Controllability of Boolean control networks with ... - Wiley Online Library

Research Article Received 8 November 2012

Published online 21 March 2013 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.2773 MOS subject classification: 93B05; 34A37; 03G05

Controllability of Boolean control networks with impulsive effects and forbidden states Yang Liu* † , Hongwei Chen and Bo Wu Communicated by Q. Wang This paper investigates the controllability of Boolean control networks (BCNs) with impulsive effects while avoiding certain forbidden states. Using semi-tensor product of matrices, the BCNs with impulsive effects can be converted into impulsive discrete-time systems. Then, some necessary and sufficient conditions for the controllability are obtained. It is interesting to find that impulsive effects play an important role in the controllability of BCNs. Finally, an example is given to show the efficiency of the obtained results. Copyright © 2013 John Wiley & Sons, Ltd. Keywords: Boolean control network; systems biology; impulsive effects; controllability; semi-tensor product

1. Introduction The Boolean network (BN) is the simplest logical dynamic system that was firstly proposed by Kauffman for modeling complex and nonlinear biological systems [1]. It was developed by [2], Farrow et al.[3], Albert and Barabasi [4], and many others, and it has become a powerful tool in describing, analyzing, and simulating the cellular networks. In BNs, the state of a gene can be described by Boolean variables: active and inactive. Then, the state of each gene is determined by the states of its neighborhood genes, using Boolean functions. BNs play an important role in modeling cell regulation, because they can represent features of living organisms [5, 6]. Hence, the study of BNs has attracted a great attention, not only from the biology community but also from physics, system science, and others. The most interesting problem concerns the topological structure of a BN, including the fixed points, cycles, basin of attractors, and transient time [7–11]. Another challenging and important topic is the controllability of Boolean control networks (BCNs) [12–14]. Recently, a new matrix product, called the semi-tensor product (STP) of matrices, was proposed. Using STP, a logical equation can be expressed as an algebraic equation and the dynamics of a Boolean (control) network can be converted into a linear (bilinear) discretetime (control) system [15]. Then, some interesting results have been achieved, which are briefly described as follows: the controllability and observability of BCNs have been investigated in [16, 17], and Laschov and Margaliot [18, 19] have studied the optimal control for multivalued logical control networks; the realization of BCNs has been presented in [20], and Cheng et al.[21] have studied the stability and stabilization of BNs; formulas for calculating fixed points and cycles have been obtained in [15]. Biological networks may experience abrupt changes of states at certain time instants. These abrupt changes of states may occur at prescribed time instants and/or triggered by specified events along a particular trajectory. To describe mathematically an evolution of a real process with a short-term perturbation, it is natural to assume that these perturbations act instantaneously, that is, in the form of impulses. There are many papers studying the systems with impulsive effects [22–27]. For BNs with impulsive effects, stability and stabilization have been studied in [28] and observability analysis has been investigated in [29]. As we know, systematic analysis of biological systems is an important issue in systems biology. Controllability is a structural property of the system, and it is one of the fundamental concepts in systematic science and control theory. Controllability analysis in biological systems model using BCNs may reveal how the structure and organization of the systems guarantee the property of controllability (see [16, 17]). For biological systems, as some states may correspond to unfavorable or dangerous situations, it is necessary to avoid certain forbidden states when one considers the problem of designing a control sequence that steers the BCNs between two states. For example, from a location corresponding to a diseased state of the biological systems to a location corresponding to a healthy state. In the context of probabilistic Boolean network, this type of problems can be casted as stochastic optimal control problems and solved numerically using dynamic programming. To the best of our knowledge, there is no result about the controllability of BCNs with impulsive effects avoiding certain forbidden states. Motivated by the aforementioned analysis, in this paper, we will study this problem and

College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, 321004, China *Correspondence to: Yang Liu, College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, 321004, China. † E-mail: [email protected]

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Copyright © 2013 John Wiley & Sons, Ltd.

Math. Meth. Appl. Sci. 2014, 37 1–9

Y. LIU, H. CHEN AND B. WU extend the results of [17] to the class of BCNs with impulsive effects. Moreover, it is interesting to find that impulsive effects play an important role in the controllability of BCNs. The rest of the paper is organized as follows. Section 2 provides a brief review for the STP of matrices and some notations. In Section 3, we first convert the BCNs with impulsive effects into impulsive discrete-time systems. Then, necessary and sufficient conditions of the controllability for the BCNs with impulsive effects avoiding forbidden states are obtained. An example is given in Section 4 to illustrate the efficiency of the obtained results. Finally, conclusions are summarized in Section 5.

2. Preliminaries In this paper, the matrix product we use is the STP of matrices. Considering an m  n matrix A and a p  q matrix B, we define the STP of A and B, denoted by A Ë B, as follows: Definition 1 ([15]) 1. Let X be a row vector of dimension np and Y be a column vector of dimension p. Then, we split X into p equal-sized blocks as X 1 , : : : , X p , which are 1  n rows. Define the STP, denoted by Ë, as 8 p X ˆ ˆ ˆ X Ë Y D X i yi 2 Rn , ˆ ˆ < iD1

p ˆ X ˆ ˆ T T ˆ Y Ë X D yi .X i /T 2 Rn . ˆ : iD1

2. Let A 2 Mmn and B 2 Mpq . If either n is a factor of p, say nt D p and denote it as A t B, or p is a factor of n, say n D pt and denote it as A t B; then, we define the STP of A and B, denoted by C D A Ë B, as the following: C consists of m  q blocks as C D .Cij /, and each block is Cij D Ai Ë Bj , i D 1, : : : , m, j D 1, : : : , q, where Ai is ith row of A and Bj is the jth column of B. Semi-tensor product of matrices is a generalization of conventional matrix product, which extends the conventional matrix product to any two matrices. All the main properties of the conventional matrix product remain true for this generalization. On the basis of this result, the symbol Ë is omitted in most places. Definition 2 ([15]) An mn  mn matrix Wm,n is called swap matrix if it is constructed in the following way: label its columns by .11, 12, : : : , 1n, : : : , m1, m2, : : : , mn/ and its rows by .11, 21, : : : , m1, : : : , 1n, 2n, : : : , mn/. Then, its element in the position ..I, J/, .i, j// is assigned as  1, I D i and J D j, w.I,J/,.i,j/ D ıi,jI,J D 0, otherwise. When m D n, we briefly denote it as WŒn D WŒm,n . Furthermore, for X 2 Rm and Y 2 Rn , WŒm,n Ë X Ë Y D Y Ë X. For statement ease, we firstly introduce some notations. ˚  1. Define a delta set as k :D ıki j i D 1, 2, : : : , k , where ıki is the ith column of the identity matrix Ik with degree k. i h 2. Let matrix M D ıni1 , ıni2 , : : : , ınir , we simply denote it as M D ın Œi1 , i2 , : : : , ir . 3. We denote the ith column (row) of matrix A by Coli .A/.Rowi .A//, and denote the set of columns (rows) of matrix A by Col.A/.Row.A//. 4. A matrix A 2 Mmn is called a logical matrix if the columns of A are elements of m , and the set of all m  n logical matrices is denoted by Lmn . 5. For an n  mn matrix A, we split it into m square blocks as A D ŒBlk1 .A/, Blk2 .A/, : : : , Blkm .A/. We denote the ith n  n block of n  mn matrix A by Blki .A/. 6. If A 2 Mmn is a real matrix, then the inequality A > 0 means that all the entries of A are positive, that is, Ai,j > 0, 8i D 1, 2, : : : , m, j D 1, 2, : : : , n. A logical domain, denoted by D, is defined as D D fTrue D 1, False D 0g. To use matrix expression, we define each element in D with a vector as: True  ı21 and False  ı22 , and then D  2 . Using STP of matrices, a logical function with n arguments L : Dn ! D can be expressed in the algebraic form as follows: Lemma 3 ([15]) Any logical function L.A1 , : : : , An / with logical arguments A1 , : : : , An 2 2 can be expressed in a multilinear form as L.A1 , : : : , An / D ML A1 A2    An ,

2

where ML 2 L22n is unique, called the structure matrix of L. Copyright © 2013 John Wiley & Sons, Ltd.

Math. Meth. Appl. Sci. 2014, 37 1–9

Y. LIU, H. CHEN AND B. WU Lemma 4 ([15]) Assume Zj D A1 A2    Aj with logical arguments A1 , A2 ,    , Aj 2 2 , then Zj2 D ˆj Zj , where ˆj D

Qj

iD1 I2i1

˝ Œ.I2 ˝ WŒ2,2ji  /Mr , Mr D ı4 Œ1, 4, and ‘˝’ is the Kronecker product.

3. Main results In the real word, many evolutionary processes such as BNs may experience abrupt changes of states at some time instants because of the sudden environment changes. Those sudden and sharp changes are often of very short duration and are assumed to occur instantaneously in the form of impulses. Motivated by the aforementioned analysis, in this paper, we consider the BCNs in the following case: the states Ai , i D 1, 2, : : : , n, experience sudden and sharp changes at time tk , where ftk g  Z C , 0 D t0 < t1 < t2 <    < tk <    , k 2 Z C . There are logical functions gi , transferring the Ai .tk / into gi .A1 .tk  1/, A2 .tk  1/, : : : , An .tk  1//. A BCN with n network nodes A1 , A2 , : : : , An describing such case is given by 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ :

A1 .t C 1/ D f1 .u1 .t/, : : : , um .t/, A1 .t/, : : : , An .t//, A2 .t C 1/ D f2 .u1 .t/, : : : , um .t/, A1 .t/, : : : , An .t//, .. . An .t C 1/ D fn .u1 .t/, : : : , um .t/, A1 .t/, : : : , An .t//, A1 .tk / D g1 .A1 .tk  1/, A2 .tk  1/, : : : , An .tk  1//, A2 .tk / D g2 .A1 .tk  1/, A2 .tk  1/, : : : , An .tk  1//, .. . An .tk / D gn .A1 .tk  1/, A2 .tk  1/, : : : , An .tk  1//,

tk1 t < tk  1, tk1 t < tk  1, tk1 t < tk  1,

(1)

k 2 ZC,

where fi : DmCn ! D, gi : Dn ! D, i D 1, 2, : : : , n are Boolean functions and uj 2 D, j D 1, : : : , m are inputs, t D 0, 1, 2, : : :. n n Using the STP of matrices, let x.t/ D ËniD1 Ai .t/, and u.t/ D Ëm jD1 uj .t/, where ËiD1 :  ! 2 is a bijective mapping pointed out by [15]. By Lemma 3, we can find structure matrices M1i D Mfi , M2i D Mgi , i D 1, 2, : : : , n, such that Ai .t C 1/ D M1i u.t/x.t/, i D 1, 2, : : : , n, tk1 t < tk  1,

(2)

Ai .tk / D M2i x.tk  1/, i D 1, 2, : : : , n, k 2 Z C .

(3)

For tk1 t < tk  1, multiplying the equations in (2) and (3), respectively, we have x.t C 1/ D A1 .t C 1/ Ë A2 .t C 1/ Ë    Ë An .t C 1/ D M11 u.t/x.t/M12 u.t/x.t/M13 u.t/x.t/    M1n u.t/x.t/ D M11 .I2mCn ˝ M12 /ˆmCn u.t/x.t/M13 u.t/x.t/    M1n u.t/x.t/

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D  D M11 .I2mCn ˝ M12 /ˆmCn .I2mCn ˝ M13 /ˆmCn    .I2mCn ˝ M1n /ˆmCn u.t/x.t/ , L1 u.t/x.t/. When t D tk , x.tk / D M21 x.tk  1/M22 x.tk  1/    M2n x.tk  1/ D M21 .I2n ˝ M22 /ˆn x.tk  1/    M2n x.tk  1/ D 

(5)

D M21 .I2n ˝ M22 /ˆn .I2n ˝ M23 /ˆn    .I2n ˝ M2n /ˆn x.tk  1/ , L2 x.tk  1/. Then, system (1) can be converted into (

x.t C 1/ D L1 u.t/x.t/, tk1 t < tk  1, x.tk / D L2 x.tk  1/, k 2 Z C ,

(6)

3

Copyright © 2013 John Wiley & Sons, Ltd.

Math. Meth. Appl. Sci. 2014, 37 1–9

Y. LIU, H. CHEN AND B. WU where L1 D M11

n Y 

  I2mCn ˝ M1j ˆmCn ,

jD2

L2 D M21

n Y 

  I2n ˝ M2j ˆn ,

jD2

ˆn D

n Y

   I2q1 ˝ I2 ˝ WŒ2,2nq  Mr .

qD1

Remark 1 In system (6), there are many differences between the dynamics of systems with impulsive effects or not. In the following, we consider the time instant t in two cases. Case 1 Assume t is not an impulsive point, then x1 .t/ D L1 u.t1/x.t1/. Because L1 u 2 L2n 2n and x.t1/ 2 2n , then x.t/ 2 Col.L1 u/.  Assume x.t  1/ D ı2i n , 1 i 2n , then 

x1 .t/ D L1 u.t  1/x.t  1/ D L1 u.t  1/ı2i n D Coli .L1 u.t  1//. Because L1 u.t  1/ 2 fBlk1 .L1 /, Blk2 .L1 /, : : : , Blk2m .L1 /g, we have x1 .t/ 2 fColi .Blk1 .L1 //, Coli .Blk2 .L1 //, : : : , Coli .Blk2m .L1 //g. 

Case 2 Assume t is an impulsive point, then x2 .t/ D L2 x.t  1/, and x2 .t/ D Coli .L2 / when x.t  1/ D ı2i n . If x2 .t/ 2 fColi .Blk1 .L1 //, : : : , Coli .Blk2m .L1 //g, we can choose a proper control u such that x1 .t/ D x2 .t/, which means there is no difference between the dynamics of the system with impulsive effects or not at time t. To focus on the controllability of nontrivial BCNs with impulsive effects, we assume the following: for any 1 i 2n , Coli .L2 / 62 fColi .Blk1 .L1 //, : : : , Coli .Blk2m .L1 //g. In the following, we consider the problem of designing a control sequence that steer the BCN (6) between two states, while avoiding certain forbidden states. n o n Given two states T0 , Ts 2 ı21n , : : : , ı22n and a set of undesirable states C, let l.s; T0 , Ts , C/ be the number of different control sequences that steer the system (6) from x.0/ D T0 to x.s/ D Ts while avoiding C (i.e., x.i/ 62 C for i D 0, 1, : : : , s). Let jCj denote the cardinality of Pm C, Q1 D 2iD1 Blki .L1 /, Q2 D L2 . Give an integer s > 0, assume there are k impulsive points t1 , : : : , tk .t1 < t2 <    < tk / between 0 and s, and s D tk C j, 0 j < tkC1  tk . For example, let tk D 4k, k 2 Z C , and s D 10, then there are two impulsive points (t1 D 4, t2 D 8) between 0 and 10, and j D 2. The following result provides a simple algebraic expression for l.s; T0 , Ts , C/. Theorem 5 Consider system (6), suppose that the forbidden states in C are ı2i1n , : : : , ı2izn , where z D jCj. Let Q1c , Q2c be the matrices obtained from Q1 , Q2 substituting the elements in the rows and columns with indexes i1 , : : : , iz by zeros, respectively. Then, O s T0 , l.s; T0 , Ts , C/ D Ts Q

(7)

O s D .Q1c /j Q2c .Q1c /tk tk1 1    Q2c .Q1c /t2 t1 1 Q2c .Q1c /t1 1 . where Q Proof We first consider s D 1 : .1/t D 1 is an impulsive point, that is, t1 D 1 and j D 0; .2/t D 1 is not an impulsive point, that is, t1 2 and j D 1. j O 1 D Q2c , and there is no control in the equation x.tk / D L2 x.tk 1/. For case (1), let l1 D l.1; T0 , T1 , C/, T0 D ı2i n , T1 D ı2n . In this case, Q If the states x.0/ D T0 , x.1/ D T1 satisfy the equation x.1/ D Q2c x.0/ with x.0/ 62 C, x.1/ 62 C, that is, T1 D Q2c T0 , we say that l.1; T0 , T1 , C/ D 1, otherwise l.1; T0 , T1 , C/ D 0. Hence, O 1 T0 . l.1; T0 , T1 , C/ D .Q2c /ji D T1T Q2c T0 D T1T Q j O 1 D Q1c , if T0 2 C or T1 2 C, then clearly l2 D 0. Because Then, we consider case (2). Let l2 D l.1; T0 , T1 , C/, T0 D ı2i n , T1 D ı2n . In this case, Q O 1 T0 . in Q1c either the row corresponding to j or the column corresponding to i is zero, T1T Q1c T0 D 0, so l.1; T0 , T1 , C/ D T1T Q1c T0 D T1T Q Now, suppose that T0 62 C and T1 62 C, let u1 , : : : , uq be the different control sequences steering system (6) from x.0/ D T0 to x.1/ D T1 , that is,

T1 D L1 Ë ui .0/ Ë T0 , i 2 f1, : : : , pg.

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4 Copyright © 2013 John Wiley & Sons, Ltd.

Math. Meth. Appl. Sci. 2014, 37 1–9

Y. LIU, H. CHEN AND B. WU Because each control value is a column of I2m , there exist q D 2m  p different control sequences v j such that T1 ¤ L1 Ë v j .0/ Ë T0 , j 2 f1, : : : , qg.

(9)

Multiplying (8) and (9) from the left by T1T yields 1 D T1T L1 Ë ui .0/ Ë T0 , i 2 f1, : : : , pg, 0 D T1T L1 Ë v j .0/ Ë T0 , j 2 f1, : : : , qg. Because each of the control values is a different column of I2m , summing up this set of p C q D 2m equations yields p D T1T L1 Ë 12m Ë T0 D T1T Q1 T0 . We conclude that when T0 62 C and T1 62 C, l.1; T0 , T1 , C/ D T1T Q1 T0 , because in this case T1T Q1 T0 D T1T Q1c T0 , l.1; T0 , T1 , C/ D T1T Q1c T0 D O 1 T0 . This proves (7) for s D 1. T1T Q j O s T0 , then we consider For the induction step, assume TsC1 D ı2n , T0 D ı2i n , Ts D ı2n , 1  2n ,  2 NC , l.s; T0 , Ts , C/ D TsT Q O sC1 T0 , for s C 1 < tkC1 , TsC1 Q  T O sC1 T0 D ı j n Q O sC1 ı i n TsC1 Q 2 2  Os D Q1c Q ji

D

2n X

 Os .Q1c /j Q

D1 2n

D

X

j

ı2n

T

i

 T O s ıi n Q1c ı2n ı2n Q 2

D1 n

D

 T T O s T0 . TsC1 Q1c ı2n ı2n Q

2 X D1

When s C 1 D tkC1 ,  T O sC1 T0 D ı j n Q O sC1 ı i n TsC1 Q 2 2  Os D Q2c Q ji

D

2n X

 Os .Q2c /j Q

D1 n

D

2  X

j

ı2n

T

i

 T O s ıi n .Q2c /c ı2n ı2n Q 2

D1 n

D

2 X

 T T O s T0 . TsC1 Q2c ı2n ı2n Q

D1

O s T0 yields Combining the aforementioned two cases and applying the induction hypothesis l.s; T0 , Ts , C/ D TsT Q O sC1 T0 D TsC1 Q

2n   X l 1; ı2n , TsC1 , C l s; T0 , ı2n , C D1

D l.s C 1; T0 , Ts C 1, C/. This is the number of control sequences that steer from T0 to TsC1 in s C 1 time steps avoiding C. The proof is completed.



Copyright © 2013 John Wiley & Sons, Ltd.

5

Remark 2 O s D .Q1c /s in Theorem 5. When there is no impulse in system (6), Q Math. Meth. Appl. Sci. 2014, 37 1–9

Y. LIU, H. CHEN AND B. WU Remark 3 For the particular case C D ;, that is, Q1c D Q1 , Q2c D Q2 , l.s; T0 , Ts , ;/ is just the number of different control sequences that steer system (6) from x.0/ D T0 to x.s/ D Ts . Definition 6 o o n n n Consider system (6), denote its undesirable states by C D ı2i1n , : : : , ı2izn , where z D jCj, and let x.0/ D T0 2 X D ı21n , : : : , ı22n n C, and s > 0. 1. x.s/ 2 X is said to be reachable from x.0/ D T0 at time s avoiding C if we can find a sequence of controls u.t/ that steers system (6) from x.0/ D T0 to x.s/ D Ts avoiding C. The reachable set from x.0/ at time s avoiding C is denoted by RsC .x.0//. The overall reachable set from x.0/ avoiding C is denoted by RC .x.0// D

1 [

RsC .x.0//.

sD1

2. The system (6) is said to be controllable at x.0/ if RC .x.0// D X . It is said to be controllable if it is controllable at every x 2 X . From Theorem 5 and Definition 6, one can have the following reachable result. Theorem 7 O s D .Q1c /j Q2c .Q1c /tk tk1 1    Q2c .Q1c /t1 1 . Consider system (6), with Q

 j Os > 0. 1. x.s/ D ı2˛n is reachable from x.0/ D ı2n at the sth step avoiding C if and only if Q ˛j

j

2. x.s/ D ı2˛n is reachable from x.0/ D ı2n avoiding C if and only if there exists a positive integer k such that

Pk

sD1



Os Q

˛j

> 0.

Q 2C be the matrices obtained from Q1 , Q2 by deleting the rows and columns with indexes fi1 , i2 , : : : , iz g, respectively. Note Q 1C , Q Let Q Q 2C 2 Mll , with l D 2n  jCj. Theorem 5 shows that all controllability information is contained in Q O s . Arguing as in the proof Q 1C , Q that Q of Theorem 5 yields the following result on the controllability. Theorem 8   tk tk1 1  t1 1  Q 1c Q 1c Q 1C j Q Q 2c Q Q 2c Q Ls D Q Q . Consider system (6), with Q

 P j L s > 0. 1. The system is controllable at x.0/ D ı2n avoiding C if and only if there exists a positive integer k such that ksD1 Colj Q Pk L s > 0. 2. The system is controllable avoiding C if and only if there exists a positive integer k such that sD1 Q

Remark 4  s Q 1c in Theorem 8. Then, system (6) is controllable if and only if there exists a positive Ls D Q When there is no impulse in system (6), Q P L s > 0. This controllability criterion is presented in [17], which deals with the controllability of BCNs with integer k such that ksD1 Q forbidden states. Hence, the obtained result here is an extension of [17]. Remark 5 Considering system (6), we divide the state space into two categories: X1 D Col.L1 /, X2 D X n X1 . Assume there is no impulse in system (6), then x.t C 1/ D L1 u.t/x.t/ for all t D 0, 1, 2, : : :. Because L1 2 L2n 2mCn and u.t/x.t/ 2 2mCn , then x.t C 1/ 2 Col.L1 / for any t D 0, 1, : : : and there is no more other possibility for x.t C 1/. Hence, the state belongs to X2 , denoted by ı2i n , and is unreachable for any t > 0. However, if the impulsive disturbances happen, that is, x.tk / D L2 x.tk  1/, for t D tk , k D 1, 2, : : :, then x.tk / 2 Col.L2 /, and it j is possible that ı2i n 2 Col.L2 /. Let x.tk  1/ D ı2n 2 Col.L1 /, and the jth column of L2 may equal to ı2i n , that is, j

x.tk / D L2 x.tk  1/ D L2 ı2n D Colj .L2 / D ı2i n .

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The state ı2i n is reachable because of the impulsive effects. Moreover, if we choose the impulses properly, states that belong to X2 may be reachable. Moreover, system (6) may be controllable because of impulsive effects, which means that impulses play an important role in the controllability of BCNs.

4. An illustrative example

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 ˚ Consider the following BCN with impulsive effects and undesirable states C D ı81 , ı85 , 8 x1 .t C 1/ D x2 .t/, ˆ ˆ ˆ ˆ ˆ x .t C 1/ D x3 .t/, ˆ ˆ ˆ 2 ˆ < x3 .t C 1/ D u.t/ _ Œx2 .t/ ^ x3 .t/, for t k1 t < tk  1, ˆ x .t / D x .t  1/, 1 k 2 k ˆ ˆ ˆ ˆ ˆ x .t / D x 2 3 .tk  1/, ˆ k ˆ ˆ : x3 .tk / D x1 .tk  1/ ^ x3 .tk  1/, tk D 4k, k 2 Z C . Copyright © 2013 John Wiley & Sons, Ltd.

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Math. Meth. Appl. Sci. 2014, 37 1–9

Y. LIU, H. CHEN AND B. WU Denote x.t/ D x1 .t/x2 .t/x3 .t/, then we can convert system (11) into ( x.t C 1/ D L1 u.t/x.t/, tk1 t < tk  1, x.tk / D L2 x.tk  1/, tk D 4k, k 2 Z C ,

(12)

where L1 D ı8 Œ1, 3, 5, 7, 1, 3,P 5, 7, 1, 4, 6, 8, 1, 4, 6, 8, L2 D ı8 Œ7, 6, 3, 2, 8, 6, 4, 2. It is easy to obtain Q1 D 2iD1 Blki .L1 /, Q2 D L2 , in detail i h Q1 D 2ı81 ı83 C ı84 ı85 C ı86 ı87 C ı88 2ı81 ı83 C ı84 ı85 C ı86 ı87 C ı88 , i h Q2 D ı87 ı86 ı83 ı82 ı88 ı86 ı84 ı82 . The matrices Q1c ,Q2c are obtained from Q1 ,Q2 substituting the elements in the rows and columns with indexes f1, 5g by zeros, Q 2c are obtained from Q1 , Q2 by deleting the rows and columns with indexes f1, 5g, respectively, that is, Q 1c , Q respectively. The matrices Q i h Q 2C D ı6 Œ4, 2, 1, 4.3, 1. Q 1C D ı62 C ı63 ı64 ı65 C ı66 ı62 C ı63 ı64 ı65 C ı66 , Q Q Let s D 3, x.3/ D ı88 , x.0/ D ı83 . After a straightforward computation, we have    O1 O2 O3 Q D Q D 0, Q D 1 > 0. 83

83

83

 ˚ at the third step avoiding ı81 , ı85 . In the following, we consider the controllability of system (11). We first assume that there are no impulsive effects in the system. In O s D .Q1c /s as Remark 2. When s D 4, we have this case, Q 3 2 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 7 7 6 6 0 1 1 1 0 1 1 1 7 7 6 6 0 1 1 1 0 1 1 1 7 7. 6 O (13) Q4 D 6 7 6 0 0 0 0 0 0 0 0 7 6 0 2 1 2 0 2 1 2 7 7 6 4 0 2 1 2 0 2 1 2 5 0 2 1 2 0 2 1 2  O 4 in (13), one can see that Row2 Q O 4 D 0, then we have From matrix Q  o n O 04 T0 D 0, T0 2 ı82 , ı83 , ı84 , ı86 , ı87 , ı88 . l 4; T0 , ı82 , C D ı82 Q

Hence, from (1) of Theorem 7, x.3/ D ı88

is reachable from x.0/ D ı83

It means that the state x.4/ D ı82 is not reachable from any initial state x.0/ at the fourth step because there is no column vector ı82 in L1 . Thus, the system is uncontrollable from Definition 6. O 4 D Q2c .Q1c /3 as Theorem 5. Now, we analyze the BCN with impulsive effects as system (11). Let s D 4, then we have Q 3 2 0 0 0 0 0 0 0 0 6 0 2 1 2 0 2 1 2 7 7 6 6 0 1 0 1 0 1 0 1 7 7 6 6 0 1 1 1 0 1 1 1 7 7. 6 O (14) Q4 D 6 7 6 0 0 0 0 0 0 0 0 7 6 0 1 1 1 0 1 1 1 7 7 6 4 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0    O 4 in (14), one can see that Row2 Q O 4 > 0 except Q L4 L 4 , then we have From matrix Q and Q l



4; T0 , ı82 , C



21

O 4 T0 D ı82 Q

25

o n > 0, T0 2 ı82 , ı83 , ı84 , ı86 , ı87 , ı88 ,

which means that x.4/ D ı82 is reachable from every initial state x.0/ 2 we have 2 2 1 2 2 6 3 1 2 3 6 4 X 6 Ls D 6 3 2 2 3 Q 6 3 3 3 3 6 sD1 4 2 1 3 2 2 1 2 2

˚

 ı82 , ı83 , ı84 , ı86 , ı87 , ı88 at the fourth step. Through calculation,

1 1 2 3 1 1

2 2 2 3 3 2

3 7 7 7 7 > 0. 7 7 5

7

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Math. Meth. Appl. Sci. 2014, 37 1–9

Y. LIU, H. CHEN AND B. WU  ˚ Hence, from .2/ of Theorem 8, system (11) is controllable avoiding states ı81 , ı85 , which shows that impulses play an important role in the controllability of BCNs as mentioned in Remark 5. Remark 6 To the aforementioned example, we have o o n n Col.L1 / D ı81 , ı83 , ı84 , ı85 , ı86 , ı87 , ı88 , Col.L2 / D ı82 , ı83 , ı84 , ı86 , ı87 , ı88 . On the basis of the analysis in Remark 5, if there are no impulsive effects in BCN, then x.tC1/ 2 Col.L1 / for any t D 0, 1, : : :, which means x.t C1/ ¤ ı82 . When impulses happen as the BCN given by (11), the fourth and eighth column of L2 equal to ı82 . Because ı84 , ı88 2 Col.L1 /, it is possible that x.tk  1/ reaches the states ı84 , ı88 ; then, the state ı22n may be reachable at the fourth step from (10).

5. Conclusion This paper investigates the controllability of BCN with impulsive effects avoiding certain forbidden states. We first derive a simple formula for the number of different control sequences that steer a BCN between two given states. Then, we derive some necessary and sufficient conditions for controllability of the BCNs with impulsive effects. The obtained results present the importance of impulsive effects on the controllability of BCNs. Finally, an example is provided to show the effectiveness of the obtained results.

Acknowledgements The authors would like to thank the editor and the anonymous reviewers for their constructive comments and suggestions to improve the quality of the paper. This work was supported by the NNSFs of China (grant nos. 11271333, 11101373, and 61074011).

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