Stabilization of Boolean control networks with stochastic impulses

doi:10.1016/j.jfranklin.2019.06.039

Journal of the Franklin Institute

Volume 356, Issue 13, September 2019, Pages 7164-7182

https://doi.org/10.1016/j.jfranklin.2019.06.039 Get rights and content

Abstract

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

Introduction

Boolean networks (BNs) were firstly introduced by Kauffman [1] in 1969 to describe the genetic circuits. In BNs, every gene is symbolized as a binary variable, 0 (inactive) or 1 (active) per time, whose next state is determined by Boolean functions. Compared with other gene regulatory models, BNs are much simpler in structure, which have become a powerful tool in describing, analyzing and simulating cellular networks. Hence, the applications of BNs have been extensively studied by biologists, chemists and system scientists [2], [3], [4], [5]. As is well known, it is a common problem to design effective intervention strategies to reach desirable cellular states when modeling gene regulatory networks as BNs. For example, a binary input may present whether a certain intervention is administered or not at each time step [6]. When control inputs and outputs are added to BNs, the concept is naturally extended to Boolean control networks (BCNs). Since BCNs share the appealing properties of BNs and also deal with the presence of control, they have become the focus of research [7], [8]. Recently, a matrix product, namely the semi-tensor product (STP) is used to analyze BNs by Cheng et al. [9]. The advantage of this method is that one can convert a BN (BCN) to a linear (bilinear) discrete-time system. Then, many challenging problems of BNs have been solved, including controllability and observability [10], [11], [12], stability and stabilization [6], [13], [14], [15], [16], disturbance decoupling [17], [18] and other problems [19], [20], [21], [22], [23]. Besides, the STP method has also been applied to other kinds of BNs, for example, probabilistic Boolean control networks (PBCNs), which are derived from BCNs and coped with the stochastic nature of the genetic regulation networks [24], [25], [26], [27], [28]. In [27], Kobayashi and Hiraishi proposed methods on optimal control of PBNs. Trairatphisan et al. [28] summarized the recent developments and biomedical applications of PBNs.

As one of the most important behavior of dynamical networks, stability and stabilization have been widely studied in ecological systems [29], [30], neural networks [31], [32], and so on. Stabilization of BCNs, which can be achieved by designing moderate controllers, such as robust controllers [33], [34], state feedback controllers [35], [36] and impulsive controllers [37], [38], is a significant research field. For example, in the treatment of disease, medicine therapy strategies need to be designed such that patients are steered to the desirable state and maintained this state afterward [6]. There are many studies of stabilization both for deterministic and stochastic BCNs. Cheng et al. [39] investigated the stabilization of BCNs via the open-loop control and the state feedback control. Later, the stabilization researches were extended to the design of controllers such that BCNs were converged to the same periodic trajectory [40]. Based on trajectory stabilization, Zheng et al. [14] studied the stabilization and set stabilization of delayed BCNs by designing state feedback controllers. Furthermore, Li and Wang provided a necessary and sufficient condition to design all possible output feedback controllers in [41]. However, many practical networks are not inherently deterministic. There exists a common requirement to deal with the probability problem. Some control mechanisms have been uncovered in the solution of this problem, which are fundamental to stabilize the whole system. Zhao and Cheng [42] designed a control sequence such that the PBCN converges to a fixed point with probability one. Li and Tang [43] analyzed the set stabilization of switched BCNs by designing two kinds of state feedback controllers.

In the real world, evolutionary processes in biological networks are often subject to instantaneous disturbances and abrupt changes at certain instants, which may be caused by switching phenomena, or other sudden changes, i.e., they exhibit impulsive effects [44]. On the other hand, impulsive effects are unavoidable in biological systems due to the mutability of external environment. Hence, different kinds of models, including neural networks [45], [46], differential equations [47], [48], dynamical networks [49] and BNs [37], [38], [50], [51], have been investigated for the stability and stabilization under the influence of impulses. Since BNs are used widely to model biological networks, the study of stabilizing BNs with impulsive effects is meaningful. In [37], the stability and stabilization of BNs with impulsive effects were investigated, and some necessary and sufficient conditions were obtained. Subsequently, the research was extended to switched BNs with impulsive effects under two types of controls [38]. The stabilization of BNs with impulsive effects and state constraints was investigated in [50]. In these results, the impulses were described as deterministic models.

As we all know, stochastic phenomena are very common in nature, and stochastic models have come to play an important role in many branches of science and industry. As an important influence factor of genetic evolution in biological network, impulses are usually generated with randomness. We call these as stochastic impulses. In the past decades, stochastic impulses have been used to solve optimal control problems, including the management of renewable resources [52], discovering optimal strategy for investment models [53] and so on. The stochastic impulses, which are used to describe the random external environment, are represented as several possible regulatory models in this study. However, to our best knowledge, there is little research on BNs with stochastic impulses.

Motivated by the above discussions, the stabilization of BCNs with stochastic impulses will be investigated. Different from [37], where the impulse model was deterministic, we study stochastic model at stochastic impulses instants. In this case, each impulse model corresponds to a certain probability, which makes the system structure more complicated. To solve this problem, a series of matrices and convergable sets are defined. A necessary and sufficient condition based on convergable sets is presented to solve the stabilization problem for BCNs with stochastic impulses. Meanwhile, state feedback controller and different impulsive strategies are designed to influence the trajectory of each state. Finally, these obtained results are extended to the stabilization of PBCNs with stochastic impulses.

The rest of this paper is organized as follows. Section 2 gives a brief introduction to some notations. The establishment of BCNs with stochastic impulses and its algebraic form are also discussed. Section 3 presents the main results of this paper. Based on the results obtained in Section 3, some conditions to analyze the stabilization of PBCNs with stochastic impulses are proposed in Section 4. Then, numerical examples are given to show the efficiency of the obtained results in Section 5. Finally, conclusions are summarized in Section 6.

Section snippets

Notations and definitions

•
$Z$ denotes the set of integer numbers.
•
Denote by $A^{T}$ the transpose of matrix A.
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I_n represents the n × n identity matrix, $1_{2^{m}} : = {(1, 1, \dots, 1)}^{T},$ $0_{2^{m}} : = {(0, 0, \dots, 0)}^{T}$ .
•
$D : = {T = 1, F = 0},$ where $T \sim {[1 0]}^{T}$ and $F \sim {[0 1]}^{T}$ .
•
Denote the ith column of matrix A by Col_i(A), Col(A) is the set of all columns of A. Row_i(A) represents the ith row of matrix A.
•
$Δ_{n} : = {δ_{n}^{i} | 1 \leq i \leq n},$ where $δ_{n}^{i} : = C o l_{i} (I_{n})$ .
•
$δ_{m} [i_{1} \dots i_{n}]$ represents matrix A with $C o l_{j} (A) = δ_{m}^{i_{j}}$ .
•
|A| represents the cardinal number of the set A.
•
The set of n × m logical matrix L is defined by $L$

Main results

In this section, the global stabilization of BCNs with stochastic impulses is addressed. If the system can be stabilized, any initial state can reach the predefined target by designing an appropriate impulsive strategy and a corresponding controller. Before formulating the problem, a series of matrices are defined as follows: $\begin{matrix} M_{0} = L \lor G, {\tilde{M}}_{0} = ⌊ M_{0} ⌋, \\ M_{i} = {\tilde{M}}_{i - 1} M_{0}, {\tilde{M}}_{i} = ⌊ M_{i} ⌋, \end{matrix} i = 1, 2, \dots$ where $L = (B) \sum_{i = 1}^{2^{m}} B l k_{i} (L_{0}) = L_{0} ⋉_{B} 1_{2^{m}},$ $G = \sum_{i = 1}^{r} p_{i} G_{k i}$ .

For the given state $x_{d} = δ_{2^{n}}^{α} \in Δ_{2^{n}}$ and any $j \in Z_{+},$ let Φ_j(x_d) be the convergable set

Extension to PBCNs with stochastic impulses

As an extension, consider the following PBCNs with stochastic impulses: ${\begin{matrix} x (t + 1) = {\begin{matrix} L_{01} u (t) x (t) \\ L_{02} u (t) x (t) \\ ⋮ \\ L_{0 s} u (t) x (t) \end{matrix}, t \neq t_{k} \\ x (t + 1) = {\begin{matrix} G_{k 1} x (t) \\ G_{k 2} x (t) \\ ⋮ \\ G_{k q} x (t) \end{matrix}, t = t_{k} \end{matrix},$ where $L_{0 i} \in L_{2^{n} \times 2^{n + m}} (i = 1, 2, \dots, s)$ and $G_{k j} \in L_{2^{n} \times 2^{n}} (j = 1, 2, \dots, q)$ . Denote the probabilities as $P {L_{0} = L_{0 i}} = p_{i}^{l}, P {G_{k} = G_{k j}} = p_{j}^{g} (i = 1, 2, \dots, s, j = 1, 2, \dots, q)$ and $\sum_{i = 1}^{s} p_{i}^{l} = 1, \sum_{j = 1}^{q} p_{j}^{g} = 1$ .

Let $\ddot{L} = \sum_{i = 1}^{s} p_{i}^{l} L_{0 i},$ $\ddot{G} = (\sum_{i = 1}^{q} p_{i}^{g} G_{k i}) (1_{2^{m}}^{T} \otimes I_{2^{n}})$ . Similarly, matrices $M_{i}^{^{'}}$ and ${\bar{M}}_{i} (i = 0, 1, 2, \dots)$ are defined as follows: $M_{0}^{^{'}} = \ddot{L} \lor \ddot{G}, {\bar{M}}_{0} = ⌊ M_{0}^{^{'}} ⌋ ⋉_{B} 1_{2^{m}},$ $M_{i}^{^{'}} = ({\bar{M}}_{i - 1} \ddot{L}) \lor ({\bar{M}}_{i - 1} \ddot{G}), {\bar{M}}_{i} = ⌊ M_{i}^{^{'}} ⌋ ⋉_{B} 1_{2^{m}} .$ Based

Illustrative examples

In this section, two numerical examples will be used to show the correctness of the previous results.

Example 1

Consider the BCN model for the λ switch under impulsive effects which was proposed in [44]. As a virus, λ phage grows by injecting its own chromosomes into the infected bacteria cell. Two possible pathways: lysogeny and lysis, are the results of expressing different sets of genes. Some environmental factors including temperature rate, concentration of nutrition and growth rate, which affecting

Conclusion

In this work, the global stabilization of BCNs to a fixed state, with stochastic impulses is discussed. Firstly, the algebraic expression of BCNs with stochastic impulses is obtained by using the STP method. Then, the definition of stabilization with probability one is presented. Based on the algebraic form, some matrices and convergable sets are defined to reduce complexity of the study. By resorting to these definitions, some criteria are derived to judge whether BCNs with stochastic impulses

Acknowledgment

This work was jointly supported by the National Natural Science Foundation of China under Grants 61603268, 61272530, 61573096 and 61573102, the Fundamental Research Funds for the Central Universities under Grant No. JBK190502.

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Stabilization of Boolean control networks with stochastic impulses

Abstract

Introduction

Section snippets

Notations and definitions

Main results

Extension to PBCNs with stochastic impulses

Illustrative examples

Conclusion

Acknowledgment

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