Static output feedback set stabilization for context-sensitive probabilistic Boolean control networks

Applied Mathematics and Computation 332 (2018) 263–275

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Static output feedback set stabilization for context-sensitive probabilistic Boolean control networks Liyun Tong a, Yang Liu a, Jungang Lou b,∗, Jianquan Lu c, Fuad E. Alsaadi d a

College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, China School of Information Engineering, Huzhou University, Huzhou 313000, China c School of Mathematics, Southeast University, Nanjing 210096, China d School of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia b

a r t i c l e

i n f o

Keywords: Context-sensitive probabilistic Boolean control network Set stabilization Semi-tensor product Static output feedback control

a b s t r a c t In this paper, we investigate the static output feedback set stabilization for contextsensitive probabilistic Boolean control networks (CS-PBCNs) via the semi-tensor product of matrices. An algorithm for finding the largest control invariant set with probability one is obtained by the algebraic representations of logical dynamics. Based on the analysis of the set stabilization, necessary and sufficient conditions for S-stabilization are obtained. Static output feedback controllers are designed to achieve S-stabilization for a CS-PBCN. At last, examples to study metastatic melanoma are given to show the effectiveness of our main results. © 2018 Elsevier Inc. All rights reserved.

1. Introduction Boolean networks (BNs) [1] as a valuable tool for bio-genetic engineering, which has attracted the interest of many experts and scholars. And it has been used for simulating, analyzing and reaching the genetic regulatory networks [2]. In BNs, 1 (or 0) corresponds to the on (or off) state of the Boolean variable. And the update of each state for the Boolean variable is governed by logical relationship at every discrete time. When referring to the Boolean control networks (BCNs), it can be seen as BNs added controllers. As for the research of BNs and BCNs, Cheng first proposed semi-tensor product (STP) in [3] and [4]. Using STP, a BN can be converted into a standard discrete-time linear system in [5–9], as well as feedback shift registers [10]. Moreover, there have been many useful and interesting results in BNs and BCNs, including optimal control [11,12], controllability [13,14], observability [15,16], output regulation [17], solvability [18], attractor transformation [19,20], periodic trajectories [21], synchronization [22–24] etc. In addition, stochastic dynamical networks [25], neural networks [26,27], nonlinear singular systems [28], and complex-valued single neuron model [29] have been further investigated. Also the impulsive disturbances or disturbance decoupling for BCNs have been also studied by Yang et al. [30], Cheng [31] and Gao et al. [32]. Unlike the classical BCNs, probabilistic Boolean control networks (PBCNs) consider the stochastic phenomena, which can not be reflected by general BCNs. Therefore, to study the PBCNs is a very meaningful expansion for classic BCNs. Investigations on PBCNs have been very mature, involving a lot of contents, such as state stabilization [33], weak reachability [34], controllability [35,36], as well as output tracking control problem [37]. In [38], the author has studied the stochastic ∗

Corresponding author. E-mail address: [email protected] (J. Lou).

https://doi.org/10.1016/j.amc.2018.03.043 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.

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L. Tong et al. / Applied Mathematics and Computation 332 (2018) 263–275

stabilization of n-person random evolutionary Boolean games. Possieri and Teel [39] combined classical deterministic BCNs and PBCNs to investigate global asymptotic stability by constructing Lyapunov functions. In this paper, we consider a more general model named context-sensitive PBCNs (CS-PBCNs), in which the deciding time is randomly selected. It represents that at each time point, there is a probability of whether to switch a new context or remain in the previous context. Generally speaking, the CS-PBCNs are more complicated than those of PBCNs, and accordingly to study control problems of CS-PBCNs will be more difficulty. However, due to its practicality in gene regulatory networks, there are still many significant studies. Pal et al. [40] have treated intervention via external control variables in CS-PBCNs and the results are applied to gene-expression data collected in a study of metastatic melanoma. The authors have used probabilistic model checking to consider optimal control in apoptosis network [41]. In [42], some sufficient conditions for the stability and stabilization of CS-PBCNs have been presented. And for the optimal control in CS-PBCNs, an integer programming approach is used in [43]. There have been some more results on CS-PBCNs, such as robust control of CS-PBCNs [44] and steady-state approximation of CS-PBCNs [45]. In the study of BNs and BCNs, the stabilization or set stabilization are the important issues in the real world which can not be ignored. Li et al. [46] have considered the stabilization problem for BCNs with normal state feedback control. The set stabilization has been investigated in [47] with switched signals and Li et al. [48] investigated the equivalence issue of two kinds of controllers in BCNs. To stabilize systems, the pinning control is used in [49] to control the dynamics of the entire network by selecting key nodes according to some specific properties of nodes. When modeling a biological regulation network, introducing a feedback loop from some specific (output) variables can be viewed either as an attempt to achieve a more complete representation of the network dynamics [50]. Also Li and Wang [51] point out that output feedback controller is one of the possible feedback stabilizers for the disease treatment, which is a meaningful topic in both theoretical developments and practical applications. Meanwhile, the output state feedback controller has been also used to investigate the disturbance decoupling problem for BCNs in [52]. Of course, the output state feedback controllers studied in the above literatures are static output output state feedback controllers. It is well known that, many experts and scholars have studied a lot of problems about CS-PBCNs, and achieved pretty good results. In the control issue of BCNs, the discussions of set stabilization have been also plentiful. The set stabilization problem of CS-PBCNs with static output feedback control is new and challenging. The main difficulties lie in the following two aspects: i) The dynamics of a CS-PBCNs with static output feedback control is much more complicated than a normal PBCNs, which makes the mathematical derivation tough. ii) In order to make S-stabilizable, finding the largest control invariance set and conditions to ensure stabilization are challenging. To overcome these difficult issues, we should solve the following key problems. First, we need to analyze the logical structure of CS-PBCNs and simplify the transition probability matrix. Then we can find the largest control invariance set for CS-PBCNs. Finally, the static output feedback law matrix K needs to be designed according to the analysis of reachable sets Rk (S∗ ). In this paper, we will study the static output feedback set stabilization of CS-PBCNs. And the main contributions of this paper lie in the following three aspects: (i) The algorithm of finding the largest control invariance set S∗ is designed. (ii) A necessary and sufficient condition obtained in Theorem 1 for CS-PBCNs to judge whether it is S-stabilizable by an static output feedback controller. (iii) The static output feedback controllers for the S-stabilizable are obtained by Theorem 2. The rest of this paper is organized as follows. Section 2 provides some preliminaries on STP and gives an algebraic expression of CS-PBCNs. Some necessary and sufficient conditions are obtained for the S-stabilization of CS-PBCNs based on static output feedback control in Section 3. An illustrative example is provided to support the efficiency of the obtained results in Section 4. Finally, a brief conclusion is presented in Section 5. 2. Preliminaries 2.1. Semi-tensor product of matrices First, some necessary notations are provided in the following. • D := {1, 0}, and D n := D × · · · × D.



 n



• Denote Mm × n as the set of all m × n matrices. • k := {δki |i = 1, 2, . . . , k}, where δki is the ith column of identity matrix Ik . For compactness, 2 = . • A matrix A ∈ Mm × n is called a logical matrix if the column set of A, denoted by Col(A), satisfies Col(A)⊆m , and let Coli (A) and Rowj (A) be the ith column and the jth row of the matrix A, respectively. i

i

in • Denote the set of n × m logical matrices by Lm×n . If A ∈ Mm × n , it can be expressed as A = [δm1 , δm2 , . . . , δm ], or simply by A = δm [ i 1 , i 2 , . . . , i n ] . • v = (v1 , v2 , · · · , vn , )T is called a probabilistic vector for in=1 vi = 1, and the set of n dimensional probabilistic vectors is denoted by Yn . • Split a matrix M into equal dimension blocks and denote by Blki (M) the ith block of matrix M. • For two Boolean vectors a = (a1 , a2 . . . an )T ∈ D n , b = (b1 , b2 . . . bn )T ∈ D n , the Boolean addition  is defined as

a  b := a ∨ b = (a1 ∨ b1 . . . an ∨ bn )T ∈ D n .

L. Tong et al. / Applied Mathematics and Computation 332 (2018) 263–275 i

265

i

in • A nonempty set η = {δm1 , δm2 , . . . , δm }, let |η| = {i1 , i2 , . . . , in }, η = n and denote by [η] a column consisting for the Boolean addition of all the column in η. • Let v = (v1 , v2 , . . . , vm , )T ∈ ϒ m , w = (w1 , w2 , . . . , wm , )T ∈ ϒ m . The element-wise multiplication of two probabilistic vectors v and w is defined as

v ◦ w := [v1 w1 v2 w2 . . . vm wm ]T . T • 1n := [1 ···1 · · · 0]T .  1 ] and 0n := [0 0  n

n

• (A)i, j denotes the (i, j)th element of matrix A. • d = (d1 , d2 , . . . , dm , )T is a m-dimensional column vector and the set of m-dimensional column vectors is denoted by Rm . Definition 1. [4] The semi-tensor product of two matrices A ∈ Mm × n and B ∈ Mp × q is defined by

A  B = (A  Iα /n )(B  Iα /p ), where α = lcm(n, p) is the least common multiple of n and p, and  is the tensor (or Kronecker) product. When n = p, A  B = (A  I1 )(B  I1 ) = AB. So STP is a generalization of the conventional matrix. We simply call it “product” and omit the symbol “” if no confusion raises. Definition 2. [3] A swap matrix W[m, n] is an mn × mn matrix, defined as follows: its rows and columns are labeled by double index (i, j), the columns are arranged by the ordered multi-index Id(j, i; m, n) and the rows are arranged by the ordered multi-index Id(j, i; n, m). Then the element at the position [(I, J), (i, j)] is

 w(I,J ),(i, j ) = δi,I,Jj =

1,

I = i and = j,

0,

otherwise.

Lemma 1. [3] Let X ∈ Rm and Y ∈ Rn be two column vectors. Then W[m,n] X Y = Y X . For a given A ∈ Mm × n , and Z ∈ Rt , one has ZA = W[m,t] AW[t,n] Z = (It  A )Z. Lemma 2. [4] Let X = x1 x2 . . . xn , where xi ∈ 2 , i = 1, 2, . . . , n, then Xn2 = n Xn , where n = where Mr is called power reducing matrix satisfying x2i = Mr xi .

n

i=1 I2i−1

 [(I2  W[2,2n−i ] )Mr ],

2.2. The Algebraic representation of CS-PBCNs Using matrix expressions, we denote a logical domain by D, in which 1 and 0 are represented by δ21 and δ22 , respectively. Then D equals to 2 , and a logical function with n arguments f : D n → D can be expressed in an algebraic form by using STP of matrices. Lemma 3. [3] Let f (x1 , x2 , . . . , xn ): D n → D be a logical function. Then there exists a unique matrix M f ∈ L2×2n , called the structural matrix of f, such that

f (x1 , x2 , . . . , xn ) = M f ni=1 xi , xi ∈ 2 , where ni=1 xi = x1  · · ·  xn . A BCN with n nodes and m inputs can be described as:

⎧ x1 (t + 1 ) = f1 (x1 (t ), . . . , xn (t ), u1 (t ), . . . , um (t )), ⎪ ⎪ ⎪ ⎪ x2 (t + 1 ) = f2 (x1 (t ), . . . , xn (t ), u1 (t ), . . . , um (t )), ⎪ ⎪ ⎨ ..

. ⎪ ⎪ ⎪ ⎪ xn (t + 1 ) = fn (x1 (t ), . . . , xn (t ), u1 (t ), . . . , um (t )), ⎪ ⎪ ⎩ y j (t ) = g j (x1 (t ), . . . , xn (t )),

(1)

where fi : D n → D, 1 ≤ i ≤ n, g j : D n → D, 1 ≤ j ≤ p are logical functions, and ui ∈ D, 1 ≤ i ≤ m are control inputs. The BCN (1) becomes a PBCN, if its logical functions fi could be one of the li possible models, such that fi ∈ i j { fi1 , fi2 , . . . , fili } with the probabilities P ( fi = fij ) = pij  0 for j = 1, 2, . . . , li , and lj=1 pi = 1 for i = 1, 2, . . . , n. There are

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L. Tong et al. / Applied Mathematics and Computation 332 (2018) 263–275

totally M =

n

⎡

i=1 li

1 ⎢1 ⎢. ⎢. ⎢. ⎢1 ⎢ ⎢1

= ⎢ K ⎢1 ⎢. ⎢. ⎢. ⎢1 ⎢ ⎢. ⎣ .. l1

to denote the index set of M models: models. Using matrix K

1 1 .. . 1 1 1 .. . 1 .. . l2

··· ··· .. . ··· ··· ··· .. . ··· .. . ···

1 1 .. . 1 2 2 .. . 2 .. . ln−1

⎤

1 2⎥ .. ⎥ ⎥ .⎥ ln ⎥ ⎥ 1⎥ ⎥ 2⎥ ⎥ .. ⎥ .⎥ ⎥ ln ⎥ .. ⎥ .⎦ ln

, denote the λth model by  = { f λ1 , f λ2 , . . . , f λn }, where λ = n−1 (λ − 1 ) n−1 l By the matrix K i λ n i=1 j=i j+1 + λn for 1 ≤ λi ≤ li . 1 2 λ

Using STP, we can present the meaning of  λ more clearly. Denote δl i by the λi th models of the logical function fi . Then i for  λ we have

δlλ11  δlλ22  · · ·  δlλnn n−1

(λi −1 )

= δni=1 l

n−1 j=i

l j+1 +λn

i=1 i

λ = δM

, then  = { f 1 , f 1 , . . . , f 1 }, λ = For example, when λ = 1, it represents the model that in the first row of the matrix K 1 n 1 2 n−1 n−1 ( λ − 1 ) l + λ = 1 for λ = 1 . n i i i=1 j=i j+1

represents a possible network with probability Each row of K  K

is the ijth entry in matrix K

. Pλ = P [network  λ is selected] = nj=1 p j i j , where K ij p

Letting x(t ) = ni=1 xi (t ), u(t ) = m u (t ) and y(t ) =  j=1 y j (t ) each BCN can further be converted into the following i=1 i discrete-time system by Lemma 3:



xi (t + 1 ) = Mi u(t )x(t ), i = 1, . . . , n, y j (t ) = N j x(t ), j = 1, . . . , p.

(2)

where Mi is the structure matrix of fi and Nj is the structure matrix of gj . Multiplying the equations in (2) together and model  λ is represented equivalently as



x(t + 1 ) = Lλ u(t )x(t ), y(t ) = Hx(t ),

(3)

where Lλ = M1 ni=2 [(I2n+m  Mi )n+m ] ∈ L2n ×2n+m , H = N1 Pj=2 [(I2n  N j )n ] ∈ L2 p ×2n . And the static output feedback law to be determined for the system (1) is in the following from,

⎧ u (t ) = k (y (t ), . . . , y (t )), n 1 1 1 ⎪ ⎪ ⎪ ⎨ u2 (t ) = k2 (y1 (t ), . . . , yn (t )), (4)

.. ⎪ ⎪ . ⎪ ⎩

um (t ) = km (y1 (t ), . . . , yn (t )),

where ki : D m → D, 1 ≤ i ≤ m are logical functions. Similar to the above analysis, (4) is equivalent to the following algebraic form

u(t ) = Ky(t ),

(5)

where K = [(I2n  Ki )n ] ∈ L2m ×2 p , K = 0, Ki is the structure matrix of ki . Hence, the overall expected value of x(t + 1 ) satisfies K1 m i=2

Ex(t + 1 ) = Lu(t )Ex(t ),

where L = M λ=1 Pλ Lλ . Now, we consider the CS-PBCNs and each model  λ is called a context. The selection of current context  λ is based on the previous one. At each time point, the random decision is made whether to switch a new context with probability

L. Tong et al. / Applied Mathematics and Computation 332 (2018) 263–275

d·

pk 1−pλ

267

(pk , pλ are the probability of context  k and  λ , k = 1, 2, . . . , M, k = λ) or remain in the previous context with

probability 1 − d (0 < d < 1). Therefore, for the CS-PBCNs, the transition probability from x(t) to x(t + 1 ) is

P {x(t + 1 )|x(t ), u(t )} = pλ ·

M 

λ=1

P {x(t + 1 )|x(t ) ∈ λ , u(t )},

where x(t) ∈  λ means that x(t) takes a value as the λth context. Based on x(t) ∈  λ , the transition probability for the context switch to a new context  k from context  λ is d · pk T T 1−p [x (t + 1 )Lk x (t )] and the another transition probability is (1 − d )[x (t + 1 )Lλ x (t )] (the context remains as  λ ). Then λ

we have

P {x(t + 1 )|x(t ), u(t )} = pλ · = pλ ·

M 

λ=1 M 

P {x(t + 1 )|x(t ) ∈ λ , u(t )}

λ=1

k =λ

M  M 

=d·

M 

{d ·

λ=1

k =λ

M



= x (t + 1 ) d ·

M  M 

λ=1

 = x (t + 1 )

 pk [xT (t + 1 )Lk x(t )] + pλ · (1 − d )[xT (t + 1 )Lλ x(t )] 1 − pλ

pλ ·

T

T

pk [xT (t + 1 )Lk x(t )] + (1 − d )[xT (t + 1 )Lλ x(t )]} 1 − pλ

M  M 

λ=1

k =λ

k =λ

λ=1







M  pk pλ · Lk x(t ) + xT (t + 1 ) pλ · (1 − d )Lλ x(t ) 1 − pλ

λ=1

pk pλ · d · L + 1 − pλ k

M 

λ=1



pλ · (1 − d )Lλ x(t ).

From the above equation, we know that the algebraic form of the CS-PBCNs can be expressed as

Ex(t + 1 ) = Lu(t )Ex(t ), where

L=

M  M 

λ=1

 pk L + pλ · ( 1 − d )Lλ . 1 − pλ k M

pλ · d ·

k =λ

λ=1

Before studying the set stabilization of CS-PBCNs, we should analyze the transition probability matrix L. That is

L=

M  M 

λ=1

k = λ



=d

 +

 pk L + pλ · ( 1 − d )Lλ 1 − pλ k M

pλ · d ·

λ=1

p2 p1 p3 p1 pM p1 L2 + L3 + · · · + LM 1 − p1 1 − p1 1 − p1

 + ···

p1 pM p2 pM pM−1 pM L1 + L2 + · · · + LM−1 1 − pM 1 − pM 1 − pM



+ (1 − d )( p1 L1 + p2 L2 + · · · + pM LM )



=d

p2 p1 p3 p1 pM p1 + + ··· + 1 − p2 1 − p3 1 − pM



+d

 +d

 L1 + ( 1 − d ) p1 L1 + · · ·

p1 p2 p3 p2 pM p2 + + ··· + 1 − p1 1 − p3 1 − pM



pM p1 pM p2 pM pM−1 + + ··· + 1 − p1 1 − p2 1 − pM−1

L2 + ( 1 − d ) p2 L2

 LM + ( 1 − d ) pM LM

= d · p1 L1 + (1 − d ) p1 L1 + · · · + d · pM LM + (1 − d ) pM LM =

M 

λ=1

= :

[d · pλ + (1 − d ) pλ ]Lλ

M 

λ=1

Qλ Lλ .

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L. Tong et al. / Applied Mathematics and Computation 332 (2018) 263–275

where pλ =

M  pλ pk , Qλ = d · pλ + (1 − d ) pλ 1 − pk k =λ

M

and λ=1 Qλ = 1. Therefore, we have

L=

M 

λ=1

Qλ Lλ .

3. Main results i

i

Let S = {δ21n , δ22n , . . . , δ2irn } be a subset of 2n . Remark 1. In this paper, we consider the static output feedback set stabilization of CS-PBCNs in terms of finite-time stability. When it comes to finite-time stability, which was defined in autonomous systems [53]. It means that there exists a time T, the system is S-stabilization with probability one for t > T. The concept is similar to that provided by Li et al. [33,37]. Definition 3. (Set Stabilization). The CS-PBCN (1) is said to be S-stabilization with probability one if, for any initial state x(0 ) ∈ 2n , there exists a control sequence U (u(0 ), u(1 ), . . . , u(T − 1 )) and an integer T ≥ 0, such that for all t ≥ T,

P [x(t ) ∈ S | x(0 ) = δ2i n , U] = 1, i = 1, 2, . . . , 2n . Definition 4. The set S ⊆ S is the control invariant set of S for CS-PBCN (1) with probability one if for all is ∈| S|

P [x(t + 1 ) ∈ S | x(t ) = δ2isn , u(t )] = 1. A set S∗ is called the largest control invariant set of S for CS-PBCN (1) with probability one, if it contains the largest number of elements among all the control invariant sets of S. In the following, we will provide an algorithm to find the largest control invariant set S∗ of S. β

j

Algorithm 1. Step 1: If for each x(t ) = δ21n ∈ S there exists u = δ2m , such that

(Col j1 (Blkβ ( L ))) ◦ (12n − [S] ) = 02n . Let

S∗

= S; else let S1 be expressed as a set of states that satisfy the above formula and then go to Step 2. j β Step 2: If for each x(t ) = δ22n ∈ S1 there exists u = δ2m , such that

(Col j2 (Blkβ ( L ))) ◦ (12n − [S1 ] ) = 02n . Let S∗ = S1 ; else let S2 be expressed as a set of states that satisfy the above formula and then go to Step 3. j β Step 3: If for each x(t ) = δ23n ∈ S2 there exists u = δ2m , such that

(Col j3 (Blkβ ( L ))) ◦ (12n − [S2 ] ) = 02n . Let S∗ = S2 ; else let S3 be expressed as a set of states that satisfy the above formula. β j Step 4: Continue the similar process until we find the set Sp such that for each x(t ) = δ2nn ∈ S p , there exists u = δ2m , we have

(Col jn (Blkβ ( L ))) ◦ (12n − [S p ] ) = 02n . Let S∗ = S p be the largest control invariant set of S. Assume that a probabilistic vector v ∈ ϒ n and denote by (v ) the set of some columns of a logical matrix Ln×n , where the position of 1 corresponds to the nonzero elements in the vector v. For example, if v = [a1 a2 0 0 a5 a6 a7 0]T ∈ ϒ 8 , then (v ) = {δ81 , δ82 , δ85 , δ86 , δ87 }. Proposition 1. Using Algorithm 1, the largest control invariant set S∗ of S can be determined for CS-PBCN (1) with probability one. β

Proof. For all x(t ) = δ2nn ∈ S p , there exists u = δ2m , we have j

Ex(t + 1 ) = Lu(t )Ex(t ) β

= L  δ2m  δ2jnn = Blkβ ( L )δ2jnn = Col jn (Blkβ ( L )).

L. Tong et al. / Applied Mathematics and Computation 332 (2018) 263–275

269

Since (Col jn (Blkβ ( L ))) ◦ (12n − [S p ] ) = 02n , we have (Col jn (Blkβ ( L ))) ⊆ S p , which implies that β

P [x(t + 1 ) ∈ S p | x(t ) = δ2isn , u = δ2m ] = 1, for is ∈ |Sp |. Thus, Sp is a control invariant set of S, and it is easy to note that Sp is the largest control invariant set S∗ of S with probability one from Algorithm 1. This completes the proof.  Remark 2. Algorithm 1 can be applied to normal BCNs, and the difference on the algorithm between PBCNs and BCNs is n that Col ( L ) ⊆ ϒ 2 and Col (L ) ⊆ 2n , respectively. a

a

a

Assume S∗ = {δ2n1 , δ2n2 , . . . , δ2nq }. Denote by Rk (S∗ ) the set consisting of all the states that can be steered to S∗ with probability one in k steps. Without lose of generality, we suppose that R0 (S∗ ) = S∗ . Then, we have the following lemmas. Lemma 4. For CS-PBCN (1), β (i) R1 (S∗ ) = {δ2i n : there exists u = δ2m such that

j∈|S∗ | (Blkβ (L )) j,i = 1}. β

(ii) Rk+1 (S∗ ) = {δ2i n : there exists u = δ2m such that

j∈|R (S∗ )| (Blkβ (L )) j,i = 1}. k

β

Proof. (i) Note that for x(t ) = δ2i n , u = δ2m , we have β Ex(t + 1 ) = Lu(t )Ex(t ) = L  δ2m  δ2i n

= Bl kβ ( L )δ2i n = Coli (Bl kβ ( L )). Hence, x(t + 1 ) ∈ S∗ a β β ⇔ ∃u = δ2m , such that a ∈|S∗ | P [x(t + 1 ) = δ2nj | x(t ) = δ2i n , u = δ2m ] = 1, j β ⇔ ∃u = δ2m , such that (Coli (Blkβ ( L ))) ⊆ S∗ , β ⇔ ∃u = δ2m , such that a ∈|R0 (S∗ )| (Blkβ ( L ))a j ,i = 1. j

β

(ii) Note that for x(t ) = δ2i n ∈ Rk+1 (S∗ ), u = δ2m , we have Ex(t + 1 ) = Coli (Blkβ ( L )). Thus, x(t + 1 ) ∈ Rk (S∗ ) aj β β ⇔ ∃u = δ2m , such that a ∈|R (S∗ )| P [x(t + 1 ) = δ2n | x(t ) = δ2i n , u = δ2m ] = 1, j

k

β ⇔ ∃u = δ2m , such that (Coli (Blkβ ( L ))) ⊆ Rk (S∗ ), β ⇔ ∃u = δ2m , such that a ∈|R (S∗ )| (Blkβ ( L ))a j ,i = 1. j

This completes the proof.



k

Lemma 5. (i) If S∗ ⊆R1 (S∗ ), then Rk (S∗ ) ⊆ Rk+1 (S∗ ) for all k ≥ 1. (ii) If R1 (S∗ ) = S∗ , then Rk (S∗ ) = S∗ for all k > 1. (iii) If R j+1 (S∗ ) = R j (S∗ ) for some j ≥ 1, then Rk (S∗ ) = R j (S∗ ) for all k ≥ j. β

Proof. Firstly, we prove conclusion (i) by induction. Suppose that δ2i n ∈ R1 (S∗ ). Then, there exists u = δ2m such that aj β i a j ∈|S∗ | P [x (t + 1 ) = δ2n | x (t ) = δ2n , u = δ2m ] = 1, ⇔ a ∈|S∗ | (Blkβ ( L ))a j ,i = 1, which together with S∗ ⊆R1 (S∗ ) implies that j aj β i a j ∈|R1 (S∗ )| P [x (t + 1 ) = δ2n | x (t ) = δ2n , u = δ2m ] = 1,

⇔ = 1. ∗ (Blk (L )) a j ∈| R 1 ( S ) |

β

a j ,i

Hence, δ2i n ∈ R2 (S∗ ) and R1 (S∗ )⊆R2 (S∗ ).

β

Assume that Rk−1 (S∗ ) ⊆ Rk (S∗ ) holds for the integer k > 1. For δ2i n ∈ Rk (S∗ ), there exists u = δ2m such that aj β i a j ∈|Rk−1 (S∗ )| P [x (t + 1 ) = δ2n | x (t ) = δ2n , u = δ2m ] = 1,

⇔ = 1. ∗ (Blk (L )) β

a j ∈|Rk−1 (S )|

a j ,i

Since Rk−1 (S∗ ) ⊆ Rk (S∗ ), then we have aj β i a j ∈|Rk (S∗ )| P [x (t + 1 ) = δ2n | x (t ) = δ2n , u = δ2m ] = 1,

⇔ = 1. ∗ (Blk (L )) a j ∈| R k ( S ) |

β

a j ,i

Hence, δ2i n ∈ Rk+1 (S∗ ) and Rk (S∗ ) ⊆ Rk+1 (S∗ ). Secondly, we shall prove conclusion (ii), the proof of conclusion (iii) is quiet similar. Then, we use induction on k. It is obvious that R1 (S∗ ) = S∗ holds for k = 1. Assume that Rk (S∗ ) = S∗ holds for the integer k ≥ 1. From R1 (S∗ ) = S∗ and β

conclusion (i), one can see that S∗ = Rk (S∗ ) ⊆ Rk+1 (S∗ ). And for δ2i n ∈ Rk+1 (S∗ ), there exists u = δ2m such that aj β i a j ∈|Rk (S∗ )| P [x (t + 1 ) = δ2n | x (t ) = δ2n , u = δ2m ] = 1,

⇔ = 1. ∗ (Blk (L )) a j ∈| R k ( S ) |

β

a j ,i

270

L. Tong et al. / Applied Mathematics and Computation 332 (2018) 263–275

Thus, ⇔



a j ∈| ( S ∗ ) |

β

a

+ 1 ) = δ2nj | x(t ) = δ2i n , u = δ2m ] = 1, (Blk ( L )) = 1, which implies that

a j ∈| (S∗ )| P [x (t

β

a j ,i

δ2i n ∈ R1 (S∗ ) = S∗ . Hence, Rk+1 (S∗ ) ⊆ S∗ . By induction, Rk (S∗ )⊆S∗ holds for any integer k ≥ 1. This completes the proof.



From system (3) and the controller (5), we know that

u(t ) =Ky(t ) = KHx(t ). Now, we construct the static output feedback control u(t ) = Ky(t ) with

K = δ2m [k1 , k2 , . . . , k2 p ]. And then, for H = δ2 p [v1 , v2 , . . . , v2n ], one can obtain that

KH = δ2m [kv1 , kv2 , . . . , kv2n ]. Theorem 1. The CS-PBCN (1) is S-stabilizable by a static output feedback controller (4) with K = 0, if and only if, (a) S∗ = ∅; (b) There exists an integer 0 ≤ T ≤ 2n − S∗ , such that RT (S∗ ) = 2n . Proof. (Sufficiency). Assume that S∗ is a nonempty set and there exists an integer 0 ≤ T ≤ 2n − S∗  such that RT (S∗ ) = 2n . Then, for all x(0 ) ∈ 2n , we have

P [x(T ) ∈ S∗ | x(0 ) ∈ 2n , u(t ) = Ky(t )] = 1, Furthermore, for all x(T) ∈ S∗ , we have a

P [x(T + 1 ) ∈ S∗ | x(T ) = δ2nj , u(t ) = Ky(t )] = 1, for aj ∈ |S∗ |. Hence, for any initial state x(0 ) ∈ 2n , we have x(t) ∈ S∗ ⊆S, for t ≥ T, and CS-PBCN (1) is S-stabilizable. (Necessity). Assume that CS-PBCN (1) is S-stabilizable by a static output feedback controller with the form (4), and there exists an integer T such that

P [x(t ) ∈ S | x(0 ) ∈ 2n , u(t ) = Ky(t )] = 1, for t ≥ T. Then, we prove the necessity by contradiction. For condition (a), if S∗ = ∅, then it means that S does not have the largest control invariant set S∗ , and all the states in CS-PBCN (1) will not be stabilized after several iterations, which contradicts the definition of S-stabilizable. As for the condition (b), first we prove RT (S∗ ) = 2n , and then explain 0 ≤ T ≤ 2n − S∗ . If S∗ = ∅, the condition (b) is not satisfied, such that for any integer T, RT (S∗ ) = 2n . That is, for an initial state kv

x(0 ) = δ2i n , there exists a control u = KH δ2i n = δ2mi , such that kv

P [x(T ) ∈ S | x(0 ) = δ2i n , u = δ2mi ] < 1. If x(T) ∈ S, which means that

P [x(T ) ∈ S | x(0 ) ∈ 2n , u(t ) = Ky(t )] < 1, which contradicts the S-stability of CS-PBCN (1). And then, we will prove that T ≤ 2n − S∗ . Assume that S∗  = p, and let T be the smallest positive integer such that RT (S∗ ) = 2n . It is enough to show that Rk (S∗ ) ≥ k + p for every 1 ≤ k ≤ T. We shall use induction on k. If k = 1, assume that R1 (S∗ ) ≥ 1 + p and R1 (S∗ ) = p. Thus, R1 (S∗ ) = S∗ which leads to RT (S∗ ) = S∗ by Lemma 5(ii) and contrary to our choice of T. Now let 1 < k ≤ T and assume that Rk−1 (S∗ ) ≥ k − 1 + p. Then we have Rk−1 (s∗ ) ⊆ Rk (s∗ ) from Lemma 5(i), which means that Rk (S∗ ) ≥ Rk−1 (S∗ ) ≥ k − 1 + p. If Rk (S∗ ) < k + p, then we have Rk (S∗ ) = k − 1 + p. That is, Rk−1 (s∗ ) = Rk (s∗ ) implies that RT (s∗ ) = Rk−1 (s∗ ) = 2n by Lemma 5(iii), which contradicts the minimality of T. Thus, Rk (S∗ ) ≥ k + p. Therefore, T ≤ 2n − p = 2n − S∗ . Therefore, if CS-PBCN (1) is S-stabilizable by an static output feedback controller (4), the conditions (a) and (b) are satisfied. This completes the proof.  Suppose that conditions (a) and (b) in Theorem 1 hold, then 2n can be represented as the union of disjoint sets

2n =R0 (S∗ ) ∪ (R1 (S∗ ) \ R0 (S∗ )) ∪ · · · ∪ (RT (S∗ ) \ RT −1 (S∗ )).

(6)

Hence, to each 1 ≤ i ≤ 2n there correspond a unique integer 0 ≤ li ≤ T such that δ2i n ∈ (Rli (S∗ ) \ Rli −1 (S∗ )) where R−1 (S∗ ) = ∅. If li = 0, then we can choose a vector kvi ∈ 2m such that

L. Tong et al. / Applied Mathematics and Computation 332 (2018) 263–275 kv

P [x(t + 1 ) ∈ S∗ | x(t ) = δ2i n , u = δ2mi ] = 1, ⇐⇒

 j∈| | S∗

271

(Blkkvi ( L )) j,i = 1,

(7)

for i ∈ |S∗ |. Otherwise (i.e., if 2 ≤ li ≤ T), we can find kvi ∈ 2m such that kv

P [x(t + 1 ) ∈ Rli −1 (S∗ ) | x(t ) = δ2i n , u = δ2mi ] = 1, ⇐⇒



j∈|Rl −1 (S∗ )|

(Blkkvi ( L )) j,i = 1.

(8)

i

Theorem 2. Consider CS-PBCN (1), if (6) holds, every 1 ≤ i ≤ 2n corresponds a unique integer 0 ≤ li ≤ T such that δ2i n ∈ (Rli (S∗ ) \ Rli −1 (S∗ )) where R−1 (S∗ ) = ∅. Let 1 ≤ kvi ≤ 2m satisfy

⎧ (Blkkvi ( L )) j,i = 1, ⎨ j∈|S∗ | ⎩



j∈|Rl −1 (S∗ )| i

(Blkkvi ( L )) j,i = 1, f or li ≥ 2.

Then the static output feedback control law (4) with the static output feedback matrix K is given by K = δ2 m [ k 1 , k 2 , · · · , k 2 p ] . Proof. Let x(0 ) = δ2i n ∈ 2n , we have

Ex(1; x(0 ); KHx(0 )) = Lu(t )Ex(0 ) = LKH δ2i n δ2i n = Blkkv ( L )δ2i n i

= Coli (Blkkv ( L )). i

If li = 1,

then

j∈|Rl −1 (S∗ )| (Blkkvi i



x(0) ∈ R1 (S∗ ), we have implies j∈|S∗ | (Blkkvi (L )) j,i = 1 ∗

(L )) j,i = 1 implies that x(1; x(0 ); KHx(0 )) ∈ Rli −1 (S ). 

that

x(1;

x(0);

KHx(0)) ∈ S∗ .

If

li ≥ 2,

Remark 3. If the logical matrix H is full row rank, vi can take all the values from 1 to 2p . So, we can get all the values of ki in KH for i = 1, . . . , 2 p from Theorem 2. That means each column in matrix K has a corresponding value. For example, we assume that H = δ4 [2, 1, 4, 3] is full row rank, and n = 2, m = 2. Then we have KH = δ4 [k2 , k1 , k4 , k3 ] with x(t) ∈ 4 . Because u(t) is known from (7) and (8) when designing the controller, so we can get the values of ki with i = 1, 2, 3, 4. Otherwise, if the output logic matrix H is not full row rank, then vi will not take all the values from 1 to 2p . For example, we assume that H = δ4 [2, 1, 1, 1] is not full row rank, and n = 2, m = 2. Then we have KH = δ4 [k2 , k1 , k1 , k1 ] with x(t) ∈ 4 . So using the analysis of (7), (8) and the application of Theorem 2, one can only get the value of k2 , k1 in the set {1, 2, 3, 4}. As for the remaining k3 and k4 , it can be arbitrarily selected from {1, 2, 3, 4}, since these columns in the matrix K do not affect the input controllers. Actually, we can see kv

u(t ) = KHx(t ) = KH δ2i n = K  Coli H = K δ2vip = Colvi K = δ2mi with x(t ) = δ2i n , for i ∈ {1, 2, . . . , 2n }. Since H is not full rank, vi can only take a subset of {1, 2, . . . , 2 p }. So according the above equation and (7), (8), we can only get some kj for j = vi . It means the values of the remaining kj does not affect S-stabilizable for the system, so the remained kj can be any values from {1, 2, . . . , 2m }. Remark 4. If considered systems are normal BCNs but not CS-PBCNs, the obtained results can also be used to study the set stabilization problem. Thus, it shows that our results are more general relatively. Remark 5. In fact, it would be possible to design an observer for a BN or a CS-PBCN, e.g. Using the tools given in [15,16,54] and stabilize the desired set by using the obtained simulate of the state. 4. Examples Example 1. Consider the following CS-PBCNs



x1 (t + 1 ) = f1 (x1 (t ), x2 (t ), u(t )), x2 (t + 1 ) = f2 (x1 (t ), x2 (t ), u(t )),

where

f11 = x2 (t ) ∨ u(t ), P [ f1 = f11 ] = 0.2, f12 = ¬x2 (t ) ∨ u(t ), P [ f1 = f12 ] = 0.8, f21 = (u(t ) ∧ x1 (t )) ∨ [¬u(t ) ∧ (¬(x1 (t ) ↔ x2 (t )))], P [ f2 = f21 ] = 0.1,

(9)

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L. Tong et al. / Applied Mathematics and Computation 332 (2018) 263–275

f22 = x1 (t ) ∨ x2 (t ), P [ f2 = f22 ] = 0.9. And the measured outputs are assumed as



y1 (t ) = x2 (t ), y2 (t ) = x1 (t ).

Let y(t ) = y1 (t )  y2 (t ), then we have H = δ4 [1, 3, 2, 4]. Assume that S = {δ41 , δ43 }, then we should construct a static output feedback controller such that S-stabilization is satis and the model probabilities are fied. Now, the index matrix K

⎡

⎤

1 ⎢

= ⎣1 K 2 2

1 2⎥ 1⎦ 2

and

1 , 50 9 p2 = 0.2 × 0.9 = 0.18 = , 50 2 p3 = 0.8 × 0.1 = 0.08 = , 25 18 p4 = 0.8 × 0.9 = 0.72 = . 25 p1 = 0.2 × 0.1 = 0.02 =

Let d =

1 2,

L=

according to 4  4 

λ=1

We have

 pk Lk + pλ · ( 1 − d )Lλ . 1 − pλ 4

pλ · d ·

k =λ

λ=1









1 p2 p1 p3 p1 p4 p1 1 p1 p2 p3 p2 p4 p2

L= + + + p1 L1 + + + + p2 L2 2 1 − p2 1 − p3 1 − p4 2 1 − p1 1 − p3 1 − p4     1 p3 p1 p3 p2 p4 p3 1 p4 p1 p4 p2 p4 p3 + + + + p3 L3 + + + + p4 L4 2 1 − p1 1 − p2 1 − p4 2 1 − p1 1 − p2 1 − p3 = Q1 L1 + Q2 L2 + Q3 L3 + Q4 L4 . By a simple calculation,

12799 ≈ 0.039, 330050 18657 Q2 = ≈ 0.331, 56350 7657 Q3 = ≈ 0.152, 50225 551799 Q4 = ≈ 0.478, 1155175 Q1 =

and 4 

λ=1

Qλ = 1.

Let x(t ) = x1 (t )  x2 (t ), and the state transition matrix of each network can be calculated as

L1 = δ4 [1, 1, 2, 2, 2, 3, 1, 4], L2 = δ4 [1, 1, 1, 2, 1, 3, 1, 4], L3 = δ4 [1, 1, 2, 2, 4, 1, 3, 2], L4 = δ4 [1, 1, 1, 2, 3, 1, 3, 2]. Then, the probability matrix L is described as

L. Tong et al. / Applied Mathematics and Computation 332 (2018) 263–275

⎡

1

⎢0

L=⎣ 0 0

0.809 0.191 0 0

1 0 0 0

0 1 0 0

0.331 0.039 0.478 0.152

0.63 0 0.37 0

0.37 0 0.63 0

0 0.63 0 0.37

273

⎤ ⎥ ⎦

By using Algorithm 1, it can be verified that (Col1 (Blk1 ( L ))) ◦ (14 − [S] ) = 04 and k1 = 1; (Col3 (Blk2 ( L ))) ◦ (14 − [S] ) = 04 and k2 = 2. Then the largest control invariant set S∗ = S = {δ41 , δ43 }. 4

From Theorem 2, it can be calculated that j∈{1,3} (Blkk3 (L )) j,2 = 1, then we have k3 = 1 or 2. For δ4 , (Blk ( L )) = 1, then we get k = 1. Therefore, one of the static output feedback control laws is given by j∈{1,3,2}

k4

4

j,4

K = δ2 [1, 2, 2, 1]. And then, the static output feedback control u(t ) = y1 (t ) ↔ y2 (t ) such that S-stabilization is satisfied for CS-PBCNs (9). Example 2. Consider a gene regulatory network containing four genes: WNT5A, pirin, S100P and STC3. The network was used to study metastatic melanoma in [40]. These genes can take value 1 or 0, which denotes the gene expressed or unexpressed. Then, we can apply the procedure given in [40] to generate a Boolean networks. Assume that WNT5A and S100P are states (denoted by x1 and x2 , respectively), and both pirin and STC3 are controls (denoted by u1 and u2 , respectively). As for outputs, WNT5A and S100P (with value 1) are expressing y1 and y2 , set x1 , x2 , u1 , u2 , y1 , y2 ∈ D := {1, 0}. It is easy to see that the dynamics of the gene regulatory network can be described as



x1 (t + 1 ) = f1 (x1 (t ), x2 (t ), u1 (t ), u2 (t )),

(10)

x2 (t + 1 ) = f2 (x1 (t ), x2 (t ), u1 (t ), u2 (t )), where

f11 = u1 (t ) ∧ [u2 (t ) ∧ (x1 (t ) → x2 (t )) ∨ (¬u2 (t ) ∧ (x1 (t ) ∧ x2 (t )))] ∨ [¬u1 (t ) ∧ u2 (t ) ∧ (x1 (t ) ↔ x2 (t ))], f12 = u1 (t ) ∧ (¬x1 (t ) ∧ x2 (t )) ∨ u2 (t ) ∧ (¬x2 (t )), f2 = ¬u1 (t ) ∧ u2 (t ) ∧ x2 (t ) ∨ (¬x1 (t )) ∨ x2 (t ). and



y1 (t ) = x1 (t ),

(11)

y2 (t ) = 1.

So after calculation with STP, we have H = δ4 [1, 1, 1, 2], and it is not full row rank. Assume that f 1 ∈ { f11 , f12 } with probability P11 = 0.6 and P12 = 0.4, the random selection probability d = 0.3, and the set S = {δ41 , δ43 , δ44 }. Let x(t ) = x1 (t )  x2 (t ), u(t ) = u1 (t )  u2 (t ), y(t ) = y1 (t )  y2 (t ), the algebraic form of the system (8) can be given as



x1 (t + 1 ) = M1 u(t )x(t ),

(12)

x2 (t + 1 ) = M2 u(t )x(t ), where M1 ∈ {M11 , M12 }, and

M11 = δ2 [1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2], M12 = δ2 [1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 2, 2], M2 = δ2 [1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 2]. According to the analysis of pages 8 and 9, we have









p2 p1 p1 p2

L= d· + ( 1 − d ) · p1 L1 + d · + ( 1 − d ) · p2 L2 = Q1 L1 + Q2 L2 . 1 − p2 1 − p1 By a simple calculation,

Q1 = 0.54, Q2 = 0.46, and 2 

λ=1

Qλ = 1,

and the state transition matrix of each network can be calculated as

L1 = δ4 [1, 2, 3, 2, 1, 2, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4], L2 = δ4 [1, 2, 1, 2, 3, 2, 3, 4, 3, 4, 1, 2, 3, 4, 3, 4].

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L. Tong et al. / Applied Mathematics and Computation 332 (2018) 263–275

Then, the probability matrix L is described as

⎡

1 0 ⎢

L=⎣ 0 0

0 1 0 0

0.46 0 0.54 0

0 1 0 0

0.54 0 0.46 0

0 1 0 0

0 0 1 0

0 0 0 1

0 0 1 0

0 0 0 1

0.46 0 0.54 0

0 0.46 0 0.54

0 0 1 0

0 0 0 1

0 0 1 0

⎤

0 0⎥ 0⎦ 1

By using Algorithm 1, it can be verified that (Col1 (Blkk1 ( L ))) ◦ (14 − [S] ) = 04 for k1 = 1, 2, 3, 4; (Col3 (Blkk1 ( L ))) ◦ (14 −

[S] ) = 04 for k1 = 1, 2, 3, 4; (Col4 (Blkk2 (L ))) ◦ (14 − [S] ) = 04 for k2 = 2, 4; Then the largest control invariant set S∗ = S = {δ41 , δ43 , δ44 }. From Theorem 2, it can be calculated that j∈{1,3,4} (Blkk1 ( L )) j,2 = 1, then we have k1 = 3 or 4. As for k3 , k4 , they can take arbitrary values in 4 . Therefore, one of the output feedback control laws is given by

K = δ4 [3, 4, 2, 4]. Then, the static output feedback control u(t) will be



u1 (t ) = ¬y1 (t ) ∧ y2 (t ), u2 (t ) = y1 (t ) ∧ y2 (t ),

such that S-stabilization is achieved for system (10). 5. Conclusions In this paper, we studied the static output feedback set stabilization of CS-PBCNs. Using the algebraic representation of logic functions, an algorithm for finding the largest control invariant set S∗ of S with probability one was provided. Moreover, we have obtained a necessary and sufficient condition for the existence of the S-stabilizable static output feedback controllers. Then, two examples have been given to show the efficiency of the proposed results. Acknowledgment This work was partially supported by the National Natural Science Foundation of China under Grant nos. 11671361, 61772199, 61573102, and the China Postdoctoral Science Foundation under Grant no. nos. 2015M580378 and 2016T90406, the Jiangsu Provincial Key Laboratory of Networked Collective Intelligence under Grant No. BM2017002, and the Natural Science Foundation of Jiangsu Province of China under grant BK20170019. References [1] S.A. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets, J. Theor. Biol. 22 (1969) 437–467. [2] E.H. Davidson, J.P. Rast, P. Oliveri, A. Ransick, C. Calestani, C.H. Yuh, T. Minokawa, G. Amore, V. Hinman, C. Arenas-Mena, et al., A genomic regulatory network for development, Science 295 (2002) 1669–1678. [3] D. Cheng, H. Qi, Z. Li, Analysis and Control of Boolean Networks: A Semi-Tensor Product Approach, Springer Science & Business Media, 2010. [4] D. Cheng, H. Qi, A linear representation of dynamics of boolean networks, IEEE Trans. Autom. Control 55 (2010) 2251–2258. [5] D. Cheng, H. Qi, Controllability and observability of boolean control networks, Automatica 45 (2009) 1659–1667. [6] D. Laschov, M. Margaliot, Controllability of boolean control networks via the perron–frobenius theory, Automatica 48 (2012) 1218–1223. [7] F. Li, J. Sun, Controllability of higher order boolean control networks, Appl. Math. Comput. 219 (2012) 158–169. [8] Q. Zhu, Y. Liu, J. Lu, J. Cao, Observability of boolean control networks, Sci. China Inf. Sci. 61 (2018) 092201. [9] K. Zhang, L. Zhang, L. Xie, Finite automata approach to observability of switched boolean control networks, Nonlinear Anal. Hybrid Syst. 19 (2016) 186–197. [10] J. Lu, M. Li, Y. Liu, D.W.C. Ho, J. Kurths, Nonsingularity of grain-like cascade FSRs via semi-tensor product, Sci. China Inf. Sci. 61 (2018) 010204. [11] D. Laschov, M. Margaliot, Minimum-time control of boolean networks, SIAM J. Control Optim. 51 (2013) 2869–2892. [12] L. Wang, Y. Liu, Z. Wu, F.E. Alsaadi, Strategy optimization for static games based on STP method, Appl. Math. Comput. 316 (2018) 390–399. [13] Y. Liu, H. Chen, B. Wu, Controllability of boolean control networks with impulsive effects and forbidden states, Math. Methods Appl. Sci. 37 (2014a) 1–9. [14] Y. Liu, J. Lu, B. Wu, Some necessary and sufficient conditions for the controllability of temporal boolean control networks, ESAIM Control Optim. Calc. Var. 20 (2014b) 158–173. [15] E. Fornasini, M.E. Valcher, Observability, reconstructibility and state observers of boolean control networks, IEEE Trans. Autom. Control 58 (2013) 1390–1401. [16] C. Possieri, A.G. Busetto, Observer design for boolean control networks with unknown inputs, IET Control Theory Appl. 11 (2017) 2116–2121. [17] H. Li, L. Xie, Y. Wang, Output regulation of boolean control networks, IEEE Trans. Autom. Control 62 (2017) 2993–2998. [18] Y. Liu, B. Li, H. Chen, J. Cao, Function perturbations on singular boolean networks, Automatica 84 (2017) 36–42. [19] B. Gao, H. Peng, D. Zhao, W. Zhang, Y. Yang, Attractor transformation by impulsive control in boolean control network, Math. Probl. Eng. (2013a) 1–8. [20] B. Gao, L. Li, H. Peng, J. Kurths, W. Zhang, Y. Yang, Principle for performing attractor transits with single control in boolean networks., Phys. Rev. E 88 (2013b) 062706. [21] E. Fornasini, M.E. Valcher, On the periodic trajectories of boolean control networks, Automatica 49 (2013) 1506–1509. [22] R. Li, M. Yang, T. Chu, Synchronization of boolean networks with time delays, Appl. Math. Comput. 219 (2012) 917–927. [23] J. Zhong, J. Lu, T. Huang, D.W.C. Ho, Controllability and synchronization analysis of identical-hierarchy mixed-valued logical control networks, IEEE Trans. Cybern. 47 (2017) 3482–3493. [24] Y. Liu, L. Sun, J. Lu, J. Liang, Feedback controller design for the synchronization of boolean control networks, IEEE Trans. Neural Netw. Learn. Syst. 27 (2016) 1991–1996. [25] J. Lu, J. Kurths, J. Cao, N. Mahdavi, C. Huang, Synchronization control for nonlinear stochastic dynamical networks: pinning impulsive strategy, IEEE Trans. Neural Netw. Learn. Syst. 23 (2012) 285–292.

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