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Controllability of Boolean control networks with time delays both in states and inputs
Neurocomputing 129 (2014) 467–475
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Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Controllability of Boolean control networks with time delays both in states and inputs Ming Han, Yang Liu n, Yanshuai Tu College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, China
art ic l e i nf o
a b s t r a c t
Article history: Received 18 December 2012 Received in revised form 27 May 2013 Accepted 1 September 2013 Communicated by Lixian Zhang Available online 20 October 2013
This paper investigates the controllability of Boolean control networks (BCNs) with time-invariant delays both in states and inputs. The necessary and sufficient conditions on the controllability via two kinds of controls are presented by providing the corresponding reachable sets. The proposed results generalize the controllability analysis on Boolean control networks without time delay. And we also consider the optimal control problem of BCNs from a given initial state to a desired state in minimal time. Three examples include a biological example are given to show the effectiveness of the proposed results. & 2013 Elsevier B.V. All rights reserved.
Keywords: Boolean (control) networks (BCNs) Controllability Time delay Semi-tensor product
1. Introduction In the recent years, a new view of biology, called the systems biology, has attracted increasing interest from various fields. Systems biology usually focuses on the system-level properties of the whole network dynamics, e.g., [1,2]. Especially, the Boolean network, a network with nodes and directed edges, becomes a powerful tool in describing, analyzing and simulating the systems biology. The Boolean networks are a class of models first introduced by Kauffman [3]. In this model, a gene expression level can be described by a binary value, 0 or 1, indicating two transcriptional states: active (on) or inactive (off). And the state of each gene is determined by the states of its neighborhood genes on the basis of logical rules. As to the structure of Boolean networks, including the fixed points, cycles, attractors and transient time, we refer [4–9] for details. In [6], the identification of Boolean control networks is addressed, and the matrix representation of the Boolean networks and its applications are discussed in [8]. Specific examples of genetic regulation networks modeled by BNs include: the network controlling the segment polarity genes in the fly Drosophila melanogaster [4]; the ABC network determining floral organ cell fate in Arabidopsis [5], and so on. BNs with binary inputs are referred to as BCNs. The controllability of BCNs
n
Corresponding author. Tel./fax: þ 86 57982298897. E-mail address: [email protected] (Y. Liu).
0925-2312/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2013.09.012
has been addressed in [10]. Akutsu et al. [7] showed that control problems for BCNs are in general NP-hard. In order to systematically analyze the Boolean network, [9] has proposed a way to convert the Boolean networks into discrete time dynamic systems based on semi-tensor product (STP) of matrices. Then using this method, [10] has studied the controllability and observability of Boolean control networks and [13] has analyzed the stability and stabilization of Boolean network. Besides, there are also many other results obtained, for example, see [14,15]. It is said in [16] that one of the major goals of systems biology is to develop a control theory for complex biological systems. Controllability as one of the fundamental concepts in control theory is becoming more and more important. In the literature, many different definitions of controllability have been given in [17–21], which strongly depend on class of dynamical control systems. Then there have appeared many results on the systematic study of controllability, e.g., [22–25]. However, when it comes to the control of Boolean network, there have been only very few results. In [10], controllability and observability of Boolean control networks have been investigated. In [11], the input–state incidence matrix of a control Boolean network has been proposed. Based on it, an easily verifiable necessary and sufficient condition for the controllability of a Boolean control network is obtained. As we all know that time delay phenomenon is very common in real world. It has been recognized that the slow processes of transcription, translation, and translocation or the finite switching speed of amplifiers will unavoidable cause time delays, which should be taken into account in the biological systems or artificial
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genetic networks in order to reduce the impaction on the dynamical behavior of models [26]. Therefore, time delays should be considered in the dynamics of genetic networks to have more accurate models. Usually, we consider the systems with time delay in state, for example, in [27–29]. For Boolean networks, there also has been some recent papers that discussed the controllability problem with delays in states. For example, Li and Sun have investigated the controllability of Boolean networks with timeinvariant delays in states [30]. Meanwhile, considering time delay in the control is also very important to the system performance. In the recent studies, the systems with time delays in the control have been analyzed widely in economic, biological and physiological industry systems [31–36]. Besides, the controllability of systems with time delays in both states and control has investigated in [37,38]. But to the best of our knowledge, there has been no results in the literature concerning the controllability of the Boolean control networks with time delays in both states and controls. So in this paper, we will extend our result to the more general case, namely, the controllability of Boolean control networks with time delays both in states and controls. Moreover, optimal controls are also important in the context of BCNs that model biological systems. For example, a natural problem is to determine a control that steers the BCN from an initial state (that corresponding to a disease state) to a desired state (that corresponding to healthy state) in minimal time. Motivated by the above discussions, a natural question is that whether we can contribute to the optimal control problem for BCNs. So we also study the time-optimal control problem in the examples. The rest of this paper is organized as follows. In Section 2, we introduce the STP of matrices and the matrix expression of logical function. The main result on the controllability problem of Boolean networks with time delays and an algorithm for the timeoptimal control of BNs is given in Section 3. Section 4 gives three examples for illustration and some conclusions are finally drawn in Section 5.
2. Preliminaries 2.1. STP of matrices
of conventional matrix product can be applied to the STP of matrix, e.g., distributive rule, Associative rule and so on. Since then, we can omit ⋉ in this paper. In the following, we introduce some special properties, for details, see [9]. Definition 2.3 (Cheng and Qi [9]). (1) Assume A g t B, then (where is the Kronecker product, It is the identity matrix) A⋉B ¼ AðB I t Þ: Assume A ! t B, then A⋉B ¼ ðA I t ÞB: (2) Assume A A M mn is given. Let Z A Rt be a row vector. Then A⋉Z ¼ Z⋉ðI t AÞ: Let Z A Rt be a column vector. Then Z⋉A ¼ ðI t AÞ⋉Z: Then we give some notations as the following: (1) Define a delta set as Δk ≔fδk ji ¼ 1; 2; …; kg, where δki is the ith column of It. (2) A matrix AA M mn is called a logical matrix if the columns of A, denoted by Col(A), satisfy ColðAÞ Δm . i i i (3) Assume a matrix A ¼ ½δm1 ; δm2 ; …; δmn , we denote it as A ¼ δm ½i1 ; i1 ; …; in . i
Next, we give an important definition as follows: Definition 2.4 (Cheng and Qi [9]). (1) An mn mn matrix W m;n is called swap matrix, if it is constructed in the following way: label its columns by ð11; 12; …; 1n; m1; m2; …; mnÞ and its rows by ð11; 21; …; m1 ; 1n; 2n; …; mnÞ. Then its element in the position ððI; JÞ; ði; jÞÞ is assigned as 1; I ¼ i and J ¼ j; I;J wðI;JÞ;ði;jÞ ¼ δi;j ¼ ð1Þ 0 otherwise: When m¼n, we briefly denote W ½n ≔W ½m;n . (2) Let X A Rm and Y A Rn be two columns. Then W ½m;n ⋉X⋉Y ¼ Y⋉X; W ½n;m ⋉Y⋉X ¼ X⋉Y:
Definition 2.1 (Cheng and Qi [9]). 2.2. Matrix expression of logic (1) Let X be a row vector of dimension np, and Y be a column vector of dimension p. Then we split X into equal-size blocks as X 1 ; …; X P , which are 1 n rows. Define the semi-tensor product (STP) of matrices, denoted by ⋉, as 8 p > > X⋉Y ¼ ∑ X i y A Rn > > i < i¼1 p > > > Y T ⋉X T ¼ ∑ yi ðX i ÞT A Rn : > : i¼1
(2) Let A A M mn and B A M pq . If either n is a factor of p, say nt ¼ p and denote it as A ! t B, or p is a factor of n, say n¼ pt and denote it as A g t B, then we define the STP of A and B, denoted by C ¼ A⋉B, as the following: C consists of m q blocks as C ¼ C ij and each block is C ij ¼ Ai ⋉Bj ;
i ¼ 1; …; m; j ¼ 1; …; q;
where Ai is the ith row of A and Bj is the jth column of B. Remark 2.2. When n ¼p, A⋉B ¼ AB. So it is a generalization of the conventional matrix product, and all the fundamental properties
In this paper, we recall the matrix expression of logic as [9] for details. Define a logical domain D as D ¼ fT ¼ 1; F ¼ 0g, then to use the STP of matrices, we give the logical values a vector form as: 1 2 T ¼ 1 δ2 ; F ¼ 0 δ2 and denote
Δ≔Δ2 ¼ fδ12 ; δ22 g D: Similar, a logical function with n arguments f : Dn -D can be expressed in the algebraic form as follows: Lemma 2.5 (Cheng and Qi [9]). Any logical function LðA1 ; …; An Þ with logical arguments A1 ; …; An A Δ can be expressed in a multilinear form as LðA1 ; …; An Þ ¼ M L A1 A2 ⋯An ; where ML is unique, called the structure matrix of L. Another important lemma for this paper is given as the following: Lemma 2.6 (Cheng and Qi [9]). Assume P k ¼ A1 A2 ⋯Ak , then P 2k ¼ Φk P k ;
M. Han et al. / Neurocomputing 129 (2014) 467–475
where k
Φk ¼ ∏ I2i 1 ½ðI2 W ½2;2k i ÞMr ; i¼1
M r ¼ δ4 ½1; 4:
3. Main results A Boolean network of a set of nodes A1 ; A2 ; …; An can be described as 8 A1 ðt þ 1Þ ¼ f 1 ðA1 ðtÞ; A2 ðtÞ; …; An ðtÞÞ; > > > > < A2 ðt þ 1Þ ¼ f 2 ðA1 ðtÞ; A2 ðtÞ; …; An ðtÞÞ; ð2Þ ⋮ > > > > : An ðt þ 1Þ ¼ f ðA1 ðtÞ; A2 ðtÞ; …; An ðtÞÞ; n where f i ; i ¼ 1; 2; …; n, are logical function and t ¼ 0; 1; 2; … . Then we assume that the Boolean networks have timeinvariant delays in states as follows: 8 A ðt þ 1Þ ¼ f 1 ðA1 ðt τÞ; A2 ðt τÞ; …; An ðt τÞÞ; > > > 1 > < A2 ðt þ 1Þ ¼ f ðA1 ðt τÞ; A2 ðt τÞ; …; An ðt τÞÞ; 2 ð3Þ ⋮ > > > > : An ðt þ 1Þ ¼ f ðA1 ðt τÞ; A2 ðt τÞ; …; An ðt τÞÞ; n
where
τ is a positive integer delay.
3.1. Controllability of Boolean control networks Define xðtÞ ¼ ⋉ni¼ 1 Ai ðtÞ, where ⋉ni¼ 1 is a bijective mapping pointed out in [9]. Using Lemma 2.5, we can find the structure matrix Mi for each function f i ; i ¼ 1; 2; …; n. So system (3) can be converted into an algebraic form as follows: Ai ðt þ 1Þ ¼ M i A1 ðt τÞA2 ðt τÞ⋯An ðt τÞ ¼ M i xðt τÞ:
ð4Þ
ð5Þ
Then the matrix form is Aðt þ 1Þ ¼ M c Bðt τÞCðt τÞ:
or
Ed W ½2 XY ¼ X:
ð8Þ
In this paper, two kinds of controls are considered: (1) The controls are logical variables satisfying certain logical rules, called input network as follows: 8 u ðt þ 1Þ ¼ g 1 ðu1 ðt τ Þ; u2 ðt τÞ⋯um ðt τÞÞ; > > > 1 > < u2 ðt þ 1Þ ¼ g 2 ðu1 ðt τ Þ; u2 ðt τÞ⋯um ðt τÞÞ; ð9Þ ⋮ > > > > : um ðt þ 1Þ ¼ g ðu1 ðt τÞ; u2 ðt τÞ⋯um ðt τÞÞ; m where g i ; i ¼ 1; 2; …; m, are logical function and the initial states uj ði τÞ; j ¼ 1; 2; …; m; i ¼ 1; 2; …; τ can be arbitrarily given. (2) The control is a free Boolean sequence. Precisely, set uðtÞ ¼ ⋉m j ¼ 1 uj ðtÞ. Then the control is a designed sequence. Similarly, let xðtÞ ¼ ⋉ni¼ 1 Ai ðtÞ, uðtÞ ¼ ⋉m j ¼ 1 uj ðtÞ, we can also find the structure matrix M1i and M2j for each logical function fi and gj. Then system (8) and (9) can be converted as Ai ðt þ 1Þ ¼ M 1i xðt τÞ;
i ¼ 1; 2; …; n;
ð10Þ
uj ðt þ 1Þ ¼ M 2j uðt τÞ;
j ¼ 1; 2; …; m:
ð11Þ
xðt þ 1Þ ¼ A1 ðt þ1ÞA2 ðt þ1Þ⋯An ðt þ 1Þ ¼ M 11 uðt τÞxðt τÞ⋯M 1n uðt τÞxðt τÞ ¼ M 11 ðI 2m þ n M 12 ÞðI 2m W ½2m ;2n ÞΦm ⋉ðI 2m Φn Þ⋯ðI 2m þ n M 1n Þ
⋉ðI 2m W ½2m ;2n ÞΦm ðI 2m Φn Þ ⋉uðt τÞxðt τÞ 9 Luðt τÞxðt τÞ: And, multiplying (11), we can have
ð6Þ
To get the form of (4), we can use a dummy matrix Ed ¼ δ2 ½1; 2; 1; 2, which is constructed in [9]. Then for any two logical variables X; Y, Ed XY ¼ Y
integer delays in both states and controls as follows: 8 A1 ðt þ 1Þ ¼ f 1 ðu1 ðt τÞ; …; um ðt τÞ; > > > > > A 1 ðt τ Þ; …; An ðt τ ÞÞ; > > > > > A > < 2 ðt þ 1Þ ¼ f 2 ðu1 ðt τÞ; …; um ðt τÞ; A1 ðt τÞ; …; An ðt τÞÞ; > > > ⋮ > > > > > > An ðt þ 1Þ ¼ f n ðu1 ðt τÞ; …; um ðt τÞ; > > : A1 ðt τÞ; …; An ðt τÞÞ;
Multiplying (10) yields
Remark 3.1. Note that usually the indegree is much less than n, that is, the right-hand side of ith equation of (3) may not have all Ai ; i ¼ 1; 2; …; n. For example, for node A: Aðt þ 1Þ ¼ Bðt τÞ 4 Cðt τÞ:
469
ð7Þ
Then we can rewrite (6) as Aðt þ 1Þ ¼ M c Ed Aðt τÞBðt τÞCðt τÞ ¼ M c Ed xðt τÞ: Multiplying all equations (4), together yields xðt þ 1Þ ¼ M 1 xðt τÞM 2 xðt τÞ⋯M n xðt τÞ: Then according to Lemma 2.6, (3) can be converted to a standard discrete-time dynamic system as xðt þ 1Þ ¼ L1 xðt τÞ; where L1 ¼ M 1 ðI 2n M 2 ÞΦn ðI 2n M 3 ÞΦn ⋯ðI 2n M n ÞΦn , Φn is defined as Φn ¼ ⋉ni¼ 1 I 2i 1 ½ðI 2 W ½2;2n i ÞM r ; M r ¼ δ4 ½1; 4. L1 is called the network transition matrix of (3). In the real word, biological systems may experience delay of states and control because of the environment changes. This maybe due to changes in environment, such as temperature, growth rate, etc. Next, we consider Boolean control network with time-invariant
uðt þ 1Þ ¼ u1 ðt þ 1Þu2 ðt þ1Þ⋯um ðt þ 1Þ ¼ M 21 uðt τÞM 22 uðt τÞ⋯M 2m uðt τÞ ¼ M 21 ðI 2m M 22 ÞΦm ðI 2m M 23 ÞΦm
⋯ðI 2m M 2m ÞΦm uðt τÞ 9 Guðt τÞ: From the above conclusion, it is easy to obtain an algebraic form of the Boolean control network (8)–(9) as ( uðt þ 1Þ ¼ Guðt τÞ; ð12Þ xðt þ 1Þ ¼ Luðt τÞxðt τÞ; where L is the network transition matrix of (8), and G is the network transition matrix of (9). Next, we consider the controllability problem of Boolean control network (8)–(9), equivalently (12). 3.1.1. Control via input Boolean network In the following, we consider the case when the controls are logical variables satisfying certain logical rules. First, we give the following refer to the Definition 1 of [10]: Definition 3.2. Consider system (8) with control (9). Given initial state xð τÞ; xð τ þ 1Þ; …; xð0Þ A D2n and the destination state xd, xd is said to be controllable from initial state xði τÞ ði A 0; 1; …; τÞ at s
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M. Han et al. / Neurocomputing 129 (2014) 467–475
steps with fixed input structure G, if we can find the initial input uði τÞ ðiA 0; 1; …; τÞ such that xðs þ iÞ ¼ xd . According to the above definition we may consider four cases: (i) fixed s and fixed G; (ii) fixed s and designable G; (iii) free s 4 0 and fixed G; (iv) free s 4 0 and designable G. First we start from case (i). Theorem 3.3. Consider system (8) with control (9), equivalently (12), where G is fixed. xd is s step reachable from xði τÞ ði A 0; 1; …; τÞ, if and only if
¼ LG2 ðI 2m LGÞðI 22m LÞðI 2m Φm Þ xð2τ þ 4Þ ¼ Luðτ þ 3Þxðτ þ 3Þ ¼ LG2 uð1 τÞLGðI 2m LÞΦm uð1 τÞ
and
8 a > LG ðI 2m LGa 1 Þ > > > < ⋉ðI LGa 2 Þ⋯ðI 2am LÞ 22m ΘG ðs þ iÞ ¼ > ⋉ðI 2ða 1Þm Φm Þ⋯ðI 2m Φm ÞΦm ; a Z 0 > > > : L; a ¼ 0;
where Φm is defined as Φm ¼ ⋉m i ¼ 1 I 2i 1 ½ðI 2 W ½2;2m i ÞM r , M r ¼ δ4 ½1; 4. Proof. First, we consider the control in (12) as follows: uð1Þ ¼ Guð τÞ; uð2Þ ¼ Guð1 τÞ; ⋮ uðτ þ 1Þ ¼ Guð0Þ; uðτ þ 2Þ ¼ Guð1Þ ¼ G2 uð τÞ; uðτ þ 3Þ ¼ Guð2Þ ¼ G2 uð1 τÞ; ⋮ uð2ðτ þ1ÞÞ ¼ Guðτ þ 1Þ ¼ G2 uð0Þ; uð2τ þ 3Þ ¼ Guðτ þ 2Þ ¼ G3 uð τÞ; uð2τ þ 4Þ ¼ Guðτ þ 3Þ ¼ G3 uð1 τÞ; ⋮ uð3ðτ þ1ÞÞ ¼ Guð2τ þ 2Þ ¼ G3 uð0Þ; ⋮: Amuse that there exist a A f0; 1; 2; …;g; b A f1; 2; …; τ þ 1g such that s þ i ¼ aðτ þ1Þ þb:
m
m
⋉Φm uð1 τÞxð1 τÞ; ⋮ xð3ðτ þ 1ÞÞ ¼ Luð2τ þ 2Þxð2τ þ 2Þ
¼ LG2 uð0ÞLGðI 2m LÞΦm uð0Þxð0Þ ¼ LG2 ðI 2m LGÞðI 22m LÞðI 2m Φm Þ
G
s þ i ¼ aðτ þ1Þ þb
⋉xð1 τÞ
¼ LG ðI 2 LGÞðI 22m LÞðI 2 Φm Þ 2
xd A ColfΘ ðs þ iÞW ½2n ;2m xðb 1 τÞg; where and hereafter “Col” is the column set, also there exist unique a A f0; 1; 2; …g and b A f1; 2; …; τ þ 1g such that s þ i satisfies
⋉Φm uð τÞxð τÞ;
⋉Φm uð0Þxð0Þ;
⋮: Similar to the analysis about the control, we can obtain that xðs þ iÞ ¼ xðaðτ þ 1Þ þ bÞ ¼ Luðða 1Þðτ þ 1Þ þ b 1Þ⋉xðða 1Þðτ þ 1Þ þ b 1Þ ¼ LGa ðI 2m LGa 1 ÞðI 22m LGa 2 Þ ⋉ðI 23m LGa 3 Þ…; ðI 2am LÞ⋉ðI 2ða 1Þm Φm Þ…ðI 2m Φm Þ ⋉Φm uðb 1 τÞxðb 1 τÞ ¼ Θ ðs þ iÞuðb 1 τÞxðb 1 τÞ G
¼ Θ ðs þ iÞW ½2n ;2m xðb 1 τÞ⋉uðb 1 τÞ: G
ð13Þ
Since Θ ðs þ iÞW ½2n ;2m xðb 1 τÞ is a 2 2 matrix, whose columns are elements in Δ2n , and uðb 1 τÞ A Δ2m , we can drive the conclusion. □ G
n
m
Remark 3.4. Compared with the result of [30], we need to consider the process of control carefully. And the calculus process is obviously more complex and difficult. Next, we consider the case (ii). m Since there are ð2m Þ2 possible distinct G, we can express each G in the condensed form and order them in “increasing order”, see [9]. For example, when m ¼1, we have G1 ¼ δ2 ½1; 1, G2 ¼ δ2 ½1; 2, G3 ¼ δ2 ½2; 1, G2 ¼ δ2 ½2; 2. Thus we may consider a subset m Λ f1; 2; …; ð2m Þ2 g and allow G be chosen from the admissible set fGλ jλ A Λg. Then the following results can be obtained. Corollary 3.5. Consider system (8) with control (9), where G A fGλ j
λ A Λg. Then xd is s step reachable from xði τÞ ðiA 0; 1; …; τÞ , if and only if there exists a Gλ such that xd A ColfΘ λ ðs þ iÞW ½2n ;2m xðb 1 τÞg G
Then from (13), we can easily get that
and there exist a A f0; 1; 2; …g and b A f1; 2; …; τ þ 1g such that s þ i satisfies
uðs þ iÞ ¼ uðaðτ þ 1Þ þ bÞ
s þi ¼ aðτ þ 1Þ þ b
¼ Ga þ 1 uðb 1 τÞ: Above all, a straightforward computation shows the following: xð1Þ ¼ Luð τÞxð τÞ; xð2Þ ¼ Luð1 τÞxð1 τÞ; ⋮ xðτ þ 1Þ ¼ Luð0Þxð0Þ; xðτ þ 2Þ ¼ Luð1Þxð1Þ ¼ LGuð τÞLuð τÞxð τÞ ¼ LGðI 2m LÞΦm uð τÞxð τ Þ; xðτ þ 3Þ ¼ Luð2Þxð2Þ ¼ LGuð1 τÞLuð1 τÞxð1 τÞ ¼ LGðI 2m LÞΦm uð1 τÞxð1 τÞ; ⋮ xð2ðτ þ 1ÞÞ ¼ Luðτ þ 1Þxðτ þ 1Þ ¼ LGuð0ÞLuð0Þxð0Þ ¼ LGðI 2m LÞΦm uð0Þxð0Þ; xð2τ þ3Þ ¼ Luðτ þ 2Þxðτ þ 2Þ ¼ LG2 uð τ ÞLGðI 2m LÞΦm uð τÞxð τÞ
and
8 a LGλ ðI 2m LGaλ 1 Þ > > > > < ⋉ðI a2 Þ⋯ðI 2am LÞ 22m LGλ ΘG ðs þ iÞ ¼ > ⋉ðI ða 1Þm Φm Þ⋯ðI 2m Φm ÞΦm ; a Z 0 > 2 > > : L; a ¼ 0: Finally, we consider cases (iii) and (iv). First, we need the following lemma.
Lemma 3.6 (Cheng and Qi [10]). For a Boolean network, if its network transition matrix is nonsingular, then every point is on a cycle. In the following, we assume that G is nonsingular. According to Lemma 2.5, starting from uði τÞ; i ¼ 0; 1; …; τ, as G is nonsingular, then we can find a minimum T i 4 0 such that GT i uði τÞ ¼ uði τÞ. Hence fuði τÞ; Guði τÞ; …; GT i 1 uði τÞg is a
M. Han et al. / Neurocomputing 129 (2014) 467–475
cycle of length Ti. Following the procedure in [12], we can construct a mapping
for fixed G ¼ Gk , the reachable set from xði τÞ is RGk ¼ ⋃ RGuðik τÞ ; uði τ Þ
Ψ i ≔½LGT i 1 uði τÞ½LGT i 2 uði τÞ⋯½LGuði τÞ ½Luði τÞ:
ð14Þ
Then for xði τÞ, we consider the sequence xði τÞ; Ψ i xði τÞ; … . Similarly, we can find the transient period ri and a minimum T i ′ 4 0 such that ′
471
′
Ψ Ti i xði τÞ ¼ Ψ ri i þ T i xði τÞ:
ð15Þ
Remark 3.7. Starting from xði τÞ, then the process is as follows: xði τÞ-xði þ 1Þ ¼ Luði τÞxði τÞxði þ 1 þ τ þ 1Þ ¼ LGuði τÞLuði τÞxði τÞ-⋯-
(3) for admissible fGλjλ A Λ g, the reachable set is G R ¼ ⋃ ⋃ Ruðiλ τÞ : λ A Λuði τÞ
3.1.2. Control via free Boolean sequence In the following, we consider the case when the controls are free Boolean sequences.
xði þ 1 þ ðT i þ 2Þðτ þ 1ÞÞ ¼ LGuði τÞLuði τÞΨ i xði τÞ-⋯-
Definition 3.9 (Cheng and Qi [10]). Given initial state xð τÞ; x ð τ þ1Þ; …; xð0Þ A D2n and the destination state xd. The Boolean control network (8) is said to be controllable from initial state xði τÞ ði A 0; 1; …; τÞ to xd at s steps, if we can find the initial input uði τÞ ðiA 0; 1; …; τÞ such that xðs þ iÞ ¼ xd .
xði þ 1 þ 2T i ðτ þ 1ÞÞ ¼ Ψ i xði τÞ⋯
Define L~ ¼ LW ½2n ;2m , then the second equation in (12) can be expressed as
xði þ 1 þ T i ðτ þ 1ÞÞ ¼ Ψ i xði τÞ-
xði þ 1 þ ðT i þ 1Þðτ þ 1ÞÞ ¼ Luði τÞΨ i xði τÞ2
T′ 1
xði þ 1 þ ððT′ 1ÞT i þ 1Þðτ þ 1ÞÞ ¼ Luði τÞΨ i
~ τÞuðt τÞ: xðt þ 1Þ ¼ Lxðt
xði τÞ-
xði þ 1 þ ððT′ 1ÞT i þ 2Þðτ þ 1ÞÞ ¼ LGuði τÞLuði τÞΨ
T′ 1 i
It yields
xði τÞ-⋯-xði þ 1 þ T′T i ðτ þ 1ÞÞ ¼ Ψ i xði τÞ⋯ T′
τ
τ ÞΨ
xði þ 1 þ ððr i þ T ′i 1ÞT i þ 1Þð þ 1ÞÞ ¼ Luði xði Þ-xði þ 1 þ ððr i þ T ′i 1ÞT i þ 2Þð þ1ÞÞ
τ
τ
~ þ i 1 τÞuðs þ i 1 τÞ xðs þ iÞ ¼ Lxðs ¼ L~ xðs þ i 2 2τÞuðs þ i 2 2τÞ⋉uðs þ i 1 τÞ 2
r i þ T ′i 1 i
¼ L~ xðs þ i 3 3τÞuðs þ i 3 3τÞ ⋉uðs þ i 2 2τÞuðs þ i 1 τÞ ¼ ⋯ 3
¼ LGuði τÞLuði τÞ
′
Ψ iri þ T i 1 xði τÞ-⋯-xði þ 1 þ ððri þ T ′i ÞT i Þðτ þ 1ÞÞ r þ T ′i
¼ Ψ ii
¼ L~ xðs þ i k kτÞuðs þ i k kτÞ ⋉uðs þ i ðk 1Þ ðk 1ÞτÞ⋯⋉uðs þ i 1 τÞ: k
xði τÞ ¼ Ψ i xði τÞ: T′
ð18Þ
Assume that Then the reachable set starting from xði τÞ with uði τÞ ði ¼ 0; 1 ; …; τÞ can be constructed as the following algorithm: Step 1 Find Ti such that uði τÞ; Guði τÞ; …; GT i 1 uði τÞ ði ¼ 0; 1; …; τÞ is a cycle in the input space. Step 2 Find the transient period ri and a minimum T ′i 4 0 satisfying (15). Step 3 Construct a sequence for i ¼ 0; 1; …; τ, xk ði τÞ ¼ Ψ i xði τÞ; k
k ¼ 0; 1; …; r i þ T ′i 1:
ð16Þ
Step 4 For each xk ði τÞ construct inductively a sequence for j ¼ 1; …; T i 1, xkj ði
τÞ ¼ LG
j1
uði τ
Þxkj 1 ði
τÞ:
ð17Þ
Notice that the above construction is a special case of the general one discussed in [12] for constructing input–state product cycle, so it is easily to see fxkj ði τÞg is the set of reachable points starting from xði τÞ with uði τÞ and fixed G. The result can be obtained as follows:
s þ i k kτ ¼ j τ;
j A f0; 1; …; τg:
Then we have the following result: Theorem 3.10. xd is reachable from xði τÞ ðiA 0; 1; …; τÞ at s steps by controls of Boolean sequences uðs þ i k kτÞ ⋉uðs þ i ðk 1Þ ðk 1Þτ Þ⋯uðs þ i 1 τÞ if and only if xd A ColfL~ xðs þ i k kτÞg; k
where there exist unique k and j such that s þ i k kτ ¼ j τ;
j A f0; 1; …; τg:
When the length of sequence s is free, the generalization consequence of Theorem 3.10 can be immediately obtained as follows: Corollary 3.11. xd is reachable from xði τÞ ðiA 0; 1; …; τÞ at s steps by controls of Boolean sequences uðs þi k kτÞ⋉uðs þ i ðk 1Þ ðk 1Þτ Þ⋯uðs þ i 1 τÞ if and only if 1 k xd A Col ⋃ L~ xðs þ i k kτÞ ; a¼1
Theorem 3.8. Consider system (8) with control (9). Assume G is nonsingular and use the above algorithm, then
where there exist unique j and k such that
(1) for given uði τÞ; i ¼ 0; 1; …; τ and G, the set of reachable states is
Remark 3.12. It is obvious that if we choose τ ¼ 0, the obtained results can deduce the conclusion in [10], which can also prove the exactness of our studies.
RGuði τÞ ¼ fxkj ði τÞjk ¼ 0; 1; …; r i þ T ′i 1; j ¼ 0; 1; …; T i 1g; where fxkj ði τÞg are constructed by (16)–(17) and the steady state reachable set is defined by G k k (2) RSuði τÞ ≔fxj ði τÞ A Ruði τÞ jk Z r i g;
s þ i k kτ ¼ j τ;
j A f0; 1; …; τg:
Remark 3.13. Notice that in the proof procedure of our result, the value of time delay is irrelevant to the state of any step. And in this paper, the time delay has no restriction at all. So our results can be applied to deal with BCNs with any time delays.
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3.2. Optimal control of BCNs In this part, we will consider the problem of designing a timeoptimal control uði τÞ that steers the BCNs from initial state xði τÞ to desired state xd in minimal time, which is relevant to biological systems. For example, to prevent λ phage spread, we need to design a control sequence that steers the corresponding systems from lysogeny state to lysis state in minimal time. Next, we will give an algorithm to obtain a control to be time-optimal. According to Theorems 3.3 and 3.10, we can get the available state of any fixed step. So for any desired state, we can give the proper input to obtain the optimal control, i.e., we can solve the control problem of BCN from a given initial state to a desired state in minimal time by some inputs. To reach the desired state, we need to get the available state by order. First, we calculate the available state of calculate of xð1Þ, denote vector Xð1Þ. Assume s ¼ 1; i ¼ 0, then we can get Xð1Þ through Theorem 3.3 or 3.10. Next, we must judge whether xd can be get at the first step or not. If xd is the jth component of Xð1Þ, we can also the proper input. We denote N1 the dimension of X(s). j If the control is BN, then let uðb 1 τÞ ¼ δN1 , where s þ i ¼ aðτ þ 1Þ þ b; if the control is free Boolean sequence, then let j uðs þ i k kτÞ⋉uðs þ i ðk 1Þ ðk 1ÞτÞ⋯uðs þ i 1 τÞ ¼ δN1 , where s þ i k kτ ¼ j τ. Otherwise, if xd is not the component of Xð1Þ, then we need to consider xð2Þ. So the optimal control from xði τÞ to xd can be constructed as the following algorithm (fix i¼0): Step 1 For s ¼1, then calculate the available state of calculate of xð1Þ. Step 2 Count the dimension of Xð1Þ. j Step 3 If xd is the jth component of Xð1Þ, let the input to be δN1 , then xð1Þ ¼ xd . Else, let s ¼2, repeat Steps 1–3. Remark 3.14. Notice that for any s and i, if s þ i is fixed, the integer a and b or k and j is same. So we can fix i ¼0 to simplified operation, and it will not affect our result.
Based on Theorem 3.3, we assume that i ¼ 2; s ¼ 9; τ ¼ 2; Að1 τÞ ¼ 1 2 2 Bð1 τÞ ¼ δ2 and Cð1 τÞ ¼ δ2 , i.e., xð1 τÞ ¼ δ8 . Then according to s þi ¼ aðτ þ 1Þ þ b, a ¼ 3; b ¼ 2. And through calculation, we can have
ΘG ðs þ iÞW ½2n ;2m xðb 1 τÞ ¼ ΘG ð11ÞW ½23 ;22 xð 1Þ ¼ LG3 ðI 22 LG2 ÞðI 24 LGÞ ⋉ðI 28 LÞðI 24 Φ2 Þ
⋉ðI 22 Φ2 ÞΦ2 W ½23 ;22 xð 1Þ
¼ δ8 ½2; 4; 6; 2:
Thus, we conclude that the reachable set at step 9 is fδ8 ; δ8 ; δ8 g. 4 Next, we choose xd ¼ δ8 for example. So we have to find the initial control uð1 τÞ to drive the trajectory to the assigned destination. G From the matrix expression of Θ ð11ÞW ½23 ;22 xð 1Þ, it is obvious 4 that the 2nd column of δ8 ½2; 4; 6; 2 is δ8 . Hence, we can choose 2 2 uð1 τÞ ¼ δ4 to drive the state to xd. Let uð1 τÞ ¼ δ4 , or equiva1 2 lently, u1 ð τÞ ¼ δ2 ; u2 ð τÞ ¼ δ2 , then 2
6
6
xd ¼ Θ ðs þ iÞW ½23 ;22 xð1 τÞuð1 τÞ ¼ δ8 : G
4
In the following, we assume that xð τÞ ¼ δ8 ; xð τ þ 1Þ ¼ δ8 ; xð τ 3 4 þ 2Þ ¼ δ8 ðτ ¼ 2Þ and the given desired state xd ¼ δ8 . Then according to the algorithm of optimal control, we can get the available state of xð1Þ; xð2Þ; …; xð11Þ as follows: δ8 ½1; 2; 1; 2, δ8 ½5; 6; 7; 8, δ8 ½5; 6; 5; 6, δ8 ½1; 5; 2; 6, δ8 ½2; 6; 6; 2, δ8 ½2; 6; 2; 6, δ8 ½2; 2; 6; 6, δ8 ½8; 8; 6; 5, δ8 ½8; 8; 6; 6, δ8 ½6; 8; 6; 8, δ8 ½2; 4; 6; 2, δ8 ½2; 4; 6; 8. It is obvious that xd appears at the 11 step for the first time and N1 1 ¼ 4; j ¼ 2. Then according to s þ i ¼ aðτ þ 1Þ þ b, we can get b¼ 2. Hence, we can 2 choose uðb 1 τÞ ¼ uð 1Þ ¼ δ4 to drive the state to xd. So, the 11 step is the minimal time of BCNs (19) from the given initial state to a desired state. 1 To simulate the process, we give the input uð τÞ ¼ δ4 ; uð τ þ 1Þ 2 3 ¼ δ4 ; uð τ þ2Þ ¼ δ4 . Then we can get the process of the BCNs, which is showed in Fig. 2. In the figure, the black block indicates 2 that the node's state is of f(or 0), i.e., the node is δ2 . On the contrary, the white block indicates that the node's state is on 1 (or 1), i.e., the node is δ2 in our paper. 1
2
Remark 3.15. In the algorithm, we can see that if the control is Boolean network, we only need to ensure the control uðb 1 τÞ, i.e., the other control uði τÞ; i ¼ 0; 1; …; τ and i ab 1, can be arbitrary. It will not affect the Optimal control of BCNs.
Remark 4.2. For the system (19), We can also consider the case (ii) in our paper. Assume that the admissible set of G is nonsingular, and denote G1 ¼ δ4 ½1; 2; 3; 4; G2 ¼ δ4 ½1; 2; 4; 3; G3 ¼ δ4 ½1; 3; 2; 4 ; …; G24 ¼ δ4 ½4; 3; 2; 1. Then the corresponding matrices
4. Examples
V i ≔Θ i ðs þ iÞW ½2n ;2m xðb 1 τÞ; G
Example 4.1. Consider the Boolean control system as follows (Fig. 1): Its dynamics is described as 8 > < Aðt þ 1Þ ¼ Bðt τÞ2Cðt τÞ; Bðt þ 1Þ ¼ Cðt τÞ 3 u1 ðt τÞ; > : Cðt þ 1Þ ¼ Aðt τÞ 4 u ðt τÞ;
i ¼ 1; 2; …; 24
are δ8 ½1; 6; 1; 4, δ8 ½1; 6; 7; 6, δ8 ½1; 6; 7; 4, δ8 ½1; 6; 6; 8, δ8 ½1; 2; 1; 8, δ8 ½5; 2; 1; 4, δ8 ½5; 2; 7; 6, δ8 ½7; 6; 2; 4, δ8 ½8; 4; 2; 6, δ8 ½8; 2; 1; 2, δ8 ½2; 2; 5; 4, δ8 ½2; 2; 6; 6, δ8 ½2; 6; 5; 2, δ8 ½1; 2; 1; 8, δ8 ½2; 2; 8; 2, δ8 ½7; 2; 5; 6,
ð19Þ
2
with controls satisfying ( u1 ðt þ1Þ ¼ g 1 ðu1 ðt τÞ; u2 ðt τÞÞ ¼ :u2 ðt τÞ; u2 ðt þ1Þ ¼ g 1 ðu1 ðt τÞ; u2 ðt τÞÞ ¼ u1 ðt τÞ:
ð20Þ
Denote xðtÞ ¼ AðtÞBðtÞCðtÞ; uðtÞ ¼ u1 ðtÞu2 ðtÞ, then we can express systems (19)–(20) by (12) with G ¼ M n W ½2 ¼ δ4 ½3; 1; 4; 2; L ¼ M e ðI 22 M d ÞðI 2 Φ1 ÞðI 23 M c ÞW ½2;23 W ½22 ;23 ¼ δ8 ½1; 5; 5; 1; 2; 6; 6; 2; 2; 6; 6; 2; 2; 6; 6; 2; 1; 7; 5; 3; 2; 8; 6; 4; 2; 8; 6; 4; 2; 8; 6; 4:
Fig. 1. Boolean control network of (19).
δ8 ½1; 4; 8; 6, δ8 ½7; 6; 5; 2, δ8 ½1; 6; 1; 4, δ8 ½6; 2; 1; 6,
M. Han et al. / Neurocomputing 129 (2014) 467–475
δ8 ½7; 6; 2; 6, δ8 ½5; 6; 1; 2, δ8 ½6; 6; 2; 8, δ8 ½5; 6; 7; 2. So the cup set of all reachable sets for all Gi, i ¼ 1; 2; …; 24 at step 5 is fδ8 ; δ8 ; δ8 ; δ8 ; δ8 ; δ8 ; δ8 g: 1
2
4
5
6
7
8
Now we assume the destination state xd ¼ δ8 for example. Since 5 the 1st column of V7 is δ8 (there are also some other choices such as V 8 ; V 10 ; V 13 ), we can choose G7 ¼ δ4 ½2; 1; 3; 4 and u1 ð τÞu2 1 5 ð τÞ ¼ δ4 to reach δ8 at 5 steps. 5
Example 4.3. Consider the Boolean control system whose logical equation is 8 > < Aðt þ 1Þ ¼ Bðt τÞ 4 u1 ðt τÞ; Bðt þ 1Þ ¼ :u2 ðt τÞ; ð21Þ > : Cðt þ 1Þ ¼ Aðt τÞ 3 u ðt τÞ; 2
Denote xðtÞ ¼ AðtÞBðtÞCðtÞ; uðtÞ ¼ u1 ðtÞu2 ðtÞ. Then we can express system (21) as ~ τÞuðt τÞ; xðt þ 1Þ ¼ Lxðt where L~ can be expressed by
decision. When B is ON [OFF] and E is OFF [ON], the phage is in the lysogenic [lytic] state. Whether or not gene B will be switched on depends on a subtle control process in which five phage genes, A,B, C,D and E, and the environmental state play a prominent role. The detail presentation can be found in the recent monograph [39]. Here we consider the network with time delays both in state and control (Fig. 3). Its dynamics is described as 8 Aðt þ1Þ ¼ ð:Aðt τÞÞ 4 ð:Eðt τ ÞÞ; > > > > > Bðt þ 1Þ ¼ ð:Eðt τÞÞ 4 Bðt τÞ 3 Cðt τÞ; > < Cðt þ 1Þ ¼ ð:Bðt τÞÞ 4 uðt τÞ 4 ðAðt τÞ 3 Dðt τÞÞ; ð22Þ > > > Dðt þ 1Þ ¼ ð:Bðt τÞÞ 4 uðt τÞ 4 Aðt τÞ; > > > : Eðt þ 1Þ ¼ ð:Bðt τÞÞ 4 ð:Cðt τÞÞ: Remark 4.6. During the processes grow on a bacterium, several environmental conditions including temperature, growth rate and concentration of nutrition may cause the time delay. Therefore, it is reasonable to consider the network with time delay. Denote xðt þ 1Þ ¼ Aðt þ 1ÞBðt þ 1ÞCðt þ 1ÞDðt þ1ÞEðt þ1Þ. Then we can express system (22) as
L~ ¼ M c ðI 22 M n ÞðI 23 M d Þ
⋉W ½2;23 ðI 23 Φ1 ÞðI 22 Ed W ½2 Þ
~ τÞuðt τÞ; xðt þ 1Þ ¼ Lxðt
¼ δ8 ½3; 1; 7; 5; 3; 1; 7; 5; 7; 5; 7; 5; 7; 5; 7; 5; 3; 2; 7; 6; 3; 2; 7; 6; 7; 6; 7; 6; 7; 6; 7; 6:
where L~ can be expressed by
Based on Theorem 3.10, assume that i ¼ 1; s ¼ 4; τ ¼ 2; Að 1Þ ¼ B 1 2 2 ð 1Þ ¼ δ2 and Cð 1Þ ¼ δ2 , i.e., xð 1Þ ¼ δ8 . We wish to know whether a designed state can be reached at 4 steps from xð 1Þ. By simple calculation, we can get 2 L~ xð1 τÞ ¼ δ8 ½7; 5; 7; 5; 3; 1; 7; 5; 7; 6; 7; 6; 3; 2; 7; 6:
Then, there exist k ¼ 2; j ¼ 1 such that s þ i k kτ ¼ j τ and 2 xðs þ iÞ ¼ xð5Þ ¼ L~ xð1 τÞuð1 τÞuð2Þ. Using Theorem 3.10, it is clear 4 8 that at 4 steps, all states but δ8 ; δ8 can be reached. Now we choose 2 1 1 destination state xd ¼ δ8 , notice that the 6th column of L~ xð1 τÞ is δ8 , 6 which means controls δ16 can drive the trajectory to the destination 1 state δ8 . Hence, the corresponding controls can be chosen by 1 2 1 2 u1 ð 1Þ ¼ δ2 ; u2 ð 1Þ ¼ δ2 ; u1 ð2Þ ¼ δ2 ; u2 ð2Þ ¼ δ2 .
L~ ¼ δ32 ½24; 24; 24; 24; 24; 24; 24; 24; 32; 32; 24; 24; 32; 32; 24; 24; 18; 24; 18; 24; 18; 24; 18; 24; 25; 31; 25; 31; 25; 31; 25; 31; 24; 24; 8; 8; 24; 24; 8; 8; 32; 32; 8; 8; 32; 32; 8; 8; 20; 24; 4; 8; 24; 24; 8; 8; 27; 31; 11; 15; 31; 31; 15; 15: Based on Theorem 3.10, assume that i ¼ 0; s ¼ 9; τ ¼ 2, there exist k ¼ 3; j ¼ 2 such that s þ i k kτ ¼ j τ. Then we wish to know whether a designed state can be reached at 9 steps from 1 xð0Þ. Also we can assume that Að0Þ ¼ Bð0Þ ¼ Dð0Þ ¼ Eð0Þ ¼ δ2 and 2 5 Cð0Þ ¼ δ2 , i.e., xð0Þ ¼ δ32 . By simple calculation, we can get 3 L~ xð2 τÞ ¼ δ32 ½25; 31; 25; 31; 25; 31; 25; 31:
Remark 4.4. Similar to Example 3.1, we assume that xð τÞ ¼ δ18 ; xð τ þ 1Þ ¼ δ28 ; xð τ þ 2Þ ¼ δ38 ðτ ¼ 2Þ and the given desired state 1 xd ¼ δ8 . Then according to the algorithm of optimal control, we first get the available state of xð1Þ is δ8 ½3; 1; 7; 5. It is obvious that xd appears at the 1 step for the first time and N 1 ¼ 4; j ¼ 2. Then according to s þ i k kτ ¼ j τ, we can get k ¼1. Hence, we can 2 choose uð τÞ ¼ uð 2Þ ¼ δ4 to drive the state to xd. So, the first step is the minimal time of BCNs (21) from the given initial state to a desired state.
Node u
6
8
10
12
14
16
18
20
22
24
Input Pin Connector A
Example 4.5 (A biological example). Consider following the Boolean model, which is a simple BCN model for the λ switch [40]. The λ phage is a virus that grows on a bacterium upon infection of the bacterium. To ensure successful propagation, the virus had to develop efficient mechanisms of precise response to changes in the physiology of its host. This was achieved by a specific genetic switch that allows this virus to choose the most effective developmental pathway for the given environmental conditions. The virus can then follow one of two different pathways: lysogeny or lysis. For example, two genes, B and E, directly affect this
4
473
D
C
E
B
Fig. 3. Boolean control network of (22).
26
28
30
32
34
36
38
Fig. 2. The process of BCNs (19) (black/white block is for 0/1).
40
42
44
46
474
M. Han et al. / Neurocomputing 129 (2014) 467–475
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
46
Generation Fig. 4. The process of BCNs (22) (black/white block is for 0/1).
3 And xðs þiÞ ¼ xð9Þ ¼ L~ xð0Þuð0Þuð3Þuð6Þ. Using Theorem 3.10, it is 25 31 clear that at 9 steps, the states δ32 and δ32 can be reached. Now we 25 choose destination state xd ¼ δ32 , notice that the 1th (or 3th, 5th, 3 11 1 3 5 7 7th) column of L~ xð2 τÞ is δ32 , which means controls δ8 ðδ8 ; δ8 ; δ8 Þ 25 can drive the trajectory to the destination state δ32 . Hence, the 1 1 corresponding controls can be chosen by uð0Þ ¼ δ2 ; uð3Þ ¼ δ2 and 1 uð6Þ ¼ δ2 . In the following, we assume that xð τÞ ¼ δ3 21 ; xð τ þ 1Þ ¼ δ3 3 2 ; xð τ þ 2Þ ¼ δ3 25 ðτ ¼ 2Þ and the given desired state xd ¼ δ3 22 5. Then according to the algorithm of optimal control, we can get the available state of xð1Þ; xð2Þ; …; xð9Þ as follows: δ32 ½24; 24, δ32 ½24; 24, δ32 ½32; 32, δ32 ½8; 8; 8; 8, δ32 ½8; 8; 8; 8, δ32 ½15; 15 ; 15; 15, δ32 ½24; 24; 24; 24; 24; 24; 24; 24, δ32 ½24; 24; 24; 24; 24; 24; 24; 24, δ32 ½25; 31; 25; 31; 25; 31; 25; 31. It is obvious that xd appears at the 9 step for the first time and N 9 ¼ 8; j ¼ 1. Then according to s þ i k kτ ¼ j τ, we can get j¼2. Hence, we can choose 1 uð0Þ ¼ uð3Þ ¼ uð6Þ ¼ δ2 to drive the state to xd. So, the 9 step is the minimal time of BCNs (22) from the given initial state to a desired state. 1 To simulate the process, we fixed the input uð0Þ ¼ uð3Þ ¼ uð6Þ ¼ δ2 and the other control is randomly selected by computer. Then we can get the process of the BCNs, which is showed in Fig. 4. In the figure, the black block also indicates that the node's state is off (or 0), i.e., the 2 node is δ2 . On the contrary, the white block indicates that the node's 1 state is on (or 1), i.e., the node is δ2 in our paper.
5. Conclusion In this paper, we have investigated the controllability of Boolean control networks with time delays both in state and control. First, we provide a brief review of the STP of matrices and the matric expression of logic. Second, using the method of STP of matrices and the matric expression of logic, the Boolean networks with time delays both in state and control have been converted into discrete time delay systems. Then, we have considered the controllability via two kinds of controls. In both cases, according to different input structure and step, four situations have been studied and the necessary and sufficient conditions have been obtained. Then, we consider the optimal control problem of BCNs from a given initial state to a desired state in minimal time. Finally, we have given three examples to show the efficiency of the obtained results.
Acknowledgments We thank Prof. Jinde Cao for his constructive suggestions. Furthermore, the authors wish to thank the editor and the reviewers for a number of constructive comments and suggestions that have improved the quality of the paper. This work was supported by NNSF of China (Grant nos. 11271333, 11011373 and 61074011) and National Undergraduate Training Programs for Innovation and Entrepreneurship (Grant no. 201310345010).
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[31] J. Klamka, Stochastic controllability of systems with multiple delays in control, Int. J. Appl. Math. Comput. Sci. 19 (2009) 39–47. [32] S. Zhao, J. Sun, Controllability and observability for time-varying switched impulsive controlled systems, Int. J. Robust Nonlinear Control 20 (2010) 1313–1325. [33] R.A. Umana, Null controllability of nonlinear infinite neutral systems with multiple delays in control, J. Comput. Anal. Appl. 10 (2008) 509–522. [34] Y. Liu, S. Zhao, Controllability for a class of linear time varying impulsive systems with time delay in control input, IEEE Trans. Autom. Control 56 (2011) 395–399. [35] H. Zhang, G. Duan, L. Xie, Linear quadratic regulation for linear time-varying systems with multiple input delays, Automatica 42 (2006) 1465–1476. [36] P. Cui, C. Zhang, H. Zhang, H. Zhao, Indefinite linear quadratic optimal control problem for singular discrete-time system with multiple input delays, Automatica 45 (2009) 2458–2461. [37] P. Yang, G. Xie, L. Wang, Controllability of linear discrete-time systems with time-delay in state and control, Int. J. Control 82 (2009) 1288–1296. [38] J.P. Dauer, R.D. Gahl, Controllability of nonlinear delay systems, J. Optim. Theory Appl. 21 (1977) 59–70. [39] D. Laschov, M. Margaliot, Minimum-time control of Boolean networks [Online]. Available: 〈http://paloma.eng.tau.ac.il/ michaelm/min time bcn. pdf〉. [40] D. Thieffry, R. Thomas, Dynamical behaviour of biological regulatory networks II: immunity control in bacteriophage lambda, Bull. Math. Biol. 57 (1995) 277–297.
Ming Han was born in Hangzhou, Zhejiang Province, China, in 1990. She is a MS candidate in the College of Mathematics, Physics and Information Engineering, Zhejiang Normal University. Her research interests include impulsive systems, delayed systems, neural networks, graph theory.
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Yang Liu received the B.S. degree in mathematics from Zhejiang Normal University, Zhejiang, China, in 2003, and the Ph.D. degree from Tongji University, Shanghai, in 2008. He is currently an associate professor at the Department of Mathematics, Zhejiang Normal University. His research interest covers complex analysis, Clifford analysis, switched system, delayed system and nonlinear control. He has published over 30 papers in refereed international journals. He services as the reviewer for Mathematical Reviews. He is the recipient of Shanghai Outstanding Ph.D's Thesis Award in 2011.
Yanshuai Tu was born in Jinhua, Zhejiang Province, China, in 1990. He is a MS candidate in the College of Mathematics, Physics and Information Engineering, Zhejiang Normal University. His research interests include nonlinear systems, neural networks, computational physics, three-dimensional sensing.
Contents lists available at ScienceDirect
Neurocomputing journal homepage: www.elsevier.com/locate/neucom
Controllability of Boolean control networks with time delays both in states and inputs Ming Han, Yang Liu n, Yanshuai Tu College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, China
art ic l e i nf o
a b s t r a c t
Article history: Received 18 December 2012 Received in revised form 27 May 2013 Accepted 1 September 2013 Communicated by Lixian Zhang Available online 20 October 2013
This paper investigates the controllability of Boolean control networks (BCNs) with time-invariant delays both in states and inputs. The necessary and sufficient conditions on the controllability via two kinds of controls are presented by providing the corresponding reachable sets. The proposed results generalize the controllability analysis on Boolean control networks without time delay. And we also consider the optimal control problem of BCNs from a given initial state to a desired state in minimal time. Three examples include a biological example are given to show the effectiveness of the proposed results. & 2013 Elsevier B.V. All rights reserved.
Keywords: Boolean (control) networks (BCNs) Controllability Time delay Semi-tensor product
1. Introduction In the recent years, a new view of biology, called the systems biology, has attracted increasing interest from various fields. Systems biology usually focuses on the system-level properties of the whole network dynamics, e.g., [1,2]. Especially, the Boolean network, a network with nodes and directed edges, becomes a powerful tool in describing, analyzing and simulating the systems biology. The Boolean networks are a class of models first introduced by Kauffman [3]. In this model, a gene expression level can be described by a binary value, 0 or 1, indicating two transcriptional states: active (on) or inactive (off). And the state of each gene is determined by the states of its neighborhood genes on the basis of logical rules. As to the structure of Boolean networks, including the fixed points, cycles, attractors and transient time, we refer [4–9] for details. In [6], the identification of Boolean control networks is addressed, and the matrix representation of the Boolean networks and its applications are discussed in [8]. Specific examples of genetic regulation networks modeled by BNs include: the network controlling the segment polarity genes in the fly Drosophila melanogaster [4]; the ABC network determining floral organ cell fate in Arabidopsis [5], and so on. BNs with binary inputs are referred to as BCNs. The controllability of BCNs
n
Corresponding author. Tel./fax: þ 86 57982298897. E-mail address: [email protected] (Y. Liu).
0925-2312/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2013.09.012
has been addressed in [10]. Akutsu et al. [7] showed that control problems for BCNs are in general NP-hard. In order to systematically analyze the Boolean network, [9] has proposed a way to convert the Boolean networks into discrete time dynamic systems based on semi-tensor product (STP) of matrices. Then using this method, [10] has studied the controllability and observability of Boolean control networks and [13] has analyzed the stability and stabilization of Boolean network. Besides, there are also many other results obtained, for example, see [14,15]. It is said in [16] that one of the major goals of systems biology is to develop a control theory for complex biological systems. Controllability as one of the fundamental concepts in control theory is becoming more and more important. In the literature, many different definitions of controllability have been given in [17–21], which strongly depend on class of dynamical control systems. Then there have appeared many results on the systematic study of controllability, e.g., [22–25]. However, when it comes to the control of Boolean network, there have been only very few results. In [10], controllability and observability of Boolean control networks have been investigated. In [11], the input–state incidence matrix of a control Boolean network has been proposed. Based on it, an easily verifiable necessary and sufficient condition for the controllability of a Boolean control network is obtained. As we all know that time delay phenomenon is very common in real world. It has been recognized that the slow processes of transcription, translation, and translocation or the finite switching speed of amplifiers will unavoidable cause time delays, which should be taken into account in the biological systems or artificial
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genetic networks in order to reduce the impaction on the dynamical behavior of models [26]. Therefore, time delays should be considered in the dynamics of genetic networks to have more accurate models. Usually, we consider the systems with time delay in state, for example, in [27–29]. For Boolean networks, there also has been some recent papers that discussed the controllability problem with delays in states. For example, Li and Sun have investigated the controllability of Boolean networks with timeinvariant delays in states [30]. Meanwhile, considering time delay in the control is also very important to the system performance. In the recent studies, the systems with time delays in the control have been analyzed widely in economic, biological and physiological industry systems [31–36]. Besides, the controllability of systems with time delays in both states and control has investigated in [37,38]. But to the best of our knowledge, there has been no results in the literature concerning the controllability of the Boolean control networks with time delays in both states and controls. So in this paper, we will extend our result to the more general case, namely, the controllability of Boolean control networks with time delays both in states and controls. Moreover, optimal controls are also important in the context of BCNs that model biological systems. For example, a natural problem is to determine a control that steers the BCN from an initial state (that corresponding to a disease state) to a desired state (that corresponding to healthy state) in minimal time. Motivated by the above discussions, a natural question is that whether we can contribute to the optimal control problem for BCNs. So we also study the time-optimal control problem in the examples. The rest of this paper is organized as follows. In Section 2, we introduce the STP of matrices and the matrix expression of logical function. The main result on the controllability problem of Boolean networks with time delays and an algorithm for the timeoptimal control of BNs is given in Section 3. Section 4 gives three examples for illustration and some conclusions are finally drawn in Section 5.
2. Preliminaries 2.1. STP of matrices
of conventional matrix product can be applied to the STP of matrix, e.g., distributive rule, Associative rule and so on. Since then, we can omit ⋉ in this paper. In the following, we introduce some special properties, for details, see [9]. Definition 2.3 (Cheng and Qi [9]). (1) Assume A g t B, then (where is the Kronecker product, It is the identity matrix) A⋉B ¼ AðB I t Þ: Assume A ! t B, then A⋉B ¼ ðA I t ÞB: (2) Assume A A M mn is given. Let Z A Rt be a row vector. Then A⋉Z ¼ Z⋉ðI t AÞ: Let Z A Rt be a column vector. Then Z⋉A ¼ ðI t AÞ⋉Z: Then we give some notations as the following: (1) Define a delta set as Δk ≔fδk ji ¼ 1; 2; …; kg, where δki is the ith column of It. (2) A matrix AA M mn is called a logical matrix if the columns of A, denoted by Col(A), satisfy ColðAÞ Δm . i i i (3) Assume a matrix A ¼ ½δm1 ; δm2 ; …; δmn , we denote it as A ¼ δm ½i1 ; i1 ; …; in . i
Next, we give an important definition as follows: Definition 2.4 (Cheng and Qi [9]). (1) An mn mn matrix W m;n is called swap matrix, if it is constructed in the following way: label its columns by ð11; 12; …; 1n; m1; m2; …; mnÞ and its rows by ð11; 21; …; m1 ; 1n; 2n; …; mnÞ. Then its element in the position ððI; JÞ; ði; jÞÞ is assigned as 1; I ¼ i and J ¼ j; I;J wðI;JÞ;ði;jÞ ¼ δi;j ¼ ð1Þ 0 otherwise: When m¼n, we briefly denote W ½n ≔W ½m;n . (2) Let X A Rm and Y A Rn be two columns. Then W ½m;n ⋉X⋉Y ¼ Y⋉X; W ½n;m ⋉Y⋉X ¼ X⋉Y:
Definition 2.1 (Cheng and Qi [9]). 2.2. Matrix expression of logic (1) Let X be a row vector of dimension np, and Y be a column vector of dimension p. Then we split X into equal-size blocks as X 1 ; …; X P , which are 1 n rows. Define the semi-tensor product (STP) of matrices, denoted by ⋉, as 8 p > > X⋉Y ¼ ∑ X i y A Rn > > i < i¼1 p > > > Y T ⋉X T ¼ ∑ yi ðX i ÞT A Rn : > : i¼1
(2) Let A A M mn and B A M pq . If either n is a factor of p, say nt ¼ p and denote it as A ! t B, or p is a factor of n, say n¼ pt and denote it as A g t B, then we define the STP of A and B, denoted by C ¼ A⋉B, as the following: C consists of m q blocks as C ¼ C ij and each block is C ij ¼ Ai ⋉Bj ;
i ¼ 1; …; m; j ¼ 1; …; q;
where Ai is the ith row of A and Bj is the jth column of B. Remark 2.2. When n ¼p, A⋉B ¼ AB. So it is a generalization of the conventional matrix product, and all the fundamental properties
In this paper, we recall the matrix expression of logic as [9] for details. Define a logical domain D as D ¼ fT ¼ 1; F ¼ 0g, then to use the STP of matrices, we give the logical values a vector form as: 1 2 T ¼ 1 δ2 ; F ¼ 0 δ2 and denote
Δ≔Δ2 ¼ fδ12 ; δ22 g D: Similar, a logical function with n arguments f : Dn -D can be expressed in the algebraic form as follows: Lemma 2.5 (Cheng and Qi [9]). Any logical function LðA1 ; …; An Þ with logical arguments A1 ; …; An A Δ can be expressed in a multilinear form as LðA1 ; …; An Þ ¼ M L A1 A2 ⋯An ; where ML is unique, called the structure matrix of L. Another important lemma for this paper is given as the following: Lemma 2.6 (Cheng and Qi [9]). Assume P k ¼ A1 A2 ⋯Ak , then P 2k ¼ Φk P k ;
M. Han et al. / Neurocomputing 129 (2014) 467–475
where k
Φk ¼ ∏ I2i 1 ½ðI2 W ½2;2k i ÞMr ; i¼1
M r ¼ δ4 ½1; 4:
3. Main results A Boolean network of a set of nodes A1 ; A2 ; …; An can be described as 8 A1 ðt þ 1Þ ¼ f 1 ðA1 ðtÞ; A2 ðtÞ; …; An ðtÞÞ; > > > > < A2 ðt þ 1Þ ¼ f 2 ðA1 ðtÞ; A2 ðtÞ; …; An ðtÞÞ; ð2Þ ⋮ > > > > : An ðt þ 1Þ ¼ f ðA1 ðtÞ; A2 ðtÞ; …; An ðtÞÞ; n where f i ; i ¼ 1; 2; …; n, are logical function and t ¼ 0; 1; 2; … . Then we assume that the Boolean networks have timeinvariant delays in states as follows: 8 A ðt þ 1Þ ¼ f 1 ðA1 ðt τÞ; A2 ðt τÞ; …; An ðt τÞÞ; > > > 1 > < A2 ðt þ 1Þ ¼ f ðA1 ðt τÞ; A2 ðt τÞ; …; An ðt τÞÞ; 2 ð3Þ ⋮ > > > > : An ðt þ 1Þ ¼ f ðA1 ðt τÞ; A2 ðt τÞ; …; An ðt τÞÞ; n
where
τ is a positive integer delay.
3.1. Controllability of Boolean control networks Define xðtÞ ¼ ⋉ni¼ 1 Ai ðtÞ, where ⋉ni¼ 1 is a bijective mapping pointed out in [9]. Using Lemma 2.5, we can find the structure matrix Mi for each function f i ; i ¼ 1; 2; …; n. So system (3) can be converted into an algebraic form as follows: Ai ðt þ 1Þ ¼ M i A1 ðt τÞA2 ðt τÞ⋯An ðt τÞ ¼ M i xðt τÞ:
ð4Þ
ð5Þ
Then the matrix form is Aðt þ 1Þ ¼ M c Bðt τÞCðt τÞ:
or
Ed W ½2 XY ¼ X:
ð8Þ
In this paper, two kinds of controls are considered: (1) The controls are logical variables satisfying certain logical rules, called input network as follows: 8 u ðt þ 1Þ ¼ g 1 ðu1 ðt τ Þ; u2 ðt τÞ⋯um ðt τÞÞ; > > > 1 > < u2 ðt þ 1Þ ¼ g 2 ðu1 ðt τ Þ; u2 ðt τÞ⋯um ðt τÞÞ; ð9Þ ⋮ > > > > : um ðt þ 1Þ ¼ g ðu1 ðt τÞ; u2 ðt τÞ⋯um ðt τÞÞ; m where g i ; i ¼ 1; 2; …; m, are logical function and the initial states uj ði τÞ; j ¼ 1; 2; …; m; i ¼ 1; 2; …; τ can be arbitrarily given. (2) The control is a free Boolean sequence. Precisely, set uðtÞ ¼ ⋉m j ¼ 1 uj ðtÞ. Then the control is a designed sequence. Similarly, let xðtÞ ¼ ⋉ni¼ 1 Ai ðtÞ, uðtÞ ¼ ⋉m j ¼ 1 uj ðtÞ, we can also find the structure matrix M1i and M2j for each logical function fi and gj. Then system (8) and (9) can be converted as Ai ðt þ 1Þ ¼ M 1i xðt τÞ;
i ¼ 1; 2; …; n;
ð10Þ
uj ðt þ 1Þ ¼ M 2j uðt τÞ;
j ¼ 1; 2; …; m:
ð11Þ
xðt þ 1Þ ¼ A1 ðt þ1ÞA2 ðt þ1Þ⋯An ðt þ 1Þ ¼ M 11 uðt τÞxðt τÞ⋯M 1n uðt τÞxðt τÞ ¼ M 11 ðI 2m þ n M 12 ÞðI 2m W ½2m ;2n ÞΦm ⋉ðI 2m Φn Þ⋯ðI 2m þ n M 1n Þ
⋉ðI 2m W ½2m ;2n ÞΦm ðI 2m Φn Þ ⋉uðt τÞxðt τÞ 9 Luðt τÞxðt τÞ: And, multiplying (11), we can have
ð6Þ
To get the form of (4), we can use a dummy matrix Ed ¼ δ2 ½1; 2; 1; 2, which is constructed in [9]. Then for any two logical variables X; Y, Ed XY ¼ Y
integer delays in both states and controls as follows: 8 A1 ðt þ 1Þ ¼ f 1 ðu1 ðt τÞ; …; um ðt τÞ; > > > > > A 1 ðt τ Þ; …; An ðt τ ÞÞ; > > > > > A > < 2 ðt þ 1Þ ¼ f 2 ðu1 ðt τÞ; …; um ðt τÞ; A1 ðt τÞ; …; An ðt τÞÞ; > > > ⋮ > > > > > > An ðt þ 1Þ ¼ f n ðu1 ðt τÞ; …; um ðt τÞ; > > : A1 ðt τÞ; …; An ðt τÞÞ;
Multiplying (10) yields
Remark 3.1. Note that usually the indegree is much less than n, that is, the right-hand side of ith equation of (3) may not have all Ai ; i ¼ 1; 2; …; n. For example, for node A: Aðt þ 1Þ ¼ Bðt τÞ 4 Cðt τÞ:
469
ð7Þ
Then we can rewrite (6) as Aðt þ 1Þ ¼ M c Ed Aðt τÞBðt τÞCðt τÞ ¼ M c Ed xðt τÞ: Multiplying all equations (4), together yields xðt þ 1Þ ¼ M 1 xðt τÞM 2 xðt τÞ⋯M n xðt τÞ: Then according to Lemma 2.6, (3) can be converted to a standard discrete-time dynamic system as xðt þ 1Þ ¼ L1 xðt τÞ; where L1 ¼ M 1 ðI 2n M 2 ÞΦn ðI 2n M 3 ÞΦn ⋯ðI 2n M n ÞΦn , Φn is defined as Φn ¼ ⋉ni¼ 1 I 2i 1 ½ðI 2 W ½2;2n i ÞM r ; M r ¼ δ4 ½1; 4. L1 is called the network transition matrix of (3). In the real word, biological systems may experience delay of states and control because of the environment changes. This maybe due to changes in environment, such as temperature, growth rate, etc. Next, we consider Boolean control network with time-invariant
uðt þ 1Þ ¼ u1 ðt þ 1Þu2 ðt þ1Þ⋯um ðt þ 1Þ ¼ M 21 uðt τÞM 22 uðt τÞ⋯M 2m uðt τÞ ¼ M 21 ðI 2m M 22 ÞΦm ðI 2m M 23 ÞΦm
⋯ðI 2m M 2m ÞΦm uðt τÞ 9 Guðt τÞ: From the above conclusion, it is easy to obtain an algebraic form of the Boolean control network (8)–(9) as ( uðt þ 1Þ ¼ Guðt τÞ; ð12Þ xðt þ 1Þ ¼ Luðt τÞxðt τÞ; where L is the network transition matrix of (8), and G is the network transition matrix of (9). Next, we consider the controllability problem of Boolean control network (8)–(9), equivalently (12). 3.1.1. Control via input Boolean network In the following, we consider the case when the controls are logical variables satisfying certain logical rules. First, we give the following refer to the Definition 1 of [10]: Definition 3.2. Consider system (8) with control (9). Given initial state xð τÞ; xð τ þ 1Þ; …; xð0Þ A D2n and the destination state xd, xd is said to be controllable from initial state xði τÞ ði A 0; 1; …; τÞ at s
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steps with fixed input structure G, if we can find the initial input uði τÞ ðiA 0; 1; …; τÞ such that xðs þ iÞ ¼ xd . According to the above definition we may consider four cases: (i) fixed s and fixed G; (ii) fixed s and designable G; (iii) free s 4 0 and fixed G; (iv) free s 4 0 and designable G. First we start from case (i). Theorem 3.3. Consider system (8) with control (9), equivalently (12), where G is fixed. xd is s step reachable from xði τÞ ði A 0; 1; …; τÞ, if and only if
¼ LG2 ðI 2m LGÞðI 22m LÞðI 2m Φm Þ xð2τ þ 4Þ ¼ Luðτ þ 3Þxðτ þ 3Þ ¼ LG2 uð1 τÞLGðI 2m LÞΦm uð1 τÞ
and
8 a > LG ðI 2m LGa 1 Þ > > > < ⋉ðI LGa 2 Þ⋯ðI 2am LÞ 22m ΘG ðs þ iÞ ¼ > ⋉ðI 2ða 1Þm Φm Þ⋯ðI 2m Φm ÞΦm ; a Z 0 > > > : L; a ¼ 0;
where Φm is defined as Φm ¼ ⋉m i ¼ 1 I 2i 1 ½ðI 2 W ½2;2m i ÞM r , M r ¼ δ4 ½1; 4. Proof. First, we consider the control in (12) as follows: uð1Þ ¼ Guð τÞ; uð2Þ ¼ Guð1 τÞ; ⋮ uðτ þ 1Þ ¼ Guð0Þ; uðτ þ 2Þ ¼ Guð1Þ ¼ G2 uð τÞ; uðτ þ 3Þ ¼ Guð2Þ ¼ G2 uð1 τÞ; ⋮ uð2ðτ þ1ÞÞ ¼ Guðτ þ 1Þ ¼ G2 uð0Þ; uð2τ þ 3Þ ¼ Guðτ þ 2Þ ¼ G3 uð τÞ; uð2τ þ 4Þ ¼ Guðτ þ 3Þ ¼ G3 uð1 τÞ; ⋮ uð3ðτ þ1ÞÞ ¼ Guð2τ þ 2Þ ¼ G3 uð0Þ; ⋮: Amuse that there exist a A f0; 1; 2; …;g; b A f1; 2; …; τ þ 1g such that s þ i ¼ aðτ þ1Þ þb:
m
m
⋉Φm uð1 τÞxð1 τÞ; ⋮ xð3ðτ þ 1ÞÞ ¼ Luð2τ þ 2Þxð2τ þ 2Þ
¼ LG2 uð0ÞLGðI 2m LÞΦm uð0Þxð0Þ ¼ LG2 ðI 2m LGÞðI 22m LÞðI 2m Φm Þ
G
s þ i ¼ aðτ þ1Þ þb
⋉xð1 τÞ
¼ LG ðI 2 LGÞðI 22m LÞðI 2 Φm Þ 2
xd A ColfΘ ðs þ iÞW ½2n ;2m xðb 1 τÞg; where and hereafter “Col” is the column set, also there exist unique a A f0; 1; 2; …g and b A f1; 2; …; τ þ 1g such that s þ i satisfies
⋉Φm uð τÞxð τÞ;
⋉Φm uð0Þxð0Þ;
⋮: Similar to the analysis about the control, we can obtain that xðs þ iÞ ¼ xðaðτ þ 1Þ þ bÞ ¼ Luðða 1Þðτ þ 1Þ þ b 1Þ⋉xðða 1Þðτ þ 1Þ þ b 1Þ ¼ LGa ðI 2m LGa 1 ÞðI 22m LGa 2 Þ ⋉ðI 23m LGa 3 Þ…; ðI 2am LÞ⋉ðI 2ða 1Þm Φm Þ…ðI 2m Φm Þ ⋉Φm uðb 1 τÞxðb 1 τÞ ¼ Θ ðs þ iÞuðb 1 τÞxðb 1 τÞ G
¼ Θ ðs þ iÞW ½2n ;2m xðb 1 τÞ⋉uðb 1 τÞ: G
ð13Þ
Since Θ ðs þ iÞW ½2n ;2m xðb 1 τÞ is a 2 2 matrix, whose columns are elements in Δ2n , and uðb 1 τÞ A Δ2m , we can drive the conclusion. □ G
n
m
Remark 3.4. Compared with the result of [30], we need to consider the process of control carefully. And the calculus process is obviously more complex and difficult. Next, we consider the case (ii). m Since there are ð2m Þ2 possible distinct G, we can express each G in the condensed form and order them in “increasing order”, see [9]. For example, when m ¼1, we have G1 ¼ δ2 ½1; 1, G2 ¼ δ2 ½1; 2, G3 ¼ δ2 ½2; 1, G2 ¼ δ2 ½2; 2. Thus we may consider a subset m Λ f1; 2; …; ð2m Þ2 g and allow G be chosen from the admissible set fGλ jλ A Λg. Then the following results can be obtained. Corollary 3.5. Consider system (8) with control (9), where G A fGλ j
λ A Λg. Then xd is s step reachable from xði τÞ ðiA 0; 1; …; τÞ , if and only if there exists a Gλ such that xd A ColfΘ λ ðs þ iÞW ½2n ;2m xðb 1 τÞg G
Then from (13), we can easily get that
and there exist a A f0; 1; 2; …g and b A f1; 2; …; τ þ 1g such that s þ i satisfies
uðs þ iÞ ¼ uðaðτ þ 1Þ þ bÞ
s þi ¼ aðτ þ 1Þ þ b
¼ Ga þ 1 uðb 1 τÞ: Above all, a straightforward computation shows the following: xð1Þ ¼ Luð τÞxð τÞ; xð2Þ ¼ Luð1 τÞxð1 τÞ; ⋮ xðτ þ 1Þ ¼ Luð0Þxð0Þ; xðτ þ 2Þ ¼ Luð1Þxð1Þ ¼ LGuð τÞLuð τÞxð τÞ ¼ LGðI 2m LÞΦm uð τÞxð τ Þ; xðτ þ 3Þ ¼ Luð2Þxð2Þ ¼ LGuð1 τÞLuð1 τÞxð1 τÞ ¼ LGðI 2m LÞΦm uð1 τÞxð1 τÞ; ⋮ xð2ðτ þ 1ÞÞ ¼ Luðτ þ 1Þxðτ þ 1Þ ¼ LGuð0ÞLuð0Þxð0Þ ¼ LGðI 2m LÞΦm uð0Þxð0Þ; xð2τ þ3Þ ¼ Luðτ þ 2Þxðτ þ 2Þ ¼ LG2 uð τ ÞLGðI 2m LÞΦm uð τÞxð τÞ
and
8 a LGλ ðI 2m LGaλ 1 Þ > > > > < ⋉ðI a2 Þ⋯ðI 2am LÞ 22m LGλ ΘG ðs þ iÞ ¼ > ⋉ðI ða 1Þm Φm Þ⋯ðI 2m Φm ÞΦm ; a Z 0 > 2 > > : L; a ¼ 0: Finally, we consider cases (iii) and (iv). First, we need the following lemma.
Lemma 3.6 (Cheng and Qi [10]). For a Boolean network, if its network transition matrix is nonsingular, then every point is on a cycle. In the following, we assume that G is nonsingular. According to Lemma 2.5, starting from uði τÞ; i ¼ 0; 1; …; τ, as G is nonsingular, then we can find a minimum T i 4 0 such that GT i uði τÞ ¼ uði τÞ. Hence fuði τÞ; Guði τÞ; …; GT i 1 uði τÞg is a
M. Han et al. / Neurocomputing 129 (2014) 467–475
cycle of length Ti. Following the procedure in [12], we can construct a mapping
for fixed G ¼ Gk , the reachable set from xði τÞ is RGk ¼ ⋃ RGuðik τÞ ; uði τ Þ
Ψ i ≔½LGT i 1 uði τÞ½LGT i 2 uði τÞ⋯½LGuði τÞ ½Luði τÞ:
ð14Þ
Then for xði τÞ, we consider the sequence xði τÞ; Ψ i xði τÞ; … . Similarly, we can find the transient period ri and a minimum T i ′ 4 0 such that ′
471
′
Ψ Ti i xði τÞ ¼ Ψ ri i þ T i xði τÞ:
ð15Þ
Remark 3.7. Starting from xði τÞ, then the process is as follows: xði τÞ-xði þ 1Þ ¼ Luði τÞxði τÞxði þ 1 þ τ þ 1Þ ¼ LGuði τÞLuði τÞxði τÞ-⋯-
(3) for admissible fGλjλ A Λ g, the reachable set is G R ¼ ⋃ ⋃ Ruðiλ τÞ : λ A Λuði τÞ
3.1.2. Control via free Boolean sequence In the following, we consider the case when the controls are free Boolean sequences.
xði þ 1 þ ðT i þ 2Þðτ þ 1ÞÞ ¼ LGuði τÞLuði τÞΨ i xði τÞ-⋯-
Definition 3.9 (Cheng and Qi [10]). Given initial state xð τÞ; x ð τ þ1Þ; …; xð0Þ A D2n and the destination state xd. The Boolean control network (8) is said to be controllable from initial state xði τÞ ði A 0; 1; …; τÞ to xd at s steps, if we can find the initial input uði τÞ ðiA 0; 1; …; τÞ such that xðs þ iÞ ¼ xd .
xði þ 1 þ 2T i ðτ þ 1ÞÞ ¼ Ψ i xði τÞ⋯
Define L~ ¼ LW ½2n ;2m , then the second equation in (12) can be expressed as
xði þ 1 þ T i ðτ þ 1ÞÞ ¼ Ψ i xði τÞ-
xði þ 1 þ ðT i þ 1Þðτ þ 1ÞÞ ¼ Luði τÞΨ i xði τÞ2
T′ 1
xði þ 1 þ ððT′ 1ÞT i þ 1Þðτ þ 1ÞÞ ¼ Luði τÞΨ i
~ τÞuðt τÞ: xðt þ 1Þ ¼ Lxðt
xði τÞ-
xði þ 1 þ ððT′ 1ÞT i þ 2Þðτ þ 1ÞÞ ¼ LGuði τÞLuði τÞΨ
T′ 1 i
It yields
xði τÞ-⋯-xði þ 1 þ T′T i ðτ þ 1ÞÞ ¼ Ψ i xði τÞ⋯ T′
τ
τ ÞΨ
xði þ 1 þ ððr i þ T ′i 1ÞT i þ 1Þð þ 1ÞÞ ¼ Luði xði Þ-xði þ 1 þ ððr i þ T ′i 1ÞT i þ 2Þð þ1ÞÞ
τ
τ
~ þ i 1 τÞuðs þ i 1 τÞ xðs þ iÞ ¼ Lxðs ¼ L~ xðs þ i 2 2τÞuðs þ i 2 2τÞ⋉uðs þ i 1 τÞ 2
r i þ T ′i 1 i
¼ L~ xðs þ i 3 3τÞuðs þ i 3 3τÞ ⋉uðs þ i 2 2τÞuðs þ i 1 τÞ ¼ ⋯ 3
¼ LGuði τÞLuði τÞ
′
Ψ iri þ T i 1 xði τÞ-⋯-xði þ 1 þ ððri þ T ′i ÞT i Þðτ þ 1ÞÞ r þ T ′i
¼ Ψ ii
¼ L~ xðs þ i k kτÞuðs þ i k kτÞ ⋉uðs þ i ðk 1Þ ðk 1ÞτÞ⋯⋉uðs þ i 1 τÞ: k
xði τÞ ¼ Ψ i xði τÞ: T′
ð18Þ
Assume that Then the reachable set starting from xði τÞ with uði τÞ ði ¼ 0; 1 ; …; τÞ can be constructed as the following algorithm: Step 1 Find Ti such that uði τÞ; Guði τÞ; …; GT i 1 uði τÞ ði ¼ 0; 1; …; τÞ is a cycle in the input space. Step 2 Find the transient period ri and a minimum T ′i 4 0 satisfying (15). Step 3 Construct a sequence for i ¼ 0; 1; …; τ, xk ði τÞ ¼ Ψ i xði τÞ; k
k ¼ 0; 1; …; r i þ T ′i 1:
ð16Þ
Step 4 For each xk ði τÞ construct inductively a sequence for j ¼ 1; …; T i 1, xkj ði
τÞ ¼ LG
j1
uði τ
Þxkj 1 ði
τÞ:
ð17Þ
Notice that the above construction is a special case of the general one discussed in [12] for constructing input–state product cycle, so it is easily to see fxkj ði τÞg is the set of reachable points starting from xði τÞ with uði τÞ and fixed G. The result can be obtained as follows:
s þ i k kτ ¼ j τ;
j A f0; 1; …; τg:
Then we have the following result: Theorem 3.10. xd is reachable from xði τÞ ðiA 0; 1; …; τÞ at s steps by controls of Boolean sequences uðs þ i k kτÞ ⋉uðs þ i ðk 1Þ ðk 1Þτ Þ⋯uðs þ i 1 τÞ if and only if xd A ColfL~ xðs þ i k kτÞg; k
where there exist unique k and j such that s þ i k kτ ¼ j τ;
j A f0; 1; …; τg:
When the length of sequence s is free, the generalization consequence of Theorem 3.10 can be immediately obtained as follows: Corollary 3.11. xd is reachable from xði τÞ ðiA 0; 1; …; τÞ at s steps by controls of Boolean sequences uðs þi k kτÞ⋉uðs þ i ðk 1Þ ðk 1Þτ Þ⋯uðs þ i 1 τÞ if and only if 1 k xd A Col ⋃ L~ xðs þ i k kτÞ ; a¼1
Theorem 3.8. Consider system (8) with control (9). Assume G is nonsingular and use the above algorithm, then
where there exist unique j and k such that
(1) for given uði τÞ; i ¼ 0; 1; …; τ and G, the set of reachable states is
Remark 3.12. It is obvious that if we choose τ ¼ 0, the obtained results can deduce the conclusion in [10], which can also prove the exactness of our studies.
RGuði τÞ ¼ fxkj ði τÞjk ¼ 0; 1; …; r i þ T ′i 1; j ¼ 0; 1; …; T i 1g; where fxkj ði τÞg are constructed by (16)–(17) and the steady state reachable set is defined by G k k (2) RSuði τÞ ≔fxj ði τÞ A Ruði τÞ jk Z r i g;
s þ i k kτ ¼ j τ;
j A f0; 1; …; τg:
Remark 3.13. Notice that in the proof procedure of our result, the value of time delay is irrelevant to the state of any step. And in this paper, the time delay has no restriction at all. So our results can be applied to deal with BCNs with any time delays.
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3.2. Optimal control of BCNs In this part, we will consider the problem of designing a timeoptimal control uði τÞ that steers the BCNs from initial state xði τÞ to desired state xd in minimal time, which is relevant to biological systems. For example, to prevent λ phage spread, we need to design a control sequence that steers the corresponding systems from lysogeny state to lysis state in minimal time. Next, we will give an algorithm to obtain a control to be time-optimal. According to Theorems 3.3 and 3.10, we can get the available state of any fixed step. So for any desired state, we can give the proper input to obtain the optimal control, i.e., we can solve the control problem of BCN from a given initial state to a desired state in minimal time by some inputs. To reach the desired state, we need to get the available state by order. First, we calculate the available state of calculate of xð1Þ, denote vector Xð1Þ. Assume s ¼ 1; i ¼ 0, then we can get Xð1Þ through Theorem 3.3 or 3.10. Next, we must judge whether xd can be get at the first step or not. If xd is the jth component of Xð1Þ, we can also the proper input. We denote N1 the dimension of X(s). j If the control is BN, then let uðb 1 τÞ ¼ δN1 , where s þ i ¼ aðτ þ 1Þ þ b; if the control is free Boolean sequence, then let j uðs þ i k kτÞ⋉uðs þ i ðk 1Þ ðk 1ÞτÞ⋯uðs þ i 1 τÞ ¼ δN1 , where s þ i k kτ ¼ j τ. Otherwise, if xd is not the component of Xð1Þ, then we need to consider xð2Þ. So the optimal control from xði τÞ to xd can be constructed as the following algorithm (fix i¼0): Step 1 For s ¼1, then calculate the available state of calculate of xð1Þ. Step 2 Count the dimension of Xð1Þ. j Step 3 If xd is the jth component of Xð1Þ, let the input to be δN1 , then xð1Þ ¼ xd . Else, let s ¼2, repeat Steps 1–3. Remark 3.14. Notice that for any s and i, if s þ i is fixed, the integer a and b or k and j is same. So we can fix i ¼0 to simplified operation, and it will not affect our result.
Based on Theorem 3.3, we assume that i ¼ 2; s ¼ 9; τ ¼ 2; Að1 τÞ ¼ 1 2 2 Bð1 τÞ ¼ δ2 and Cð1 τÞ ¼ δ2 , i.e., xð1 τÞ ¼ δ8 . Then according to s þi ¼ aðτ þ 1Þ þ b, a ¼ 3; b ¼ 2. And through calculation, we can have
ΘG ðs þ iÞW ½2n ;2m xðb 1 τÞ ¼ ΘG ð11ÞW ½23 ;22 xð 1Þ ¼ LG3 ðI 22 LG2 ÞðI 24 LGÞ ⋉ðI 28 LÞðI 24 Φ2 Þ
⋉ðI 22 Φ2 ÞΦ2 W ½23 ;22 xð 1Þ
¼ δ8 ½2; 4; 6; 2:
Thus, we conclude that the reachable set at step 9 is fδ8 ; δ8 ; δ8 g. 4 Next, we choose xd ¼ δ8 for example. So we have to find the initial control uð1 τÞ to drive the trajectory to the assigned destination. G From the matrix expression of Θ ð11ÞW ½23 ;22 xð 1Þ, it is obvious 4 that the 2nd column of δ8 ½2; 4; 6; 2 is δ8 . Hence, we can choose 2 2 uð1 τÞ ¼ δ4 to drive the state to xd. Let uð1 τÞ ¼ δ4 , or equiva1 2 lently, u1 ð τÞ ¼ δ2 ; u2 ð τÞ ¼ δ2 , then 2
6
6
xd ¼ Θ ðs þ iÞW ½23 ;22 xð1 τÞuð1 τÞ ¼ δ8 : G
4
In the following, we assume that xð τÞ ¼ δ8 ; xð τ þ 1Þ ¼ δ8 ; xð τ 3 4 þ 2Þ ¼ δ8 ðτ ¼ 2Þ and the given desired state xd ¼ δ8 . Then according to the algorithm of optimal control, we can get the available state of xð1Þ; xð2Þ; …; xð11Þ as follows: δ8 ½1; 2; 1; 2, δ8 ½5; 6; 7; 8, δ8 ½5; 6; 5; 6, δ8 ½1; 5; 2; 6, δ8 ½2; 6; 6; 2, δ8 ½2; 6; 2; 6, δ8 ½2; 2; 6; 6, δ8 ½8; 8; 6; 5, δ8 ½8; 8; 6; 6, δ8 ½6; 8; 6; 8, δ8 ½2; 4; 6; 2, δ8 ½2; 4; 6; 8. It is obvious that xd appears at the 11 step for the first time and N1 1 ¼ 4; j ¼ 2. Then according to s þ i ¼ aðτ þ 1Þ þ b, we can get b¼ 2. Hence, we can 2 choose uðb 1 τÞ ¼ uð 1Þ ¼ δ4 to drive the state to xd. So, the 11 step is the minimal time of BCNs (19) from the given initial state to a desired state. 1 To simulate the process, we give the input uð τÞ ¼ δ4 ; uð τ þ 1Þ 2 3 ¼ δ4 ; uð τ þ2Þ ¼ δ4 . Then we can get the process of the BCNs, which is showed in Fig. 2. In the figure, the black block indicates 2 that the node's state is of f(or 0), i.e., the node is δ2 . On the contrary, the white block indicates that the node's state is on 1 (or 1), i.e., the node is δ2 in our paper. 1
2
Remark 3.15. In the algorithm, we can see that if the control is Boolean network, we only need to ensure the control uðb 1 τÞ, i.e., the other control uði τÞ; i ¼ 0; 1; …; τ and i ab 1, can be arbitrary. It will not affect the Optimal control of BCNs.
Remark 4.2. For the system (19), We can also consider the case (ii) in our paper. Assume that the admissible set of G is nonsingular, and denote G1 ¼ δ4 ½1; 2; 3; 4; G2 ¼ δ4 ½1; 2; 4; 3; G3 ¼ δ4 ½1; 3; 2; 4 ; …; G24 ¼ δ4 ½4; 3; 2; 1. Then the corresponding matrices
4. Examples
V i ≔Θ i ðs þ iÞW ½2n ;2m xðb 1 τÞ; G
Example 4.1. Consider the Boolean control system as follows (Fig. 1): Its dynamics is described as 8 > < Aðt þ 1Þ ¼ Bðt τÞ2Cðt τÞ; Bðt þ 1Þ ¼ Cðt τÞ 3 u1 ðt τÞ; > : Cðt þ 1Þ ¼ Aðt τÞ 4 u ðt τÞ;
i ¼ 1; 2; …; 24
are δ8 ½1; 6; 1; 4, δ8 ½1; 6; 7; 6, δ8 ½1; 6; 7; 4, δ8 ½1; 6; 6; 8, δ8 ½1; 2; 1; 8, δ8 ½5; 2; 1; 4, δ8 ½5; 2; 7; 6, δ8 ½7; 6; 2; 4, δ8 ½8; 4; 2; 6, δ8 ½8; 2; 1; 2, δ8 ½2; 2; 5; 4, δ8 ½2; 2; 6; 6, δ8 ½2; 6; 5; 2, δ8 ½1; 2; 1; 8, δ8 ½2; 2; 8; 2, δ8 ½7; 2; 5; 6,
ð19Þ
2
with controls satisfying ( u1 ðt þ1Þ ¼ g 1 ðu1 ðt τÞ; u2 ðt τÞÞ ¼ :u2 ðt τÞ; u2 ðt þ1Þ ¼ g 1 ðu1 ðt τÞ; u2 ðt τÞÞ ¼ u1 ðt τÞ:
ð20Þ
Denote xðtÞ ¼ AðtÞBðtÞCðtÞ; uðtÞ ¼ u1 ðtÞu2 ðtÞ, then we can express systems (19)–(20) by (12) with G ¼ M n W ½2 ¼ δ4 ½3; 1; 4; 2; L ¼ M e ðI 22 M d ÞðI 2 Φ1 ÞðI 23 M c ÞW ½2;23 W ½22 ;23 ¼ δ8 ½1; 5; 5; 1; 2; 6; 6; 2; 2; 6; 6; 2; 2; 6; 6; 2; 1; 7; 5; 3; 2; 8; 6; 4; 2; 8; 6; 4; 2; 8; 6; 4:
Fig. 1. Boolean control network of (19).
δ8 ½1; 4; 8; 6, δ8 ½7; 6; 5; 2, δ8 ½1; 6; 1; 4, δ8 ½6; 2; 1; 6,
M. Han et al. / Neurocomputing 129 (2014) 467–475
δ8 ½7; 6; 2; 6, δ8 ½5; 6; 1; 2, δ8 ½6; 6; 2; 8, δ8 ½5; 6; 7; 2. So the cup set of all reachable sets for all Gi, i ¼ 1; 2; …; 24 at step 5 is fδ8 ; δ8 ; δ8 ; δ8 ; δ8 ; δ8 ; δ8 g: 1
2
4
5
6
7
8
Now we assume the destination state xd ¼ δ8 for example. Since 5 the 1st column of V7 is δ8 (there are also some other choices such as V 8 ; V 10 ; V 13 ), we can choose G7 ¼ δ4 ½2; 1; 3; 4 and u1 ð τÞu2 1 5 ð τÞ ¼ δ4 to reach δ8 at 5 steps. 5
Example 4.3. Consider the Boolean control system whose logical equation is 8 > < Aðt þ 1Þ ¼ Bðt τÞ 4 u1 ðt τÞ; Bðt þ 1Þ ¼ :u2 ðt τÞ; ð21Þ > : Cðt þ 1Þ ¼ Aðt τÞ 3 u ðt τÞ; 2
Denote xðtÞ ¼ AðtÞBðtÞCðtÞ; uðtÞ ¼ u1 ðtÞu2 ðtÞ. Then we can express system (21) as ~ τÞuðt τÞ; xðt þ 1Þ ¼ Lxðt where L~ can be expressed by
decision. When B is ON [OFF] and E is OFF [ON], the phage is in the lysogenic [lytic] state. Whether or not gene B will be switched on depends on a subtle control process in which five phage genes, A,B, C,D and E, and the environmental state play a prominent role. The detail presentation can be found in the recent monograph [39]. Here we consider the network with time delays both in state and control (Fig. 3). Its dynamics is described as 8 Aðt þ1Þ ¼ ð:Aðt τÞÞ 4 ð:Eðt τ ÞÞ; > > > > > Bðt þ 1Þ ¼ ð:Eðt τÞÞ 4 Bðt τÞ 3 Cðt τÞ; > < Cðt þ 1Þ ¼ ð:Bðt τÞÞ 4 uðt τÞ 4 ðAðt τÞ 3 Dðt τÞÞ; ð22Þ > > > Dðt þ 1Þ ¼ ð:Bðt τÞÞ 4 uðt τÞ 4 Aðt τÞ; > > > : Eðt þ 1Þ ¼ ð:Bðt τÞÞ 4 ð:Cðt τÞÞ: Remark 4.6. During the processes grow on a bacterium, several environmental conditions including temperature, growth rate and concentration of nutrition may cause the time delay. Therefore, it is reasonable to consider the network with time delay. Denote xðt þ 1Þ ¼ Aðt þ 1ÞBðt þ 1ÞCðt þ 1ÞDðt þ1ÞEðt þ1Þ. Then we can express system (22) as
L~ ¼ M c ðI 22 M n ÞðI 23 M d Þ
⋉W ½2;23 ðI 23 Φ1 ÞðI 22 Ed W ½2 Þ
~ τÞuðt τÞ; xðt þ 1Þ ¼ Lxðt
¼ δ8 ½3; 1; 7; 5; 3; 1; 7; 5; 7; 5; 7; 5; 7; 5; 7; 5; 3; 2; 7; 6; 3; 2; 7; 6; 7; 6; 7; 6; 7; 6; 7; 6:
where L~ can be expressed by
Based on Theorem 3.10, assume that i ¼ 1; s ¼ 4; τ ¼ 2; Að 1Þ ¼ B 1 2 2 ð 1Þ ¼ δ2 and Cð 1Þ ¼ δ2 , i.e., xð 1Þ ¼ δ8 . We wish to know whether a designed state can be reached at 4 steps from xð 1Þ. By simple calculation, we can get 2 L~ xð1 τÞ ¼ δ8 ½7; 5; 7; 5; 3; 1; 7; 5; 7; 6; 7; 6; 3; 2; 7; 6:
Then, there exist k ¼ 2; j ¼ 1 such that s þ i k kτ ¼ j τ and 2 xðs þ iÞ ¼ xð5Þ ¼ L~ xð1 τÞuð1 τÞuð2Þ. Using Theorem 3.10, it is clear 4 8 that at 4 steps, all states but δ8 ; δ8 can be reached. Now we choose 2 1 1 destination state xd ¼ δ8 , notice that the 6th column of L~ xð1 τÞ is δ8 , 6 which means controls δ16 can drive the trajectory to the destination 1 state δ8 . Hence, the corresponding controls can be chosen by 1 2 1 2 u1 ð 1Þ ¼ δ2 ; u2 ð 1Þ ¼ δ2 ; u1 ð2Þ ¼ δ2 ; u2 ð2Þ ¼ δ2 .
L~ ¼ δ32 ½24; 24; 24; 24; 24; 24; 24; 24; 32; 32; 24; 24; 32; 32; 24; 24; 18; 24; 18; 24; 18; 24; 18; 24; 25; 31; 25; 31; 25; 31; 25; 31; 24; 24; 8; 8; 24; 24; 8; 8; 32; 32; 8; 8; 32; 32; 8; 8; 20; 24; 4; 8; 24; 24; 8; 8; 27; 31; 11; 15; 31; 31; 15; 15: Based on Theorem 3.10, assume that i ¼ 0; s ¼ 9; τ ¼ 2, there exist k ¼ 3; j ¼ 2 such that s þ i k kτ ¼ j τ. Then we wish to know whether a designed state can be reached at 9 steps from 1 xð0Þ. Also we can assume that Að0Þ ¼ Bð0Þ ¼ Dð0Þ ¼ Eð0Þ ¼ δ2 and 2 5 Cð0Þ ¼ δ2 , i.e., xð0Þ ¼ δ32 . By simple calculation, we can get 3 L~ xð2 τÞ ¼ δ32 ½25; 31; 25; 31; 25; 31; 25; 31:
Remark 4.4. Similar to Example 3.1, we assume that xð τÞ ¼ δ18 ; xð τ þ 1Þ ¼ δ28 ; xð τ þ 2Þ ¼ δ38 ðτ ¼ 2Þ and the given desired state 1 xd ¼ δ8 . Then according to the algorithm of optimal control, we first get the available state of xð1Þ is δ8 ½3; 1; 7; 5. It is obvious that xd appears at the 1 step for the first time and N 1 ¼ 4; j ¼ 2. Then according to s þ i k kτ ¼ j τ, we can get k ¼1. Hence, we can 2 choose uð τÞ ¼ uð 2Þ ¼ δ4 to drive the state to xd. So, the first step is the minimal time of BCNs (21) from the given initial state to a desired state.
Node u
6
8
10
12
14
16
18
20
22
24
Input Pin Connector A
Example 4.5 (A biological example). Consider following the Boolean model, which is a simple BCN model for the λ switch [40]. The λ phage is a virus that grows on a bacterium upon infection of the bacterium. To ensure successful propagation, the virus had to develop efficient mechanisms of precise response to changes in the physiology of its host. This was achieved by a specific genetic switch that allows this virus to choose the most effective developmental pathway for the given environmental conditions. The virus can then follow one of two different pathways: lysogeny or lysis. For example, two genes, B and E, directly affect this
4
473
D
C
E
B
Fig. 3. Boolean control network of (22).
26
28
30
32
34
36
38
Fig. 2. The process of BCNs (19) (black/white block is for 0/1).
40
42
44
46
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2
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6
8
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20
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24
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30
32
34
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42
44
46
Generation Fig. 4. The process of BCNs (22) (black/white block is for 0/1).
3 And xðs þiÞ ¼ xð9Þ ¼ L~ xð0Þuð0Þuð3Þuð6Þ. Using Theorem 3.10, it is 25 31 clear that at 9 steps, the states δ32 and δ32 can be reached. Now we 25 choose destination state xd ¼ δ32 , notice that the 1th (or 3th, 5th, 3 11 1 3 5 7 7th) column of L~ xð2 τÞ is δ32 , which means controls δ8 ðδ8 ; δ8 ; δ8 Þ 25 can drive the trajectory to the destination state δ32 . Hence, the 1 1 corresponding controls can be chosen by uð0Þ ¼ δ2 ; uð3Þ ¼ δ2 and 1 uð6Þ ¼ δ2 . In the following, we assume that xð τÞ ¼ δ3 21 ; xð τ þ 1Þ ¼ δ3 3 2 ; xð τ þ 2Þ ¼ δ3 25 ðτ ¼ 2Þ and the given desired state xd ¼ δ3 22 5. Then according to the algorithm of optimal control, we can get the available state of xð1Þ; xð2Þ; …; xð9Þ as follows: δ32 ½24; 24, δ32 ½24; 24, δ32 ½32; 32, δ32 ½8; 8; 8; 8, δ32 ½8; 8; 8; 8, δ32 ½15; 15 ; 15; 15, δ32 ½24; 24; 24; 24; 24; 24; 24; 24, δ32 ½24; 24; 24; 24; 24; 24; 24; 24, δ32 ½25; 31; 25; 31; 25; 31; 25; 31. It is obvious that xd appears at the 9 step for the first time and N 9 ¼ 8; j ¼ 1. Then according to s þ i k kτ ¼ j τ, we can get j¼2. Hence, we can choose 1 uð0Þ ¼ uð3Þ ¼ uð6Þ ¼ δ2 to drive the state to xd. So, the 9 step is the minimal time of BCNs (22) from the given initial state to a desired state. 1 To simulate the process, we fixed the input uð0Þ ¼ uð3Þ ¼ uð6Þ ¼ δ2 and the other control is randomly selected by computer. Then we can get the process of the BCNs, which is showed in Fig. 4. In the figure, the black block also indicates that the node's state is off (or 0), i.e., the 2 node is δ2 . On the contrary, the white block indicates that the node's 1 state is on (or 1), i.e., the node is δ2 in our paper.
5. Conclusion In this paper, we have investigated the controllability of Boolean control networks with time delays both in state and control. First, we provide a brief review of the STP of matrices and the matric expression of logic. Second, using the method of STP of matrices and the matric expression of logic, the Boolean networks with time delays both in state and control have been converted into discrete time delay systems. Then, we have considered the controllability via two kinds of controls. In both cases, according to different input structure and step, four situations have been studied and the necessary and sufficient conditions have been obtained. Then, we consider the optimal control problem of BCNs from a given initial state to a desired state in minimal time. Finally, we have given three examples to show the efficiency of the obtained results.
Acknowledgments We thank Prof. Jinde Cao for his constructive suggestions. Furthermore, the authors wish to thank the editor and the reviewers for a number of constructive comments and suggestions that have improved the quality of the paper. This work was supported by NNSF of China (Grant nos. 11271333, 11011373 and 61074011) and National Undergraduate Training Programs for Innovation and Entrepreneurship (Grant no. 201310345010).
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Ming Han was born in Hangzhou, Zhejiang Province, China, in 1990. She is a MS candidate in the College of Mathematics, Physics and Information Engineering, Zhejiang Normal University. Her research interests include impulsive systems, delayed systems, neural networks, graph theory.
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Yang Liu received the B.S. degree in mathematics from Zhejiang Normal University, Zhejiang, China, in 2003, and the Ph.D. degree from Tongji University, Shanghai, in 2008. He is currently an associate professor at the Department of Mathematics, Zhejiang Normal University. His research interest covers complex analysis, Clifford analysis, switched system, delayed system and nonlinear control. He has published over 30 papers in refereed international journals. He services as the reviewer for Mathematical Reviews. He is the recipient of Shanghai Outstanding Ph.D's Thesis Award in 2011.
Yanshuai Tu was born in Jinhua, Zhejiang Province, China, in 1990. He is a MS candidate in the College of Mathematics, Physics and Information Engineering, Zhejiang Normal University. His research interests include nonlinear systems, neural networks, computational physics, three-dimensional sensing.