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Controllability and optimal control of a temporal Boolean network
Neural Networks 34 (2012) 10–17
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Neural networks letter
Controllability and optimal control of a temporal Boolean network Fangfei Li a,b , Jitao Sun a,∗ a
Department of Mathematics, Tongji University, Shanghai, China
b
Department of Mathematics, East China University of Science and Technology, Shanghai, China
article
info
Article history: Received 28 November 2011 Received in revised form 21 May 2012 Accepted 12 June 2012 Keywords: Temporal Boolean networks Controllability Optimal control Semi-tensor product of matrices
abstract This paper investigates the controllability and optimal control of a temporal Boolean network, where the time delays are time variant. First, using the theory of semi-tensor product of matrices, the logical systems can be converted into a discrete time variant system. Second, necessary and sufficient conditions for the controllability via two types of controls are provided respectively. Third, optimal control design algorithms are presented. Finally, examples are given to illustrate the proposed results. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Genetic regulatory networks are widely used in the research of systems biology. These models include Bayesian networks (Martínez-Rodríguez, May, & Vargas, 2008), differential equations (Shen, Wang, Liang, & Liu, 2011; Wang, Lam, Wei, Fraser, & Liu, 2008; Wang, Liu, Liu, Liang, & Vinciotti, 2009a; Xu, Venayagamoorthy, & Wunsch, 2007) and Boolean networks (Drossel, Mihaljev, & Greil, 2005; Farrow, Heidel, Maloney, & Rogers, 2004; Kauffman, 1969) etc. A Boolean network consists of a group of Boolean variables: active (1) or inactive (0), and a set of Boolean functions. Interactions between the states of each Boolean variables are determined by Boolean functions, which calculate the state of a Boolean variable from the activation of other Boolean variables. Boolean networks have attracted much attention in recent years. Recently, the control problem of Boolean networks is a hot topic. Using the semi-tensor product of matrices, a Boolean network can be transformed into a conventional discrete time system. Based on this method, many basic problems in control theory, such as controllability, observability, stabilization and optimal control are studied, for example, see Cheng and Qi (2009), Cheng, Qi, Li, and Liu (2011) and Zhao, Li, and Cheng (2011); Zhao, Qi, and Cheng (2010). Moreover, some control problems for more general Boolean networks, such as k-valued logical networks and probabilistic Boolean networks are investigated too Ching, Zhang,
∗
Corresponding author. Tel.: +86 21 65983241x1307; fax: +86 21 65981985. E-mail address: [email protected] (J. Sun).
0893-6080/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.neunet.2012.06.002
Ng, and Akutsu (2007), Li and Cheng (2010) and Shmulevich, Dougherty, Kim, and Zhang (2002). As is well known, time delay phenomena are very common in the real world. Time delay behaviors happen frequently in biological and physiological systems. In many cases, the time delay phenomenon in control systems has become an important topic in control theory. There are many results concerning basic problems for time delay systems in control theory, (see e.g. Li, Sun, & Sun, 2010, Wang, Liu, & Liu, 2009b). For Boolean networks, the controllability and observability of Boolean networks with time delays was investigated in Li and Sun (2011) and Li, Sun, and Wu (2011). One kind of Boolean networks, called temporal Boolean networks are developed to model regulatory delays, for example, see Silvescu and Honavar (2001). A temporal Boolean network has more complex structure than the Boolean networks investigated, in Li and Sun (2011), because the temporal Boolean networks allow the time delays in each Boolean variable to be different. Though a temporal Boolean network can describe a rather general Boolean network with time delays, it does not consider the case when the time delays are time variant. In fact, a Boolean network with time variant delays, appears as a natural description of several real world problems. It can approximate a real cellular regulatory network better than a Boolean network with time invariant delays. A temporal Boolean network with time-variant time delays has rather complex dynamic characteristics, hence to investigate it is more challenging. Controllability and optimal control are structure properties of a system, and they are fundamental concepts in control theory. There have been many results studying the controllability or the optimal control problems of dynamics, (see e.g. Vrabie & Lewis, 2009). For Boolean networks, the controllability was studied in Cheng and Qi (2009). The Mayer-type optimal control problem
F. Li, J. Sun / Neural Networks 34 (2012) 10–17
of Boolean networks was considered in Laschov and Margaliot (2011), and a maximum principle was put forward for the systems with single control input. The infinite horizon optimal control problem of k-valued logical control networks was investigated in Zhao et al. (2011). But to the best of our knowledge, there has been no result studying the controllability and optimal control of temporal Boolean networks with time-variant delays. Temporal Boolean networks with time-variant delays can describe more general time delay phenomena for Boolean networks, studying them is meaningful and challenging. Motivated by the above, in this paper we consider the controllability and optimal control problem of temporal Boolean networks with time variant delays. Based on the theory of semi-tensor product of matrices (also called Cheng product), we show how to convert temporal Boolean networks with time variant delays into discrete time variant systems. The controllability via two types of controls is considered. One kind of control is input networks. Another kind of control is free Boolean sequences. Necessary and sufficient conditions are provided for each case. Furthermore, based on the results we obtained, optimal control design algorithms are presented. In this paper, we provide a control design algorithm that minimizes (or maximizes) the objective functional at a fixed termination time. Meanwhile, we assume the system we studied is controllable at the initial state from an initial time instant and find a control sequence that minimizes (or maximizes) the objective functional in the shortest time. The system in this paper is time variant, which is rather complicated. The results in our paper are not trivial extensions of existing results (Cheng & Qi, 2009; Zhao et al., 2010). The rest of this paper is organized as follows: Section 2 introduces some preliminaries about the semi-tensor product of matrices (Cheng product). In Section 3, we first convert a temporal Boolean network with time variant delays into discrete time variant systems. Then necessary and sufficient conditions about the controllability via two types of controls are obtained respectively. Finally, we give the optimal control design algorithm based on the results obtained in Section 3.2. Section 4 provides examples to illustrate the proposed results. Finally, concluding remarks are provided in Section 5. 2. Preliminaries In this section, we introduce the semi-tensor product of matrices (also called Cheng product) and the vector form of Boolean variables, which are recapitulations of results mainly from Cheng and Qi (2010). 2.1. Semi-tensor product of matrices In this paper, the matrix product we use is the Cheng product. Definition 2.1. 1. Cheng and Qi (2010). 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-size blocks as X 1 , . . . , X p , which are 1 × n rows. Define the semi-tensor product (STP), denoted by n, as
p X nY = X i y i ∈ Rn , i =1
p T T y i ( X i ) T ∈ Rn . Y n X = i =1
2. Let A ∈ Mm×n and B ∈ Mp×q . If either n is a factor of p, say nt = p and denote it as A ≺t B, or p is a factor of n, say n = pt and denote it as A ≻t B, then we define the STP of A and B, denoted
11
by C = A n B, as the following: C consists of m × q blocks as C = (C ij ) and each block is C ij = Ai n Bj ,
i = 1, . . . , m, j = 1, . . . , q,
where Ai is the ith row of A and Bj is the jth column of B. The Cheng product is a generalization of the conventional matrix product. All the fundamental properties of the conventional matrix product remain true. Based on this, we can omit the symbol n. There are also some basic properties of the Cheng product, for details see Cheng and Qi (2010). 2.2. Vector form of Boolean variables In this subsection, we introduce the vector form of Boolean variables and some concerning results. First, let ‘‘1’’ and ‘‘0’’ represent the logical ‘‘True’’ and ‘‘False’’ respectively, and D := {1, 0}. We define a delta set as ∆k := {δki | i = 1, 2, . . . , k}, where δki is the ith column of the identity i
i
matrix Ik . Assume there is a matrix M = [δn1 , δn2 , . . . , δnis ], we simply denote it as M = δn [i1 , i2 , . . . , is ]. We give Boolean variables a vector form as: T = 1 ∼ δ21 ,
F = 0 ∼ δ22 ,
then the logical variable A(t ) takes values from these two vectors, i.e. A(t ) ∈ ∆ := ∆2 = {δ21 , δ22 }. The ‘‘D ’’ and ‘‘∆’’ will be used freely according to the variable types. Second, we give some notations. (1) Denote by Mm×n the set of all m × n matrices. (2) Denote by Col(A) the set of columns of a matrix A, and by Coli (A) the ith column of A. (3) Let a matrix A ∈ Mn×mn and denote by Blki (A) the ith n × n square block of A, i = 1, 2, . . . , m. (4) Denote by Ai,j the ith element of the jth column of a matrix A. Third, we define the logical matrix and the set of logical matrices. A matrix A ∈ Mm×n is called a logical matrix, if the columns of A are elements of ∆m . Denote the set of logical matrices by L, the set of n × s logical matrices is defined by Ln×s . Next, we will introduce the swap matrix and the dummy matrix. A swap matrix, W[m,n] , is an mn × mn matrix 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
w(IJ ),(ij) = δiI,,jJ =
1, 0,
I = i and J = j, otherwise.
When m = n, we briefly denote W[n] := W[n,n] . To see the properties of the swap matrix, refer to Cheng and Qi (2010). The dummy matrix is defined as Ed := δ2 [1, 2, 1, 2]. From Cheng and Qi (2010), we have for any two logical variables u, v , Ed uv = v,
or Ed W[2] uv = u.
Finally, we give a lemma that will be used in the following paper. According to Cheng and Qi (2010), for each logical function L(A1 , . . . , An ), there exists a structure matrix ML such that: Lemma 2.1. Any logical function L(A1 , . . . , An ) with logical arguments A1 , . . . , An ∈ ∆ can be expressed in a multi-linear form as L(A1 , . . . , An ) = ML A1 A2 · · · An , where ML ∈ L2×2n is unique, called the structure matrix of L.
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F. Li, J. Sun / Neural Networks 34 (2012) 10–17
3. Main results
In the following, we consider the temporal Boolean network with controls as follows
3.1. Converting a temporal Boolean network with time variant delays into a discrete time variant system
A1 (t + 1) = f1t (u1 (t ), . . . , um (t ), A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ (t )), . . . , An (t − τ (t ))) A2 (t + 1) = f2t (u1 (t ), . . . , um (t ), A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ (t )), . . . , An (t − τ (t ))) .. . t An (t + 1) = fn (u1 (t ), . . . , um (t ), A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ (t )), . . . , An (t − τ (t ))),
Recall that a Boolean network can be described as
A1 (t + 1) = f1 (A1 (t ), A2 (t ), . . . , An (t )), A2 (t + 1) = f2 (A1 (t ), A2 (t ), . . . , An (t )), .. . An (t + 1) = fn (A1 (t ), A2 (t ), . . . , An (t )),
(1)
where fi are logical functions, t = 0, 1, 2, . . . . But in the real world, time delay phenomena are very common, Silvescu and Honavar (2001) introduced a temporal Boolean network as follows
A (t + 1) = f (A (t ), . . . , A (t ), A (t − 1), . . . , 1 1 1 n 1 An (t − 1), . . . , A1 (t − τ ), . . . , An (t − τ )) A2 (t + 1) = f2 (A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ ), . . . , An (t − τ )) .. . An (t + 1) = fn (A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ ), . . . , An (t − τ )).
(2)
(3)
We give an example to illustrate this kind of system. This example is a special case of temporal Boolean networks. It is a biochemical network of coupled oscillations in the cell cycle (Goodwin, 1963).
A(t + 3) = ¬(A(t ) ∧ B(t + 1)), B(t + 3) = ¬(A(t + 1) ∧ B(t )).
It can be converted into the form
A(t + 1) = ¬(A(t − 2) ∧ B(t − 1)), B(t + 1) = ¬(A(t − 1) ∧ B(t − 2)),
where time delay is invariant.
t ≥ 2.
i = 1, 2, . . . .
(5)
= M1t
n
[(I2m+n(τ (t )+1) ⊗ Mit )Φm+n(τ (t )+1) ]
i=2
u(t )x(t )x(t − 1) · · · x(t − τ (t )) , L˜ t u(t )x(t )x(t − 1) · · · x(t − τ (t )).
Remark 3.1. The time delays in system (3) are time variant. It is a rather complicated system. Moreover, we note that time variant dynamics draw much attention in control theories, (see e.g. Zhao, Sun, & Liu, 2008), similarly, we consider a time variant temporal Boolean network with time variant delays.
Example 3.1. Consider the following Boolean network
Ai (t + 1) = Mit u(t )x(t )x(t − 1) · · · x(t − τ (t )),
x(t + 1) = A1 (t + 1)A2 (t + 1) · · · An (t + 1)
where fit means that the logical functions fi are time variant and τ (t ) are time-variant time delays which are positive integers, and there exists the smallest integer τ satisfying maxt ≥0 {τ (t )} ≤ τ < +∞, t = 0, 1, . . . .
where ui are controls (or inputs), ui ∈ ∆, i = 1, . . . , m. In order to convert (4) into algebraic form, we define x(t ) = m nni=1 Ai (t ) ∈ ∆2n , u(t ) = nm i=1 ui (t ) ∈ ∆2 . Assume that the structure matrix of fit is Mit ∈ L2×2m+n(τ (t )+1) , we can express (4) as
Multiplying the equations (5) together, yields
Note that in many cases, time delays are not time invariant but time variant. Motivated by this, in this paper, we consider a more general temporal Boolean network
A (t + 1) = f t (A (t ), . . . , A (t ), A (t − 1), . . . , 1 1 n 1 1 An (t − 1), . . . , A1 (t − τ (t )), . . . , An (t − τ (t ))) t A2 (t + 1) = f2 (A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ (t )), . . . , An (t − τ (t ))) .. . An (t + 1) = fnt (A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ (t )), . . . , An (t − τ (t ))),
(4)
Let z (t ) = nτi=0 zi (t ), where z0 (t ) = x(t ), z1 (t ) = x(t − 1), . . . , zτ (t ) (t ) = x(t − τ (t )), we have x(t + 1) = L˜ t u(t )x(t )x(t − 1) · · · x(t − τ (t ))
= L˜ t u(t )Edn(τ −τ (t )) W[2n(τ (t )+1) ,2n(τ −τ (t )) ] z (t ) = L˜ t (I2m ⊗ Edn(τ −τ (t )) W[2n(τ (t )+1) ,2n(τ −τ (t )) ] )u(t )z (t ) , L¯ t u(t )z (t ).
z (t + 1) = nτi=0 zi (t + 1) = x(t + 1)x(t ) · · · x(t − τ + 1)
= =
L¯ t u(t )z (t )x(t ) · · · x(t − τ + 1)
L¯ t u(t )W[2nτ ,2n(τ +1) ]
(x(t ) · · · x(t − τ + 1))2 x(t − τ ) = L¯ t u(t )W[2nτ ,2n(τ +1) ] Φnτ z (t ) = L¯ t (I2m ⊗ W[2nτ ,2n(τ +1) ] Φnτ )u(t )z (t ) , Lt u(t )z (t ). 3.2. Controllability of temporal Boolean network In this subsection, we will consider the controllability of the temporal Boolean network (4). Without loss of generality, in this paper, we assume that the j initial state of system (4) is nτi=0 x(−i) = δ2n(τ +1) . Moreover, let X = (A1 , . . . , An )T .
Definition 3.1. Consider the temporal Boolean network (4). Given its initial state sequence x(0) ∈ ∆2n ∼ X (0) ∈ Dn , . . . , x(−τ ) ∈ ∆2n ∼ X (−τ ) ∈ Dn , and destination state xd ∈ ∆2n ∼ Xd ∈ Dn , xd is said to be reachable from the initial state sequence nτi=0 x(−i) from time instant 0 at the sth step, if we can find control sequence {u(0), . . . , u(s − 1)}, such that the initial state sequence nτi=0 x(−i) can be driven to the destination state xd = x(u, s).
F. Li, J. Sun / Neural Networks 34 (2012) 10–17
Definition 3.2. (1) Consider the temporal Boolean network (4). The reachable set from time instant 0 at time s is the set of states which are reachable from the initial state sequence nτi=0 x(−i) from time instant 0 at time s, denoted by Rs,0 (nτi=0 x(−i)). (2) The overall reachable set from time instant 0 is the set of states which are reachable from the initial state sequence nτi=0 x(−i) from time instant 0, denoted by R0 (nτi=0 x(−i)). Definition 3.3. (1) System (4) is said to be controllable at initial state sequence nτi=0 x(−i) from time instant 0, if R0 (nτi=0 x(−i)) = ∆2n . (2) System (4) is said to be controllable from time instant 0, if for any initial state sequence nτi=0 x(−i), we have R0 (nτi=0 x(−i)) = ∆2n . We consider the controllability of temporal Boolean network (4) with two kinds of controls. (I) The controls are logical variables satisfying certain logical rules, called the input networks as:
Proof. (i) Assume that u(0) = δ z (1) = L0 u(0)z (0) =
(6)
z (t + 1) = Lt u(t )z (t ), u(t + 1) = Gu(t ).
(7)
Theorem 3.1. Consider system (4) with input network control (6), equivalently (7), we have
nτi=0
j
(i) xd = δ is reachable from x(−i) = δ2n(τ +1) from time instant 0 at the sth step, if and only if
t =1
z (3) = L2 u(2)z (2) = L2 G2 u(0)z (2)
= Blki0 (L2 G2 )
Blki0 (Lt Gt )
=
j
x(−i) = δ2n(τ +1) from time (ii) xd = δ is reachable from instant 0, if and only if there exists positive integer N, such that 2m 0
Blki0 (Lt G )
> 0.
t
i0 =1 t =N −1
t =2
z (s) =
j
Ednτ W[2n ,2nτ ]
N
2m
0
Blki0 (Lt Gt )
k=1 i0 =1 t =k−1
> 0. i,j
(iv) System (7) is controllable from time instant 0, if and only if there exists positive integer N, such that all the entries of Ednτ W[2n ,2nτ ]
0
Blki0 (Lt Gt )z (0).
x(s) = z0 (s) = Ednτ W[2n ,2nτ ] z (s) 0
= Ednτ W[2n ,2nτ ]
N 2m 0 k=1 i0 =1 t =k−1
0
j
Blki0 (Lt Gt )δ2n(τ +1)
t =s−1
= Colj
0
Ednτ W[2n ,2nτ ]
Blki0 (Lt G ) . t
t =s−1
From the above equality, we can get the sum of all the elements of j Rs,0 (δ2n(τ +1) ) as 2m
0
Ednτ W[2n ,2nτ ]
i 0 =1
j
Blki0 (Lt Gt )δ2n(τ +1)
t =s −1
= Ednτ W[2n ,2nτ ]
2m 0
j
Blki0 (Lt Gt )δ2n(τ +1)
i0 =1 t =s−1
Ednτ W[2n ,2nτ ]
2m 0
Blki0 (Lt G ) . t
This equality implies that xd = δ2i n is reachable from nτi=0 x(−i) =
δ2j n(τ +1) from time instant 0 at the sth step, if and only if 2m 0 nτ t n n τ Ed W[2 ,2 ] Blki0 (Lt G ) > 0. i,j
(ii)–(iv). Similar to the discussion of (i), the conclusions follow. (II) The control is free Boolean sequences. Theorem 3.2. Consider system (4) with free Boolean sequences control, we have (i) xd = δ2i n is reachable from nτi=0 x(−i) = δ2n(τ +1) from time instant 0 at the sth step, if and only if j
Blki0 (Lt Gt ) > 0.
Blki0 (Lt Gt )z (0)
t =s−1
i0 =1 t =s−1
i,j
(iii) System (7) is controllable at nτi=0 x(−i) = δ2n(τ +1) from time instant 0, if and only if there exists positive integer N, such that all the entries of Colj
Blki0 (Lt Gt )z (0),
i0 =1 t =s−1
i ,j
nτi=0
Ednτ W[2n ,2nτ ]
Blki0 (Lt Gt )z (0)
.. .
> 0.
i0 =1 t =s−1
0 t =1
0
= Colj
i 2n
Blki0 (Lt Gt )z (0),
= Ednτ W[2n ,2nτ ]
2m 0
(0) = Blki0 (L0 )z (0),
z (s) = nτi=0 zi (s), yields
where G is the transition matrix of (6). (II) The control is free Boolean sequences. Precisely, set u(t ) = nm j=1 uj (t ). Then the control is a designed sequence. We consider case (I) first. Using the structure matrix approach to the temporal Boolean network, the algebraic form of (4), (6) can be obtained as
Ednτ W[2n ,2nτ ]
0
=
u(t + 1) = Gu(t ),
δ
through calculation, we have
z (2) = L1 u(1)z (1) = L1 Gu(0)z (1) = Blki0 (L1 G)Blki0 (L0 )z (0)
where gi , i = 1, 2, . . . , m are logical functions. We can express (6) as
i 2n
i L0 20m z
t =s−1
u1 (t + 1) = g1 (u1 (t ), u2 (t ), . . . , um (t )), u2 (t + 1) = g2 (u1 (t ), u2 (t ), . . . , um (t )), .. . um (t + 1) = gm (u1 (t ), u2 (t ), . . . , um (t )),
13 i0 , 2m
Ednτ W[2n ,2nτ ]
0 2m t =s−1 it =1
Blkit (Lt )
> 0. i,j
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F. Li, J. Sun / Neural Networks 34 (2012) 10–17
(ii) xd = δ2i n is reachable from nτi=0 x(−i) = δ2n(τ +1) from time instant 0, if and only if there exists positive integer N, such that j
0 2m
Ednτ W[2n ,2nτ ]
Blkit (Lt ) i ,j
j
Ednτ W[2n ,2nτ ]
2m
N 0
Blkit (Lt )
> 0.
(iv) System (4) is controllable from time instant 0, if and only if there exists positive integer N, such that all the entries of N 0 2m
Proof. Assume that u(t ) = δ2tm , through calculation, we have i
i
z (1) = L0 u(0)z (0) = L0 δ20m z (0) = Blki0 (L0 )z (0), i
z (2) = L1 u(1)z (1) = L1 δ21m Blki0 (L0 )z (0) = Blki1 (L1 )Blki0 (L0 )z (0)
=
Blkit (Lt )z (0),
t =1 i
z (3) = L2 u(2)z (2) = L2 δ22m z (2)
= Blki2 (L2 )
0
Blkit (Lt )z (0)
t =1
=
0 t =2
z (s) =
Blkit (Lt )z (0).
Remark 3.2. As the number of elements of Lt is finite, the transition matrix Lt of (3) has finite modes. Hence, the upper bound of N is 2n(τ +1) + 1.
A (t + 1) = f t (u (t ), . . . , u (t ), u (t − 1), . . . , 1 1 m 1 um (t − 1), .1. . , u1 (t − τ (t )), . . . , um (t − τ (t )), A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ (t )), . . . , An (t − τ (t ))), A2 (t + 1) = f2t (u1 (t ), . . . , um (t ), u1 (t − 1), . . . , um (t − 1), . . . , u1 (t − τ (t )), . . . , um (t − τ (t )), A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ (t )), . . . , An (t − τ (t ))), .. . An (t + 1) = fnt (u1 (t ), . . . , um (t ), u1 (t − 1), . . . , um (t − 1), . . . , u1 (t − τ (t )), . . . , um (t − τ (t )), A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ (t )), . . . , An (t − τ (t ))).
(8)
x(t + 1) = L˜ ′t u(t )u(t − 1) · · · u(t − τ (t ))x(t ) x(t − 1) · · · x(t − τ (t ))
x(s) = z0 (s) = Ednτ W[2n ,2nτ ] z (s)
= Ednτ W[2n ,2nτ ]
0
= L˜ ′t Edm(τ −τ (t )) W[2m(τ (t )+1) ,2m(τ −τ (t )) ] v(t )Edn(τ −τ (t )) W[2n(τ (t )+1) ,2n(τ −τ (t )) ] z (t ) = L˜ ′t Edm(τ −τ (t )) W[2m(τ (t )+1) ,2m(τ −τ (t )) ] (I2m(τ +1) ⊗ Edn(τ −τ (t )) W[2n(τ (t )+1) ,2n(τ −τ (t )) ] )v(t )z (t ) , L¯ ′t u(t )z (t ).
Blkit (Lt )z (0)
t =s−1
= Ednτ W[2n ,2nτ ]
0
j
Blkit (Lt )δ2n(τ +1)
z (t + 1) = nτi=0 zi (t + 1) = x(t + 1)x(t ) · · · x(t − τ + 1) = L¯ ′t v(t )z (t )x(t ) · · · x(t − τ + 1)
t =s−1
0
= Colj Ednτ W[2n ,2nτ ]
Blkit (Lt ) .
t =s−1
From the above equality, we can get the sum of all the elements of j Rs,0 (δ2n(τ +1) ) as
i0 =1 i1 =1
Let z (t ) = nτi=0 zi (t ), where z0 (t ) = x(t ), z1 (t ) = x(t − 1), . . . , zτ (t ) (t ) = x(t −τ (t )) and v(t ) = nτi=0 Ui (t ), where U0 (t ) = u(t ), U1 (t ) = u(t − 1), . . . , Uτ (t ) (t ) = u(t − τ (t )), we have
z (s) = nτi=0 zi (s), yields
···
i,j
x(t + 1) = L˜ ′t u(t )u(t − 1) · · · u(t − τ (t ))x(t )x(t − 1) · · · x(t − τ (t )).
t =s−1
2m 2m
t =s−1 it =1
Then, by defining x(t ) = nni=1 Ai (t ) ∈ ∆2n , u(t ) = nm i=1 ui (t ) ∈ ∆2m , we can convert (8) into
Blkit (Lt )z (0),
.. . 0
δ2j n(τ +1) from time instant 0 at the sth step, if and only if 0 2m nτ Blkit (Lt ) > 0. Ed W[2n ,2nτ ]
Remark 3.3. In the real world, the control always has time delays. The system describing such a case is as follows:
Blkit (Lt ) > 0.
k=1 t =k−1 it =1
0
Blkit (Lt ) .
(ii)–(iv). Similar to the discussion of (i), the conclusions follow.
k=1 t =k−1 it =1
Ednτ W[2n ,2nτ ]
This equality implies that xd = δ2i n is reachable from nτi=0 x(−i) =
> 0.
(iii) System (4) is controllable at nτi=0 x(−i) = δ2n(τ +1) from time instant 0, if and only if there exists positive integer N, such that all the entries of Colj
= Colj Ednτ W[2n ,2nτ ]
0 2m t =s −1 i t =1
t =N −1 i t =1
2m
Ednτ W[2n ,2nτ ]
is−1 =1
= Ednτ W[2n ,2nτ ]
j
Blkit (Lt )δ2n(τ +1)
t =s−1 2m
2m
···
i0 =1 i1 =1
= Ednτ W[2n ,2nτ ]
0
0 2m t =s−1 it =1
2m 0
j
Blkit (Lt )δ2n(τ +1)
= = =
L¯ ′t v(t )W[2nτ ,2n(τ +1) ] (x(t ) · · · x(t − τ + 1))2 x(t − τ )
L¯ ′t v(t )W[2nτ ,2n(τ +1) ] Φnτ z (t )
L¯ ′t (I2m(τ +1) ⊗ W[2nτ ,2n(τ +1) ] Φnτ )v(t )z (t ) , L′t v(t )z (t ).
Similar to the proof of Theorems 3.1 and 3.2, we can get the results of controllability. Here we only give a rough discussion, detailed analysis remains for further study. 3.3. Optimal control problem
is−1 =1 t =s−1 j
Blkit (Lt )δ2n(τ +1)
In this subsection, we consider the optimal control problem of a temporal Boolean network, in other words, a control that maximizes (or minimizes) a given objective functional.
F. Li, J. Sun / Neural Networks 34 (2012) 10–17
15
Fix a vector r ∈ R , and consider the objective functional (Laschov & Margaliot, 2011),
From Theorem 3.2, we have, if (Ednτ W[2n ,2nτ ] Blkis−1 (Ls−1 ))i,k > 0, then x(s) = δ2i n can be reached from δ2kn(τ +1) at one step with
J (u) = r T x(N ; u),
u(s − 1) = δ2sm−1 . If (
2n
i
(9)
n
In this paper, based on the results of Theorem 3.2, the Mayertype optimal control is studied under free Boolean sequence control, which is arbitrary for a given m. j Case (a). Given an initial state δ2n(τ +1) , find a control sequence {u(0), . . . , u(s − 1)} that minimizes (or maximizes) the objective functional J (u) = r T x(t ; u) at t = s, where s > 1 is a fixed final time.
Step 1. Minimize (or maximize) the objective functional J (u) j under the constraints x(s) ∈ Rs,0 (δ2n(τ +1) ), and obtain the ∗ minimum (or maximum) value J and the corresponding x(s) = δ2i n . Step 2. If s = 1, find one i0 such that (Ednτ W[2n ,2nτ ] Blki0 (L0 ))i,j
> 0. Set u(0) = δ stop. Else, find one k, such that (Ednτ 0 2m W[2n ,2nτ ] Blkis−1 (Ls−1 ))i,k > 0, ( t =s−2 it =1 Blkit (Lt ))k,j i0 , 2m
Note that x(s − 1) = Ednτ W[2n ,2nτ ] δ2k(n+1)τ = δ2ns−1 , similarly, we can find two integers k′ and is−2 , such that (Ednτ W[2n ,2nτ ] Blkis−2
m k (Ls−2 ))ks−1 ,k′ > 0, ( 0t =s−3 2it =1 Blkit (Lt ))k′ ,j > 0. That is δ2ns−1 ′
δ
k′ 2n(τ +1)
δ
j
Case (b). Assume that the system is controllable at δ2n(τ +1) from time instant 0. We are interested in finding a control sequence that minimizes (or maximizes) the objective functional at shortest time t = s.
Step 1. Minimize (or maximize) the objective functional J (u) and obtain the minimum (maximum) value J ∗ and the corresponding x(s) = δ2i n . Step 2. Find the smallest s, such that
Ednτ W[2n ,2nτ ]
,δ
j
is−1 =1 t =s−1 j
Blkit (Lt )δ2n(τ +1)
t =s−1 it =1
= Colj
0 2m
Set u(0) = δ20m , stop. Else, find one k, such that (Ednτ W[2n ,2nτ ] i
Blkit (Lt ) . j
0 2m
t =s−1 it =1
2m
i t =1
Blkit (Lt ))k,j > 0. Set
and
Ednτ W[2n ,2nτ ] 2k(n+1)τ
δ
ks−1
, δ2n .
Step 4. If s − 1 = 1, stop. Else, set s = s − 1 and i = ks−1 (that is, replace s by s − 1 and replace i by ks−1 ) and go back to step 3. Remark 3.5. The control design in the algorithm is determined by judging whether some of the elements of the matrices are positive or not. So one of the advantages of these results is that the algorithm is easy to implement. But we note that, when time delays are large, the dimension of the system is high. So further research is need to reduce the complexity. 4. Examples
Then δ2i n(τ +1) is reachable from δ2n(τ +1) at the sth step if and only
x(s − 1) =
is−1 2m
t =s−2
t =s−1 it =1
if
i,j
Step 3. If s = 1, find one i0 such that (Ednτ W[2n ,2nτ ] Blki0 (L0 ))i,j > 0.
u( s − 1 ) = δ
Blkit (Lt )δ2n(τ +1)
> 0.
Blkit (Lt )
t =s−1 it =1
.
j
Blkis−1 (Ls−1 ))i,k > 0, (
From the proof of Theorem 3.2, we assume that z (0) = δ2n(τ +1) , thus
=
0 2m
0
ks−1 2n
Step 3. If s − 1 = 1, stop. Else, set s = s − 1 and i = ks−1 (that is, replace s by s − 1 and replace i by ks−1 ) and go back to step 2.
2m 0
j
can be reached from δ2n(τ +1) at s − 2 step. Continuing this process, the sequence of controls {u(0), . . . , u(s − 1)} can be obtained which can force the trajectory from x(0) to x(s).
i
Ednτ W[2n ,2nτ ] 2k(n+1)τ
i
can be reached from δ2kn(τ +1) at one step with u(s − 2) = δ2sm−2 , and
> 0. Set u(s − 1) = δ2sm−1 and x(s − 1) =
Blkit (Lt ))k,j > 0, then δ2kn(τ +1)
Algorithm 3.2.
Algorithm 3.1.
0 2m
i t =1
k
R2 , x(N ; u) ∈ ∆2n that J (u) = r T x(N ; u) = Coli (r T ). Hence, to minimize (or maximize) the cost functional (9) is equivalent to finding the minimum (or maximum) component of r T = [r1 , r2 , . . . , r2n ]T under the constraint x(s) ∈ Rs,0 (δ2j n(τ +1) ).
i0 =1 i1 =1
2m
can be reached from δ2n(τ +1) at the s − 1 step.
Remark 3.4. Assume that x(N ; u) = δ2i n . It can be seen from r ∈
···
t =s−2
j
where N is the final time.
2m 2m
0
Blkit (Lt )
> 0. i,j
Hence, we have the following proposition.
Example 4.1. Consider the following temporal Boolean network for t = 0, A(t + 1) = u(t ) ∧ (¬(A(t − 1) ∧ B(t − 1))) B(t + 1) = u(t ) ∨ (¬(A(t − 1) ∧ B(t − 1)))
for t > 0, A(t + 1) = u(t ) ↔ (¬(A(t − 2) ∧ B(t − 1))) B(t + 1) = u(t ) ∧ (¬(A(t − 1) ∧ B(t − 2))),
Proposition 3.1. The control sequence {u(0), . . . , u(s − 1)} obtained by Algorithm 3.1 can minimize (or maximize) the cost functional (9) at the fixed termination time t = s.
where u(t ) ∈ ∆.
Proof. We only have to prove that the control sequence {u(0), . . . , j u(s − 1)} can force the trajectory from x(0) = δ2n(τ +1) to x(s) = δ2i n .
By letting x(t ) = A(t )B(t ), z (t ) = x(t )x(t − 1)x(t − 2), for t = 0, we have
16
F. Li, J. Sun / Neural Networks 34 (2012) 10–17
x(t + 1) = A(t + 1)B(t + 1) = Mc (I2 ⊗ Mn Mc )(I8 ⊗ Md )(I16 ⊗ Mn Mc )
× Φ3 (I2 ⊗ Ed4 W[16,4] )u(t )x(t )x(t − 1)x(t − 2) , L˜ 0 u(t )x(t )x(t − 1)x(t − 2), z (t + 1) = x(t + 1)x(t )x(t − 1) = L0 (I32 ⊗ W[16,4] )(I2 ⊗ Φ4 )u(t )z (t ) , L0 u(t )z (t ),
Step 2. Find the smallest s = 2 such that
Ed4 W[4,16]
0 2
Blkit (Lt )
t =1 it =1
> 0. 2,4
Step 3. Find one k = 33, such that
(
Ed4 W[4,16] Blk2
(L1 ))2,33 > 0,
where
2
Blkit (L0 )
it =1
L0 = δ64 [33, 33, 33, 33, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 37, 37, 37, 37, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 41, 41, 41, 41,
33 Set u(1) = δ22 and x(1) = Ed4 W[4,16] δ64 = δ43 . Step 4. Find one i0 = 1, such that
10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 45, 45, 45,
(Ed4 W[4,16] Blk1 (L0 ))3,4 > 0.
45, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 49, 49,
Set u(0) = δ21 .
49, 49, 34, 34, 34, 34, 35, 35, 35, 35, 36, 36, 36, 36, 53,
> 0. 33,4
57, 57, 57, 57, 42, 42, 42, 42, 43, 43, 43, 43, 44, 44, 44,
Hence, we find the control sequence {u(0) = δ21 , u(1) = δ22 } 4 that drives δ64 to δ42 at the smallest time to minimize the objective functional.
44, 61, 61, 61, 61, 46, 46, 46, 46, 47, 47, 47, 47,
5. Conclusions
53, 53, 53, 38, 38, 38, 38, 39, 39, 39, 39, 40, 40, 40, 40,
48, 48, 48, 48]. For t > 0, we have x(t + 1) = A(t + 1)B(t + 1) = Me (I2 ⊗ Mn Mc )(I8 ⊗ Mc )(I16 ⊗ Mn Mc )
× (I2 ⊗ W[2,4] )Φ1 (I2 ⊗ Ed2 )(I8 ⊗ W[2,4] ) × (I16 ⊗ W[2] )u(t )x(t )x(t − 1)x(t − 2) , L˜ 1 u(t )x(t )x(t − 1)x(t − 2), z (t + 1) = x(t + 1)x(t )x(t − 1)
= L˜ 1 u(t )x(t )x(t − 1)x(t − 2)x(t )x(t − 1) = L˜ 1 (I32 ⊗ W[16,4] )(I2 ⊗ Φ4 )u(t )z (t ) , L1 u(t )z (t ),
This paper has studied the controllability and optimal control problem of a temporal Boolean network, where the time delays are time variant. First, the logical system has been converted into a discrete time variant system by using the theory of semi-tensor product of matrices. Second, this paper has provided necessary and sufficient conditions for the controllability via two types of controls respectively, where one kind of controls is input networks, the other kind of controls is free Boolean sequences. Third, optimal control design algorithms have been given. Finally, examples have been presented to illustrate the proposed results. There are many control related problems for Boolean control systems, such as infinite-horizon optimal control, filtering etc., which remain for further study.
where L1 = δ64 [49, 33, 17, 1, 18, 2, 18, 2, 35, 35, 3, 3, 4, 4, 4, 4, 53, 37, 21, 5, 22, 6, 22, 6, 39, 39, 7, 7, 8, 8, 8, 8, 57, 41, 25, 9, 26, 10, 26, 10, 43, 43, 11, 11, 12, 12, 12, 12, 61, 45, 29, 13, 30, 14, 30, 14, 47, 47, 15, 15, 16, 16, 16, 16, 12, 17, 49, 49, 50, 50, 50, 50, 19, 19, 51, 51, 52, 52, 52, 52, 21, 21, 53, 53, 54, 54, 54, 54, 23, 23, 55, 55, 56, 56, 56, 56, 25, 25, 57, 57, 58, 58, 58, 58, 27, 27, 59, 59, 60, 60, 60, 60, 29, 29, 61, 61, 62, 62, 62, 62, 31, 31, 63, 63, 64, 64, 64, 64]. 4 , through calculation, we have Assume that the initial state is δ64
Col4
Ed4 W[4,16]
3 0 2
Blkit (Lt )
> 0.
k=1 t =k−1 it =1 4 Theorem 3.2 implies that this system is controllable at δ64 from time instant 0.
Example 4.2. Consider the system in Example 4.1. Assume that the objective functional is J (u) = r T x(N , u), where r T = [4, 2, 5, 6]. We want to find the smallest time s to minimize this objective functional. From Algorithm 3.2, we have the following algorithm. Step 1. Minimize the objective functional J (u) and obtain the minimum value J ∗ and the corresponding x(s) = δ42 .
Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant 61174039, and by the Fundamental Research Funds for the Central Universities of China. References Cheng, D., & Qi, H. (2009). Controllability and observability of Boolean control networks. Automatica, 45, 1659–1667. Cheng, D., & Qi, H. (2010). A linear representation of dynamics of Boolean networks. IEEE Transactions on Automatic Control, 55, 2251–2258. Cheng, D., Qi, H., Li, Z., & Liu, J. (2011). Stability and stabilization of Boolean networks. International Journal of Robust and Nonlinear Control, 21, 134–156. Ching, W., Zhang, S., Ng, M., & Akutsu, T. (2007). An approximation method for solving the steady-state probability distribution of probabilistic Boolean networks. Bioinformatics, 23, 1511–1518. Drossel, B., Mihaljev, T., & Greil, F. (2005). Number and length of attractors in a critical Kauffman model with connectivity one. Physical Review Letters, 94, 88701. Farrow, C., Heidel, J., Maloney, J., & Rogers, J. (2004). Scalar equations for synchronous Boolean networks with biological applications. IEEE Transactions on Neural Networks, 15, 348–354. Goodwin, B. (1963). Temporal organization in cells. Academic Press. Kauffman, S. (1969). Metabolic stability and epigenesis in randomly constructed genetic nets. Journal of Theoretical Biology, 22, 437–467. Laschov, D., & Margaliot, M. (2011). A maximum principle for single-input Boolean control networks. IEEE Transactions on Automatic Control, 56, 913–917. Li, Z., & Cheng, D. (2010). Algebraic approach to dynamics of multi-valued networks. International Journal of Bifurcation and Chaos, 20, 561–582. Li, F., & Sun, J. (2011). Controllability of Boolean control networks with time delays in states. Automatica, 47, 603–607. Li, C., Sun, J., & Sun, R. (2010). Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects. Journal of the Franklin Institute, 347, 1186–1198. Li, F., Sun, J., & Wu, Q. (2011). Observability of Boolean control networks with state time delays. IEEE Transactions on Neural Networks, 22, 948–954. Martínez-Rodríguez, A., May, J., & Vargas, L. (2008). An optimization-based approach for the design of Bayesian networks. Mathematical and Computer Modelling, 48, 1265–1278.
F. Li, J. Sun / Neural Networks 34 (2012) 10–17 Shen, B., Wang, Z., Liang, J., & Liu, X. (2011). Sampled-data h∞ filtering for stochastic genetic regulatory networks. International Journal of Robust and Nonlinear Control, 21, 1759–1777. Shmulevich, I., Dougherty, E., Kim, S., & Zhang, W. (2002). Probabilistic Boolean networks: a rule-based uncertainty model for gene regulatory networks. Bioinformatics, 18, 261–274. Silvescu, A., & Honavar, V. (2001). Temporal Boolean network models of genetic networks and their inference from gene expression time series. Complex Systems, 13, 54–70. Vrabie, D., & Lewis, F. (2009). Neural network approach to continuous-time direct adaptive optimal control for partially unknown nonlinear systems. Neural Networks, 22, 237–246. Wang, Z., Lam, J., Wei, G., Fraser, K., & Liu, X. (2008). Filtering for nonlinear genetic regulatory networks with stochastic disturbances. IEEE Transactions on Automatic Control, 53, 2448–2457.
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Wang, Z., Liu, Y., & Liu, X. (2009b). State estimation for jumping recurrent neural networks with discrete and distributed delays. Neural Networks, 22, 41–48. Wang, Z., Liu, X., Liu, Y., Liang, J., & Vinciotti, V. (2009a). An extended Kalman filtering approach to modeling nonlinear dynamic gene regulatory networks via short gene expression time series. IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB), 6, 410–419. Xu, R., Venayagamoorthy, G., & Wunsch, D., II (2007). Modeling of gene regulatory networks with hybrid differential evolution and particle swarm optimization. Neural Networks, 20, 917–927. Zhao, Y., Li, Z., & Cheng, D. (2011). Optimal control of logical control networks. IEEE Transactions on Automatic Control, 56, 1766–1776. Zhao, Y., Qi, H., & Cheng, D. (2010). Input-state incidence matrix of Boolean control networks and its applications. Systems & Control Letters, 59, 767–774. Zhao, S., Sun, J., & Liu, L. (2008). Finite-time stability of linear time-varying singular systems with impulsive effects. International Journal of Control, 81, 1824–1829.
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Neural Networks journal homepage: www.elsevier.com/locate/neunet
Neural networks letter
Controllability and optimal control of a temporal Boolean network Fangfei Li a,b , Jitao Sun a,∗ a
Department of Mathematics, Tongji University, Shanghai, China
b
Department of Mathematics, East China University of Science and Technology, Shanghai, China
article
info
Article history: Received 28 November 2011 Received in revised form 21 May 2012 Accepted 12 June 2012 Keywords: Temporal Boolean networks Controllability Optimal control Semi-tensor product of matrices
abstract This paper investigates the controllability and optimal control of a temporal Boolean network, where the time delays are time variant. First, using the theory of semi-tensor product of matrices, the logical systems can be converted into a discrete time variant system. Second, necessary and sufficient conditions for the controllability via two types of controls are provided respectively. Third, optimal control design algorithms are presented. Finally, examples are given to illustrate the proposed results. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Genetic regulatory networks are widely used in the research of systems biology. These models include Bayesian networks (Martínez-Rodríguez, May, & Vargas, 2008), differential equations (Shen, Wang, Liang, & Liu, 2011; Wang, Lam, Wei, Fraser, & Liu, 2008; Wang, Liu, Liu, Liang, & Vinciotti, 2009a; Xu, Venayagamoorthy, & Wunsch, 2007) and Boolean networks (Drossel, Mihaljev, & Greil, 2005; Farrow, Heidel, Maloney, & Rogers, 2004; Kauffman, 1969) etc. A Boolean network consists of a group of Boolean variables: active (1) or inactive (0), and a set of Boolean functions. Interactions between the states of each Boolean variables are determined by Boolean functions, which calculate the state of a Boolean variable from the activation of other Boolean variables. Boolean networks have attracted much attention in recent years. Recently, the control problem of Boolean networks is a hot topic. Using the semi-tensor product of matrices, a Boolean network can be transformed into a conventional discrete time system. Based on this method, many basic problems in control theory, such as controllability, observability, stabilization and optimal control are studied, for example, see Cheng and Qi (2009), Cheng, Qi, Li, and Liu (2011) and Zhao, Li, and Cheng (2011); Zhao, Qi, and Cheng (2010). Moreover, some control problems for more general Boolean networks, such as k-valued logical networks and probabilistic Boolean networks are investigated too Ching, Zhang,
∗
Corresponding author. Tel.: +86 21 65983241x1307; fax: +86 21 65981985. E-mail address: [email protected] (J. Sun).
0893-6080/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.neunet.2012.06.002
Ng, and Akutsu (2007), Li and Cheng (2010) and Shmulevich, Dougherty, Kim, and Zhang (2002). As is well known, time delay phenomena are very common in the real world. Time delay behaviors happen frequently in biological and physiological systems. In many cases, the time delay phenomenon in control systems has become an important topic in control theory. There are many results concerning basic problems for time delay systems in control theory, (see e.g. Li, Sun, & Sun, 2010, Wang, Liu, & Liu, 2009b). For Boolean networks, the controllability and observability of Boolean networks with time delays was investigated in Li and Sun (2011) and Li, Sun, and Wu (2011). One kind of Boolean networks, called temporal Boolean networks are developed to model regulatory delays, for example, see Silvescu and Honavar (2001). A temporal Boolean network has more complex structure than the Boolean networks investigated, in Li and Sun (2011), because the temporal Boolean networks allow the time delays in each Boolean variable to be different. Though a temporal Boolean network can describe a rather general Boolean network with time delays, it does not consider the case when the time delays are time variant. In fact, a Boolean network with time variant delays, appears as a natural description of several real world problems. It can approximate a real cellular regulatory network better than a Boolean network with time invariant delays. A temporal Boolean network with time-variant time delays has rather complex dynamic characteristics, hence to investigate it is more challenging. Controllability and optimal control are structure properties of a system, and they are fundamental concepts in control theory. There have been many results studying the controllability or the optimal control problems of dynamics, (see e.g. Vrabie & Lewis, 2009). For Boolean networks, the controllability was studied in Cheng and Qi (2009). The Mayer-type optimal control problem
F. Li, J. Sun / Neural Networks 34 (2012) 10–17
of Boolean networks was considered in Laschov and Margaliot (2011), and a maximum principle was put forward for the systems with single control input. The infinite horizon optimal control problem of k-valued logical control networks was investigated in Zhao et al. (2011). But to the best of our knowledge, there has been no result studying the controllability and optimal control of temporal Boolean networks with time-variant delays. Temporal Boolean networks with time-variant delays can describe more general time delay phenomena for Boolean networks, studying them is meaningful and challenging. Motivated by the above, in this paper we consider the controllability and optimal control problem of temporal Boolean networks with time variant delays. Based on the theory of semi-tensor product of matrices (also called Cheng product), we show how to convert temporal Boolean networks with time variant delays into discrete time variant systems. The controllability via two types of controls is considered. One kind of control is input networks. Another kind of control is free Boolean sequences. Necessary and sufficient conditions are provided for each case. Furthermore, based on the results we obtained, optimal control design algorithms are presented. In this paper, we provide a control design algorithm that minimizes (or maximizes) the objective functional at a fixed termination time. Meanwhile, we assume the system we studied is controllable at the initial state from an initial time instant and find a control sequence that minimizes (or maximizes) the objective functional in the shortest time. The system in this paper is time variant, which is rather complicated. The results in our paper are not trivial extensions of existing results (Cheng & Qi, 2009; Zhao et al., 2010). The rest of this paper is organized as follows: Section 2 introduces some preliminaries about the semi-tensor product of matrices (Cheng product). In Section 3, we first convert a temporal Boolean network with time variant delays into discrete time variant systems. Then necessary and sufficient conditions about the controllability via two types of controls are obtained respectively. Finally, we give the optimal control design algorithm based on the results obtained in Section 3.2. Section 4 provides examples to illustrate the proposed results. Finally, concluding remarks are provided in Section 5. 2. Preliminaries In this section, we introduce the semi-tensor product of matrices (also called Cheng product) and the vector form of Boolean variables, which are recapitulations of results mainly from Cheng and Qi (2010). 2.1. Semi-tensor product of matrices In this paper, the matrix product we use is the Cheng product. Definition 2.1. 1. Cheng and Qi (2010). 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-size blocks as X 1 , . . . , X p , which are 1 × n rows. Define the semi-tensor product (STP), denoted by n, as
p X nY = X i y i ∈ Rn , i =1
p T T y i ( X i ) T ∈ Rn . Y n X = i =1
2. Let A ∈ Mm×n and B ∈ Mp×q . If either n is a factor of p, say nt = p and denote it as A ≺t B, or p is a factor of n, say n = pt and denote it as A ≻t B, then we define the STP of A and B, denoted
11
by C = A n B, as the following: C consists of m × q blocks as C = (C ij ) and each block is C ij = Ai n Bj ,
i = 1, . . . , m, j = 1, . . . , q,
where Ai is the ith row of A and Bj is the jth column of B. The Cheng product is a generalization of the conventional matrix product. All the fundamental properties of the conventional matrix product remain true. Based on this, we can omit the symbol n. There are also some basic properties of the Cheng product, for details see Cheng and Qi (2010). 2.2. Vector form of Boolean variables In this subsection, we introduce the vector form of Boolean variables and some concerning results. First, let ‘‘1’’ and ‘‘0’’ represent the logical ‘‘True’’ and ‘‘False’’ respectively, and D := {1, 0}. We define a delta set as ∆k := {δki | i = 1, 2, . . . , k}, where δki is the ith column of the identity i
i
matrix Ik . Assume there is a matrix M = [δn1 , δn2 , . . . , δnis ], we simply denote it as M = δn [i1 , i2 , . . . , is ]. We give Boolean variables a vector form as: T = 1 ∼ δ21 ,
F = 0 ∼ δ22 ,
then the logical variable A(t ) takes values from these two vectors, i.e. A(t ) ∈ ∆ := ∆2 = {δ21 , δ22 }. The ‘‘D ’’ and ‘‘∆’’ will be used freely according to the variable types. Second, we give some notations. (1) Denote by Mm×n the set of all m × n matrices. (2) Denote by Col(A) the set of columns of a matrix A, and by Coli (A) the ith column of A. (3) Let a matrix A ∈ Mn×mn and denote by Blki (A) the ith n × n square block of A, i = 1, 2, . . . , m. (4) Denote by Ai,j the ith element of the jth column of a matrix A. Third, we define the logical matrix and the set of logical matrices. A matrix A ∈ Mm×n is called a logical matrix, if the columns of A are elements of ∆m . Denote the set of logical matrices by L, the set of n × s logical matrices is defined by Ln×s . Next, we will introduce the swap matrix and the dummy matrix. A swap matrix, W[m,n] , is an mn × mn matrix 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
w(IJ ),(ij) = δiI,,jJ =
1, 0,
I = i and J = j, otherwise.
When m = n, we briefly denote W[n] := W[n,n] . To see the properties of the swap matrix, refer to Cheng and Qi (2010). The dummy matrix is defined as Ed := δ2 [1, 2, 1, 2]. From Cheng and Qi (2010), we have for any two logical variables u, v , Ed uv = v,
or Ed W[2] uv = u.
Finally, we give a lemma that will be used in the following paper. According to Cheng and Qi (2010), for each logical function L(A1 , . . . , An ), there exists a structure matrix ML such that: Lemma 2.1. Any logical function L(A1 , . . . , An ) with logical arguments A1 , . . . , An ∈ ∆ can be expressed in a multi-linear form as L(A1 , . . . , An ) = ML A1 A2 · · · An , where ML ∈ L2×2n is unique, called the structure matrix of L.
12
F. Li, J. Sun / Neural Networks 34 (2012) 10–17
3. Main results
In the following, we consider the temporal Boolean network with controls as follows
3.1. Converting a temporal Boolean network with time variant delays into a discrete time variant system
A1 (t + 1) = f1t (u1 (t ), . . . , um (t ), A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ (t )), . . . , An (t − τ (t ))) A2 (t + 1) = f2t (u1 (t ), . . . , um (t ), A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ (t )), . . . , An (t − τ (t ))) .. . t An (t + 1) = fn (u1 (t ), . . . , um (t ), A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ (t )), . . . , An (t − τ (t ))),
Recall that a Boolean network can be described as
A1 (t + 1) = f1 (A1 (t ), A2 (t ), . . . , An (t )), A2 (t + 1) = f2 (A1 (t ), A2 (t ), . . . , An (t )), .. . An (t + 1) = fn (A1 (t ), A2 (t ), . . . , An (t )),
(1)
where fi are logical functions, t = 0, 1, 2, . . . . But in the real world, time delay phenomena are very common, Silvescu and Honavar (2001) introduced a temporal Boolean network as follows
A (t + 1) = f (A (t ), . . . , A (t ), A (t − 1), . . . , 1 1 1 n 1 An (t − 1), . . . , A1 (t − τ ), . . . , An (t − τ )) A2 (t + 1) = f2 (A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ ), . . . , An (t − τ )) .. . An (t + 1) = fn (A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ ), . . . , An (t − τ )).
(2)
(3)
We give an example to illustrate this kind of system. This example is a special case of temporal Boolean networks. It is a biochemical network of coupled oscillations in the cell cycle (Goodwin, 1963).
A(t + 3) = ¬(A(t ) ∧ B(t + 1)), B(t + 3) = ¬(A(t + 1) ∧ B(t )).
It can be converted into the form
A(t + 1) = ¬(A(t − 2) ∧ B(t − 1)), B(t + 1) = ¬(A(t − 1) ∧ B(t − 2)),
where time delay is invariant.
t ≥ 2.
i = 1, 2, . . . .
(5)
= M1t
n
[(I2m+n(τ (t )+1) ⊗ Mit )Φm+n(τ (t )+1) ]
i=2
u(t )x(t )x(t − 1) · · · x(t − τ (t )) , L˜ t u(t )x(t )x(t − 1) · · · x(t − τ (t )).
Remark 3.1. The time delays in system (3) are time variant. It is a rather complicated system. Moreover, we note that time variant dynamics draw much attention in control theories, (see e.g. Zhao, Sun, & Liu, 2008), similarly, we consider a time variant temporal Boolean network with time variant delays.
Example 3.1. Consider the following Boolean network
Ai (t + 1) = Mit u(t )x(t )x(t − 1) · · · x(t − τ (t )),
x(t + 1) = A1 (t + 1)A2 (t + 1) · · · An (t + 1)
where fit means that the logical functions fi are time variant and τ (t ) are time-variant time delays which are positive integers, and there exists the smallest integer τ satisfying maxt ≥0 {τ (t )} ≤ τ < +∞, t = 0, 1, . . . .
where ui are controls (or inputs), ui ∈ ∆, i = 1, . . . , m. In order to convert (4) into algebraic form, we define x(t ) = m nni=1 Ai (t ) ∈ ∆2n , u(t ) = nm i=1 ui (t ) ∈ ∆2 . Assume that the structure matrix of fit is Mit ∈ L2×2m+n(τ (t )+1) , we can express (4) as
Multiplying the equations (5) together, yields
Note that in many cases, time delays are not time invariant but time variant. Motivated by this, in this paper, we consider a more general temporal Boolean network
A (t + 1) = f t (A (t ), . . . , A (t ), A (t − 1), . . . , 1 1 n 1 1 An (t − 1), . . . , A1 (t − τ (t )), . . . , An (t − τ (t ))) t A2 (t + 1) = f2 (A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ (t )), . . . , An (t − τ (t ))) .. . An (t + 1) = fnt (A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ (t )), . . . , An (t − τ (t ))),
(4)
Let z (t ) = nτi=0 zi (t ), where z0 (t ) = x(t ), z1 (t ) = x(t − 1), . . . , zτ (t ) (t ) = x(t − τ (t )), we have x(t + 1) = L˜ t u(t )x(t )x(t − 1) · · · x(t − τ (t ))
= L˜ t u(t )Edn(τ −τ (t )) W[2n(τ (t )+1) ,2n(τ −τ (t )) ] z (t ) = L˜ t (I2m ⊗ Edn(τ −τ (t )) W[2n(τ (t )+1) ,2n(τ −τ (t )) ] )u(t )z (t ) , L¯ t u(t )z (t ).
z (t + 1) = nτi=0 zi (t + 1) = x(t + 1)x(t ) · · · x(t − τ + 1)
= =
L¯ t u(t )z (t )x(t ) · · · x(t − τ + 1)
L¯ t u(t )W[2nτ ,2n(τ +1) ]
(x(t ) · · · x(t − τ + 1))2 x(t − τ ) = L¯ t u(t )W[2nτ ,2n(τ +1) ] Φnτ z (t ) = L¯ t (I2m ⊗ W[2nτ ,2n(τ +1) ] Φnτ )u(t )z (t ) , Lt u(t )z (t ). 3.2. Controllability of temporal Boolean network In this subsection, we will consider the controllability of the temporal Boolean network (4). Without loss of generality, in this paper, we assume that the j initial state of system (4) is nτi=0 x(−i) = δ2n(τ +1) . Moreover, let X = (A1 , . . . , An )T .
Definition 3.1. Consider the temporal Boolean network (4). Given its initial state sequence x(0) ∈ ∆2n ∼ X (0) ∈ Dn , . . . , x(−τ ) ∈ ∆2n ∼ X (−τ ) ∈ Dn , and destination state xd ∈ ∆2n ∼ Xd ∈ Dn , xd is said to be reachable from the initial state sequence nτi=0 x(−i) from time instant 0 at the sth step, if we can find control sequence {u(0), . . . , u(s − 1)}, such that the initial state sequence nτi=0 x(−i) can be driven to the destination state xd = x(u, s).
F. Li, J. Sun / Neural Networks 34 (2012) 10–17
Definition 3.2. (1) Consider the temporal Boolean network (4). The reachable set from time instant 0 at time s is the set of states which are reachable from the initial state sequence nτi=0 x(−i) from time instant 0 at time s, denoted by Rs,0 (nτi=0 x(−i)). (2) The overall reachable set from time instant 0 is the set of states which are reachable from the initial state sequence nτi=0 x(−i) from time instant 0, denoted by R0 (nτi=0 x(−i)). Definition 3.3. (1) System (4) is said to be controllable at initial state sequence nτi=0 x(−i) from time instant 0, if R0 (nτi=0 x(−i)) = ∆2n . (2) System (4) is said to be controllable from time instant 0, if for any initial state sequence nτi=0 x(−i), we have R0 (nτi=0 x(−i)) = ∆2n . We consider the controllability of temporal Boolean network (4) with two kinds of controls. (I) The controls are logical variables satisfying certain logical rules, called the input networks as:
Proof. (i) Assume that u(0) = δ z (1) = L0 u(0)z (0) =
(6)
z (t + 1) = Lt u(t )z (t ), u(t + 1) = Gu(t ).
(7)
Theorem 3.1. Consider system (4) with input network control (6), equivalently (7), we have
nτi=0
j
(i) xd = δ is reachable from x(−i) = δ2n(τ +1) from time instant 0 at the sth step, if and only if
t =1
z (3) = L2 u(2)z (2) = L2 G2 u(0)z (2)
= Blki0 (L2 G2 )
Blki0 (Lt Gt )
=
j
x(−i) = δ2n(τ +1) from time (ii) xd = δ is reachable from instant 0, if and only if there exists positive integer N, such that 2m 0
Blki0 (Lt G )
> 0.
t
i0 =1 t =N −1
t =2
z (s) =
j
Ednτ W[2n ,2nτ ]
N
2m
0
Blki0 (Lt Gt )
k=1 i0 =1 t =k−1
> 0. i,j
(iv) System (7) is controllable from time instant 0, if and only if there exists positive integer N, such that all the entries of Ednτ W[2n ,2nτ ]
0
Blki0 (Lt Gt )z (0).
x(s) = z0 (s) = Ednτ W[2n ,2nτ ] z (s) 0
= Ednτ W[2n ,2nτ ]
N 2m 0 k=1 i0 =1 t =k−1
0
j
Blki0 (Lt Gt )δ2n(τ +1)
t =s−1
= Colj
0
Ednτ W[2n ,2nτ ]
Blki0 (Lt G ) . t
t =s−1
From the above equality, we can get the sum of all the elements of j Rs,0 (δ2n(τ +1) ) as 2m
0
Ednτ W[2n ,2nτ ]
i 0 =1
j
Blki0 (Lt Gt )δ2n(τ +1)
t =s −1
= Ednτ W[2n ,2nτ ]
2m 0
j
Blki0 (Lt Gt )δ2n(τ +1)
i0 =1 t =s−1
Ednτ W[2n ,2nτ ]
2m 0
Blki0 (Lt G ) . t
This equality implies that xd = δ2i n is reachable from nτi=0 x(−i) =
δ2j n(τ +1) from time instant 0 at the sth step, if and only if 2m 0 nτ t n n τ Ed W[2 ,2 ] Blki0 (Lt G ) > 0. i,j
(ii)–(iv). Similar to the discussion of (i), the conclusions follow. (II) The control is free Boolean sequences. Theorem 3.2. Consider system (4) with free Boolean sequences control, we have (i) xd = δ2i n is reachable from nτi=0 x(−i) = δ2n(τ +1) from time instant 0 at the sth step, if and only if j
Blki0 (Lt Gt ) > 0.
Blki0 (Lt Gt )z (0)
t =s−1
i0 =1 t =s−1
i,j
(iii) System (7) is controllable at nτi=0 x(−i) = δ2n(τ +1) from time instant 0, if and only if there exists positive integer N, such that all the entries of Colj
Blki0 (Lt Gt )z (0),
i0 =1 t =s−1
i ,j
nτi=0
Ednτ W[2n ,2nτ ]
Blki0 (Lt Gt )z (0)
.. .
> 0.
i0 =1 t =s−1
0 t =1
0
= Colj
i 2n
Blki0 (Lt Gt )z (0),
= Ednτ W[2n ,2nτ ]
2m 0
(0) = Blki0 (L0 )z (0),
z (s) = nτi=0 zi (s), yields
where G is the transition matrix of (6). (II) The control is free Boolean sequences. Precisely, set u(t ) = nm j=1 uj (t ). Then the control is a designed sequence. We consider case (I) first. Using the structure matrix approach to the temporal Boolean network, the algebraic form of (4), (6) can be obtained as
Ednτ W[2n ,2nτ ]
0
=
u(t + 1) = Gu(t ),
δ
through calculation, we have
z (2) = L1 u(1)z (1) = L1 Gu(0)z (1) = Blki0 (L1 G)Blki0 (L0 )z (0)
where gi , i = 1, 2, . . . , m are logical functions. We can express (6) as
i 2n
i L0 20m z
t =s−1
u1 (t + 1) = g1 (u1 (t ), u2 (t ), . . . , um (t )), u2 (t + 1) = g2 (u1 (t ), u2 (t ), . . . , um (t )), .. . um (t + 1) = gm (u1 (t ), u2 (t ), . . . , um (t )),
13 i0 , 2m
Ednτ W[2n ,2nτ ]
0 2m t =s−1 it =1
Blkit (Lt )
> 0. i,j
14
F. Li, J. Sun / Neural Networks 34 (2012) 10–17
(ii) xd = δ2i n is reachable from nτi=0 x(−i) = δ2n(τ +1) from time instant 0, if and only if there exists positive integer N, such that j
0 2m
Ednτ W[2n ,2nτ ]
Blkit (Lt ) i ,j
j
Ednτ W[2n ,2nτ ]
2m
N 0
Blkit (Lt )
> 0.
(iv) System (4) is controllable from time instant 0, if and only if there exists positive integer N, such that all the entries of N 0 2m
Proof. Assume that u(t ) = δ2tm , through calculation, we have i
i
z (1) = L0 u(0)z (0) = L0 δ20m z (0) = Blki0 (L0 )z (0), i
z (2) = L1 u(1)z (1) = L1 δ21m Blki0 (L0 )z (0) = Blki1 (L1 )Blki0 (L0 )z (0)
=
Blkit (Lt )z (0),
t =1 i
z (3) = L2 u(2)z (2) = L2 δ22m z (2)
= Blki2 (L2 )
0
Blkit (Lt )z (0)
t =1
=
0 t =2
z (s) =
Blkit (Lt )z (0).
Remark 3.2. As the number of elements of Lt is finite, the transition matrix Lt of (3) has finite modes. Hence, the upper bound of N is 2n(τ +1) + 1.
A (t + 1) = f t (u (t ), . . . , u (t ), u (t − 1), . . . , 1 1 m 1 um (t − 1), .1. . , u1 (t − τ (t )), . . . , um (t − τ (t )), A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ (t )), . . . , An (t − τ (t ))), A2 (t + 1) = f2t (u1 (t ), . . . , um (t ), u1 (t − 1), . . . , um (t − 1), . . . , u1 (t − τ (t )), . . . , um (t − τ (t )), A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ (t )), . . . , An (t − τ (t ))), .. . An (t + 1) = fnt (u1 (t ), . . . , um (t ), u1 (t − 1), . . . , um (t − 1), . . . , u1 (t − τ (t )), . . . , um (t − τ (t )), A1 (t ), . . . , An (t ), A1 (t − 1), . . . , An (t − 1), . . . , A1 (t − τ (t )), . . . , An (t − τ (t ))).
(8)
x(t + 1) = L˜ ′t u(t )u(t − 1) · · · u(t − τ (t ))x(t ) x(t − 1) · · · x(t − τ (t ))
x(s) = z0 (s) = Ednτ W[2n ,2nτ ] z (s)
= Ednτ W[2n ,2nτ ]
0
= L˜ ′t Edm(τ −τ (t )) W[2m(τ (t )+1) ,2m(τ −τ (t )) ] v(t )Edn(τ −τ (t )) W[2n(τ (t )+1) ,2n(τ −τ (t )) ] z (t ) = L˜ ′t Edm(τ −τ (t )) W[2m(τ (t )+1) ,2m(τ −τ (t )) ] (I2m(τ +1) ⊗ Edn(τ −τ (t )) W[2n(τ (t )+1) ,2n(τ −τ (t )) ] )v(t )z (t ) , L¯ ′t u(t )z (t ).
Blkit (Lt )z (0)
t =s−1
= Ednτ W[2n ,2nτ ]
0
j
Blkit (Lt )δ2n(τ +1)
z (t + 1) = nτi=0 zi (t + 1) = x(t + 1)x(t ) · · · x(t − τ + 1) = L¯ ′t v(t )z (t )x(t ) · · · x(t − τ + 1)
t =s−1
0
= Colj Ednτ W[2n ,2nτ ]
Blkit (Lt ) .
t =s−1
From the above equality, we can get the sum of all the elements of j Rs,0 (δ2n(τ +1) ) as
i0 =1 i1 =1
Let z (t ) = nτi=0 zi (t ), where z0 (t ) = x(t ), z1 (t ) = x(t − 1), . . . , zτ (t ) (t ) = x(t −τ (t )) and v(t ) = nτi=0 Ui (t ), where U0 (t ) = u(t ), U1 (t ) = u(t − 1), . . . , Uτ (t ) (t ) = u(t − τ (t )), we have
z (s) = nτi=0 zi (s), yields
···
i,j
x(t + 1) = L˜ ′t u(t )u(t − 1) · · · u(t − τ (t ))x(t )x(t − 1) · · · x(t − τ (t )).
t =s−1
2m 2m
t =s−1 it =1
Then, by defining x(t ) = nni=1 Ai (t ) ∈ ∆2n , u(t ) = nm i=1 ui (t ) ∈ ∆2m , we can convert (8) into
Blkit (Lt )z (0),
.. . 0
δ2j n(τ +1) from time instant 0 at the sth step, if and only if 0 2m nτ Blkit (Lt ) > 0. Ed W[2n ,2nτ ]
Remark 3.3. In the real world, the control always has time delays. The system describing such a case is as follows:
Blkit (Lt ) > 0.
k=1 t =k−1 it =1
0
Blkit (Lt ) .
(ii)–(iv). Similar to the discussion of (i), the conclusions follow.
k=1 t =k−1 it =1
Ednτ W[2n ,2nτ ]
This equality implies that xd = δ2i n is reachable from nτi=0 x(−i) =
> 0.
(iii) System (4) is controllable at nτi=0 x(−i) = δ2n(τ +1) from time instant 0, if and only if there exists positive integer N, such that all the entries of Colj
= Colj Ednτ W[2n ,2nτ ]
0 2m t =s −1 i t =1
t =N −1 i t =1
2m
Ednτ W[2n ,2nτ ]
is−1 =1
= Ednτ W[2n ,2nτ ]
j
Blkit (Lt )δ2n(τ +1)
t =s−1 2m
2m
···
i0 =1 i1 =1
= Ednτ W[2n ,2nτ ]
0
0 2m t =s−1 it =1
2m 0
j
Blkit (Lt )δ2n(τ +1)
= = =
L¯ ′t v(t )W[2nτ ,2n(τ +1) ] (x(t ) · · · x(t − τ + 1))2 x(t − τ )
L¯ ′t v(t )W[2nτ ,2n(τ +1) ] Φnτ z (t )
L¯ ′t (I2m(τ +1) ⊗ W[2nτ ,2n(τ +1) ] Φnτ )v(t )z (t ) , L′t v(t )z (t ).
Similar to the proof of Theorems 3.1 and 3.2, we can get the results of controllability. Here we only give a rough discussion, detailed analysis remains for further study. 3.3. Optimal control problem
is−1 =1 t =s−1 j
Blkit (Lt )δ2n(τ +1)
In this subsection, we consider the optimal control problem of a temporal Boolean network, in other words, a control that maximizes (or minimizes) a given objective functional.
F. Li, J. Sun / Neural Networks 34 (2012) 10–17
15
Fix a vector r ∈ R , and consider the objective functional (Laschov & Margaliot, 2011),
From Theorem 3.2, we have, if (Ednτ W[2n ,2nτ ] Blkis−1 (Ls−1 ))i,k > 0, then x(s) = δ2i n can be reached from δ2kn(τ +1) at one step with
J (u) = r T x(N ; u),
u(s − 1) = δ2sm−1 . If (
2n
i
(9)
n
In this paper, based on the results of Theorem 3.2, the Mayertype optimal control is studied under free Boolean sequence control, which is arbitrary for a given m. j Case (a). Given an initial state δ2n(τ +1) , find a control sequence {u(0), . . . , u(s − 1)} that minimizes (or maximizes) the objective functional J (u) = r T x(t ; u) at t = s, where s > 1 is a fixed final time.
Step 1. Minimize (or maximize) the objective functional J (u) j under the constraints x(s) ∈ Rs,0 (δ2n(τ +1) ), and obtain the ∗ minimum (or maximum) value J and the corresponding x(s) = δ2i n . Step 2. If s = 1, find one i0 such that (Ednτ W[2n ,2nτ ] Blki0 (L0 ))i,j
> 0. Set u(0) = δ stop. Else, find one k, such that (Ednτ 0 2m W[2n ,2nτ ] Blkis−1 (Ls−1 ))i,k > 0, ( t =s−2 it =1 Blkit (Lt ))k,j i0 , 2m
Note that x(s − 1) = Ednτ W[2n ,2nτ ] δ2k(n+1)τ = δ2ns−1 , similarly, we can find two integers k′ and is−2 , such that (Ednτ W[2n ,2nτ ] Blkis−2
m k (Ls−2 ))ks−1 ,k′ > 0, ( 0t =s−3 2it =1 Blkit (Lt ))k′ ,j > 0. That is δ2ns−1 ′
δ
k′ 2n(τ +1)
δ
j
Case (b). Assume that the system is controllable at δ2n(τ +1) from time instant 0. We are interested in finding a control sequence that minimizes (or maximizes) the objective functional at shortest time t = s.
Step 1. Minimize (or maximize) the objective functional J (u) and obtain the minimum (maximum) value J ∗ and the corresponding x(s) = δ2i n . Step 2. Find the smallest s, such that
Ednτ W[2n ,2nτ ]
,δ
j
is−1 =1 t =s−1 j
Blkit (Lt )δ2n(τ +1)
t =s−1 it =1
= Colj
0 2m
Set u(0) = δ20m , stop. Else, find one k, such that (Ednτ W[2n ,2nτ ] i
Blkit (Lt ) . j
0 2m
t =s−1 it =1
2m
i t =1
Blkit (Lt ))k,j > 0. Set
and
Ednτ W[2n ,2nτ ] 2k(n+1)τ
δ
ks−1
, δ2n .
Step 4. If s − 1 = 1, stop. Else, set s = s − 1 and i = ks−1 (that is, replace s by s − 1 and replace i by ks−1 ) and go back to step 3. Remark 3.5. The control design in the algorithm is determined by judging whether some of the elements of the matrices are positive or not. So one of the advantages of these results is that the algorithm is easy to implement. But we note that, when time delays are large, the dimension of the system is high. So further research is need to reduce the complexity. 4. Examples
Then δ2i n(τ +1) is reachable from δ2n(τ +1) at the sth step if and only
x(s − 1) =
is−1 2m
t =s−2
t =s−1 it =1
if
i,j
Step 3. If s = 1, find one i0 such that (Ednτ W[2n ,2nτ ] Blki0 (L0 ))i,j > 0.
u( s − 1 ) = δ
Blkit (Lt )δ2n(τ +1)
> 0.
Blkit (Lt )
t =s−1 it =1
.
j
Blkis−1 (Ls−1 ))i,k > 0, (
From the proof of Theorem 3.2, we assume that z (0) = δ2n(τ +1) , thus
=
0 2m
0
ks−1 2n
Step 3. If s − 1 = 1, stop. Else, set s = s − 1 and i = ks−1 (that is, replace s by s − 1 and replace i by ks−1 ) and go back to step 2.
2m 0
j
can be reached from δ2n(τ +1) at s − 2 step. Continuing this process, the sequence of controls {u(0), . . . , u(s − 1)} can be obtained which can force the trajectory from x(0) to x(s).
i
Ednτ W[2n ,2nτ ] 2k(n+1)τ
i
can be reached from δ2kn(τ +1) at one step with u(s − 2) = δ2sm−2 , and
> 0. Set u(s − 1) = δ2sm−1 and x(s − 1) =
Blkit (Lt ))k,j > 0, then δ2kn(τ +1)
Algorithm 3.2.
Algorithm 3.1.
0 2m
i t =1
k
R2 , x(N ; u) ∈ ∆2n that J (u) = r T x(N ; u) = Coli (r T ). Hence, to minimize (or maximize) the cost functional (9) is equivalent to finding the minimum (or maximum) component of r T = [r1 , r2 , . . . , r2n ]T under the constraint x(s) ∈ Rs,0 (δ2j n(τ +1) ).
i0 =1 i1 =1
2m
can be reached from δ2n(τ +1) at the s − 1 step.
Remark 3.4. Assume that x(N ; u) = δ2i n . It can be seen from r ∈
···
t =s−2
j
where N is the final time.
2m 2m
0
Blkit (Lt )
> 0. i,j
Hence, we have the following proposition.
Example 4.1. Consider the following temporal Boolean network for t = 0, A(t + 1) = u(t ) ∧ (¬(A(t − 1) ∧ B(t − 1))) B(t + 1) = u(t ) ∨ (¬(A(t − 1) ∧ B(t − 1)))
for t > 0, A(t + 1) = u(t ) ↔ (¬(A(t − 2) ∧ B(t − 1))) B(t + 1) = u(t ) ∧ (¬(A(t − 1) ∧ B(t − 2))),
Proposition 3.1. The control sequence {u(0), . . . , u(s − 1)} obtained by Algorithm 3.1 can minimize (or maximize) the cost functional (9) at the fixed termination time t = s.
where u(t ) ∈ ∆.
Proof. We only have to prove that the control sequence {u(0), . . . , j u(s − 1)} can force the trajectory from x(0) = δ2n(τ +1) to x(s) = δ2i n .
By letting x(t ) = A(t )B(t ), z (t ) = x(t )x(t − 1)x(t − 2), for t = 0, we have
16
F. Li, J. Sun / Neural Networks 34 (2012) 10–17
x(t + 1) = A(t + 1)B(t + 1) = Mc (I2 ⊗ Mn Mc )(I8 ⊗ Md )(I16 ⊗ Mn Mc )
× Φ3 (I2 ⊗ Ed4 W[16,4] )u(t )x(t )x(t − 1)x(t − 2) , L˜ 0 u(t )x(t )x(t − 1)x(t − 2), z (t + 1) = x(t + 1)x(t )x(t − 1) = L0 (I32 ⊗ W[16,4] )(I2 ⊗ Φ4 )u(t )z (t ) , L0 u(t )z (t ),
Step 2. Find the smallest s = 2 such that
Ed4 W[4,16]
0 2
Blkit (Lt )
t =1 it =1
> 0. 2,4
Step 3. Find one k = 33, such that
(
Ed4 W[4,16] Blk2
(L1 ))2,33 > 0,
where
2
Blkit (L0 )
it =1
L0 = δ64 [33, 33, 33, 33, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 37, 37, 37, 37, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 41, 41, 41, 41,
33 Set u(1) = δ22 and x(1) = Ed4 W[4,16] δ64 = δ43 . Step 4. Find one i0 = 1, such that
10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 45, 45, 45,
(Ed4 W[4,16] Blk1 (L0 ))3,4 > 0.
45, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 49, 49,
Set u(0) = δ21 .
49, 49, 34, 34, 34, 34, 35, 35, 35, 35, 36, 36, 36, 36, 53,
> 0. 33,4
57, 57, 57, 57, 42, 42, 42, 42, 43, 43, 43, 43, 44, 44, 44,
Hence, we find the control sequence {u(0) = δ21 , u(1) = δ22 } 4 that drives δ64 to δ42 at the smallest time to minimize the objective functional.
44, 61, 61, 61, 61, 46, 46, 46, 46, 47, 47, 47, 47,
5. Conclusions
53, 53, 53, 38, 38, 38, 38, 39, 39, 39, 39, 40, 40, 40, 40,
48, 48, 48, 48]. For t > 0, we have x(t + 1) = A(t + 1)B(t + 1) = Me (I2 ⊗ Mn Mc )(I8 ⊗ Mc )(I16 ⊗ Mn Mc )
× (I2 ⊗ W[2,4] )Φ1 (I2 ⊗ Ed2 )(I8 ⊗ W[2,4] ) × (I16 ⊗ W[2] )u(t )x(t )x(t − 1)x(t − 2) , L˜ 1 u(t )x(t )x(t − 1)x(t − 2), z (t + 1) = x(t + 1)x(t )x(t − 1)
= L˜ 1 u(t )x(t )x(t − 1)x(t − 2)x(t )x(t − 1) = L˜ 1 (I32 ⊗ W[16,4] )(I2 ⊗ Φ4 )u(t )z (t ) , L1 u(t )z (t ),
This paper has studied the controllability and optimal control problem of a temporal Boolean network, where the time delays are time variant. First, the logical system has been converted into a discrete time variant system by using the theory of semi-tensor product of matrices. Second, this paper has provided necessary and sufficient conditions for the controllability via two types of controls respectively, where one kind of controls is input networks, the other kind of controls is free Boolean sequences. Third, optimal control design algorithms have been given. Finally, examples have been presented to illustrate the proposed results. There are many control related problems for Boolean control systems, such as infinite-horizon optimal control, filtering etc., which remain for further study.
where L1 = δ64 [49, 33, 17, 1, 18, 2, 18, 2, 35, 35, 3, 3, 4, 4, 4, 4, 53, 37, 21, 5, 22, 6, 22, 6, 39, 39, 7, 7, 8, 8, 8, 8, 57, 41, 25, 9, 26, 10, 26, 10, 43, 43, 11, 11, 12, 12, 12, 12, 61, 45, 29, 13, 30, 14, 30, 14, 47, 47, 15, 15, 16, 16, 16, 16, 12, 17, 49, 49, 50, 50, 50, 50, 19, 19, 51, 51, 52, 52, 52, 52, 21, 21, 53, 53, 54, 54, 54, 54, 23, 23, 55, 55, 56, 56, 56, 56, 25, 25, 57, 57, 58, 58, 58, 58, 27, 27, 59, 59, 60, 60, 60, 60, 29, 29, 61, 61, 62, 62, 62, 62, 31, 31, 63, 63, 64, 64, 64, 64]. 4 , through calculation, we have Assume that the initial state is δ64
Col4
Ed4 W[4,16]
3 0 2
Blkit (Lt )
> 0.
k=1 t =k−1 it =1 4 Theorem 3.2 implies that this system is controllable at δ64 from time instant 0.
Example 4.2. Consider the system in Example 4.1. Assume that the objective functional is J (u) = r T x(N , u), where r T = [4, 2, 5, 6]. We want to find the smallest time s to minimize this objective functional. From Algorithm 3.2, we have the following algorithm. Step 1. Minimize the objective functional J (u) and obtain the minimum value J ∗ and the corresponding x(s) = δ42 .
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