Stabilization of Boolean control networks with stochastic impulses

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Stabilization of Boolean control networks with stochastic impulses Xin Hu a,b, Chi Huang a,c,∗, Jianquan Lu c, Jinde Cao c a School

of Economic Information Engineering, Southwestern University of Finance and Economics, Chengdu 611130, China b College of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China c School of Mathematics, Southeast University, Nanjing 210096, China Received 27 June 2018; received in revised form 10 June 2019; accepted 22 June 2019 Available online xxx

Abstract This paper studies the stabilization problem of Boolean control networks with stochastic impulses, where stochastic impulses model is described as a series of possible regulatory models with corresponding probabilities. The stochastic impulses model makes the research more realistic. The global stabilization problem is trying to drive all states to reach the predefined target with probability 1. A necessary and sufficient condition is presented to judge whether a given system is globally stabilizable. Meanwhile, an algorithm is proposed to stabilize the given system by designing a state feedback controller and different impulses strategies. As an extension, these results are applied to analyze the global stabilization to a fixed state of probability Boolean control networks with stochastic impulses. Finally, two examples are given to demonstrate the effectiveness of the obtained results. © 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.

1. Introduction Boolean networks (BNs) were firstly introduced by Kauffman [1] in 1969 to describe the genetic circuits. In BNs, every gene is symbolized as a binary variable, 0 (inactive) or 1 (active) per time, whose next state is determined by Boolean functions. Compared with other ∗ Corresponding author at: School of Economic Information Engineering, Southwestern University of Finance and Economics, Chengdu 611130, China. E-mail address: [email protected] (C. Huang).

https://doi.org/10.1016/j.jfranklin.2019.06.039 0016-0032/© 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.

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gene regulatory models, BNs are much simpler in structure, which have become a powerful tool in describing, analyzing and simulating cellular networks. Hence, the applications of BNs have been extensively studied by biologists, chemists and system scientists [2–5]. As is well known, it is a common problem to design effective intervention strategies to reach desirable cellular states when modeling gene regulatory networks as BNs. For example, a binary input may present whether a certain intervention is administered or not at each time step [6]. When control inputs and outputs are added to BNs, the concept is naturally extended to Boolean control networks (BCNs). Since BCNs share the appealing properties of BNs and also deal with the presence of control, they have become the focus of research [7,8]. Recently, a matrix product, namely the semi-tensor product (STP) is used to analyze BNs by Cheng et al. [9]. The advantage of this method is that one can convert a BN (BCN) to a linear (bilinear) discrete-time system. Then, many challenging problems of BNs have been solved, including controllability and observability [10–12], stability and stabilization [6,13– 16], disturbance decoupling [17,18] and other problems [19–23]. Besides, the STP method has also been applied to other kinds of BNs, for example, probabilistic Boolean control networks (PBCNs), which are derived from BCNs and coped with the stochastic nature of the genetic regulation networks [24–28]. In [27], Kobayashi and Hiraishi proposed methods on optimal control of PBNs. Trairatphisan et al. [28] summarized the recent developments and biomedical applications of PBNs. As one of the most important behavior of dynamical networks, stability and stabilization have been widely studied in ecological systems [29,30], neural networks [31,32], and so on. Stabilization of BCNs, which can be achieved by designing moderate controllers, such as robust controllers [33,34], state feedback controllers [35,36] and impulsive controllers [37,38], is a significant research field. For example, in the treatment of disease, medicine therapy strategies need to be designed such that patients are steered to the desirable state and maintained this state afterward [6]. There are many studies of stabilization both for deterministic and stochastic BCNs. Cheng et al. [39] investigated the stabilization of BCNs via the openloop control and the state feedback control. Later, the stabilization researches were extended to the design of controllers such that BCNs were converged to the same periodic trajectory [40]. Based on trajectory stabilization, Zheng et al. [14] studied the stabilization and set stabilization of delayed BCNs by designing state feedback controllers. Furthermore, Li and Wang provided a necessary and sufficient condition to design all possible output feedback controllers in [41]. However, many practical networks are not inherently deterministic. There exists a common requirement to deal with the probability problem. Some control mechanisms have been uncovered in the solution of this problem, which are fundamental to stabilize the whole system. Zhao and Cheng [42] designed a control sequence such that the PBCN converges to a fixed point with probability one. Li and Tang [43] analyzed the set stabilization of switched BCNs by designing two kinds of state feedback controllers. In the real world, evolutionary processes in biological networks are often subject to instantaneous disturbances and abrupt changes at certain instants, which may be caused by switching phenomena, or other sudden changes, i.e., they exhibit impulsive effects [44]. On the other hand, impulsive effects are unavoidable in biological systems due to the mutability of external environment. Hence, different kinds of models, including neural networks [45,46], differential equations [47,48], dynamical networks [49] and BNs [37,38,50,51], have been investigated for the stability and stabilization under the influence of impulses. Since BNs are used widely to model biological networks, the study of stabilizing BNs with impulsive effects is meaningful. In [37], the stability and stabilization of BNs with impulsive effects were Please cite this article as: X. Hu, C. Huang and J. Lu et al., Stabilization of Boolean control networks with stochastic impulses, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.06.039

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investigated, and some necessary and sufficient conditions were obtained. Subsequently, the research was extended to switched BNs with impulsive effects under two types of controls [38]. The stabilization of BNs with impulsive effects and state constraints was investigated in [50]. In these results, the impulses were described as deterministic models. As we all know, stochastic phenomena are very common in nature, and stochastic models have come to play an important role in many branches of science and industry. As an important influence factor of genetic evolution in biological network, impulses are usually generated with randomness. We call these as stochastic impulses. In the past decades, stochastic impulses have been used to solve optimal control problems, including the management of renewable resources [52], discovering optimal strategy for investment models [53] and so on. The stochastic impulses, which are used to describe the random external environment, are represented as several possible regulatory models in this study. However, to our best knowledge, there is little research on BNs with stochastic impulses. Motivated by the above discussions, the stabilization of BCNs with stochastic impulses will be investigated. Different from [37], where the impulse model was deterministic, we study stochastic model at stochastic impulses instants. In this case, each impulse model corresponds to a certain probability, which makes the system structure more complicated. To solve this problem, a series of matrices and convergable sets are defined. A necessary and sufficient condition based on convergable sets is presented to solve the stabilization problem for BCNs with stochastic impulses. Meanwhile, state feedback controller and different impulsive strategies are designed to influence the trajectory of each state. Finally, these obtained results are extended to the stabilization of PBCNs with stochastic impulses. The rest of this paper is organized as follows. Section 2 gives a brief introduction to some notations. The establishment of BCNs with stochastic impulses and its algebraic form are also discussed. Section 3 presents the main results of this paper. Based on the results obtained in Section 3, some conditions to analyze the stabilization of PBCNs with stochastic impulses are proposed in Section 4. Then, numerical examples are given to show the efficiency of the obtained results in Section 5. Finally, conclusions are summarized in Section 6. 2. Preliminaries 2.1. Notations and definitions • • • • • • • • • •

Z denotes the set of integer numbers. Denote by AT the transpose of matrix A. In represents the n × n identity matrix, 12m := (1, 1, . . . , 1)T , 02m := (0, 0, . . . , 0)T . D := {T = 1, F = 0}, where T ∼ [1 0]T and F ∼ [0 1]T . Denote the ith column of matrix A by Coli (A), Col(A) is the set of all columns of A. Rowi (A ) represents the ith row of matrix A. n := {δni |1 ≤ i ≤ n}, where δni := Coli (In ). i δm [i1 . . . in ] represents matrix A with Col j (A ) = δmj . |A| represents the cardinal number of the set A. The set of n × m logical matrix L is defined by Ln×m , where Col(L) ⊆ n , Bn×m represents the set of n × m Boolean matrix B and (B)ij ∈ {0, 1}. Suppose that A = (ai j ), B = (bi j ) ∈ Bm×n . Then A ∨ B = (ai j ∨ bi j ), A ∧ B = (ai j ∧ bi j ). The symbols “∨” and “∧” represent the logical operators OR and AND, respectively.

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• Denote the Kronecker product of two matrices by . • The Boolean addition is defined as follows: n  (B) αi := α1 ∨ α2 ∨ . . . ∨ αn , ∀αi ∈ D. k=1

• For any X ∈ Bn×m and Y ∈ B p×q , if m = p, the Boolean product X B Y is defined by m  (X B Y )i j = (B) xik ∧ yk j . If m = t p or p = tm, the Boolean product is defined as k=1

X B Y = X B (Y  It ) and X B Y = (X  It ) B Y , respectively. • Denote by (E) the set of all of the logical sub-matrices for any Boolean matrix E ∈ Bm×n , i.e., (E ) = {e|E ∧ e = e, e ∈ Lm×n }. It is worth noting that  T (E ) = (E T ). • Coefficient matrix L ∈ L2n ×2n+m can be split into 2m logic square blocks, denoting the ith block as Blki (L), i = 1, 2, . . . , 2m . • “ ·” represents the Boolean under floor. In particular, the positive integers greater than or equal to 1 become 1 in this paper. For an m × n matrix A = (ai j ), A = ( ai j  ). Definition 1 [14]. The semi-tensor product (STP) of two matrices L ∈ Rm×n and G ∈ Rq×p is introduced as follows: L  G = (L  Ic/n )(G  Ic/q ), c = lcm(n, q), where lcm(n, q) denotes the least common multiple of n and q. Note that the STP of matrices becomes the conventional product when n = q, that is L  G = LG. For convenience, “ is omitted in this paper. Lemma 1 [54]. Any logical function f (x1 , . . . , xn ) with logical arguments x1 , . . . , xn ∈ D can be expressed in a multi-linear form as f (x1 , . . . , xn ) = M f x1 x2 . . . xn , where M f ∈ L2×2n is unique, called the structural matrix of f. 2.2. Algebraic form of BCNs with stochastic impulses A BCN can be expressed as: ⎧ x1 (t + 1) = f1 (u1 (t ), . . . , um (t ), x1 (t ), . . . , xn (t )) ⎪ ⎪ ⎪ ⎨x2 (t + 1) = f2 (u1 (t ), . . . , um (t ), x1 (t ), . . . , xn (t )) , .. ⎪ . ⎪ ⎪ ⎩ xn (t + 1) = fn (u1 (t ), . . . , um (t ), x1 (t ), . . . , xn (t ))

(1)

where fi : D n+m → D(i = 1, 2, . . . , n) are network logical functions; xi ∈ D(i = 1, 2, . . . , n) are state variables; u j ∈ D( j = 1, 2, . . . , m) are control input variables. Using Lemma 1, the structural matrix of each logical function fi (i = 1, 2, . . . , n) in the BCN (1) can be obtained as Li ∈ L2×2n+m . In order to convert the system into its equivalent algebraic form, the following lemma is recalled. Lemma 2 [55]. Consider a system described as Eq. (1). Let x(t ) = ni=1 xi (t ) and u(t ) = mj=1 u j (t ). Then, there exists a unique matrix M ∈ L2n ×2n+m such that x(t + 1) = Mu(t )x(t ). Please cite this article as: X. Hu, C. Huang and J. Lu et al., Stabilization of Boolean control networks with stochastic impulses, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.06.039

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Next, based on Lemma 2, the BCN (1) can be converted into its algebraic form as x(t + 1) = L0 u(t )x(t ).

(2)

For the BCN (2), u(t) is a state feedback controller in the form of u(t ) = K x(t ),

(3)

where K ∈ B2m ×2n is a state feedback gain matrix to be designed. In some realistic situations, the behavior of networks may be heavily affected by some abrupt environment changes. For example, some genes in biological systems may experience sudden changes and deviate from their original growth trajectory. However, impulses can also have a positive impact on the control problems, such as shortening the minimal time between the given initial state and desired state [44]. Thus, it is worthy to investigate impulsive effects which can be modeled as below: ⎧ x1 (t + 1) = g1 (x1 (t ), x2 (t ), . . . , xn (t )) ⎪ ⎪ ⎪ ⎨x2 (t + 1) = g2 (x1 (t ), x2 (t ), . . . , xn (t )) , (4) .. ⎪ . ⎪ ⎪ ⎩ xn (t + 1) = gn (x1 (t ), x2 (t ), . . . , xn (t )) where gi : D n → D(i = 1, 2, . . . , n) are impulse functions. Similarly, the network (4) is converted into x(t + 1) = Gk x(t ), where Gk is the structure matrix of network (4). Combining the two models, the algebraic form of BCNs with impulse can be constructed as follows:  x(t + 1) = L0 u(t )x(t ), t = tk , x(t + 1) = Gk x(t ), t = tk where {tk }∞ k=1 is the impulse instants sequence, satisfying tk ∈ Z+ , tk < tk+1 , and lim k→+∞ tk = +∞. The BCN with impulse can capture regular network behaviors and provide effective tool of many real systems. However, sometimes it is hard to realize accurate modelling due to the stochastic nature of impulse. Thus, the stochastic impulses need to be considered. Specially, a BCN with stochastic impulses is built as follows: ⎧ x(t + 1) = L ⎪ ⎪ ⎧0 u(t )x(t ), t = tk ⎪ ⎪ Gk1 x(t ) ⎪ ⎪ ⎨ ⎪ ⎪ ⎨ Gk2 x(t ) , (5) x(t + 1) = . , t = tk ⎪ ⎪ . ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ Gkr x(t ) where Gki is the structure matrix of the ith impulses model. The impulse model (4) becomes a stochastic impulses model, if the impulse model attime t satisfies Gk ∈ {Gk1 , . . . , Gkr }, denoted as P{Gk = Gki } = pi > 0(i = 1, . . . , r), then ri=1 pi = 1. Considering the positive effects of impulses, a state can be influenced to reach the predefined target by designing reasonable impulsive strategies. The impulse instants need to be designed in this paper, which is different from the advance setting in [44]. That is, for different initial states, the corresponding impulsive strategy is proposed. At every moment, the Please cite this article as: X. Hu, C. Huang and J. Lu et al., Stabilization of Boolean control networks with stochastic impulses, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.06.039

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impulsive strategies determine the states evolved through the network function or the stochastic impulses function. Our aim is to schedule an impulse-instant sequence {tk (x0 )} such that the problem of global stabilization, introduced below, is solvable. Definition 2. (Global stabilization) For a given state xd ∈ 2n , the BCN (5) with stochastic impulses is said to be global stabilized to xd , if there exists a fixed positive integer T ≥ 1, a corresponding impulsive strategy {tk (x0 )} and a feedback controller u such that for any initial state x0 ∈ 2n , one has P{x(t ) = xd |x(0) = x0 , u(t ) = K x(t )} = 1,

t ≥ T,

where K is a feedback gain matrix that needs to be designed. 3. Main results In this section, the global stabilization of BCNs with stochastic impulses is addressed. If the system can be stabilized, any initial state can reach the predefined target by designing an appropriate impulsive strategy and a corresponding controller. Before formulating the problem, a series of matrices are defined as follows: 0 = M0 , M0 = L ∨ G, M i = 1, 2, . . . Mi = Mi−1 M0 , Mi = Mi , where L = (B)

2 m i=1

Blki (L0 ) = L0 B 12m , G =

r 

pi Gki .

i=1

For the given state xd = δ2αn ∈ 2n and any j ∈ Z+ , let j (xd ) be the convergable set containing states that can be steered to xd at the jth step. Then, a series of convergable sets i are defined as: based on matrices M 0 (xd ) = {xd }, j−1 )}( j = 1, 2, . . . ).  j (xd ) =  T {Rowα (M  j−1 ) = x ∈ (x ) xi is obtained. The following lemma From the above definition, RowαT (M i j d gives an equivalent condition to illustrate that the states in convergable sets can reach the given state with probability 1. Lemma 3. Given a fixed state xd = δ2αn , for any initial state x0 = δ2jn ∈ 2n and ∀t ∈ Z>0 , x0 ∈ t (xd ),

(6)

if and only if P{x(t ) = xd |x(0) = x0 , u(t ) = K x(t )} = 1.

(7)

Proof. (Necessity) The proof will be given by induction. 0 )α j = 1 which means xd ∈ (M 0 x0 ). Consider the case t = 1. If x0 ∈ 1 (xd ), then (M According to the definition of M0 , at least one of xd ∈ (Lx0 ) and xd ∈ (Gx0 ) is true. Thus, x0 can transform to xd at one step with probability 1, i.e., P{x(1) = xd |x(0) = x0 , u(0) = K x0 } = 1. Suppose that Eq. (7) holds for any integer t ≥ 1. For the case of t + 1, x(0) ∈ t+1 (xd ) t )α j = 1 and (M t−1 M0 )α j ≥ 1. Therefore, state x0 can be steered to the set means that (M Please cite this article as: X. Hu, C. Huang and J. Lu et al., Stabilization of Boolean control networks with stochastic impulses, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.06.039

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 t (xd ) with probability 1 in one step, i.e., a1 ∈t (xd ) P{x(1) = a1 |x(0) = x0 , u(0) = K x(0)} = 1. Since Eq. (7) holds for the states x(1) ∈ t (xd ), then =

P {x(t + 1) = xd |x(0) = x0 , u(t ) = K x(t )} P{x(t + 1) = xd |x(1) = a1 , u(t ) = K x(t )}

a1 ∈t (xd )

×P{x(1) = a1 |x(0) = x0 , u(0) = K x0 } = 1.

By induction, P{x(t ) = xd |x(0) = x0 , u(t ) = K x(t )} = 1 holds for any initial state x0 ∈ t (xd ). (Sufficiency) When t = 1, it follows from Eq. (7) that the state x0 can reach the given state xd in one step with probability 1. There are two available choice models for x0 . One is the stochastic impulses model, then P{x(1) = xd |x(0) = x0 , u(0) = K x0 } =

r 

P{xd = Gki x0 }.

(8)

i=1

If Eq. (8) holds, xd ∈ (Gki x0 ) is obtained for any i = 1, 2, . . . , r, which means xd ∈ (Gx0 ). For another choice, one has xd = L0 u(0)x(0). Since L0 u(0) is a 2n × 2n block, there exists at least one square block Blki (L0 ) (i ∈ {1, 2, . . . , 2m }) such that xd = Blki (L0 )x0 . According to 0 together, the definition of L, it can be got that xd ∈ (Lx0 ). Considering two choices and M

xd ∈ (M0 x0 ) can be concluded to show Eq. (6) holds. Now, supposed that Eq. (6) holds for any integer t ≥ 1. For the case of t + 1, the proof will be given by contradiction. In fact, if x0 satisfies Eq. (7) and x0 ∈ / t+1 (xd ), then, according to the definition of t+1 (xd ), one has t )α j = ( M t−1 M0  )α j < 1. 0  (M It follows from the proof of necessity that  P{x(1) = a1 |x(0) = x0 , u(0) = K x0 } = 1. a1 ∈t (xd )

Then =

P {x(t + 1) = xd |x(0) = x0 , u(t ) = K x(t )} P{x(t + 1) = xd |x(1) = a1 , u(t ) = K x(t )}

a1 ∈t (xd )

×P{x(1) = a1 |x(0) = x0 , u(0) = K x0 } = 1, which is a contradiction to Eq. (7). It means x0 ∈ t+1 (xd ), which proves the case of t + 1. This completes the proof.  Some significant properties of convergable sets i (i = 1, 2, . . . ) used in the following research are given in Lemma 4. Lemma 4. (i) If  j (xd ) =  j+1 (xd ) holds for j ≥ 0, then k (xd ) =  j (xd ) holds for any integer k ≥ j; (ii) If 0 (xd ) ⊆ 1 (xd ), then k (xd ) ⊆ k+1 (xd ) holds for any integer k ≥ 1. Proof. Firstly, (i) is proved by induction on j. When j = 0, 0 (xd ) = 1 (xd ) implies that only xd can reach itself. Hence, 1 (xd ) = 2 (xd ) = . . .. Please cite this article as: X. Hu, C. Huang and J. Lu et al., Stabilization of Boolean control networks with stochastic impulses, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.06.039

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Obviously, k (xd ) =  j (xd ) when k = j. Assume that k (xd ) =  j (xd ) holds for any integer k > j, i.e.,  j (xd ) =  j+1 (xd ) = · · · = k (xd ). Next,  j (xd ) = k+1 (xd ) will be proved. i−1 )}(i = 1, 2, . . . ), According to the definition of the sets i (xd ) =  T {Rowα (M j−1 ) = Rowα (M j ) = · · · = Rowα (M k−1 ) can be obtained. Then, Rowα (M i ) = Rowα (M i−1 M0  ) = Rowα (M i−1 )M0  is derived from the definition of M i (i = 1, 2, . . . ). Rowα ( M k ) = Rowα (M k−1 )M0  = Rowα (M j−1 )M0  = Rowα (M j ), which implies Thus, Rowα (M k+1 (xd ) =  j+1 (xd ) =  j (xd ). Hence, if  j (xd ) =  j+1 (xd ), then k (xd ) =  j (xd ) holds for any integer k ≥ j. (ii) When k = 1, the proof will be claimed by considering two cases: 0 (xd ) = 1 (xd ) and 0 (xd )  1 (xd ). For the case of 0 (xd ) = 1 (xd ), it follows from (i) that 0 (xd ) = 1 (xd ) = 2 (xd ) holds. If 0 (xd )  1 (xd ), there are some other points δ2jn ∈ 1 (xd )(1 ≤ j ≤ 2n ) except xd . Then, ) = 1 and (M0 )αj ≥ 1 are obtained. Together with (M 0 )αα = 1, one can get (M1 )α j = (M 20n α j j k=1 (M0 )αk (M0 )k j ≥ 1, which means δ2n ∈ 2 (xd ) and 1 (xd ) ⊆ 2 (xd ). Two cases together show that 1 (xd ) ⊆ 2 (xd ). Suppose that k−1 (xd ) ⊆ k (xd ) holds for any positive integer k. Then, for the integer k )} =  T {Rowα ( M k−1 M0  )}. Due to k−1 (xd ) ⊆ k (xd ), k + 1, k+1 (xd ) =  T {Rowα (M T T k−1 )}. Hence,  T {Rowα ( M k−2 M0  )} ⊆ it can be got that  {Rowα (Mk−2 )} ⊆  {Rowα (M k−1 M0  )}, which shows k (xd ) ⊆ k+1 (xd ). By induction, k (xd ) ⊆ k+1 (xd )  T {Rowα ( M holds for any integer k ≥ 1.  Based on Lemmas 3 and 4, the following results establish a necessary and sufficient condition for the global stabilization of system (5). Theorem 1. Consider the BCN (5) and a given state xd ∈ 2n , where Gk is chosen from the set {Gk1 , Gk2 , . . . , Gkr } and L0 is fixed. The system is global stabilization to xd , if and only if 0 (xd ) ⊆ 1 (xd ) and there exists an integer 1 ≤ T ≤ 2n − 1 such that T (xd ) = 2n . Proof. (Necessity) Since the BCN (5) is globally stabilization to the given state xd ∈ 2n , there exists impulsive strategies, a state feedback gain matrix K ∈ L2m ×2n and a positive integer T ≥ 1 such that for any initial state x0 ∈ 2n and any integer t ≥ T, one has 1 = P{x(t ) = xd |x(0) = x0 , u(t ) = K x(t )}  = P{x(t ) = xd |x(t − 1) = at−1 , u(t − 1) = K at−1 } a1 ,...,at−1 ∈2n

× . . . × P{x(1) = a1 |x(0) = x0 , u(0) = K x0 }, and 1 = P{x(t + 1) = xd |x(0) = x0 , u(t ) = K x(t )} = P{x(t + 1) = xd |x(t ) = at , u(t ) = K at } a1 ,...,at ∈2n

×P{x(t ) = at |x(t − 1) = at−1 , u(t − 1) = K at−1 } × . . . × P{x(1) = a1 |x(0) = x0 , u(0) = K x0 }.

(9)

(10)

Considering Eqs. (9) and (10), one can see P{x(t + 1) = xd |x(t ) = xd , u(t ) = K xd } = 1.

(11)

From Lemma 3, one has xd ∈ 1 (xd ), or equivalently, 0 (xd ) ⊆ 1 (xd ). In the following, T (xd ) = 2n will be proved. Since the BCN is globally stabilizable to xd , for any initial Please cite this article as: X. Hu, C. Huang and J. Lu et al., Stabilization of Boolean control networks with stochastic impulses, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.06.039

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state x0 ∈ 2n , there exists a positive integer T ≥ 1 such that P{x(t ) = xd |x(0) = x0 , u(t ) = K x(t )} = 1,

t ≥ T.

It follows from Lemma 3 that x0 ∈ t (xd ). Hence, t (xd ) = 2n , ∀t ≥ T and T (xd ) = 2n . At last, the proof of T ≤ 2n − 1 will be given. Assume that T is the smallest positive integer such that T (xd ) = 2n . It can be proved that |t (xd )| ≥ t + |0 (xd )| = t + 1, ∀1 ≤ t ≤ T .

(12)

When t = 1, if |1 (xd )| < 2, then 1 (xd ) = 0 (xd ). From Lemma 4, T (xd ) = 0 (xd ) = 2n , which is a contradiction. Assume that Eq. (12) holds for any 1 ≤ t < T. Since Eq. (12) holds for t = 1, then 0 (xd ) ⊆ 1 (xd ) and t (xd ) ⊆ t+1 (xd ). If |t+1 (xd )| < t + 2, then t (xd ) = t+1 (xd ). Therefore, by Lemma 4, t (xd ) = T (xd ) = 2n is obtained, which is a contradiction to that T is the smallest positive integer. Thus, Eq. (12) holds for t + 1. In summary, Eq. (12) holds. Taking t = T , then 2n = |T (xd )| ≥ T + 1, which implies that T ≤ 2n − 1. (Sufficiency) Firstly, a state feedback gain matrix K ∈ B2m ×2n and a series of sets are constructed. Since 0 (xd ) ⊆ 1 (xd ), one can see from Lemma 4 that 0 (xd ) ⊆ 1 (xd ) ⊆ · · · ⊆ T (xd ) = 2n . Let i (xd ) = i (xd )\i−1 (xd )(i = 1, . . . , T ). 



(13)

 (x ) = 0 (xd ) = {xd }. Obviously,  i (xd )   j (xd ) = ∅, ∀i, j ∈ {0, . . . , T }, i = j, Define 

T 0 d  and i=0 i (xd ) = 2n . According to Eq. (13), for any state δ2i n ∈ 2n , there exists a unique integer 0 ≤ ti ≤ T such ti (xd ). Since the system may experience stochastic impulses or not at any time, that δ2i n ∈  it needs to consider two cases separately. Case 1: Suppose that the state δ2i n reaches the next state by stochastic impulses at time T − ti . According to the definition of system (5), it can be learned that there is no inputs at stochastic impulses instants. So the controller is defined as u(T − ti ) = 02m , thus Coli (K ) = 02m . Additionally, the impulsive strategy of δ2i n is {tk (δ2i n )} = {0, 1, . . . , T − ti }. Case 2: If the stochastic impulses do not happen at time T − ti , the impulsive strategy of δ2i n is defined as {tk (δ2i n )} = ∅. Therefore, only the feedback controller settings need to be considered. From the definition of ti (xd ), one can obtain that ti −1 )}. δ2i n ∈  T {Rowα (M

(14)

Set K = δ2m [v1 v2 . . . v2n ] ∈ B2m ×2n . For the system (5) with the controller u(t ) = K x(t ), the proof of P{x(t ) = xd |x(0) = δ2i n , u(t ) = K x(t )} = 1 holding for any t ≥ ti (i = 1, 2, . . . , 2n ) will be proved. When 1 ≤ ti ≤ T, it is obtained from Lemma 3 and Eq. (14) that 1 = P{x(ti ) = xd |x(0) = δ2i n , u(t ) = K x(t )} = P{x(ti ) = xd |x(ti − 1) = ati −1 , u(ti − 1) = K ati −1 }  × P{a(ti − 1) = ati −1 |x(ti − 2) = ati −2 , u(ti − 2) = K ati −2 } ati −1 ∈1 (xd )

Please cite this article as: X. Hu, C. Huang and J. Lu et al., Stabilization of Boolean control networks with stochastic impulses, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.06.039

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×... ×

P{x(2) = a2 |x(1) = a1 , u(1) = K a1 }

a2 ∈ti −2 (xd )

×



P{x(1) = a1 |x(0) = δ2i n , u(0) = δ2vin }.

a1 ∈ti −1 (xd )

Suppose that P{x(t ) = xd |x(0) = δ2i n , u(t ) = K x(t )} = 1 holds for any integer t ≥ ti . For t + 1, one has ( ) := P{x(t + 1) = xd |x(0) = δ2i n , u(t ) = K x(t )}  = P{x(t + 1) = xd |x(t ) = at , u(t ) = K at } at ∈2n

×P{x(t ) = at |x(0) = δ2i n , u(t ) = K x(t )}. Together with Eq. (11), one can see ( ) := P{x(t + 1) = xd |x(0) = δ2i n , u(t ) = K x(t )} = P{x(t + 1) = xd |x(t ) = xd , u(t ) = K xd } ×P{x(t ) = xd |x(0) = δ2i n , u(t ) = K x(t )} = 1. By induction, when 1 ≤ ti ≤ T, P{x(t ) = xd |x0 = δ2i n , u(t ) = K x(t )} holds for any t ≥ ti . It means that the global stabilization problem is solvable. This completes the proof.  The proof of Theorem 1 provides an algorithm (Algorithm 1) to design a state feedback controller and different impulsive strategies to stabilize the system (5). Algorithm 1 The design of a state feedback controller and impulsive strategies. Step 1. Based on the structure matrices of system (5), calculate a series of matrices i (i = 0, 1, . . . ). M i−1 , calculate the sets i (xd )(i = 1, 2, . . . ). Find an integer Step 2. By using matrices M n i (xd )(i = 0, 1, . . . , T ) 1 ≤ T ≤ 2 − 1 such that T (xd ) = 2n . Additionally, calculate  according to Eq. (13). i Step 3. Find the states δ2qn , 1 ≤ iq ≤ 2n , 0 ≤ q ≤ 2n satisfying Case 1 for 0 ≤ ti ≤ T , rei i spectively. The impulsive strategy for δ2qn is {tk (δ2qn )} = {0, 1, . . . , T − ti }. Meanwhile, the corresponding controller is u(T − ti ) = 02m and Coliq (K ) = 02m . Step 4. Find v jk such that Eq. (14) holding for the states δ2jkn (1 ≤ jk ≤ 2n , 0 ≤ k ≤ 2n ) satisfying Case 2 for 0 ≤ ti ≤ T . Step 5. The state feedback gain matrix K is designed as   K = δ2 m . . . 0 . . . v j k . . . . iq

jk

Please cite this article as: X. Hu, C. Huang and J. Lu et al., Stabilization of Boolean control networks with stochastic impulses, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.06.039

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4. Extension to PBCNs with stochastic impulses As an extension, consider the following PBCNs with stochastic impulses: ⎧ ⎧ L01 u(t )x(t ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨L02 u(t )x(t ) ⎪ ⎪ ⎪ ⎪ x(t + 1) = . , t = tk ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎨ L0s u(t )x(t ) ⎧ , ⎪ Gk1 x(t ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨Gk2 x(t ) ⎪ ⎪ ⎪ ⎪ x(t + 1) = , t = tk ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ Gkq x(t )

(15)

where L0i ∈ L2n ×2n+m (i = 1, 2, . . . , s) and Gk j ∈ L2n ×2n ( j = 1, 2, . . . , q). Denote theprobabilities as P{L0 = L0i } = pli , P{Gk = Gk j } = pgj (i = 1, 2, . . . , s, j = 1, 2, . . . , q) and si=1 pli =  1, qj=1 pgj = 1.   Let L¨ = si=1 pli L0i , G¨ = ( qi=1 pgi Gki )(1T2m  I2n ). Similarly, matrices Mi and M i (i = 0, 1, 2, . . . ) are defined as follows: M0 = L¨ ∨ G¨ ,



M 0 = M0  B 12 m ,

Mi = (M i−1 L¨ ) ∨ (M i−1 G¨ ),



M i = Mi  B 12 m .

Based on these matrices, for PBCNs (15) with a given state xd = δ2dn ∈ 2n , a series of convergable sets are defined as follows: ∗0 (xd ) = {xd }, ∗k (xd ) =  T {Rowα (M k−1 )}(k = 1, 2, . . . ).

Remark 1. Although Mi and Mi are defined in a similar form, they are matrices with 2n × 2n i and M i are 2n × 2n Boolean matriand 2n × 2n+m dimensions, respectively. However, both M ∗ n ces. Moreover, the elements in i and i are 2 × 1 dimensional vectors, which are generated i and M i , respectively. By this meaning, the properties of i given in Lemma 4 can from M also be applied to the sets ∗i . Lemma 5 presents a necessary and sufficient condition to ensure that the states in the sets ∗i (xd ) can be transferred with probability 1. Lemma 5. For a given state xd ∈ 2n , x0 ∈ t∗ (xd ) if and only if P{x(t ) = xd |x(0) = x0 , u(t ) = K x(t )} = 1. The following results reveal the global stabilization to xd for the PBCN (15) whose proof is quite similar to that of Theorem 1 and, hence, omitted here. Theorem 2. Consider the PBCN (15) and a given state xd , where L0 and Gk are chosen from sets {L01 , L02 , . . . , L0s } and {Gk1 , Gk2 , . . . , Gkq }, respectively. The system is global stabilization, if and only if ∗0 (xd ) ⊆ ∗1 (xd ) and exists an integer 1 ≤ T ≤ 2n − 1 such that ∗T (xd ) = 2n . Please cite this article as: X. Hu, C. Huang and J. Lu et al., Stabilization of Boolean control networks with stochastic impulses, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.06.039

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5. Illustrative examples In this section, two numerical examples will be used to show the correctness of the previous results. Example 1. Consider the BCN model for the λ switch under impulsive effects which was proposed in [44]. As a virus, λ phage grows by injecting its own chromosomes into the infected bacteria cell. Two possible pathways: lysogeny and lysis, are the results of expressing different sets of genes. Some environmental factors including temperature rate, concentration of nutrition and growth rate, which affecting gene expression were considered in this model. The BCN model in [44] is ⎧ x1 (t + 1) = [¬x2 (t )] ∧ [¬x5 (t )], ⎪ ⎪ ⎪ ⎪ ⎨x2 (t + 1) = [¬x5 (t )] ∧ [x2 (t ) ∨ x3 (t )], x3 (t + 1) = [¬x2 (t )] ∧ u(t ) ∧ [x1 (t ) ∨ x4 (t )], t ≥ 0. (16) ⎪ ⎪ x (t + 1) = [ ¬ x (t )] ∧ u(t ) ∧ x (t ) , ⎪ 4 2 1 ⎪ ⎩ x5 (t + 1) = [¬x2 (t )] ∧ [¬x3 (t )], In addition to environment conditions where phage genes grow inside bacteria, the expression of many genes is affected by external physical factors including: Xray, laser, ultraviolet rays etc. These influence factors were simulated as very short duration period impulse model by Chen et al. [44]. Considering the random nature of impulse, in this example, the stochastic impulses model is constructed as: ⎧ ⎧ x1 (t + 1) = [x1 (t ) ∨ x3 (t )] ↔ x5 (t ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x2 (t + 1) = [¬x1 (t )] ∨ x3 (t ), ⎨ ⎪ ⎪ ⎪ ⎪ x3 (t + 1) = ¬x4 (t ), im pulses m odel 1 : ⎪ ⎪ ⎪ ⎪ ⎪x4 (t + 1) = [x1 (t ) ∧ x2 (t )] ∨ x3 (t ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎨ x5 (t + 1) = x1 (t ) ↔ [x3 (t ) ∨ x4 (t )], ⎧ t = tk . (17) ⎪ x1 (t + 1) = [x1 (t ) ∨ x3 (t )] → x4 (t ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨x2 (t + 1) = ¬x5 (t ), ⎪ ⎪ ⎪ ⎪ x3 (t + 1) = x1 (t )] ↔ [x4 (t ) ∨ x5 (t )], im pulses m odel 2 : ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x4 (t + 1) = [¬x2 (t )] ∧ x3 (t ), ⎪ ⎪ ⎪ ⎩ ⎩ x5 (t + 1) = [x1 (t ) ∨ x2 (t )] ∧ x3 (t ), The BCN with stochastic impulses is established by combining Eqs. (16) and (17), and 24 given the fixed state δ32 to study the global stabilization problem. By converting the systems (16) and (17) into its algebraic form, the structure matrix of each model is as follows: . L0 = δ32 [32 24 32 24 .. 32 . 32 24 32 24 .. 32 . 32 24 32 24 .. 32 . 32 24 32 24 .. 32

. 24 32 24 .. 26 . 24 32 24 .. 28 . 24 32 24 .. 32 . 24 32 24 .. 32

. 2 26 2 .. 25 . 4 32 8 .. 27 . 8 32 8 .. 31 . 8 32 8 .. 31

9 25 9 11 31 15 15 31 15 15 31 15],

. . . Gk1 = δ32 [5 21 1 17 .. 13 29 10 26 .. 5 21 1 17 .. 15 31 12 28 . . . 6 22 2 18 .. 24 8 19 3 .. 6 22 2 18 .. 24 8 19 3], Please cite this article as: X. Hu, C. Huang and J. Lu et al., Stabilization of Boolean control networks with stochastic impulses, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.06.039

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. . . Gk2 = δ32 [11 3 27 23 .. 12 4 28 24 .. 9 1 25 21 .. 12 4 28 24 . . . 15 7 31 19 .. 16 8 16 4 .. 14 6 30 18 .. 16 8 16 4]. Moreover, the probabilities for Gk1 and Gk2 are P{Gk = Gk1 } = 0.25, P{Gk = Gk2 } = 0.75, respectively. Then, we will use Algorithm 1 to solve the stabilization problem of system (17). i (i = 1, 2, . . . ). Step 1: Calculate the matrices M The matrices L0 , Gk1 and Gk2 are preprocessed as follows: L = L0  12 ,

G = 0.25 ∗ Gk1 + 0.75 ∗ Gk2 .

i (i = 0, 1, 2, . . . ) using the following formulas Then, calculate the matrices M M0 = L ∨ G, i−1 M0 , Mi = M

0 = M0  M i = Mi , M

i = 1, 2, . . .

i (xd )(i = 0, 1, 2, . . . ) and find the integer T. Step 2: Calculate the sets i (xd ) and  2 4 6 8 18 20 22 24 0 )} = {δ32 1 (xd ) =  T {Row24 (M , δ32 , δ32 , δ32 , δ32 , δ32 , δ32 , δ32 }, T 2 4 6 8 10 12 18 20 22 24 26 28 30 1 )} = {δ32 , δ32 , δ32 , δ32 , δ32 , δ32 2 (xd ) =  {Row24 (M , δ32 , δ32 , δ32 , δ32 , δ32 , δ32 , δ32 }, .. . 1 2 3 4 5 28 29 30 31 32 5 )} = {δ32 6 (xd ) =  T {Row24 (M , δ32 , δ32 , δ32 , δ32 , . . . , δ32 , δ32 , δ32 , δ32 , δ32 } = 25 . 24 For the given state xd = δ32 , it is easy to see that 0 (xd ) ⊆ 1 (xd ) and 6 (xd ) = 25 . Hence, the system (17) achieves global stabilization at T = 6. i (xd )(i = 0, 1, . . . , 6) are calculated as follows: According to Eq. (13), 

0 (xd ) = {δ 24 },  32 2 (xd ) = {δ 10 , δ 12 , δ 26 , δ 28 , δ 30 },  32 32 32 32 32 4 (xd ) = {δ 13 , δ 14 , δ 17 , δ 21 , δ 29 , δ 32 },  32 32 32 32 32 32 6 (xd ) = {δ 31 }.  32

1 (xd ) = {δ 2 , δ 4 , δ 6 , δ 8 , δ 18 , δ 20 , δ 22 },  32 32 32 32 32 32 32 3 (xd ) = {δ 7 , δ 9 , δ 11 , δ 15 , δ 16 , δ 25 , δ 27 },  32 32 32 32 32 32 32 5 (xd ) = {δ 1 , δ 3 , δ 5 , δ 19 , δ 23 },  32 32 32 32 32

Step 3: The design of impulsive strategies. i (xd )(i = 0, 1, . . . , 6) show the time when each state arrives at given A series of sets  24 state δ32 in the dynamic process of the system (17). In this process, the impulsive strategies 15 17 29 31 of some states need to be designed, such as δ32 , δ32 , δ32 , δ32 and so on. 31 31 The design of impulsive strategies is illustrated by the state δ32 . It follows from δ32 ∈ 31  6 (xd ) that δ32 satisfies Case 1 at t = T − 6 = 0. Thus, the corresponding control and im31 pulsive strategy are Col31 (K ) = 02 and {tk (δ32 )} = {0}, respectively. The control and impulsive strategies of the remaining states are designed as follows: 15 δ32 16 δ32 17 δ32 21 δ32 25 δ32 27 δ32 29 δ32 30 δ32

: : : : : : : :

Col15 (K ) = 02 , Col16 (K ) = 02 , Col17 (K ) = 02 , Col21 (K ) = 02 , Col25 (K ) = 02 , Col27 (K ) = 02 , Col29 (K ) = 02 , Col30 (K ) = 02 ,

15 {tk (δ32 )} = {0, 1, 2, 3}, 16 {tk (δ32 )} = {0, 1, 2, 3}, 17 {tk (δ32 )} = {0, 1, 2}, 21 {tk (δ32 )} = {0, 1, 2}, 25 {tk (δ32 )} = {0, 1, 2, 3}, 27 {tk (δ32 )} = {0, 1, 2, 3}, 29 {tk (δ32 )} = {0, 1, 2}, 30 {tk (δ32 )} = {0, 1, 2, 3, 4}.

Step 4: The design of the state feedback controller. Please cite this article as: X. Hu, C. Huang and J. Lu et al., Stabilization of Boolean control networks with stochastic impulses, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.06.039

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In addition to states in Step 3, all other states in the system (17) satisfy Case 2. Taking the 1 state δ32 as an example to illustrate the design of the controller. By analyzing the structure matrix L, whether v = 1 or v1 = 0 at t = 1, one can obtain 1 4 )}. δ32 ∈  T {Row24 (M

Thus, the column 1 of the matrix K can be either [1 0]T or [0 1]T . Step 5: The design of the state feedback control gain matrix K. Repeat Step 4 for other states, a state feedback control gain matrix K is designed as follows: ⎛ . . . 1 1 1 1 .. 1 0 0 0 .. 1 0 1 1 .. 1 1 0 0 ⎝ K= . . . 0 0 0 0 .. 0 1 1 1 .. 0 1 0 0 .. 0 0 0 0 ⎞ . . . 0 1 1 1 .. 0 0 0 0 .. 0 0 0 1 .. 0 0 0 1 ⎠ . . . . 0 0 0 0 .. 0 1 1 1 .. 0 1 0 0 .. 0 0 0 0 Example 2. Consider the alternative BCN models presented in [56]. Veliz-Cuba and Stigler [56] proposed a BCN as a discrete model for the lac operon, which includes the glucose control mechanisms of catabolite repression and inducer exclusion. Based on the BCN model, alternative BCN models were considered without inducer exclusion and catabolite repression. If the two alternative models occur with probabilities p1 and p2 respectively, where p1 + p2 = 1, the alternative BCN models can be considered as the PBCN. The model is given as follows: ⎧ ⎧ ⎨x1 (t + 1) = ¬u1 (t ) ∧ (x2 (t ) ∨ x3 (t )), ⎪ ⎪ ⎪ ⎪ x2 (t + 1) = x1 (t ) ∧ u2 (t ), P BCN 1 : ⎪ ⎪ ⎩ ⎨ x3 (t + 1) = (u3 (t ) ∧ x1 (t )) ∨ u2 (t ), ⎧ t = tk . (18) ⎪ ⎨x1 (t + 1) = x2 (t ) ∨ x3 (t ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎩PBCN 2 : ⎩x2 (t + 1) = x1 (t ) ∧ u2 (t ) ∧ (¬u1 (t )), x3 (t + 1) = [(u3 (t ) ∧ x1 (t )) ∨ u2 (t )] ∧ (¬u1 (t )), Considering the effects and the random nature of impulses, the stochastic impulses model is established as follows: ⎧ ⎧ ⎨x1 (t + 1) = ¬x1 (t ) → x3 (t )), ⎪ ⎪ ⎪ ⎪ x2 (t + 1) = x3 (t ) ↔ x2 (t ), im pulses m odel 1 : ⎪ ⎪ ⎩ ⎨ x3 (t + 1) = ¬x1 (t ), ⎧ t = tk . (19) ⎪ ⎨x1 (t + 1) = x1 (t ) ∧ x2 (t ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎩im pulses model 2 : ⎩x2 (t + 1) = ¬x2 (t )), x3 (t + 1) = x2 (t )) ∨ x3 (t ), The fixed state is xd = δ83 . L0 and Gk are chosen from the sets {L01 , L02 } and {Gk1 , Gk2 }, respectively, at every step. By some calculation, the structural matrix of each network is obtained as  . . . . . . . . L01 = δ8 5 5 5 5 .. 7 7 7 7 .. 5 5 5 5 .. 7 7 7 7 .. 7 7 7 7 .. 8 8 8 8 .. 8 8 8 8 .. 8 8 8 8 ..  . . . . . . . 1 1 1 5 .. 3 3 3 7 .. 1 1 1 5 .. 3 3 3 7 .. 3 3 3 7 .. 4 4 4 8 .. 4 4 4 8 .. 4 4 4 8 . Please cite this article as: X. Hu, C. Huang and J. Lu et al., Stabilization of Boolean control networks with stochastic impulses, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.06.039

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 . . . . . . . . L02 = δ8 4 4 4 8 .. 4 4 4 8 .. 4 4 4 8 .. 4 4 4 8 .. 4 4 4 8 .. 4 4 4 8 .. 4 4 4 8 .. 4 4 4 8 ..  . . . . . . . 1 1 1 5 .. 3 3 3 7 .. 1 1 1 5 .. 3 3 3 7 .. 3 3 3 7 .. 4 4 4 8 .. 4 4 4 8 .. 4 4 4 8 . The structure matrices of stochastic impulses are . Gk1 = δ8 [2 4 4 2 .. 1 7 3 5],

. Gk2 = δ8 [3 3 5 6 .. 7 7 5 6].

Additionally, the probabilities of L0i and Gki are P{L0 = L01 } = 0.1, P{L0 = L02 } = 0.9 and P{Gk = Gk1 } = 0.2, P{Gk = Gk2 } = 0.8, respectively. Step 1: Calculate M i , i = 1, 2, . . . based on {L01 , L02 } and {Gk1 , Gk2 }. The matrix L01 , L02 , Gk1 and Gk2 are preprocessed as follows: L¨ = 0.1 ∗ L01 + 0.9 ∗ L02 ,

G¨ = (0.2 ∗ Gk1 + 0.8 ∗ Gk2 ) B 12m .

Then, calculate the matrices M i (i = 0, 1, 2, . . . ) M0 = L¨ ∨ G¨ , Mi = (M i−1 L¨ ) ∨ (M i−1 G¨ ),

M 0 = M0 , M i = Mi  B 12 m .

 i (xd )(i = 0, 1, 2, . . . ). Step 2: Calculate the matrices ∗i (xd ) and  By a simple calculation as Example 1, compute ∗i (xd ) and find the integer T as follows: ∗1 (xd ) = {δ81 , δ82 , δ83 , δ85 , δ86 , δ87 }, ∗2 (xd ) = {δ81 , δ82 , δ83 , δ84 , δ85 , δ86 , δ87 , δ88 } = 2n . It is easy to obtain that both ∗0 (δ83 ) ⊆ ∗1 (δ83 ) and ∗2 (δ83 ) = 2n hold for the state δ83 . Hence, the system is global stabilization to the state δ83 .  i (xd ) = ∗i (xd ) − ∗ (xd ). Then Similar to Eq. (13), let  i−1  0 (xd ) = {δ83 }, 

 1 (xd ) = {δ81 , δ82 , δ85 , δ86 , δ87 }, 

 2 (xd ) = {δ84 , δ88 }. 

Step 3: The design of impulsive strategies. Fig. 1 shows one situation of the dynamic transformation process of the states in the PBCN with stochastic impulses. From it, one can obtain that only the state δ84 satisfies Case 1 when t = 0. Thus, the control of δ84 is Col4 (K ) = 08 . The impulsive strategy of δ84 is {tk (δ84 )} = {0}. Step 4: The design of the state feedback controller. One can get that all other states except δ84 satisfy Case 2. Taking the state δ82 as an example, we can find v2 = 7 such that δ82 ∈  T {Row3 (M 0 )}. Step 5: The design of the state feedback control gain matrix K. Please cite this article as: X. Hu, C. Huang and J. Lu et al., Stabilization of Boolean control networks with stochastic impulses, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.06.039

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Fig. 1. The dynamic transformation process for the whole state space. The Gk1 (resp.Gk2 ) is used to represent the first (second) stochastic model at stochastic impulses instants. Moreover, L represents the transformation among states through network function.

Considering the ⎛ 0 0 0 ⎜0 0 0 ⎜ ⎜0 0 0 ⎜ ⎜0 0 0 K =⎜ ⎜0 0 0 ⎜ ⎜0 0 0 ⎜ ⎝1 1 1 0 0 0

previous results, a boolean matrix K is designed as ⎞ 0 0 0 0 0 0 0 0 0 0⎟ ⎟ 0 0 0 0 0⎟ ⎟ 0 0 0 0 0⎟ ⎟. 0 0 0 0 1⎟ ⎟ 0 1 1 1 0⎟ ⎟ 0 0 0 0 0⎠ 0 0 0 0 0

Thus, every state can reach the given state through the corresponding impulsive strategy and controller. 6. Conclusion In this work, the global stabilization of BCNs to a fixed state, with stochastic impulses is discussed. Firstly, the algebraic expression of BCNs with stochastic impulses is obtained by using the STP method. Then, the definition of stabilization with probability one is presented. Based on the algebraic form, some matrices and convergable sets are defined to reduce complexity of the study. By resorting to these definitions, some criteria are derived to judge whether BCNs with stochastic impulses can be stabilized or not. For a stabilizable system, an algorithm is given to design a feedback controller and different impulsive strategies for Please cite this article as: X. Hu, C. Huang and J. Lu et al., Stabilization of Boolean control networks with stochastic impulses, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.06.039

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Please cite this article as: X. Hu, C. Huang and J. Lu et al., Stabilization of Boolean control networks with stochastic impulses, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.06.039