Yifeng LI ,Lan WANG

1National Center for Applied Mathematics in Chongqing,Chongqing Normal University,Chongqing 401331,China

2Institute of Mathematics,School of Mathematical Sciences,Nanjing Normal University,Nanjing 210023,China

Abstract: In this paper,the problem of controllability of Boolean control networks (BCNs) with multiple time delays in both states and controls is investigated.First,the controllability problem of BCNs with multiple time delays in controls is considered.For this controllability problem,a controllability matrix is constructed by defining a new product of matrices,based on which a necessary and sufficient controllability condition is obtained.Then,the controllability of BCNs with multiple time delays in states is studied by giving a necessary and sufficient condition.Subsequently,based on these results,a controllability matrix for BCNs with multiple time delays in both states and controls is proposed that provides a concise controllability condition.Finally,two examples are given to illustrate the main results.

Key words: Boolean control networks;Semi-tensor product of matrices;Controllability;Time delay

1 Introduction

In system biology,genetic regulatory networks(GRNs) are essential networks.Boolean networks(BNs),first proposed by Kauffman (1969),are an effective tool in modeling,analyzing,and simulating GRNs,wherein each gene is characterized by a Boolean variable(active (1)or inactive (0)),and interactions between the states of each gene are determined by logical functions composed of logical operators.BNs with external inputs are called Boolean control networks (BCNs).BNs and BCNs have attracted much attention from biologists,physicists,and systems scientists (Albert and Othmer,2003;Chaves et al.,2005;Klmat et al.,2006;Cheng and Qi,2009).In particular,Cheng and Qi (2010)proposed a generalized matrix product,called the semi-tensor product (STP),based on which an algebraic statespace representation framework has been established for the analysis and control of BNs or BCNs(Cheng et al.,2011).The framework makes it relatively easy to formulate and solve classical control-theoretic problems for BNs or BCNs,and thereby many fundamental results of BNs or BCNs have been obtained,such as controllability and observability(Zhao et al.,2010;Fornasini and Valcher,2013;Liang et al.,2017;Weiss and Margaliot,2019;Zhou et al.,2019;Zhang X et al.,2021;Zhu et al.,2021),optimal control(Fornasini and Valcher,2014;Wu et al.,2021;Gao et al.,2022),stability and stabilization (Li R et al.,2013;Zhong et al.,2020;Acernese et al.,2021;Guo et al.,2021;Li HT et al.,2021;Shen et al.,2021),system decomposition and decoupling (Zou and Zhu,2015;Li YF and Zhu,2020,2022,2023;Li YF et al.,2021;Feng et al.,2022),output regulation (Li HT et al.,2017),and synchronization(Zhong et al.,2014;Chen HW et al.,2018).

As is commonly known,the time delay phenomenon encountered in the real world should be taken into account to reduce its impact on the dynamic behavior of models (Wang WQ and Zhong,2012;Wang ZD et al.,2018).For BNs,based on the STP,many results concerning BCNs with time delays in states have been presented (Li FF and Sun,2011;Li FF et al.,2011;Li R et al.,2012;Zheng and Feng,2020),and the controllability of BCNs with delays in states has been investigated widely.In several works (Li FF and Sun,2011;Li R et al.,2012;Han et al.,2014;Lu et al.,2016),the controllability problem of BCNs with a constant time delay in states has been investigated.For BCNs with timevarying delays in states,the controllability problem has been investigated (Zhang LJ and Zhang,2013;Ding et al.,2018).Because the information transmissions between different pairs of nodes in a complex network are,in general,unsynchronized,BCNs with multiple time delays (high-order BCNs) have been proposed as a better model for GRNs (Chen H and Sun,2013).Moreover,necessary and sufficient conditions for the controllability of BCNs with multiple constant delays (i.e.,high-order BCNs) have been obtained (Li FF and Sun,2012;Chen H and Sun,2013;Ding et al.,2018).

System performance is greatly impacted by time delays in controls,and systems with such delays have been studied widely in economic,biological,and physiological industry fields (Cui et al.,2009;Liu and Zhao,2011;Klamka,2019).The controllability of systems with time delays in both states and controls has been investigated(Dauer and Gahl,1977;Yang et al.,2009).The controllability of BCNs with time delays in both states and controls has been studied in Han et al.(2014),in which the considered system has a constant delay.A natural question is,what is the controllability condition for BCNs with multiple time delays in both states and controls? In addition,controllability matrices play an essential role in controllability analysis of delay-free BCNs,and provide a concise criterion for controllability;i.e.,a delay-free BCN is controllable if and only if all elements of its controllability matrix are nonzero(Zhao et al.,2010).An interesting issue to be considered is how to construct controllability matrices for BCNs with multiple time delays in both states and controls.Furthermore,is there a similar concise criterion for the controllability of BCNs with multiple time delays in both states and controls with the help of controllability matrices? These questions motivated us to study the controllability of BCNs with multiple time delays in both states and controls.The main contributions of this paper are summarized as follows:

1.To our knowledge,we are investigating controllability of BCNs with multiple time delays in both states and controls for the first time.In our study,this problem is transformed into two relatively simple problems to be solved: BCN controllability with multiple time delays in controls and with multiple time delays in states.

2.To solve the controllability problem of BCNs with multiple time delays in controls,a relatively simple and clear controllability matrix is constructed by defining a new product of matrices.A necessary and sufficient condition is then obtained with the help of the controllability matrix for the controllability problem.

3.A necessary and sufficient condition is proposed for the controllability problem of BCNs with multiple time delays in states.Based on these results of controllability of BCNs with multiple time delays in states and BCNs with multiple time delays in controls,the controllability matrix for BCNs with multiple time delays in both states and controls is constructed to provide a concise controllability criterion similar to that for delay-free BCNs.

2 Preliminaries and problem setting

Throughout the paper,we use the following notations:

Definition 1(Cheng and Qi,2010) SetA=(aij)∈Rm×nandB=(bij)∈Rp×q.Letα=lcm(n,p) be the least common multiple ofnandp.Then the STP ofAandBis defined as

In Definition 1,whenn=p,the STP degenerates to the conventional matrix product.Moreover,the STP keeps most properties of the conventional product,e.g.,associativity,distributivity,and the transpose and inverse of products (Cheng et al.,2011).Hence,in the following discussion,A■Bis denoted byABfor simplicity of presentation.

Definition 2(Cheng and Qi,2010) Thekdimensional power-reducing matrix is defined as

Definition 3(Cheng and Qi,2010) The swap matrixW[m,n]is defined as

Letx ∈Δmandy ∈Δn.Thenxx=Mr,mxandW[m,n]xy=yx.In addition,for any matrixMand column vectorx ∈Rt,it holds thatxM=(It ?M)x(Cheng et al.,2011).

whereL ∈L2n×2(μ+1)m+(λ+1)n.

Remark 1The conversion process between logical form (3) and algebraic form (4) can be found in Cheng and Qi(2010).

Definition 5BCN (4) is said to be controllable atx0ifR(x0)=Δ2n.BCN (4) is said to be controllable,if for anyx0∈Δ2n(λ+1),BCN (4) is controllable atx0.

A natural question is whether we can construct a similar controllability matrix to obtain a concise criterion for the controllability of BCN (4).This is exactly the problem to be discussed in this study.

3 Main results

3.1 Controllability of BCNs with multiple time delays in controls

In this subsection,we consider the case of time delays only in controls;i.e.,λ=0 in BCN(4).Then the considered system becomesTo simplifyMs,we rearrangeLas

whereLi1i2···iμ+1∈L2n×2n,i1,i2,···,iμ+1∈[1,2m].

Definition 7Given a matrixA ∈Rp×q,lets|pandt|qwithsm=pandtn=q.ThenAis expressed as a block decomposed form asA=(Aij)m×n,whereAij ∈Rs×t.We define

It is obvious that diags×t(A)∈Rms×mntin Definition 7.

Definition 8GivenA ∈Ru×vandB ∈Rx×y,leta|uandb|vwithmpa=uandnb=v,andx=npb.ThenAis expressed as a block decomposed form asA=(Aij)mp×n,whereAij ∈Ra×b.We define a new product ofAandBdenoted byA ?Bas

Specifically,denoteA(2):=A ?AandA(s) :=A ?A(s?1).

Proposition 1Consider a matrixA ∈Rpnk×nk,which is expressed as a block decomposed form asA=(Aij)pn×nwithAij ∈Rk×k.Let

wheres=2,3,···.

ProofAccording to the definition of “?,” with a straightforward computation,we have

Remark 2Obviously,if matricesAijandBstare replaced byaij ∈R andbst ∈R respectively,in Definitions 7 and 8 and Proposition 1,Proposition 1 still holds.

Lemma 2

Suppose that whens=k,Eq.(10) holds.In the following,we considers=k+1.

Therefore,for anys=2,3,···,Eq.(10)holds.

From Lemma 2 and the meaning ofMs,it follows that

and we call it the controllability matrix of BCN(6).

Theorem 1BCN(6)is controllable if and only ifM>0.

3.2 Controllability of BCNs with multiple time delays in states

In this subsection,we consider the case of time delays only in states;i.e.,μ=0 in BCN (4).Then,BCN(4)degenerates to

which is a delay-free BCN.From Lemma 1,BCN(15)is controllable if and only if ?C >0,where

Denote the initial state of Eq.(13)x(0)x(?1)···x(?λ) byx0.Considering Eq.(14),we havex0=z(0) and

Consideringz(s) andx(s),from Eq.(16),we have

Combining inequalities (17)and(18),we have

3.3 Controllability of BCNs with multiple time delays in both states and controls

Based on the results obtained in Sections 3.1 and 3.2,we investigate the controllability of BCNs with multiple time delays in both states and controls in this subsection.

Consider BCN(4).Let

Omitting the same process as that in Section 3.2,one can obtain

whereLi1i2···iμ+1∈L2(λ+1)n×2(λ+1)n.

According to Theorem 1,BCN (22) is controllable if and only if

By Eq.(21),we have

4 Examples

Example 1Consider the following BCN with time delays in controls:

whereu ∈Δ2,x(t)∈Δ4,

A straightforward calculation shows that

Example 2Consider a BCN with time delays in both states and controls:

whereL=δ2[1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1],u,x ∈Δ2.Letz(t)=x(t)x(t ?1).Then we have

A straightforward calculation yields that

It follows from Theorem 4 that system (25) is controllable.

5 Conclusions

The controllability problem of BCNs with multiple time delays in both states and controls has been studied using the STP of matrices,which is transformed into two problems: the controllability problem of BCNs with multiple time delays in controls and the controllability problem of BCNs with multiple time delays in states.For these two controllability problems,necessary and sufficient conditions have been given,and subsequently,based on them,a controllability matrix and a necessary and sufficient condition have been proposed for the controllability of BCNs with multiple time delays in both states and controls.In future work,we will investigate some observability problems for BCNs with multiple time delays in both states and controls.

Contributors

Yifeng LI designed the research and drafted the paper.Lan WANG helped organize the paper.Yifeng LI and Lan WANG revised and finalized the paper.

Compliance with ethics guidelines

Yifeng LI and Lan WANG declare that they have no conflict of interest.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.