A minimum-time control for Boolean control networks with impulsive disturbances

Applied Mathematics and Computation 273 (2016) 477–483

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

A minimum-time control for Boolean control networks with impulsive disturbances✩ Hongwei Chen a,b, Bo Wu a,c,∗, Jianquan Lu b a

College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua 321004, China Department of Mathematics, Southeast University, Nanjing 210096, China c Graduate School, Zhejiang Normal University, Jinhua 321004, China b

a r t i c l e

i n f o

Keywords: Optimal control Boolean control network Impulsive disturbance Semi-tensor product

a b s t r a c t This paper investigates a Mayer-type optimal control and minimum-time control of a Boolean control network (BCN) with impulsive disturbances. Using the semi-tensor product, the BCN with impulsive disturbances is converted into algebraic discrete-time impulsive dynamic systems, and several necessary conditions for optimality are derived. Then we consider the problem of steering a BCN with impulsive disturbances from a given initial state to a desired state in minimal time. And a necessary condition, stated in the form of maximum principle, is obtained for a control to be time-optimal. It shows that the impulsive disturbances play an important role in the optimal control problem for BCNs. At last, a biological example is given to illustrate the effectiveness and advantage of the obtained results. © 2015 Elsevier Inc. All rights reserved.

1. Introduction The research of complex networks has grown steadily because of its potential to represent, characterize and model a wide range of intricate natural systems and phenomena [1]. One kind of complex networks called genetic regulatory network (GRN) has become a new research area in the biological science and has attracted great attention over the last few years [2,3]. In a GRN, a number of genes interact and regulate the expression of other genes by proteins. There are various types of network models that have been developed for GRNs such as Boolean models, extended probability Boolean networks models, etc. Boolean network (BN) was proposed by Kauffman for modeling complex and nonlinear biological systems, see [4,5]. So far, there are only few results on BNs because of the shortage of systematic tool to deal with logical dynamic systems [6]. BNs with binary inputs variables are referred to as Boolean control networks (BCNs). In the biological viewpoint, an input may represent the dosage that is administered to a patient. Recently, the semi-tensor product (STP) of matrices was proposed by Cheng et al. [7]. By resorting to STP, a logical equation can be expressed as an algebraic equation and the dynamics of a BCN can be converted into a linear discrete-time control system. Consequently, many interesting results on analyzing and synthesizing of BCN have been obtained [7–22]. Optimal control problem is an interesting topic in system control theory. Based on the STP technique, the optimal control of BCNs has been studied in a few contributions. In [21], the problem of finding the input sequence that maximizes an average payoff has been investigated. And the minimum energy control and optimal-satisfactory control of BCN have been considered in

✩ This work was partially supported by the NSF of Zhejiang Province of China under grant LY14A010008 and the NNSF of China under grants 61374077, 61573102, 61175119, and 11101373. ∗ Corresponding author. Tel.: +86 15067991513. E-mail address: [email protected] (B. Wu).

http://dx.doi.org/10.1016/j.amc.2015.09.075 0096-3003/© 2015 Elsevier Inc. All rights reserved.

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[22]. Also, [23] has addressed the infinite horizon optimal control problem and provide necessary and sufficient conditions for its solvability. A Mayer-type optimal control problem for BCNs with multi-input and single-input has been studied in [24] and [25], respectively. The states of biological networks and electronic networks are often subject to instantaneous disturbances and experience abrupt changes at certain instants, which may be caused by switching phenomenon or other sudden noise, i.e., they exhibit impulsive effects. Impulsive dynamical networks have attracted the interest of many researchers for their various applications in information science, biological science and automated control systems, see [26–30]. On the other hand, time-optimal controls are important in the context of BCNs. For example, a natural problem is to determine a control that steers the BCN from an initial state (that correspond 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 the impulses contribute to the optimal control and time-optimal control problem for BCNs. We will study a Mayer-type optimal control problem and time-optimal control problem for BCNs with impulses. The proof of our main result is motivated by the Pontryagin maximum principle (PMP) used in variational analysis of switched systems and optimal control analysis for BCNs in [24,25]. Our main results are several necessary conditions for a control to be optimal as well as time-optimal. It shows that the impulsive disturbances play an important role in the optimal control problem for BCNs. A biological example is given to demonstrate the theoretical results. The rest of this paper is organized as follows. Section 2 provides a brief review for the STP of matrices and some notations. In Section 3, we convert BCNs with impulses into linear discrete impulsive systems. Section 4 presents our main results on Mayertype optimal control problem and time-optimal control problem for BCNs with impulsive disturbances. A biological example is given to illustrate the efficiency of the proposed results in Section 5. Finally, a brief conclusion is drawn in Section 6. 2. Preliminaries Throughout this paper, N and N+ are the sets of nonnegative integers and positive integers, respectively. k is used to represent the delta set {δki |i = 1, 2, . . . , k}, where δki is the ith column of identity matrix Ik × k . Let Coli (A) denote the ith column of A. A matrix A ∈ Mm × n is called a logical matrix if the columns set of A, denoted by Col(A), satisfies Col(A) ⊂ m . The set of all m × n i i in ] ∈ Lm×n , denote it by A = δm [i1 , i2 , . . . , in ] for simplicity. logical matrices is denoted by Lm×n . For A = [δm1 ∼ δm2 . . . δm Definition 2.1 [8]. The STP of two matrices A ∈ Mm × n and B ∈ Mp × q is defined as

A  B = (A ⊗ Iα /n )(B ⊗ Iα /p ), where α = lcm(n, p) is the least common multiple of n and p, and ⊗ is the tensor (or Kronecker) product. Lemma 2.2 [8]. Let n denote the 22n × 2n logical matrix n = δ22n [1, 2n + 2, . . . , (2n − 2) · 2n + 2n − 1, 22 ], and let Z ∈ 2n . Then Z  Z = n Z. n

A logical domain, denoted by D, is defined as D = {True = 1, False = 0}. To use matrix expression we identify each element in D with a vector as True equals with δ21 and False equals with δ22 , and then D equals with 2 . Using STP of matrices, a logical function with n arguments f : Dn → D can be expressed in the algebraic form. Lemma 2.3 [8]. Any logical function f (x1 , . . . , xn ) with logical arguments x1 , . . . , xn ∈ D can be expressed in a multi-linear form as f (x1 , . . . , xn ) = M f x1 x2 · · · xn , where M f ∈ L2×2n is unique, called the structure matrix of f. 3. Algebraic form of BCNs with impulsive disturbances Definition 3.1 [7]. A BCN is a set of nodes, x1 , x2 , . . . , xn and u1 , u2 , . . . , um , which simultaneously interact with each other. At each given time t = 0, 1, 2, . . . , a node has only one of two different values: 1 or 0. Thus, the network can be described by a system of equations

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

. ⎪ ⎪ ⎩.

xn (t + 1) = fn (u1 (t ), . . . , um (t ), x1 (t ), . . . , xn (t )),

where fi , i = 1, 2, . . . , n, are Boolean functions, u j (t ) ∈ 2 , j = 1, . . . , m, are inputs. In the real word, biological systems may experience abrupt changes of states at some time instants because of the sudden environment changes, which are often of very short duration and are assumed to occur instantaneously in the form of impulses. In this paper we consider a BCN with impulsive disturbances as follows:

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

. ⎪ ⎪ ⎩.

xn (t + 1) = fn (u1 (t ), . . . , um (t ), x1 (t ), . . . , xn (t )),

t = tk ,

(1a)

H. Chen et al. / Applied Mathematics and Computation 273 (2016) 477–483

⎧ x (t + 1) = g1,k (x1 (t ), . . . , xn (t )), ⎪ ⎪ 1 ⎨ x2 (t + 1) = g2,k (x1 (t ), . . . , xn (t )), .

. ⎪ ⎪ ⎩.

479

t = tk ,

(1b)

xn (t + 1) = gn,k (x1 (t ), . . . , xn (t )),

where gi,k , k = 1, 2, . . . , are time-varying impulsive functions at time t = tk , {tk , k = 1, 2, . . .} ⊆ N is the set of impulsive instants with 0 ≤ t1 < t2 <  < tk < , tk → ∞ for k → ∞. Using Lemma 2.3, for each logical function fi and gi,k , i = 1, 2, . . . , n, k = 1, 2, . . . , we can find its structure matrix Fi ∈ L2×2m+n and Gi,k ∈ L2×2n , respectively. Let x(t ) = ni=1 xi (t ), u(t ) = m u (t ). Then system (1) can be expressed as j=1 j



x(t + 1) = L0 u(t )x(t ), t = tk , x(t + 1) = Lk x(t ), t = tk ,

(2)

with L0 := F1 ni=2 [(I2n+m ⊗ Fi )n+m ] and Lk := G1,k ni=2 [(I2n ⊗ Gi,k )n ]. In other worlds, there are two transition matrices L0 ∈ L2n ×2m+n and Lk ∈ L2n ×2n for a BCN with impulsive disturbances. Specific algorithms for converting a BN in the form of (1) to its algebraic representation (2) can be found in [7]. 4. Optimal control for BCNs 4.1. Mayer-type optimal control for BCNs Consider a BCN in the algebraic form (2). Let x(0) ∈ 2n be the (arbitrary) initial condition. For a final time N ∈ N+ , denote U as the set of all the control sequences {u(0), . . . , u(N − 1)} with u(i) ∈ 2m . For a control u ∈ U, denote the solution of (2) by x(t; u) with x(0) = x0 at time t. A fundamental problem for all dynamical control systems is to determine a control that maximizes n (or minimizes) a given cost-functional. Fix a vector r ∈ R2 , we consider the cost-functional

J(u) = r x(N; u).

(3)

Since x(N; u) ∈ 2n , any Boolean function f may be represented in the form (3), i.e., as f = In this case, J(u) can attain only two values, namely, zero and one, see [24,25].

r f x(N; u) with rf

being a binary vector.

Remark 4.1. In this paper, the existences of impulses make the reachability problem for BCNs (1) different. In fact, if there is no impulse in (2), then x(t + 1) = L0 u(t )x(t ) for all t = 0, 1, 2, . . . . Since L0 ∈ L2n ×2n+m and u(t )x(t ) ∈ 2n+m , then x(t + 1) ∈ Col (L0 ) for any t = 0, 1, 2, . . . and there is no more other possibility for x(t + 1). However, if the impulsive disturbances happen, i.e., x(t + 1) = Lk x(t ), for t = tk , k = 1, 2, . . . , then x(tk + 1) ∈ Col (Lk ) and it is possible that x(tk + 1) do not belong to Col(L0 ). Hence, the reachable set of x(N; u) could be different in this case. Remark 4.2. Suppose that r = δ2νn ∈ Col (L0 ) and r = δ2νn ∈ Col (Lk ). If there is no impulsive disturbance in (2), then x(t + 1) = δ2νn for any t. Thus, there exists no optimal control u∗ such that J(u∗ ) = 1 for any final time N ∈ N+ . However, when impulses happen as BCNs given by (2), then x(t + 1) ∈ Col (Lk ), and we could find corresponding optimal control u∗ such that J(u∗ ) = 1 for some final time N ∈ N+ . Hence, one can conclude that the impulsive disturbances play an important role in the optimal control problem for BCNs. β

In the following, we aim to find a control u∗ ∈ U that maximizes J given by (3), and u∗ is called optimal control. Let i=α L0 u(i) = L0 u(α)L0 u(α − 1) · · · L0 u(β), β ≤ α . From the expression of Eq. (2), it is easy to obtain the following lemma. Lemma 4.3. Given any a, b ∈ N, such that there are l + 2 impulsive instants tk0 , tk1 , . . . , tkl , tkl+1 satisfying tk0 < a ≤ tk1 < tk2 < · · · < tkl < b ≤ tkl+1 , then

x(b; u) = C (b, a; u)  x(a; u), tk +1

(4) +1

tk

l L0 u(i)]Lkl [i=tl−1 −1 L0 u(i)]Lkl−1 · · · [ai=t where C (b, a; u) = [i=b−1

k1 −1

kl

L0 u(i)]. We refer to the matrix C(b, a; u) as the transition ma-

trix from time a to time b corresponding to the control u. Proof. By Eq. (2), when b > tkl + 1, tk j > tk j−1 + 1 and tk1 > a for j = 2, . . . , l, we have

x(b; u) = L0 u(b − 1)x(b − 1)



tk +1



l = i=b−1 L0 u(i) x(tkl + 1)



tk +1



l = i=b−1 L0 u(i) Lkl x(tkl )



tk +1

 

tk

+1



+1



l = i=b−1 L0 u(i) Lkl i=tl−1 −1 L0 u(i) x(tkl−1 + 1)



tk +1

 

kl

tk



l = i=b−1 L0 u(i) Lkl i=tl−1 −1 L0 u(i) Lkl−1 · · · ai=tk kl

1

−1 L0 u

 (i) x(a; u).

(5)

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For the other cases with b = tkl + 1, or tk j = tk j−1 + 1, or tk1 = a for j = 2, . . . , l, we have the same expressions as (5). This complete the proof of Lemma 4.3. 

Remark 4.4. From Lemma 4.3, it is clear that C (a, a; u) = I2n . Furthermore, for a ≤ c ≤ b, we have C (b, a; u) = C (b, c; u)C (c, a; u). Fix a final time N ∈ N+ . Let S = {0, 1, 2, . . . , N − 1}, S1 = {t |t ∈ S, t = tk for any k = 1, 2, . . .} and S2 = {t |t ∈ S, t = tk for any k = 1, 2, . . .}. Then we have the following result. Theorem 4.5. Consider the BCNs (2). Assume that u∗ = {u∗ (0), . . . , u∗ (N − 1)} ∈ U is an optimal control that maximizes J, and x∗ is n the corresponding trajectory of (2). Let the adjoint λ : {0, 1, 2, . . . , N} → R2 be the solution of

λ ( p) = λ ( p + 1)C ( p + 1, p; u∗ ), ∀ p ∈ S, λ (N) = r , p; u∗ )

with C ( p + 1,

(6)

defined by (4), and define switching functions αi (s), i ∈

αi (s) = λ (s + 1) δ

L0 2i m x∗

{1, 2, . . . , 2m }

for any s ∈ S1 by

(s).

(7)

If for some index i, αi (s) > α j (s) for all j = i, j ∈ {1, 2, . . . , 2m }, then u∗ (s) = δ2i m . Proof. Let x∗ ( · ) = x(·; u∗ ) denote the solution of (2) corresponding to the optimal control u∗ . Then for an arbitrary time p with p ∈ S, there are two cases: p = tk for some k = 1, 2, . . . , or p = tk . For the first case, we have

x∗ (N) = C (N, p + 1; u∗ )x∗ ( p + 1) = C (N, p; u∗ )x∗ ( p).

(8)

Hence, u∗ (p) is not existent when p = tk . For the second case, we can obtain that

x∗ (N) = C (N, p + 1; u∗ )x∗ ( p + 1) = C (N, p + 1; u∗ )L0 u∗ ( p)x∗ ( p).

(9)

For an arbitrary time s ∈ S1 and an arbitrary vector v ∈ 2m , we define a new control u ∈ U by a perturbation of

u( j ) =

 v,

u∗ ( j),

u∗

as follows:

if j = s, if j =  s, j ∈ S1 .

(10)

Similarly, for x( · ) = x(·, u) with u defined by (10), we have

x(N) = C (N, s + 1; u)x(s + 1) = C (N, s + 1; u∗ )L0 vx∗ (s). Thus x∗ (N) − x(N) = C (N, s + 1; u∗ )L0 (u∗ (s) − v)x∗ (s), and

J(u∗ ) − J(u) = r (x∗ (N) − x(N)) = r C (N, s + 1; u∗ )L0 (u∗ (s) − v)x∗ (s). To simplify this expression, let

λ ( p + 1)

=

r C (N,

p + 1; u∗ )

(11) for all p ∈ S. Then one has

λ (N)

λ ( p) = r C (N, p; u ) = r [C (N, p + 1; u∗ )C ( p + 1, p; u∗ )] = λ ( p + 1)C ( p + 1, p; u∗ ).



=

r C (N, N; u∗ )

=

r ,

and

∗

(12)

With the definition of λ, we can rewrite (11) by

J(u∗ ) − J(u) = λ (s + 1)L0 (u∗ (s) − v)x∗ (s).

(13)

We now define the scalar function for all s ∈ S1 and i ∈ {1, 2, . . . , 2m } as

αi (s) := λ (s + 1)L0 δ2i m x∗ (s). Suppose that there exists an index i such that α i (s) > α j (s) for all j = i with s ∈ S1 . We claim that u∗ (s) = δ2i m . In fact, if

u∗ (s) = δ2m for some j ∈ {1, 2, . . . , 2m }, then for v = δ2i m the right-hand side of (13) satisfies j

λ (s + 1)L0 (δ2i m − δ2j m )x∗ (s) = αi (s) − α j (s) < 0. Thus, J(u∗ ) − J(u) < 0, which contradicts the optimality of u∗ (s). Then we have u∗ (s) = δ2i m . Since s is arbitrary in S1 , we give the expression of all u∗ (s). The proof of Theorem 4.5 is completed.  Remark 4.6. Theorem 4.5 is a necessary condition for optimality stated in terms of switching functions (7) that depend on the unknown x∗ (s). Nevertheless, it provides implicit information for determining a control that maximizes the given cost-functional (3). In a recent work [34], a control design algorithm that maximizes the cost functional (3) at a fixed termination time has

H. Chen et al. / Applied Mathematics and Computation 273 (2016) 477–483

481

been provided. Based on the Algorithm 3.1 in [34], one can obtain a optimal control design algorithm for BCN with impulsive disturbances. Specializing Theorem 4.5 by considering the BCNs (2) without impulsive disturbances, i.e., for any t = 0, 1, 2, . . . , x(t + 1) can be expressed by the first equation of (2), then S = S1 in Theorem 4.5 and main theorems for optimal control problem in [24,25] could be derived. In Theorem 4.5, we assume the existence of some i ∈ {1, 2, . . . , 2m } such that α i (s) > α j (s) for all j = i with s ∈ S1 . In this case, the optimal control u∗ is determined uniquely. In the following, we consider the singular case of complementary situation. Theorem 4.6. Consider the BCNs (2). Assume that for some time s ∈ S1 there exists a subset of indexes I = {i1 , . . . , il } ⊂ {1, 2, . . . , 2m } i

i

such that αi1 (s) = · · · = αil (s) and αi1 (s) > α j (s) for all j ∈ {1, 2, . . . , 2m }\I. Then u∗ (s) ∈ {δ21m , . . . , δ2lm }. Furthermore, any control in the form



w( j ) = i

z, u∗ ( j),

if j = s, if j =  s, j ∈ S1 ,

(14)

i

with z ∈ {δ21m , . . . , δ2lm }, is also an optimal control. Proof. Without loss of generality, we assume I = {i1 , i2 }. From the theorem, αi1 = αi2 > α j , for all j ∈ {1, 2, . . . , 2m }\I. Similar i

i

with the proof of Theorem 4.5, we have u∗ (s) = δ21m or u∗ (s) = δ22m , i.e., w( j) defined in (14) is the optimal control. 

4.2. Maximum principle for time-optimal control Consider the problem of designing a time-optimal control u∗ that steers the BCN (2) from initial state x(0) = x0 to desired state z = x(N; u, x0 ) in minimal time, which is relevant to biological systems. For example, to prevent λ phage spread, we should design a control sequence that steers the corresponding systems from lysogeny state to lysis state in minimal time see [31,32]. Let S ∗ = {0, 1, . . . , N∗ − 1}, then we have the following result. ∗

∗

Theorem 4.7. Suppose that u∗ ∈ U N is a minimum-time control, and let X N denote the corresponding trajectory of (2). Define the n adjoint λ : {0, 1, 2, . . . , N} → R2 as in (6). Then for any p ∈ S ∗ ,

λ ( p)x∗ ( p) = 1,

(15)

and

z C ( p, 0; u)x(0) = 0, Proof. Assume that u∗ ∈ U

N∗

∀u ∈ U.

(16)

is a minimum-time control, then for an arbitrary p ∈ S ∗ , one can obtain

z = x∗ (N∗ ) = C (N∗ , p + 1; u∗ )C ( p + 1, p; u∗ )x∗ ( p). Similar to the proof of Theorem 4.5, let λ ( p + 1) = z C (N∗ , p + 1; u∗ ). Then

1 = z C (N∗ , p + 1; u∗ )C ( p + 1, p; u∗ )x∗ ( p) =

λ ( p + 1)C ( p + 1, p; u∗ )x∗ ( p).

(17) u∗ .

This together with (12) shows that condition (15) holds. To prove (16), we use the time optimality of Seeking a contradiction, assume that there exist p such that z C(p, 0; u)x(0) = 0. Since the transition matrix must map any column of I2n to a column of I2n , this implies that

z C ( p, 0; u)x(0) = 1. That is to say, there exist a control sequence that steer the BCN from x(0) = x0 to x( p) = z. Since p < N∗ , and this contradicts the fact that N∗ is the minimal time required to steer the BCN from x0 to z. This proves (16). Now the proof is completed.  Remark 4.8. Theorems 4.5 and 4.7 give necessary conditions for a control to be optimal in the form of maximum principle. One can observe that the transition matrix C(b, a; u) defined in Lemma 4.3 is a 2n × 2n matrix, thus the criterion is applicable only to small BNs. Further study is still necessary for improving the optimality criteria. 5. A biological example Let us consider the Boolean model for the λ switch which was proposed in [31]. The λ phage is a virus that grows on a bacterium upon infection of the bacterium, the phage injects its chromosome into the bacterium cell. The virus can then follow one of two different pathways: lysogeny or lysis. The molecular mechanism responsible for the lysogeny/lysis decision is known as the λ switch. Laschov and Margaliot [32] derived a simple BCN model for the λ switch. The two possible pathways are the result of expressing different sets of genes. Several environmental conditions including temperature, growth rate and multiplicity of infection can influence the genes expressing.

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Laschov and Margaliot [32] used the Boolean input to represent whether the total environmental conditions is “favorable” or not. The BCN model is

⎧ x1 (t + 1) = [¬x2 (t )] ∧ [¬x5 (t )], ⎪ ⎪ ⎨x2 (t + 1) = [¬x5 (t )] ∧ [x2 (t ) ∨ x3 (t )], x3 (t + 1) = [¬x2 (t )] ∧ u(t ) ∧ [x1 (t ) ∨ x4 (t )], t ≥ 0. ⎪ ⎪ ⎩x4 (t + 1) = [¬x2 (t )] ∧ u(t ) ∧ x1 (t ), x5 (t + 1) = [¬x2 (t )] ∧ [¬x3 (t )].

(18a)

The input u(t) is δ21 [δ22 ] if the environmental conditions are favorable [not favorable] at time t. During the processes of gene express in the bacterium, many genes may experience abrupt changes of states at some time instants tk due to physical factor including X ray, laser, ultraviolet rays etc [33]. Those sudden and sharp changes are often of very short duration and are assumed to occur instantaneously in the form of impulses. There are a great deal of literatures studying the systems with impulsive effects, for example, see [26,29]. It is worth noting that those states of genes would change suddenly or randomly. Due to the abruptness or randomness of the states’ change, in this example, we consider impulsive Boolean network in some tk , with tk = 2k, k = 1, 2, . . . ,

⎧ x1 (t + 1) = [x1 (t ) ∨ x3 (t )] ↔ x5 (t ), ⎪ ⎪ ⎨x2 (t + 1) = [¬x1 (t )] ∨ x3 (t ), x3 (t + 1) = ¬x4 (t ), t = tk . ⎪ ⎪ ( t + 1 ) = [x ( t ) ∧ x ( t ) ] ∨ x ( t ) , x 4 1 2 3 ⎩ x5 (t + 1) = x1 (t ) ↔ [x3 (t ) ∨ x4 (t )],

(18b)

Denote x(t ) = 5i=1 xi (t ), then we can convert the system (18) into its algebraic form as (2) with

L0 =

δ32 [32 24 32 24 32 24 32 24 26 2 26 2 25 9 25 9 32 24 32 24 32 24 32 24 28 4 32 8 27 11 31 15 32 24 32 24 32 24 32 24 32 8 32 8 31 15 31 15 32 24 32 24 32 24 32 24 32 8 32 8 31 15 31 15],

Lk =

δ32 [5 21 1 17 13 29 10 26 5 21 1 17 15 31 12 28 6 22 2 18 24 8 19 3 6 22 2 18 24 8 19 3].

32 , N = 4, and r = δ 24 . Note that δ 24 = Suppose that the initial state is x1 (0) = x2 (0) = x3 (0) = x4 (0) = x5 (0) = δ22 , i.e. x(0) = δ32 32 32 δ22  δ21  δ22  δ22  δ22 corresponding to the lysogenic state. 24 ) x(4). Intuitively, this amounts to finding a control steering First, we aim to determine a control that maximizes J(u) = (δ32 32 to x(4) = δ 24 if it exists. From the expressions of r and L , we consider the functions α given by (7) for the BCN from x(0) = δ32 0 i 32 i = 1, 2,



αi (3) =

P1 x∗ (3), P2 x∗ (3),

i = 1, i = 2,

(19)

where P1 = P2 = [0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0]. 6 . Then α (3) = α (3) = 1. From Theorem 4.6, we have u∗ (3) ∈ {δ 1 , δ 2 }. Let u∗ (3) = δ 1 , using (6) yields Case 1. Assume x∗ (3) = δ32 1 2 2 2 2 25 , or x∗ (2) = δ 17 . λ (3) = λ (4)C (4, 3; u∗ ) = P1 . Since x∗ (3) = L2 x∗ (2), we have x∗ (2) = δ32 32 25 . Using (6) yields λ (2) = λ (3)C (3, 2, u∗ ) = λ (3)L , by calculation, we have λ (2) = Case 1.1 Assume x∗ (2) = δ32 2 [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 0].

Hence,

αi (1) =



P3 x∗ (3), 0,

i = 1, i = 2,

(20)

where P3 = [0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0], thus from Theorem 4.5, u∗ (1) = δ21 . 15 . Using (6) yields λ (1) = P . Thus α (0) = λ (1)L δ i x∗ (0) = 1, i = 1, 2. It is clear from Case 1.1.1 Assume x∗ (1) = δ32 3 0 2 i ∗ 1 Theorem 4.6 that u (0) ∈ {δ2 , δ22 }. Hence, in this case we conclude that {u∗ (0), u∗ (1), u∗ (3)} = {δ21 , δ21 , δ21 } and {δ22 , δ21 , δ21 } are the control sequences that satisfy the necessary condition for optimality. A calculation shows the corresponding trajectory is 32 , x∗ (1) = δ 15 , x∗ (2) = δ 25 , x∗ (3) = δ 6 , x∗ (4) = δ 24 . x∗ (0) = δ32 32 32 32 32 If we are interested in finding all controls that steer the BCN from initial state to the desired state, the remaining cases can be similarly considered. Hence, for this particular BCN with impulsive disturbances, Theorems 4.5 and 4.6 provide enough information to explicitly determine the optimal controls. 17 ∈ Col (L ), δ 17 ∈ Col (L ). Based on the analysis in Remark 4.2, if there is no impulsive Remark 5.1. It follows from (18) that δ32 0 k 32 17 . Hence, let r = δ 17 , there disturbance in such BCN, then x(t + 1) ∈ Col (L0 ) for any t = 0, 1, 2, . . . , which means x(t + 1) = δ32 32 exists no optimal control u∗ such that J(u∗ ) = 1 for any final time N ∈ N+ , i.e., for any input u(t) defined on t = 0, 1, 2, . . . , J(u) = 0.

H. Chen et al. / Applied Mathematics and Computation 273 (2016) 477–483

483

17 . Thus, for any t = t , x(t + 1) ∈ Col {L }. Let r = δ 17 as above, and When impulses happen as BCNs given by (18), Col4 (Lk ) = δ32 k k 32 ∗ N = 3. Then similar to the above analysis, we can obtain {u (0), u∗ (1)} = {δ21 , δ21 } is the control such that J(u∗ ) = 1 for N = 3. A ∗ 9 , δ 26 , δ 4 , δ 17 }. Hence, the impulsive disturbances contribute to calculation shows that the corresponding trajectory is xN = {δ32 32 32 32 the optimal control problem for this particular BCN. 32 to the lysogenic state in minimal time. Now we consider the problem of steering the BCN from the initial state x(0) = δ32 ∗ ∗ 1 15 , x∗ (2) = We already know that N = 4 and that u (k) = δ2 , k = 0, 1, 3 is a time-optimal control. A calculation yields x∗ (1) = δ32 25 , x∗ (3) = δ 6 , x∗ (4) = δ 24 , then we can verify that conditions (15) and (16) hold. Thus from Theorem 4.7, N ∗ = 4 is the minimal δ32 32 32 32 to δ 24 . time required to steer the BCN x(0) = δ32 32

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