Delayed State-Flipped Control for Stabilization of Boolean Networks with Time Delays

Abstract

This paper addresses the stabilization problem in Boolean networks with time delays (TBNs) under state-flipped control. First, the algebraic form of TBNs under state-flipped control is derived utilizing the semi-tensor product (STP) of matrices. Based on their periodic properties, the TBNs are then partitioned into subsystems. Subsequently, leveraging these developments, the equivalence between the stabilization of the original TBNs and that of the derived subsystems under state-flipped control is established. Furthermore, an equivalent stabilization criterion for the TBNs under state-flipped control is provided. An algorithm is also presented for constructing a flipped-set sequence to achieve stabilization. Finally, illustrative examples are provided to demonstrate the effectiveness of the theoretical results.

1. Introduction

Systems biology is the study of complex interactions within biological systems and their regulatory mechanisms. One key modeling framework in systems biology is gene regulatory networks (GRNs). GRNs focus on interactions among genes and their response to the external environment to control gene expression, providing an important tool for systems biology research.

Boolean networks (BNs), proposed by Kauffman in 1969 to study GRNs, are simplified mathematical models [1]. In a BN, nodes represent the expression levels of genes and proteins. All nodes are updated by only two values: $0,1$, where 1 denotes ON (active) and 0 denotes OFF (inactive). When external inputs are considered, the BNs become Boolean control networks (BCNs), which are used to model dynamic systems with control mechanisms [2]. Today, BNs and BCNs are effectively applied in diverse fields such as pharmacology, neurology, and systems science [3].

In recent years, a novel matrix product called the semi-tensor product (STP) of matrices has been proposed [4] for the representation and analysis of BNs and Boolean control networks (BCNs). This methodology has enabled breakthroughs in observability, synchronization, Ledley solution methods, and output state feedback stabilizers of BNs/BCNs [5,6,7,8,9,10,11,12,13].In addition, a reasonable and effective control methods are continuously being proposed. Ref. [14] investigates asymptotic synchronization in coupled Boolean and probabilistic Boolean networks with delays. Network synchronization under distributed delayed impulsive control is considered by [15]. Robust stability in distribution of Boolean networks under multi-bit stochastic function perturbations is discussed by [16]. Ref. [17] studies logical dynamic games. Ref. [18] considers self-triggered control for approximate synchronization of singular logical networks. Impulsive logical dynamic networks and stochastic state-heritable control are investigated by [19,20].

Time delays are a very common phenomenon, occurring in processes such as drug therapy, chemical reactions, long transmission lines in pneumatic systems, and evolutionary games. BNs and BCNs can effectively model systems whose state updates depend solely on the current state. However, when the system state update depends on the past $\tau$$(\tau \in {\mathbb{Z}}^{+})$ states (i.e., when time delays occur), TBNs are considered to be the appropriate mathematical model [4]. To date, numerous studies have been conducted on TBNs, including ref. [21], investigating pinning control for stabilizing delayed BNs; ref. [22], discussing the stabilization and controllability of hybrid switching and impulsive higher-order BNs with delays; and [23,24,25], analyzing the controllability and observability of delayed BCNs.

In real cellular networks, genes can be activated or inhibited by external stimuli such as mutagens and heat stress [26]. Specifically, ref. [26] investigated the impact of random single-gene perturbation on network evolutionary behavior. This phenomenon can be modeled in BNs by flipping node values (from 1 to 0 or 0 to 1). This type of control is called state-flipped control. The concept of state-flipped control was introduced in Rafimanzelat and Bahrami [27,28]. Similarly, state-flipped control has been introduced into BNs as a minimally invasive control strategy [27]. Due to its operational simplicity, refs. [27,28] explored the attractor controllability and stability achieved by flipping specific node states only once. Readers can see more details in [27,28]. More generally, weak stabilization of BNs under state-flipped control, involving state-flips at every step, was studied in [29]. Furthermore, researchers have extended this approach to address fundamental problems in BCNs.

The preceding discussion underscores the ubiquity of time-delay phenomena and the operational simplicity inherent in state-flipped control. Prompted by these observations, we naturally consider introducing state-flipped control to the theory of TBNs. Notably, seminal contributions by [23,24] established the foundations for controllability and observability analysis in BNs with time-invariant integer state delays. Building upon this work, this paper investigates the trajectory dynamics and stabilization of the TBNs in [23,24] under state-flipped control.

This paper establishes foundational results for the TBNs in [23,24] under state-flipped control through three key contributions: First, we will derive the algebraic form of TBNs under state-flipped control using the STP approach, then partition them into subsystems using periodic properties. Secondly, we will establish equivalence between the stabilization of the original system and that of the derived subsystems under state-flipped control. Thirdly, a stabilization criterion and a corresponding algorithm for finding flipped-set sequences will be developed. Numerical examples will be provided to validate the results.

We organize this paper as follows: Section 2 presents some preliminaries. Section 3 gives the main results of the paper. Section 4 shows the effectiveness of our main results via two examples. Section 5 presents a brief conclusion.

2. Preliminaries

In this section, we first introduce the definition and fundamental properties of STP. Next, we present the concept of state-flipped control along with its matrix representation. Finally, we derive the algebraic representations of TBNs under state-flipped control.

2.1. Notation

• $Z^{+}$ denotes the set of positive integers;
• $\mathcal{N}:=\{1,2,\cdots ,n\}$ denotes the set of nodes of a BN;
• $\mathcal{D}$ denotes the set $\{0,1\}$ and ${\mathcal{D}}^n:=\underset{\underset{n}{︸}}{\mathcal{D}\times \cdots \times \mathcal{D}}$.
• ${\Delta }_n:=\left\{{\delta }_n^i|i=1,2,\cdots ,n\right\}$, where ${\delta }_n^i$ is the i-th column of the n-dimensional identity matrix $I_n$;
• ${\delta }_n^{1,2,\cdots ,k}$ is a Boolean vector, which is equal to ${\sum }_{i=1}^k{\delta }_n^i$;
• An $n\times t$ matrix M is called a logical matrix if $M=[{\delta }_n^{i_1}{\delta }_n^{i_2}\cdots {\delta }_n^{i_t}]$, and briefly, M is denoted by $M={\delta }_n[i_1{i}_2\cdots i_t]$;
• ${\mathbb{R}}_{m\times n}$, ${\mathcal{L}}_{m\times n}$, and ${\mathcal{B}}_{m\times n}$ denote the set of $m\times n$ real matrices, logical matrices, and Boolean matrices, respectively. ${\mathbb{R}}^m$ denotes the set of m-dimensional column vectors;
• $Co{l}_i\left(L\right)\left(Ro{w}_i\left(L\right)\right)$ is the i-th column (row) of matrix L;
• ${\left(L\right)}_{ij}$ is the $(i,j)$-th entry in the matrix L;
• ${\left[L\right]}_{\mathcal{B}}$ represents converting the matrix L into a Boolean matrix, which implies that ${\left({\left[L\right]}_{\mathcal{B}}\right)}_{ij}=1$ iff ${\left(L\right)}_{ij}\ge 1$, and otherwise, ${\left({\left[L\right]}_{\mathcal{B}}\right)}_{ij}=0$;
• $E\left({\delta }_n^{1,2,\cdots ,k}\right):=\{{\delta }_n^1,{\delta }_n^2,\cdots ,{\delta }_n^k\}$ denotes the function for extracting the positions of non-zero elements in the Boolean vector;
• $2^A$ and $\left|A\right|$ represent the power set of the set A and its cardinality, respectively.

2.2. STP and Its Properties

Definition 1

([4]). Given two matrices, $A\in {\mathbb{R}}_{m\times n}$ and $B\in {\mathbb{R}}_{p\times q}$, the STP of A and B, denoted by $A⋉B$, is given

$$A⋉B=(A\otimes I_{{\displaystyle \frac{\alpha }{n}}})(B\otimes I_{{\displaystyle \frac{\alpha }{p}}}),$$

where $\alpha =lcm(n,p)$ is the least-common multiple of n and p, and ⊗ is the Kronecker product. For convenience, the symbol “⋉” can be omitted without confusion in this paper.

Lemma 1

([4]). 

(1) 

Assume that we have two column vectors, $X\in {\mathbb{R}}^m$ and $Y\in {\mathbb{R}}^n$. Then, $W_{[m,n]}XY=YX$, where $W_{[m,n]}:=[I_n\otimes {\delta }_m^1,I_n\otimes {\delta }_m^2,\cdots ,I_n\otimes {\delta }_m^m]$ is called the swap matrix. In particular, $W_{[m,m]}=W_{\left[m\right]}$.

(2) 

[Pseudo-commutative property] Assume that we have $X\in {\mathbb{R}}^m$, $A\in {\mathbb{R}}_{p\times q}$. Then, $XA=(I_m\otimes A)X$.

Lemma 2

([4]). For any logical function $f(x_1,x_2,\cdots ,x_n):{\Delta }_{2^n}\to {\Delta }_2$, $x_i\in {\Delta }_2$, there exists a unique matrix $L_f\in {\mathcal{L}}_{2\times 2^n}$, called the structural matrix of logical function f, such that

$$f(x_1,x_2,\cdots ,x_n)=L_f·{⋉}_{i=1}^n{x}_i,$$

where ${⋉}_{i=1}^n{x}_i=x_1⋉x_2⋉\cdots ⋉x_n$.

2.3. Algebraic Forms of TBNs

Due to the universality of time-delay phenomena, BNs with n nodes are modeled with constant time delays $\tau \ge 0$ in their states [23]:

$$\left\{\begin{array}{c}{X}_1(t+1)=f_1(X_1(t-\tau ),X_2(t-\tau ),\cdots ,X_n(t-\tau )),\\ X_2(t+1)=f_2(X_1(t-\tau ),X_2(t-\tau ),\cdots ,X_n(t-\tau )),\\ ⋮\\ X_n(t+1)=f_n(X_1(t-\tau ),X_2(t-\tau ),\cdots ,X_n(t-\tau )),\end{array}\right.$$

(1)

where $f_i:{\mathcal{D}}^n\to \mathcal{D}$ are Boolean logical functions, $X_i\left(t\right)\in \mathcal{D}$ is the state of node i at time t, $i=1,2,\cdots ,n$, $t=-\tau ,-\tau +1,\cdots$, and $X\left(t\right)=(X_1\left(t\right),X_2\left(t\right),\cdots ,X_n\left(t\right))\in {\mathcal{D}}^n$ is called the state of system (1) at time t. In particular, the initial state of a TBN (1) is defined as $X_0:=\{X\left(0\right),$$X(-1),\cdots ,X(-\tau )\}$.

Let ${\delta }_2^1$ and ${\delta }_2^2$ represent the equivalent logical vectors of the states ON (1) and OFF (0), respectively, i.e., ${\delta }_2^1\sim 1$ and ${\delta }_2^2\sim 0$. The equivalent vector form of the state $X\left(t\right)\in {\mathcal{D}}^n$ is denoted by $x\left(t\right)={⋉}_{i=1}^n{x}_i\left(t\right)\in {\Delta }_{2^n}$, and $X_i\left(t\right)$ can be represented by $x_i\left(t\right)$ for $i=1,2,\cdots ,n$. Applying Lemma 2, system (1) can be converted into the following matrix form:

$$\left\{\begin{array}{c}{x}_1(t+1)=L_{1}x(t-\tau ),\\ x_2(t+1)=L_{2}x(t-\tau ),\\ ⋮\\ x_n(t+1)=L_{n}x(t-\tau ),\end{array}\right.$$

(2)

where $L_i\in {\mathcal{L}}_{2\times 2^n}$ are the structural matrices of Boolean functions $f_i$ in system (1), and $i=1,2,\cdots ,n$.

Multiplying all equations in (2), we obtain

$$x(t+1)=Lx(t-\tau ),$$

(3)

where $L=L_1\ast L_2\ast \cdots \ast L_n\in {\mathcal{L}}_{2^n\times 2^n}$ is called the state transition matrix of the TBN (1), and ∗ is Khatri–Rao product of matrices. In addition, the initial state of system (3) can be denoted by $z_0=\{x\left(0\right),x(-1),\cdots ,x(-\tau )\}$.

2.4. State-Flipped Control

Here, the logical dynamics of a BN with n nodes are represented by

$$\begin{array}{c}\hfill X_i(t+1)=f_i\left(X\left(t\right)\right),i=1,2,\dots ,n,\end{array}$$

(4)

where $X_i\left(t\right)\in \mathcal{D}=\{0,1\}$ is the state of the i-th node in (4) at time t, $X\left(t\right):=(X_1\left(t\right),X_2\left(t\right),\cdots ,X_n\left(t\right))\in {\mathcal{D}}^n$ is the state of system (4) at time t, $f_i:{\mathcal{D}}^n\to \mathcal{D}$ are Boolean functions, and the initial state $X_0$ is $X\left(0\right)$.

Applying the STP, [4] converts system (4) into the following form

$$\begin{array}{c}\hfill x_i(t+1)=L_i{⋉}_{i=1}^n{x}_i\left(t\right),\end{array}$$

and

$$\begin{array}{c}\hfill x(t+1)=Lx\left(t\right),\end{array}$$

(5)

where $x_i\left(t\right)\in {\Delta }_2$ is the state of node i at time t, $x\left(t\right)={⋉}_{i=1}^n{x}_i\left(t\right)\in {\Delta }_{2^n}$ is the state of system (5) at time t, $L_i\in {\mathcal{L}}_{2\times 2^n}$ are the structural matrices of Boolean functions $f_i$ in (4), $L=L_1\ast L_2\ast \cdots \ast L_n\in {\mathcal{L}}_{2^n\times 2^n}$ is the structural matrix of BN (5), and $i=1,2,\cdots ,n$.

In the following, we begin to introduce the concept of state-flipped control for BNs (4) along with its matrix representation.

Definition 2

([27]). Let set S be $\{i_1,i_2,\cdots ,i_s\}\subseteq \mathcal{N}$. The state-flipped function with respect to S, denoted by $g_S^{\updownarrow }$, is defined as

$$g_S^{\updownarrow }\left(X\right)={\overline{X}}_S=(X_1,\cdots ,{\overline{X}}_{i_1},\cdots ,{\overline{X}}_{i_s},\cdots ,X_n),$$

where ${\overline{X}}_{i_j}$, $i_j\in S$, which implies that $X_{i_j}=1$ (or $0)$ is flipped to 0 (or $1)$, ${\overline{X}}_S$ can be achieved by flipping the $i_1$-th, $i_2$-th, ⋯, and $i_s$-th components of X, which is denoted by $X\stackrel{g_S^{\updownarrow }}{\to }{\overline{X}}_S$. S is called a flipped set of state X.

Due to the equivalence between logical variables and their vector forms, we present a matrix representation of the state-flipped function.

Definition 3

([27]). Let $S:=\{i_1,i_2,\cdots ,i_s\}\subseteq \mathcal{N}$. The matrix $G_s\in {\mathcal{L}}_{2^n\times 2^n}$, called a flipped matrix with respect to S, is given by

$$Co{l}_j\left(G_s\right)={\delta }_{2^n}^i,ifX\sim {\delta }_{2^n}^j\stackrel{g_s^{\updownarrow }}{\to }{\overline{X}}_S\sim {\delta }_{2^n}^i,$$

where $j=1,2,\cdots ,2^n$.

Thus, according to Definitions 2 and 3, the algebraic equation of state-flipped function $g_s^{\updownarrow }\left(X\right)$ is given by ${\overline{x}}_s=G_{s}x=G_s{⋉}_{i=1}^n{x}_i$. Furthermore, ${\left(G_s\right)}_{ij}=1$ implies that state ${\overline{x}}_s={\delta }_{2^n}^i$ is reachable from $x={\delta }_{2^n}^j$ by flipping the $i_1$-th, $i_2$-th, …, and $i_s$-th components of system state X. Here we call the transition $x\to {\overline{x}}_s$ a state-flipped transition of system (5). Moreover, for any two flipped sets $S_1,S_2\subseteq \mathcal{N}$ (i.e., $S_1\ne S_2$), the corresponding columns satisfy $Co{l}_j\left(G_{S_1}\right)\ne Co{l}_j\left(G_{S_2}\right)$, for all $j=1,2,\dots ,2^n$.

Next, we introduce the concept of a combinatorial flipped matrix. This matrix encompasses all possible state-flipped operations corresponding to every subset of a given maximum flipped set $A\subseteq \mathcal{N}$.

Definition 4

([27]). Let $A=\{i_1,i_2,\cdots ,i_A\}\subseteq \mathcal{N}$. The combinatorial flipped matrix of A, denoted by ${\mathcal{G}}_A\in {\mathcal{B}}_{2^n\times 2^n}$, is defined as

$${\left({\mathcal{G}}_A\right)}_{ij}=\left\{\begin{array}{cc}1,\hfill & if\exists S\in 2^A,s.t.X\sim {\delta }_{2^n}^j\stackrel{g_S^{\updownarrow }}{\to }{\overline{X}}_S\sim {\delta }_{2^n}^i,\hfill \\ 0,\hfill & otherwise.\hfill \end{array}\right.$$

(6)

From (6), ${\left({\mathcal{G}}_A\right)}_{ij}=1$ if and only if there exists a subset $S\subseteq A$ such that ${\left(G_S\right)}_{ij}=1$. Consequently, ${\mathcal{G}}_A$ is a Boolean matrix and can be expressed as ${\mathcal{G}}_A={\sum }_{S\in 2^A}{G}_S$. Moreover, the j-th column of ${\mathcal{G}}_A$ contains all states reachable from $x={\delta }_{2^n}^j$ by applying state-flipped operations corresponding to every subset of A.

3. Main Result

First, the theory of state-flipped control is applied to TBN (3), and the periodicity of its dynamic transitions under this control is analyzed. Second, we investigate the reachability of this system with the combinatorial flipped set A. Finally, we present conclusions regarding the stabilization of TBN (3) under state-flipped control.

3.1. Algebraic Representation of TBNs Under State-Flipped Control

First, we introduce the dynamic equations and the algebraic representations of the TBNs under state-flipped control. Specifically, state-flipped control defined by a given flipped set is applied to system (1) at time t, with its effect taking place at $t+\tau$, where $t=1,2,\cdots$. Additionally, state-flipped control cannot be applied to system (1) at times $t=-1,\cdots ,-\tau$.

Then, based on system (3) and Definition 2, the dynamics of the TBN under state-flipped control are described by

$$X_i(t+1)=f_i\left(g_S^{\updownarrow }\left(X_1(t-\tau ),X_2(t-\tau ),\dots ,X_n(t-\tau )\right)\right),$$

(7)

where S is the flipped set, $t=1+\tau ,2+\tau ,\dots$, and $i=1,2,\dots ,n$.

Subsequently, applying Lemma 2 and Definition 3, we obtain the algebraic representation of the given TBN (7) with flipped set S as follows:

$$x(t+1)=L{G}_{s}x(t-\tau ).$$

Here, matrices L and $G_S$ represent the state transition matrix of system (3) and state-flipped matrix of set S, respectively.

Define ${\overline{L}}_S:=L{G}_S\in {\mathcal{L}}_{2^n\times 2^n}$. This matrix ${\overline{L}}_S$ is termed the flipped-transition matrix (FTM) of system (7).

Similarly, we can define the combinatorial flipped-transition matrix (CFTM) ${\tilde{L}}_A:=L{\sum }_{S\in 2^A}{G}_S\in {\mathbb{R}}_{2^n\times 2^n}$. Using ${\tilde{L}}_A$, we obtain the algebraic representation of a TBN under combinatorial state-flipped control as follows:

$$b(t+1)={\left[{\tilde{L}}_{A}b(t-\tau )\right]}_{\mathcal{B}},$$

(8)

where $b\left(t\right)$ is a Boolean vector, and $b\left(t\right)$ records all reachable states at time t. Specifically, for $t\in [-\tau ,0]\cap \mathbb{Z}$, $b\left(t\right)=x\left(t\right)$, where $[-\tau ,0]$ is the continuous closed interval in ${\mathbb{R}}^1$.

Based on the analysis presented above, we apply the theory of state-flipped control to TBNs. It yields the algebraic representation (8) for a TBN with combinatorial flipped set A. Equation (8) provides the foundation for investigating the trajectories and stabilization of system (7).

3.2. Trajectory Analysis of TBNs

First, we examine the periodicity of TBN (3).

In TBN (3), the initial state is denoted by $z_0=\{x\left(0\right),x(-1),\cdots ,x(-\tau )\}$. Furthermore, the state of system (3) at time t depends exclusively on its state at time $t-\tau$. Consequently, the state at $t-\tau$ uniquely influences the state at time t.

For any positive integer a, we construct $\tau +1$ distinct transition paths:

• (1): $x\left(0\right)\to x(\tau +1)\to x\left(2\right(\tau +1\left)\right)\to \cdots \to x\left(a\right(\tau +1\left)\right)\to \cdots$
• (2): $x(-1)\to x\left(\tau \right)\to x\left(2\right(\tau +1)-1)\to \cdots \to x\left(a\right(\tau +1)-1)\to \cdots$
•   ⋮
• ($\tau$): $x(-\tau +1)\to x\left(2\right)\to x\left(2\right(\tau +1)-\tau +1)\to \cdots \to x\left(a\right(\tau +1)-\tau +1)\to \cdots$
• ($\tau +1$): $x(-\tau )\to x\left(1\right)\to x\left(2\right(\tau +1)-\tau )\to \cdots \to x\left(a\right(\tau +1)-\tau )\to \cdots$

Based on these paths, states separated by $\tau +1$ time intervals evolve from the same initial state within each path. Therefore, $\tau +1$ is defined as the period of system (3). Furthermore, it is evident that these $\tau +1$ paths collectively encompass all states of system (3) at any given time.

Given the initial state $z_0=\{x\left(0\right),x(-1),\cdots ,x(-\tau )\}$ of TBN (3), it is evident that for any time $t\in {\mathbb{Z}}^{+}$, there exist unique ${\tau }_i\in \{0,1,2,\cdots ,\tau \}$ and $k\in {\mathbb{Z}}^{+}$ such that $t=-{\tau }_i+k(\tau +1)$.

Thus, system (3) is partitioned into $\tau +1$ subsystems. The algebraic representation of the ${\tau }_i$-th subsystem is

$$x(-{\tau }_i+k(\tau +1))=Lx(-{\tau }_i+(k-1)(\tau +1))=\cdots =L^{k}x(-{\tau }_i),$$

(9)

where ${\tau }_i\in \{0,1,2,\cdots ,\tau \}$, $k\in {\mathbb{Z}}^{+}$, L is defined in TBN (3), and $L^k$ is the k-th power of L.

Next, we introduce the state-flipped control with a combinatorial flipped set $A\subseteq \mathcal{N}$ to subsystem (9). Thus, ${\tilde{L}}_A=L{\mathcal{G}}_A$ is the CFTM of the ${\tau }_i$-th subsystem with combinatorial flipped set A.

Therefore, the ${\tau }_i$-th subsystem (9) with combinatorial flipped set A can be expressed as

$$b(-{\tau }_i+k(\tau +1))={\left[{\tilde{L}}_{A}b(-{\tau }_i+(k-1)(\tau +1))\right]}_B=\cdots ={\left[{\left({\tilde{L}}_A\right)}^{k}b(-{\tau }_i)\right]}_B,$$

(10)

where $b\left(t\right)$ is the Boolean vector, ${\tilde{L}}_A=L{\mathcal{G}}_A$, and $b\left(t\right)=x\left(t\right)$, when $-\tau \le t\le 0$.

Herein, note that matrix L is given a new name as the periodic transition matrix of subsystem (9), and $L^k$ is called the k-periodic transition matrix. Similarly, ${\tilde{L}}_A$ and ${\left({\tilde{L}}_A\right)}^k$ are called the periodic flipped-transition matrix and k-periodic flipped-transition matrix.

3.3. Reachability of the TBNs Under State-Flipped Control

In this subsection, we investigate the reachability of each subsystem of TBN (3) via the reachability of system (10).

First, we present the definition of reachable points for TBN (3).

Definition 5

(k-step reachable point (set)). Given a combinatorial flipped set A and an initial state $z_0=\{x\left(0\right),x(-1),\cdots ,x(-\tau )\}$ of TBN (3), for any ${\tau }_i\in \{0,1,2,\cdots ,\tau \}$ and $k\in {\mathbb{Z}}^{+}$,

1. 

$x(k;x(-{\tau }_i))$ denotes the state of subsystem (9) reached from the initial point $x(-{\tau }_i)$ after k periodic transitions.

2. 

$\ddot{x}(k;x(-{\tau }_i))$ denotes the set of states of subsystem (10) reached from the initial point $x(-{\tau }_i)$ after k periodic flipped-transition steps.

The following result presents methods for analyzing the sets of k-step reachable points for systems (9) and (10).

Theorem 1.

Let $x(-{\tau }_i)={\delta }_{2^n}^i$ for any ${\tau }_i\in \{0,1,2,\cdots ,\tau \}$ and $k\in {\mathbb{Z}}^{+}$.

(1) 

The k-step reachable point $x(k;x(-{\tau }_i))$ of subsystem (9) is ${\delta }_{2^n}^j$ if and only if ${\left(L^k\right)}_{j,i}>0$.

(2) 

The set of k-step reachable points of subsystem (10) is given by

$$\ddot{x}(k;x(-{\tau }_i))=E\left(b(-{\tau }_i+k(\tau +1))\right)=E\left({\left[{\mathrm{Col}}_i{\left({\tilde{L}}_A\right)}^k\right]}_{\mathcal{B}}\right),$$

where $b(-{\tau }_i)=x(-{\tau }_i)={\delta }_{2^n}^i$.

Proof of Theorem 1.

(1) Since $x(-{\tau }_i+k(\tau +1))=L^{k}x(-{\tau }_i)$ and $x(-{\tau }_i)={\delta }_{2^n}^i$ hold, it follows that $x(-{\tau }_i+k(\tau +1))=x(k;x(-{\tau }_i))={\delta }_{2^n}^j$ iff $Co{l}_i\left(L^k\right)={\delta }_{2^n}^j$, i.e.,  ${\left(L^k\right)}_{ji}>0$.

(2) 

The first equality is proven by mathematical induction. For the base case $k=1$,

$$\begin{array}{cc}\hfill \ddot{x}(1;x(-{\tau }_i))& =E\left({\left[{\tilde{L}}_{A}x(-{\tau }_i)\right]}_{\mathcal{B}}\right)\hfill \\ \hfill & =E\left({\left[{\tilde{L}}_{A}b(-{\tau }_i)\right]}_{\mathcal{B}}\right)\hfill \\ \hfill & =E\left(b(-{\tau }_i+\tau +1)\right).\hfill \end{array}$$

Suppose that for $k=m$, $\ddot{x}(m;x(-{\tau }_i))=E\left(b(-{\tau }_i+m(\tau +1))\right)$. Let

$$\ddot{x}(m;x(-{\tau }_i)):=\{{\delta }_{2^n}^{j_1},{\delta }_{2^n}^{j_2},\cdots {\delta }_{2^n}^{j_p}\},$$

where $1\le p\le 2^n$. Then, for $k=m+1$,

$$\begin{array}{cc}\hfill \ddot{x}(m+1;x(-{\tau }_i))& =\bigcup _{{\delta }_{2^n}^{j_q}\in \ddot{x}(m;x(-{\tau }_i))}\ddot{x}\left(1;{\delta }_{2^n}^{j_q}\right)\hfill \\ \hfill & =\bigcup _{{\delta }_{2^n}^{j_q}\in \ddot{x}(m;x(-{\tau }_i))}E\left({\left[{\tilde{L}}_A{\delta }_{2^n}^{j_q}\right]}_{\mathcal{B}}\right)\hfill \\ \hfill & =E({[{\tilde{L}}_A\sum _{{\delta }_{2^n}^{j_q}\in \ddot{x}(m;x(-{\tau }_i))}{\delta }_{2^n}^{j_q}]}_{\mathcal{B}})\hfill \\ \hfill & =E\left({\left[{\tilde{L}}_{A}b(-{\tau }_i+m(\tau +1))\right]}_{\mathcal{B}}\right)\hfill \\ \hfill & =E\left(b(-{\tau }_i+(m+1)(\tau +1))\right).\hfill \end{array}$$

Moreover, we prove the second equality.

$b(-{\tau }_i+k(\tau +1))={\left[{\left({\tilde{L}}_A\right)}^{k}b(-{\tau }_i)\right]}_B$ and $b(-{\tau }_i)=x(-{\tau }_i)={\delta }_{2^n}^i$ imply that $b(-{\tau }_i+k(\tau +1))={\left[Co{l}_i\left(L^k\right)\right]}_{\mathcal{B}}$.

The conclusion is explained as follows.

   □

Next, we investigate the meaning of each element in matrix $L^k$ and matrix ${\left({\tilde{L}}_A\right)}^k$ in the following theorem.

Theorem 2.

Given a combinatorial flipped set $A\subseteq \mathcal{N}$, the k-periodic transition matrix of system (9) is $L^k$, and the k-periodic flipped-transition of system (10) is ${\left({\tilde{L}}_A\right)}^k$. For any $k\in {\mathbb{Z}}^{+}$, the matrix elements satisfy

(1) 

${\left(L^k\right)}_{i,j}=p({\delta }_{2^n}^j,k,{\delta }_{2^n}^i)$;

(2) 

${\left[{\left({\tilde{L}}_A\right)}^k\right]}_{i,j}=\tilde{p}({\delta }_{2^n}^j,k,{\delta }_{2^n}^i)$,

where $p({\delta }_{2^n}^j,k,{\delta }_{2^n}^i)$ and $\tilde{p}({\delta }_{2^n}^j,k,{\delta }_{2^n}^i)$ denote the number of distinct k-step paths from the initial state ${\delta }_{2^n}^j$ to the target state ${\delta }_{2^n}^j$ after k-periodic steps in system (9) and system (10), respectively.

Proof of Theorem 2.

Given $x(-{\tau }_i+k(\tau +1))=L^{k}x(-{\tau }_i)$, conclusion (1) can be drawn.

Next, we prove conclusion (2) through mathematical induction.

For the base case $k=1$,

$$\begin{array}{cc}\hfill {\left({\tilde{L}}_A\right)}_{ij}& =Ro{w}_i\left(Co{l}_j\left({\tilde{L}}_A\right)\right)\hfill \\ \hfill & =Ro{w}_i\left({\left[{\tilde{L}}_A{\delta }_{2^n}^j\right]}_{\mathcal{B}}\right)\hfill \\ \hfill & =Ro{w}_i\left({\Sigma }_{{\delta }_{2^n}^p\in \ddot{x}(1;{\delta }_{2^n}^j)}{\delta }_{2^n}^p\right)\hfill \\ \hfill & ={\Sigma }_{{\delta }_{2^n}^p\in \ddot{x}(1;{\delta }_{2^n}^j)}Ro{w}_i\left({\delta }_{2^n}^p\right)\hfill \\ \hfill & ={\Sigma }_{{\delta }_{2^n}^p\in \ddot{x}(1;{\delta }_{2^n}^j),i=p}1\hfill \\ \hfill & =\tilde{p}({\delta }_{2^n}^j,k,{\delta }_{2^n}^i).\hfill \end{array}$$

Suppose that for $k=m$, we have ${\left[{\left({\tilde{L}}_A\right)}^t\right]}_{ij}=\tilde{p}({\delta }_{2^n}^j,t,{\delta }_{2^n}^i)$.

Then, for $k=m+1$,

$$\begin{array}{cc}\hfill {\left[{\left({\tilde{L}}_A\right)}^{t+1}\right]}_{ij}& ={\left[{\left({\tilde{L}}_A\right)}^t{\tilde{L}}_A\right]}_{ij}\hfill \\ \hfill & =Ro{w}_i{\left({\tilde{L}}_A\right)}^{t}Co{l}_j\left({\tilde{L}}_A\right)\hfill \\ \hfill & ={\Sigma }_{p=1}^{2^n}{\left[{\left({\tilde{L}}_A\right)}^t\right]}_{ip}{\left({\tilde{L}}_A\right)}_{pj}\hfill \\ \hfill & ={\Sigma }_{p=1}^{2^n}\tilde{p}({\delta }_{2^n}^p,t,{\delta }_{2^n}^i)\tilde{p}({\delta }_{2^n}^j,1,{\delta }_{2^n}^p)\hfill \\ \hfill & =\tilde{p}({\delta }_{2^n}^j,t+1,{\delta }_{2^n}^i).\hfill \end{array}$$

Thus, ${\left[{\left({\tilde{L}}_A\right)}^k\right]}_{ij}=\tilde{p}({\delta }_{2^n}^j,k,{\delta }_{2^n}^i)$, for any $k\in {\mathbb{Z}}^{+}$.

The conclusion is explained as follows.    □

3.4. Stabilization of a TBN Under State-Flipped Control

First, we propose a definition of stabilization of TBN (3) under state-flipped control.

Definition 6

(Stabilization of TBNs under state-flipped control). Given a flipped set $A\subseteq \mathcal{N}$ and a target state $x^{\ast }={\delta }_{2^n}^c\in {\Delta }_{2^n}$, TBN (3) under state-flipped control is said to be globally stabilized to $x^{\ast }$, if there exist an integer $T\in {\mathbb{Z}}^{+}$ and a flipped-set sequence $\Gamma =\{S_0,S_1,\cdots ,S_t,\cdots \}$ such that

$$x\left(t\right)={\delta }_{2^n}^c,\mathrm{for}\mathrm{any}t\ge T,$$

where $S_t\subseteq A$ is the flipped set at time $t\in {\mathbb{Z}}^{+}$ and for any initial state sequence $z_0=\{x\left(0\right),x(-1),\cdots ,x(-\tau )\}$.

Next, we prove the equivalence between the stabilizations of TBN (3) and subsystems (9) in the following theorem.

Theorem 3.

TBN (3) achieves global stabilization to state $x^{\ast }$ under state-flipped control if and only if its $\tau +1$ subsystems in (9) reach global stabilization to $x^{\ast }$ under state-flipped control.

Proof of Theorem 3.

(Sufficiency) All subsystems (9) are globally stabilized to $x^{\ast }$ under state-flipped control. Based on Definition 6, for any ${\tau }_i\in \{0,1,\cdots ,\tau \}$ and initial state $x(-{\tau }_i)\in {\Delta }_{2^n}$, there exist $k_{{\tau }_i}\in {\mathbb{Z}}^{+}$ and flipped set sequences ${\Gamma }_i=\{S_{t_{i,0}},S_{t_{i,1}},\cdots ,S_{t_{i,j}},\cdots \}$ such that $x(-{\tau }_i+k(\tau +1))=x^{\ast }$, for any $k\ge k_{{\tau }_i}$, where $S_{t_{i,j}}$ are the flipped sets at time $t_{i,j}$, $t_{i,j}=-{\tau }_i+j(\tau +1)$, and $j\in \{0,1,2,\cdots \}$.

For any initial state sequence $z_0=\{x\left(0\right),x(-1),\cdots ,x(-\tau )\}$ of system (3), we construct $k={max}_{{\tau }_i\in \{0,1,\cdots ,\tau \}}{k}_{{\tau }_i}$, $T=k·(\tau +1)$, and a flipped sequence $\Gamma =\{S_0,S_1,\cdots ,$$S_t,\cdots \}$, where $S_t\in {\Gamma }_i$, $t=0,1,2,\cdots$, and $i\in \{0,1,2,\cdots ,\tau \}$.

Next, we prove through contradiction that the TBN (3) achieves global stabilization to $x^{\ast }$ under state-flipped control.

Suppose the system does not globally stabilize to $x^{\ast }$. There exists a positive integer $\widehat{t}\ge T$ such that $x\left(\widehat{t}\right)\ne x^{\ast }$. For the given $\widehat{t}$, we can find unique ${\widehat{\tau }}_i\in \{0,1,2,\cdots ,\tau \}$ and $\widehat{k}\in {\mathbb{Z}}^{+}$, such that $\widehat{t}=-{\widehat{\tau }}_i+\widehat{k}(\tau +1)$. We have $x(-{\widehat{\tau }}_i+\widehat{k}(\tau +1))=x\left(\widehat{t}\right)\ne x^{\ast }$, which contradicts the fact that the ${\widehat{\tau }}_i$-th subsystem in (9) is globally stabilized to $x^{\ast }$ under state-flipped control. The conclusion follows.

[Necessity] Since TBN (3) achieves global stabilization to state $x^{\ast }$ under state-flipped control, for any initial state sequence $z_0=\{x\left(0\right),x(-1),\cdots ,x(-\tau )\}$, there exist an integer $T\in {\mathbb{Z}}^{+}$ and a flipped set sequence $\Gamma =\{S_0,S_1,\cdots \}$ such that $x\left(t\right)=x^{\ast }$, $\forall t\ge T$.

Divide TBN (3) into $\tau +1$ subsystems (9), where $x(-{\tau }_i)$ is the initial state of the ${\tau }_i$-th subsystem. Let $k_{{\tau }_i}=k$. Divide the flipped set sequence $\Gamma =\{S_0,S_1,\cdots \}$ into $\tau +1$ flipped set sequences ${\Gamma }_i=\{S_{t_{i,0}},S_{t_{i,1}},\cdots ,S_{t_{i,j}},\cdots \}$ such that $x(-{\tau }_i+k(\tau +1))=x^{\ast }$, for any $k\ge k_{{\tau }_i}$, where $S_{t_{i,j}}$ are the flipped sets at time $t_{i,j}$, $t_{i,j}=-{\tau }_i+j(\tau +1)$, $j=0,1,,2,\cdots$, and $i=0,1,\cdots ,\tau$.

For T, there exists ${\tau }_i^{\ast }\in \{0,1,2,\cdots ,\tau \}$ and $k^{\ast }\in {\mathbb{Z}}^{+}$ such that $T=-{\tau }_i^{\ast }+k^{\ast }(\tau +1)$. Without loss of generality, we assume that ${\tau }_i^{\ast }=\tau$. Then, when $t\ge T$, one has $x\left(t\right)=x(-{\tau }_i+k(\tau +1))=x^{\ast }$, where ${\tau }_i\in \{0,1,2,\cdots ,\tau \}$ and $k\ge k^{\ast }$.

Thus, it is evident that that the ${\tau }_i\text{-}$th subsystem in (9) reaches global stabilization to $x^{\ast }$ under state-flipped control, where its corresponding flipped-set sequence is ${\mathrm{\Gamma }}_i=\{S_{t_{i,0}},S_{t_{i,1}},\cdots ,S_{t_{i,j}},\cdots \}$ and $i=0,1,2,\cdots ,\tau$.    □

From Definition 6 and Theorem 3, we reach an interesting result. The key goal of the global stabilization of TBN (3) under state-flipped control is to obtain the corresponding flipped-set sequence $\mathrm{\Gamma }$.

In the following, we provide an equivalent criterion for system (3) to achieve global stabilization under state-flipped control. In addition to this, an algorithm is designed to find the specific flipped sequence $\mathrm{\Gamma }$ for system (3) to achieve stabilization.

Theorem 4.

Given a flipped set $A\subseteq \mathcal{N}$ and a target state $x^{\ast }={\delta }_{2^n}^c$, TBN (3) achieves global stabilization to $x^{\ast }$ under state-flipped control if and only if

(1) 

${\left({\tilde{L}}_A\right)}_{c,c}>0$;

(2) 

there exists a positive integer $k\in [1,2^n-1]$ such that $Ro{w}_c\left[{\tilde{L}}_A^k\right]>\mathbf{0}$;

where ${\tilde{L}}_A$ and ${\tilde{L}}_A^k$ are as defined in Equation (10).

Proof of Theorem 4.

[Sufficiency] Based on Theorem 2, if ${\left({\tilde{L}}_A\right)}_{cc}=\tilde{p}(x^{\ast },1,x^{\ast })>0$, we can determine that $x^{\ast }\in \ddot{x}(1;x^{\ast })$, i.e., there exists a flipped set $S\subseteq A$, such that $L{G}_S{x}^{\ast }=x^{\ast }$.

Since there exists a positive integer $1\le k\le 2^n-1$ such that $Ro{w}_c{\left({\tilde{L}}_A\right)}^k>0$, we know that for any ${\tau }_i\in \{0,1,\cdots ,\tau \}$ and initial state $x(-{\tau }_i)\in {\Delta }_{2^n}$, $x^{\ast }\in \ddot{x}(k;x(-{\tau }_i))$, i.e., there exists a k-step flipped-set sequence ${\mathrm{\Gamma }}_i^k=\{S_{t_{i,0}},S_{t_{i,1}},\cdots ,S_{t_{i,k}}\}$, such that $x^{\ast }=x(-{\tau }_i+k(\tau +1))$.

Combining $L{G}_S{x}^{\ast }=x^{\ast }$ and flipped sequence ${\mathrm{\Gamma }}_i^k$, we make $k_{{\tau }_i}=k$ and construct a new flipped-set sequence ${\mathrm{\Gamma }}_i=\{S_{t_{i,0}},S_{t_{i,1}},\cdots ,S_{t_{i,k}},S,S,\cdots \}$, such that $x(-{\tau }_i+j(\tau +1))=x^{\ast }$, for every $j\ge k$. Therefore, the ${\tau }_i\text{-}$th subsystem with the initial state $x(-{\tau }_i)$ in (9) reaches stabilization to $x^{\ast }$ under state-flipped control. Based on Theorem 3, we conclude that TBN (3) reaches global stabilization to state $x^{\ast }$ under state-flipped control.

[Necessity] According to Theorem 3, system (3) is globally stabilized to $x^{\ast }$, which implies that all subsystems in (9) achieve global stabilization to $x^{\ast }$ under state-flipped control. Based on Definition 6, the ${\tau }_i\text{-}$th subsystem in (9) is globally stabilized to $x^{\ast }$. This implies that, for any ${\tau }_i\in \{0,1,\cdots ,\tau \}$ and initial state $x(-{\tau }_i)\in {\Delta }_{2^n}$, there exist a positive integer $k_{{\tau }_i}$ and a flipped-set sequence ${\mathrm{\Gamma }}_i=\{S_{t_{i,0}},S_{t_{i,1}},\cdots ,S_{t_{i,j}},\cdots \}$, such that $x(-{\tau }_i+k(\tau +1))=x^{\ast }$, for every $k\ge k_{{\tau }_i}$, where $t_{i,j}=-{\tau }_i+j(\tau +1)$ and $j\in \{0,1,,2,\cdots \}$.

On the one hand, for a fixed $\widehat{k}\ge k_{{\tau }_i}$, we have $x(-{\tau }_i+\widehat{k}(\tau +1))=x(-{\tau }_i+(\widehat{k}+1)(\tau +1))=x^{\ast }$, which implies that $x^{\ast }\in \ddot{x}(1;x^{\ast })$. According to Theorem 1, we have $x^{\ast }\in E{\left(\left[{\tilde{L}}_A{x}^{\ast }\right]\right)}_{\mathcal{B}}$, i.e., ${\left({\tilde{L}}_A\right)}_{cc}>0$.

On the other hand, since the total number of states in the ${\tau }_i$-th subsystem is $2^n$, we can find a value of $k_{{\tau }_i}$ that satisfies the constraint condition $1\le k_{{\tau }_i}\le 2^n$. If we let $T=k_{{\tau }_i}$, then we determine that $1\le T\le 2^n$. Furthermore, we obtain the following conclusion:

$$\begin{array}{cc}\hfill x^{\ast }& \in \ddot{x}(T;x(-{\tau }_i))\hfill \\ \hfill & =E\left({\left[{\left({\tilde{L}}_A\right)}^{T}x(-{\tau }_i)\right]}_{\mathcal{B}}\right)\hfill \\ \hfill & =E\left({\left[Co{l}_i\left({\left({\tilde{L}}_A\right)}^T\right)\right]}_{\mathcal{B}}\right),\hfill \end{array}$$

for any $x(-{\tau }_i)={\delta }_{2^n}^i\in {\Delta }_{2^n}$. Based on Theorem 4 and $x^{\ast }={\delta }_{2^n}^c$, for any $i\in \{1,2,\cdots ,2^n\}$, ${\left[{\left({\tilde{L}}_A\right)}^T\right]}_{ci}>0$, i.e., $Ro{w}_c\left[{\left({\tilde{L}}_A\right)}^T\right]>0$, for any integer $T\in [1,2^n]$.

The conclusion is explained as follows.   □

Theorem 4 provides a necessary and sufficient condition to determine whether system (3) can achieve stabilization. To enhance the applicability of the theoretical results, we present an algorithm below.

In Algorithm 1, we first search for the flipped sequence ${\mathrm{\Gamma }}_i$ of the ${\tau }_i$-th subsystem in (9), and then combine all sequences ${\mathrm{\Gamma }}_i$ alternately to obtain the flipped set sequence $\mathrm{\Gamma }$ of system (3). Note that there may be many different flipped sequences that can achieve system (3) stabilization, and Algorithm 1 provides a method to find one of the feasible flipped-set sequences.

Algorithm 1 An algorithm for finding a proper flipped-set sequence $\mathrm{\Gamma }=\{S_0,S_1,\cdots ,S_t,\cdots \}$ to steer TBN (3) to reach global stabilization

Input: structural matrix L, time delay $\tau$, flipped set $A\subseteq \mathcal{N}$, target state $x^{\ast }$ Output: flipped set sequence $\mathrm{\Gamma }$, T   1: System (3) is partitioned into $\tau +1$ subsystems in (9).   2: For each ${\tau }_i$-th subsystem in (9), obtain step $k_{{\tau }_i}$ and find flipped set sequence ${\mathrm{\Gamma }}_i=\{S_{t_{i,0}},S_{t_{i,1}},\cdots ,S_{t_{i,j}},\cdots \}$ by Algorithm 3 in [29], where ${\tau }_i=\tau -i$ and $i\in \{0,1,\cdots ,\tau \}$.   3: Construct $T=ma{x}_{{\tau }_i\in \{0,1,\cdots ,\tau \}}\{-{\tau }_i+k_{{\tau }_i}(\tau +1)\}$ and a flipped set sequence $\mathrm{\Gamma }=\{S_{-\tau ,0},\cdots ,S_{-\tau +i,0},\cdots ,S_{0,0},S_{-\tau ,1},\cdots ,S_{-\tau +i,j},\cdots \}$, where $i\in \{0,1,\cdots ,\tau \}$, $j\in {\mathbb{Z}}^{+}\cup \left\{0\right\}$, $S_{-\tau +i,j}=S_{t_{i,j}}$ and $S_{t_{i,j}}$, $k_{{\tau }_i}$ are generated in step 2.

4. Examples

In this section, we consider two examples to demonstrate the advantages of state-flipped control. Furthermore, we can illustrate the validity of the main conclusions drawn in Section 3. The following example is a classical mathematical model for Boolean networks with time delays.

Example 1.

We consider a reduced Boolean model for the $lac$ operon in the bacterium $Escherichia$ $coil$ [30] with a time delay below:

$$\left\{\begin{array}{c}{x}_1(t+1)=x_2(t-2)\vee x_3(t-2),\hfill \\ x_2(t+1)=x_1(t-2),\hfill \\ x_3(t+1)=x_1(t-2),\hfill \end{array}\right.$$

(11)

where $x_i\in {\Delta }_2$, and ∨ is an “OR” operation between logical variables.

The above Boolean model can be regarded as a TBN with $n=3$ nodes and a $\tau =2$ delay. Let $x\left(t\right)={⋉}_{i=1}^3{x}_i\left(t\right)\in {\Delta }_{2^3}$ and the initial state be $z_0=\{x\left(0\right),x(-1),x(-2)\}$. Next, we consider whether it is possible to achieve global stabilization to state ${\delta }_8^1$ of system (11) by designing state-flipped control. Firstly, we can obtain the algebraic form of system (11) using the STP method as follows:

$$x(t+1)=Lx(t-2),$$

(12)

where $L={\delta }_8[1,1,1,5,4,4,4,8]$.

According to the analysis of periodicity on TBNs in SubSection 3.2, we can divide system (12) into three independent subsystems:

(1):

$x\left(0\right)\to x\left(3\right)\to x\left(6\right)\cdots \to x\left(3a\right)\cdots$,

(2):

$x(-1)\to x\left(2\right)\to x\left(5\right)\cdots \to x(3a-1)\cdots$,

(3):

$x(-2)\to x\left(1\right)\to x\left(4\right)\cdots \to x(3a-2)\cdots$,

where $a\in {\mathbb{Z}}^{+}$.

Note that the state transition of the above subsystems only depends on matrix L mentioned in system (12). Subsequently, a state transition graph of each subsystem can be obtained, as depicted in Figure 1.

From Figure 1, we determine that the target state ${\delta }_8^1$ can never be reached from states of systems ${\delta }_8^4$, ${\delta }_8^5$, ${\delta }_8^6$, ${\delta }_8^7$, and ${\delta }_8^8$ after k transition steps, for any $k\in {\mathbb{Z}}^{+}$. After verification, the above conclusion is consistent with the results displayed in the matrices $L=L^3=L^5=L^7={\delta }_8[1,1,1,5,4,4,4,8]$ and $L^2=L^4=L^6=L^8={\delta }_8[1,1,1,4,5,5,5,8]$.

However, when we consider an appropriate flipped set, system (12) can be globally stabilized to the target state ${\delta }_8^1$. Below, we will explain this conclusion.

Firstly, after introducing combinatorial state-flipped control, the reachable point of the system (12) will change from a single fixed state to a set of states, as mentioned in SubSection 3.3. Let the largest flipped set be $A=\left\{3\right\}\subseteq \mathcal{N}=\{1,2,3\}$, which allows each subsystem to have more state transitions. Please refer to Figure 2.

Then we determine the expressions of three subsystems under combinatorial state-flipped control, as shown below:

$$\left\{\begin{array}{c}b(3a-2)={\left[{\tilde{L}}_{A}b(3(a-1)-2)\right]}_B=\cdots ={\left[{\left({\tilde{L}}_A\right)}^{a}b(-2)\right]}_B,\hfill \\ b(3a-1)={\left[{\tilde{L}}_{A}b(3(a-1)-1)\right]}_B=\cdots ={\left[{\left({\tilde{L}}_A\right)}^{a}b(-1)\right]}_B,\hfill \\ b\left(3a\right)={\left[{\tilde{L}}_{A}b\left(3(a-1)\right)\right]}_B=\cdots ={\left[{\left({\tilde{L}}_A\right)}^{a}b\left(0\right)\right]}_B,\hfill \end{array}\right.$$

where $b\left(t\right)$ is the Boolean vector, ${\tilde{L}}_A:=L{\mathcal{G}}_A$ is the combinatorial flipped-transition matrix defined in system (8) and when $-2\le t\le 0$, $b\left(t\right)=x\left(t\right)$.

Through calculation, we obtain the matrix

$${\left({\tilde{L}}_A\right)}^2=[4{\delta }_8^1,4{\delta }_8^1,2{\delta }_8^{1,4},2{\delta }_8^{1,4},2{\delta }_8^{1,5},2{\delta }_8^{1,5},{\delta }_8^{1,4,5,8},{\delta }_8^{1,4,5,8}].$$

According to Theorem 4, since ${\left({\tilde{L}}_A\right)}_{11}>0$ and $Ro{w}_1\left[{\left({\tilde{L}}_A\right)}^2\right]>0$ hold, we conclude that system (12) can achieve stabilization to ${\delta }_8^1$ under state-flipped control. Based on Theorems 1 and 2, if $x(-2)=x(-1)=x\left(0\right)={\delta }_8^6$, we determine that $\ddot{x}(2;{\delta }_8^6)=E\left({\left[{\left({\tilde{L}}_A\right)}^2{\delta }_8^6\right]}_{\mathcal{B}}\right)=\{{\delta }_8^1,{\delta }_8^5\}$, $\tilde{p}({\delta }_8^6,2,{\delta }_8^1)=2$ and $\tilde{p}({\delta }_8^6,2,{\delta }_8^5)=2$. Combining this with Algorithm 1, we obtain $k_0=k_1=k_2=2$, $T=7$, and a feasible flipped-set sequence $\mathrm{\Gamma }=\{\varnothing ,\varnothing ,\varnothing ,\{3\},\{3\},\{3\},\varnothing ,\varnothing ,\cdots \}$. Please refer to Figure 3 and Figure 4 for details.

Example 2

([31]).

$$\left\{\begin{array}{c}{x}_1(t+1)=x_2(t-1)\vee x_3(t-1),\hfill \\ x_2(t+1)=x_1(t-1)\vee x_2(t-1),\hfill \\ x_3(t+1)=\neg x_1(t-1),\hfill \end{array}\right.$$

(13)

where $x_i\in {\Delta }_2$, and ∨ and ¬ are “OR” and “NOT” operations on logical variables, respectively.

Let $x\left(t\right)={⋉}_{i=1}^3{x}_i\left(t\right)\in {\Delta }_{2^3}$ and the initial state be $z_0=\{x\left(0\right),x(-1)\}$. Then, we can obtain the algebraic form of system (13) using the STP method as follows:

$$x(t+1)=Lx(t-1),$$

(14)

where $L={\delta }_8[2,2,2,6,1,1,3,7]$. According to the analysis in Section 3.2, we can divide system (13) into two independent subsystems:

(1):

$x\left(0\right)\to x\left(2\right)\to x\left(4\right)\cdots \to x\left(2a\right)\cdots$,

(2):

$x(-1)\to x\left(1\right)\to x\left(3\right)\cdots \to x(2a-1)\cdots$,

where $a\in {\mathbb{Z}}^{+}$. A state transition graph of each subsystem is depicted in Figure 5.

After simple calculations, we determine that $L^3={\delta }_8[2,2,2,2,2,2,2,2]$. Based on Figure 5, it can be seen that all subsystems of system (14) can achieve global stability to state ${\delta }_8^2$ without any control (equivalent to the flipped set $S_t=\varnothing$ at time $t\in {\mathbb{Z}}^{+}$). According to Theorem 3, we determine that system (14) achieves global stabilization to ${\delta }_8^2$. However, introducing state-flipped control, we not only enable system (14) to achieve global stabilization to ${\delta }_8^2$ faster, but also enable system (14) to stabilize to other state ${\delta }_8^1$. Next, we will explain this conclusion. Let the largest flipped set be $A=\{1,2\}\subseteq \mathcal{N}=\{1,2,3\}$. Then we can calculate the combinatorial flipped-transition matrix

$$\left({\tilde{L}}_A\right)=[{\delta }_8^{1,3}+2{\delta }_8^2,{\delta }_8^{1,2,6,7},{\delta }_8^{1,3}+2{\delta }_8^2,{\delta }_8^{1,2,6,7},{\delta }_8^{1,3}+2{\delta }_8^2,{\delta }_8^{1,2,6,7},{\delta }_8^{1,3}+2{\delta }_8^2,{\delta }_8^{1,2,6,7}].$$

According to Theorem 4, since ${\left({\tilde{L}}_A\right)}_{11}={\left({\tilde{L}}_A\right)}_{22}=1>0$, $Ro{w}_1\left[\left({\tilde{L}}_A\right)\right]>0$ and $Ro{w}_2\left[\left({\tilde{L}}_A\right)\right]>0$, it can be seen that system 14 can achieve stabilization to ${\delta }_8^1$ and ${\delta }_8^2$ under state-flipped control.

Taking the subsystem $x\left(2a\right)=L^{a}x\left(0\right)$ as an example, we can see the effectiveness of state-flipped control by comparing the two pictures in Figure 6.

5. Conclusions

This paper addresses the stabilization problem in TBNs under state-flipped control. First, the algebraic form of TBNs under state-flipped control is derived using the STP method. The TBNs are then partitioned into subsystems based on their periodic properties. Subsequently, leveraging these developments, the equivalence between the stabilization of the TBNs and that of the derived subsystems under state-flipped control is established. Furthermore, an equivalent stabilization criterion for the TBNs under state-flipped control is provided. An algorithm is also presented to construct a flipped-set sequence for stabilization. Finally, illustrative examples are provided to demonstrate the effectiveness of the theoretical results.

This is the first time that state-flipped control is introduced to TBNs. Unlike [21,23,24,29], this work addresses a gap in the theoretical research on logical control networks with time delays. Also, this work contributes to enriching the theory of state-flipped control.