{d}\) is reachable from \(\widehat{x}_{0}\) at time k.
3.3 Feedback stabilization pinning control design
In this part, we study the pinning control design for the feedback stabilization of constrained BCNs. By Proposition 5 we consider the feedback stabilization of system (25) based on the reachability analysis.
Firstly, we give the definition of stabilization for system (10) with \(S_{x}\) and \(S_{u}\).
Definition 3
System (10) with \(S_{x}\) and \(S_{u}\) is said to be stabilizable to a given equilibrium \(x_{e}\in S_{x}\) if there exists a pinning control sequence \(\{(u_{1}(t), u_{2}(t),\ldots,u_{r}(t)):t\in\mathbb{N}\}\subset S_{u}\) under which the trajectory initialized at any \(x_{0}\in S_{x}\) converges to \(x_{e}\) and \(x(t;x_{0},(u_{1}(t), u_{2}(t),\ldots, u_{r}(t)))\in S_{x}\), \(\forall t \in\mathbb{N}\).
In this paper, we study the following two kinds of feedback pinning controls:

1.
State feedback pinning control:
$$ u_{i}(t)=K_{i}x(t), $$
(28)
where \(K_{i}\in\mathcal{L}_{2\times2^{n}}\), \(i=1,\ldots,r\).

2.
Output feedback pinning control:
$$ u_{i}(t)=G_{i}y(t), $$
(29)
where \(G_{i}\in\mathcal{L}_{2\times2^{p}}\), \(i=1,\ldots,r\).
In the following, we consider the pinning control design for the state feedback stabilization of system (10) with \(S_{x}\) and \(S_{u}\) based on the constrained algebraic form.
For system (25), let \(\widehat{x}_{e}=\Psi_{x}x_{e}=\delta _{n_{1}}^{\alpha}\in\widehat{S}_{x}\setminus\{\delta_{n_{1}}^{0}\} \). For any integer \(k>0\), define
$$\begin{aligned} \Lambda_{k}(\widehat{x}_{e}) =& \bigl\{ \delta_{n_{1}}^{\beta}\in \widehat{S}_{x} \setminus\bigl\{ \delta_{n_{1}}^{0}\bigr\} : \mbox{there exists a control sequence} \\ &\widehat{u}(0), \widehat{u}(1),\ldots,\widehat{u}(k1)\in \widehat{S}_{u}\setminus\bigl\{ \delta_{r_{1}}^{0} \bigr\} \mbox{ such that} \\ &\widehat{x}\bigl(k;\delta_{n_{1}}^{\beta}, \widehat{u}(0), \ldots ,\widehat{u}(k1)\bigr)=\delta_{n_{1}}^{\alpha} \mbox{ and} \\ &\widehat{x}\bigl(l;\delta_{n_{1}}^{\beta},\widehat{u}(0), \ldots ,\widehat{u}(l1)\bigr)\in\widehat{S}_{x}\setminus\bigl\{ \delta_{n_{1}}^{0}\bigr\} , \forall l\in\{1,\ldots,k\} \bigr\} . \end{aligned}$$
(30)
Proposition 6
System (25) is stabilized to
\(\widehat{x}_{e}=\delta_{n_{1}}^{\alpha}\)
by a state feedback control if and only if there exists a positive integer
\(\sigma\leq n_{1}\)
such that

1.
\((\overline{Q})_{\alpha,\alpha}>0\),

2.
\(\operatorname{Row}_{\alpha}(\overline{Q}^{\sigma})>0\).
Proof
(Sufficiency) We can see from Condition 1 and Theorem 1 that \(\widehat{x}_{e}\in\Lambda_{1}(\widehat{x}_{e})\), which implies that \(\Lambda_{k}(\widehat{x}_{e})\neq\emptyset\), \(\forall k=1,\ldots,\sigma\).
Denote
$$ \Lambda_{k}^{\circ}(\widehat {x}_{e})= \Lambda_{k}(\widehat{x}_{e})\setminus\Lambda _{k1}(\widehat{x}_{e}), $$
(31)
where \(\Lambda_{0}(\widehat{x}_{e}):=\emptyset\). Then, we obtain \(\Lambda_{k_{1}}^{\circ}(\widehat{x}_{e})\cap\Lambda _{k_{2}}^{\circ}(\widehat{x}_{e})=\emptyset\), \(\forall k_{1},k_{2}\in\{1,\ldots,\sigma\}\), \(k_{1}\neq k_{2}\). Moreover, by Condition 2 and Theorem 1 we have \(\bigcup_{k=1}^{\sigma}\Lambda_{k}^{\circ}(\widehat{x}_{e})=\widehat {S}_{x}\setminus\{\delta_{n_{1}}^{0}\}\). Thus, for any integer i satisfying \(1\leq i\leq n_{1}\), we can find the unique integer \(1\leq k_{i}\leq\sigma\) such that \(\delta _{n_{1}}^{i}\in\Lambda_{k_{i}}^{\circ}(\widehat{x}_{e})\).
For system (25), set \(\widehat{Q}=\delta_{n_{1}}[\mu _{1},\mu_{2},\ldots,\mu_{n_{1}r_{1}}]\). When \(k_{i}=1\), we can find an integer \(1\leq\omega_{i}\leq r_{1}\) such that \(\widehat{Q}\ltimes\delta_{r_{1}}^{\omega_{i}}\ltimes \delta_{n_{1}}^{i}=\delta_{n_{1}}^{\mu_{(\omega _{i}1)n_{1}+i}}=\widehat{x}_{e}\). When \(2\leq k_{i}\leq\sigma\), we can find an integer \(1\leq\omega _{i}\leq r_{1}\) such that \(\widehat{Q}\ltimes\delta_{r_{1}}^{\omega _{i}}\ltimes\delta_{n_{1}}^{i}=\delta_{n_{1}}^{\mu_{(\omega _{i}1)n_{1}+i}}\in\Lambda_{k_{i}1}(\widehat{x}_{e})\).
Let
$$ \widehat{K}=\delta_{r_{1}}[\omega _{1}, \omega_{2},\ldots,\omega_{n_{1}}]\in\mathcal{L}_{r_{1}\times n_{1}}. $$
(32)
Then, for any initial state \(\widehat{x}_{0}=\delta_{n_{1}}^{i}\in \widehat{S}_{x}\setminus\{\delta_{n_{1}}^{0}\}\), if \(k_{i}=1\), then we obtain
$$\widehat{x}(1;\widehat{x}_{0},\widehat{u})=\widehat{Q}\widehat {u} \widehat{x}_{0}=\widehat{Q}\widehat{K}\widehat{x}_{0} \widehat {x}_{0}=\delta_{n_{1}}^{\mu_{(\omega_{i}1)n_{1}+i}}= \widehat{x}_{e}; $$
if \(2\leq k_{i}\leq\sigma\), we have
$$\widehat{x}(1;\widehat{x}_{0},\widehat{u})=\widehat{Q}\widehat {u} \widehat{x}_{0}=\widehat{Q}\widehat{K}\widehat{x}_{0} \widehat {x}_{0}=\delta_{n_{1}}^{\mu_{(\omega_{i}1)n_{1}+i}}\in\Lambda _{k_{i}1}(\widehat{x}_{e}). $$
Hence, \(\widehat{x}(k_{i};\widehat{x}_{0},\widehat{u})=\widehat {x}_{e}\), \(\forall1\leq i\leq n_{1}\), and \(\widehat{x}(t;\widehat {x}_{0},\widehat{u})\in\widehat{S}_{x}\setminus\{\delta _{n_{1}}^{0}\}\), \(\forall0\leq t\leq k_{i}1\). Since \(\widehat {x}_{e}\in\Lambda_{1}(\widehat{x}_{e})\), we obtain
$$\widehat{x}(t;\widehat{x}_{0},\widehat{u})=\widehat{x}_{e},\quad \forall t\geq\sigma,\forall\widehat{x}_{0}\in\widehat{S}_{x} \setminus\bigl\{ \delta_{n_{1}}^{0}\bigr\} , $$
which implies that system (25) is stabilized to \(\widehat {x}_{e}=\delta_{n_{1}}^{\alpha}\) by the state feedback control \(\widehat{u}(t)=\widehat{K}\widehat{x}(t)\).
(Necessity) The proof of this part is based on a straightforward calculation, and thus we omit it here. □
Based on Propositions 5 and 6, we have the following result.
Theorem 2
System (6) with
\(S_{x}\)
and
\(S_{u}\)
is stabilized to
\(x_{e}\)
by a state feedback control if and only if there exists a positive integer
\(\sigma\leq n_{1}\)
such that
\((\overline{Q})_{\alpha,\alpha}>0\)
and
\(\operatorname{Row}_{\alpha}(\overline {Q}^{\sigma})>0\).
From the proof of Proposition 6 we get the following procedure for the pinning control design of state feedback stabilization of constrained BCNs.
Remark 4
The procedure contains the following steps:

1.
Calculate \(\Lambda_{k}(\widehat{x}_{e})\) and \(\Lambda _{k}^{\circ}(\widehat{x}_{e})\), \(k=1,\ldots,\sigma\).

2.
For every integer \(1\leq i\leq n_{1}\), find the unique integer \(1\leq k_{i}\leq\sigma\) satisfying \(\delta_{n_{1}}^{i}\in\Lambda_{k_{i}}^{\circ}(\widehat{x}_{e})\).

3.
Find an integer \(1\leq\omega_{i}\leq r_{1}\) such that if \(k_{i}=1\), then \(\delta_{n_{1}}^{\mu_{(\omega _{i}1)n_{1}+i}}=\widehat{x}_{e}\); if \(k_{i}\geq2\), then \(\delta _{n_{1}}^{\mu_{(\omega_{i}1)n_{1}+i}}\in\Lambda_{k_{i}1}(\widehat {x}_{e})\).

4.
The state feedback pinning control can be designed as \(u_{i}(t)=K_{i}x(t)\), \(i=1,\ldots,r\), with \(K_{1}\ast K_{2}\ast\cdots\ast K_{r}=K\), where \(K=\delta_{2^{r}}[p_{1},\ldots,p_{2^{n}}]\), and
$$ \textstyle\begin{cases} p_{t}=j_{\omega_{\rho}} &\mbox{if }t=i_{\rho}, \rho\in\{1,\ldots,n_{1}\} , \\ p_{t}\in\{j_{1},\ldots,j_{r_{1}}\} &\mbox{otherwise}. \end{cases} $$
(33)
Finally, we discuss the pinning control design for the output feedback stabilization of constrained BCNs. To this end, we recall the definition of a nilpotent matrix.
Definition 4
A nilpotent matrix N is a square matrix such that \(N^{k}=0\) for some positive integer k. The smallest such k is called the degree of N.
For system (25) with an output feedback control \(\widehat {u}(t)=\widehat{G}\widehat{y}(t)\), \(\widehat{G}\in\mathcal {L}_{r_{1}\times2^{p}}\), we have
$$ \widehat{x}(t+1)=\widehat {Q}\widehat{u}(t)\widehat{x}(t)= \widehat{Q}\widehat{G}\widehat {y}(t)\widehat{x}(t)=\widehat{Q}\widehat{G}\widehat {H}M_{r,n_{1}}\widehat{x}(t), $$
(34)
where \(M_{r,n_{1}}=\operatorname{diag}\{\delta_{n_{1}}^{1},\delta_{n_{1}}^{2},\ldots ,\delta_{n_{1}}^{n_{1}}\}\). Then, we have the following result on the output feedback stabilization of system (25).
Theorem 3
System (25) is stabilizable to
\(\widehat{x}_{e}=\delta_{n_{1}}^{\alpha}\)
by an output feedback control if and only if there exist a logical matrix
\(\widehat{G}\in \mathcal{L}_{r_{1}\times2^{p}}\)
and an integer
\(1\leq\tau\leq n_{1}\)
such that
$$ \widehat{Q}\widehat{G}\widehat{H}M_{r,n_{1}}=\left [ \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} A_{1}& \zeta_{3} &A_{2}\\ \zeta_{1}& 1& \zeta_{2}\\ A_{3} &\zeta_{4}& A_{4} \end{array}\displaystyle \right ] $$
(35)
and
$$ \left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} A_{1} &A_{2}\\ A_{3} &A_{4} \end{array}\displaystyle \right ] $$
(36)
is a nilpotent matrix of degree
τ, where
\(A_{1}\in\mathcal {B}_{(\alpha1)\times(\alpha1)}\), \(A_{2}\in\mathcal{B}_{(\alpha 1)\times(n_{1}\alpha)}\), \(A_{3}\in \mathcal{B}_{(n_{1}\alpha )\times(\alpha1)}\), \(A_{4}\in\mathcal{B}_{(n_{1}\alpha)\times (n_{1}\alpha)}\), \(\zeta_{1}\)
and
\(\zeta_{2}\)
are some proper Boolean row vectors, and
\(\zeta_{3}\)
and
\(\zeta_{4}\)
are zero column vectors.
Proof
(Sufficiency) Since
$$ \left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} A_{1} &A_{2}\\ A_{3} &A_{4} \end{array}\displaystyle \right ] $$
(37)
is a nilpotent matrix of degree τ, we see that
$$ \left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} A_{1}& A_{2}\\ A_{3}& A_{4} \end{array}\displaystyle \right ]^{t}=0 $$
(38)
for any integer \(t\geq\tau\), which, together with a simple calculation, shows that
$$ (\widehat{Q}\widehat{G}\widehat {H}M_{r,n_{1}})^{t}=\left [ \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 0& 0& \cdots&0\\ 1& 1& \cdots&1\\ 0 &0 &\cdots&0 \end{array}\displaystyle \right ]=\delta_{n_{1}}[\alpha,\alpha, \ldots,\alpha] $$
(39)
for any integer \(t\geq\tau\).
Hence, we obtain from (34) that
$$ \widehat{x}(t)=(\widehat{Q}\widehat {G}\widehat{H}M_{r,n_{1}})^{t} \widehat{x}(0)=\widehat{x}_{e} $$
(40)
for any \(\widehat{x}(0)\in\widehat{S}_{x}\setminus\{\delta _{n_{1}}^{0}\}\) and any integer \(t\geq\tau\).
Therefore, system (25) is stabilizable to \(\widehat {x}_{e}=\delta_{n_{1}}^{\alpha}\) by the output feedback control \(\widehat{u}(t)=\widehat{G}\widehat{y}(t)\).
(Necessity) Suppose that system (25) is stabilizable to \(\widehat{x}_{e}=\delta_{n_{1}}^{\alpha}\) by an output feedback control, say, \(\widehat{u}(t)=\widehat{G}\widehat{y}(t)\). Then, we can find the smallest integer \(1\leq\tau\leq n_{1}\) such that (40) holds for any \(\widehat{x}(0)\in\widehat{S}_{x}\setminus\{ \delta_{n_{1}}^{0}\}\) and any integer \(t\geq\tau\). Hence, \((\widehat {Q}\widehat{G}\widehat{H}M_{r,n_{1}})^{\tau}=\delta_{n_{1}}[\alpha ,\alpha,\ldots,\alpha]\).
Split \(\widehat{Q}\widehat{G}\widehat{H}M_{r,n_{1}}\in\mathcal {L}_{n_{1}\times n_{1}}\) into the following blocks:
$$\widehat{Q}\widehat{G}\widehat{H}M_{r,n_{1}}=\left [ \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} A_{1}& \zeta_{3} &A_{2}\\ \zeta_{1}& \lambda&\zeta_{2}\\ A_{3}& \zeta_{4}& A_{4} \end{array}\displaystyle \right ], $$
where \(\lambda\in\{0,1\}\), \(A_{1}\in\mathcal{B}_{(\alpha1)\times (\alpha1)}\), \(A_{2}\in\mathcal{B}_{(\alpha1)\times(n_{1}\alpha )}\), \(A_{3}\in\mathcal{B}_{(n_{1}\alpha)\times(\alpha1)}\), \(A_{4}\in\mathcal{B}_{(n_{1}\alpha)\times(n_{1}\alpha)}\), \(\zeta_{1}\) and \(\zeta_{2}\) are some proper Boolean row vectors, and \(\zeta_{3}\) and \(\zeta_{4}\) are some proper Boolean column vectors.
It is easy to see from \(\widehat{Q}\widehat{G}\widehat {H}M_{r,n_{1}}\widehat{x}_{e}=\widehat{x}_{e}\) that \(\lambda=1\), and \(\zeta_{3}\) and \(\zeta_{4}\) are zero column vectors.
In the following, we prove that
$$\widehat{A}:=\left [ \textstyle\begin{array}{@{}c@{\quad}c@{}} A_{1}& A_{2}\\ A_{3}& A_{4} \end{array}\displaystyle \right ] $$
is a nilpotent matrix of degree τ.
We can easily see from \((\widehat{Q}\widehat{G}\widehat {H}M_{r,n_{1}})^{\tau}=\delta_{n_{1}}[\alpha,\alpha,\ldots,\alpha ]\) that \(\widehat{A}^{\tau}=0\), which shows that Â is a nilpotent matrix. If its degree is less than τ, then there exists a positive integer \(\tau'<\tau\) such that \(\widehat{A}^{\tau '}=0\), and thus
$$(\widehat{Q}\widehat{G}\widehat{H}M_{r,n_{1}})^{\tau'}=\delta _{n_{1}}[\alpha,\alpha,\ldots,\alpha], $$
which is a contradiction to the minimality of τ. This completes the proof. □
Remark 5
Based on Theorem 3 and Algorithm 1 presented in [32], we can design an output feedback gain matrix, say, \(\widehat{G}=\delta _{r_{1}}[w_{1},w_{2},\ldots,w_{2^{p}}]\), under which system (25) is stabilizable to \(\widehat{x}_{e}=\delta _{n_{1}}^{\alpha}\). Then, the output feedback pinning control can be designed as \(u_{i}(t)=G_{i}y(t)\), \(i=1,\ldots,r\), with \(G_{1}\ast G_{2}\ast \cdots\ast G_{r}=G\), where \(G=\delta_{2^{r}}[j_{w_{1}},\ldots,j_{w_{2^{p}}}]\).
Remark 6
It should be pointed out that the pinning control design for the state feedback stabilization of Boolean networks was studied in [34], and a novel design procedure was established. Compared with [34], our main results have the following advantages: (i) we established some necessary and sufficient conditions for the pinning control design of both state feedback and output feedback stabilization problems, whereas [34] only considered the state feedback stabilization problem; (ii) our results are applicable to the pinning control design for the state feedback stabilization of constrained BCNs.