Boolean Control Networks Research Articles

Observability of singular Boolean control networks with state delays

This paper investigates the observability of singular Boolean control networks (SBCNs) with state delays. First, via Cheng product, the considered system is converted into its algebraic form. Then, under the uniqueness of the solution, two methods are proposed to transform the system into a general Boolean control network (BCN). Therefore, the considered system’s observability is equivalent to the BCN’s observability. Subsequently, the restricted input-state incidence matrix is introduced and some necessary and sufficient conditions are presented for the observability of BCNs with state delays. Finally, two illustrative examples are presented to show that the main results obtained are effective in analyzing the observability of SBCNs with state delays.

Optimal control of singular boolean control networks via Ledley solution method

This paper investigates the optimal control problem for a class of Boolean control networks, called singular Boolean control networks (SBCNs), which consist of two parts: difference equations and algebraic equations. By constructing the truth matrix of Ledley solution, necessary and sufficient conditions are provided for the solvability of SBNs (or SBCNs). Then an effective algorithm is presented to design an optimal control sequence by using the controllability matrix of normalized Boolean control networks.

State feedback stabilization of generic logic systems via Ledley antecedence solution

In this paper, the application of Ledley antecedence solutions in designing state feedback stabilizers of generic logic systems has been proposed. To make the method feasible, two modifications are made to the original Ledley antecedence solution theory: (i) the preassigned logical functions have been extended from being a set of equations to an admissible set; (ii) the domain of arguments has been extended from the whole state space to a restricted subset. In the proposed method, state feedback controls are considered as a set of extended Ledley antecedence solutions for a designed iterative admissible sets over their corresponding restricted subsets. Based on this, an algorithm has been proposed to verify the solvability and simultaneously provide all possible time‐optimal state feedback stabilizers when the problem is solvable. Demonstrations of the proposed method on both the Boolean control networks and mix‐valued logical systems have validated the effectiveness of the proposed method in stabilizing logic systems in general.

Topological Structure, Reachability, and Stabilization of Constrained Boolean Control Networks via Event-Triggered Control

This paper is concerned with the topological structure, reachability, and stabilization for Boolean control networks (BCNs) with state constraints via event-triggered control (ETC) scheme. In the first part, the topological structure of BCNs with state constraints is studied. Under the framework of state constrain, the definitions of fixed point and cycle are first defined. A novel phenomenon is that there may exist two kinds constrained fixed points, which are, respectively, named as the livelock and deadlock ones. It is different with the traditional fixed point. Accordingly, a formula is presented to calculate the number of constrained fixed points and constrained cycles. In the second part, the constrained reachability and stabilization problem of the event-triggered controlled BCNs are investigated. Two necessary and sufficient criteria are, respectively, obtained. Furthermore, an algorithm is developed to design all feasible controllers. Finally, a reduced model of the lac operon in the Escherichia coli is shown to illustrate the efficiency of the obtained results.

Finite automata approach to reconstructibility of switched Boolean control networks

This paper investigates the reconstructibility of switched Boolean control networks (SBCNs). Several new types of definition about reconstructibility are proposed where the existence and arbitrariness of the switching signal and the input sequence are inconsistent. A weighted pair graph describing all pairs of states which are indistinguished is defined. A deterministic finite automata method is applied into the reconstructibility analysis. Accordingly, several algorithms are designed to determine these types of reconstructibility. Moreover, for a given reconstructible SBCN, an algorithm for determining the current state is provided. Finally, an example is provided to demonstrate the effectiveness of the proposed algorithms and results.

Stabilization and set stabilization of switched Boolean control networks via flipping mechanism

Stabilization and set stabilization of switched Boolean control networks are investigated by using flipping mechanism in this paper. Firstly, with the help of Warshall algorithm, an explicit criterion for the stabilization of switched Boolean control networks is derived. Secondly, the necessary and sufficient condition for the solvability of stabilization of switched Boolean control networks, by flipping some elements of perturbation set once, is presented. Thirdly, a search algorithm is proposed to calculate the minimum number of stabilization flipped nodes and what exactly they are. Furthermore, a necessary and sufficient condition is established for the solvability of set stabilization of switched Boolean control networks by flipping some elements of perturbation set once. Analogously, an algorithm is given to find the minimum number of set stabilization flipped nodes. Finally, examples are shown to demonstrate the feasibility of the above results.

Controllability and stabilization of periodic switched Boolean control networks with application to asynchronous updating

This paper investigates the periodic switching point controllability and stabilization of periodic switched Boolean control networks (PSBCNs), and applies the obtained results to the stabilization of deterministic asynchronous Boolean control networks (DABCNs). Firstly, using the algebraic state space representation of PSBCNs, a kind of periodic switching point controllability matrix is constructed, based on which, a necessary and sufficient condition is presented for the periodic switching point reachability and controllability of PSBCNs. Secondly, using the reachable set of PSBCNs, a constructive procedure is proposed to design time-variant state feedback controllers for the periodic switching point stabilization of PSBCNs. Finally, by converting the dynamics of DABCNs into the form of PSBCNs, the time-variant state feedback stabilization problem of DABCNs is solved.

Generalized cluster synchronization of Boolean control networks with delays in both the states and the inputs

In this paper, we study the generalized cluster synchronization problem for the Boolean control networks (BCNs) with delays in both the states and the inputs. First, by using the method of semi-tensor product of matrices, the original network is transformed into an equivalent extended system. Next, based on the updated iterative equation of the system, two types of generalized cluster synchronization are investigated: 1) generalized internal cluster synchronization within the BCN, and 2) generalized cluster synchronization between the BCN and the target reference network. Some necessary and sufficient conditions are proposed guaranteeing the realization of the generalized cluster synchronization. What is more, the gain matrices of the state-feedback controllers are explicitly designed. Numerical simulations are also given to illustrate effectiveness of the theoretical results obtained.

Asynchronous Stabilization of Boolean Control Networks With Stochastic Switched Signals

This paper includes some results of switched Boolean control networks. The switched signals considered in this paper is time variant and it follows a certain probabilistic distribution vector. Given a concept of stabilization with stochastic switched signals, we obtained a necessary and sufficient condition for stabilization. Then a state feedback control depending on switched signals is designed to stabilize the system considered. Later we investigate a case when the switched signal θ(t) is unknown at time t, and what we have is the prediction switched signal ∧θ(t). For a given prediction matrix, a necessary and sufficient condition is given to preserve the stabilization. Except state feedback control, asynchronous pinning control for switched Boolean networks (BNs) is also considered. A necessary and sufficient condition is extended for the stability of BNs with stochastic switched signals. Moreover, algorithms are presented to find the minimal number of pinned nodes based on controlling minimal number of subsystems. Examples are shown to illustrate the effectiveness of the obtained results.

Time-variant Feedback Stabilization of Constrained Delayed Boolean Networks Under Nonuniform Sampled-data Control

This paper investigates the time-variant state feedback stabilization of constrained delayed Boolean control networks under nonuniform sampled-data control. Firstly, using the augmented algebraic form of constrained delayed Boolean control networks, the constrained nonuniform sampled-data controllability matrix is constructed. Secondly, based on the controllability matrix, a necessary and sufficient condition is proposed for the nonuniform sampled-data reachability of constrained delayed Boolean control networks. Thirdly, by virtue of the nonuniform sampled-data reachability, a new procedure is established to design time-variant state feedback sampled-data stabilizers via reachable set approach. The obtained results are applied to the cell survival regulation of apoptosis networks.

The set stabilization problem for Markovian jump Boolean control networks: An average optimal control approach

In this paper, the authors address the set stabilization problem for Markovian jump Boolean control networks (MJBCNs) by state feedback control. Based on the semi-tensor product, a necessary and sufficient condition for the set stabilization of MJBCNs is put forward via optimal control approach. Moreover, a policy iterative algorithm is deduced for designing the state feedback control. Finally, the proposed method and results are supported by the example of an apoptosis network.

Self-Triggered Scheduling for Boolean Control Networks

It has been shown that self-triggered control has the ability to deal with cases with constrained resources by properly setting up the rules for updating the system control when necessary. In this article, self-triggered stabilization of the Boolean control networks (BCNs), including the deterministic BCNs, probabilistic BCNs, and Markovian switching BCNs, is first investigated via the semitensor product of matrices and the Lyapunov theory of the Boolean networks. The self-triggered mechanism with the aim to determine when the controller should be updated is provided by the decrease of the corresponding Lyapunov functions between two consecutive samplings. Rigorous theoretical analysis is presented to prove that the designed self-triggered control strategy for BCNs is well defined and can make the controlled BCNs be stabilized at the equilibrium point.

Stabilization of Aperiodic Sampled-Data Boolean Control Networks: A Delay Approach

In this article, a novel method for the global stochastic stability analysis of aperiodic sampled-data Boolean control networks (BCNs) is introduced. In our article, the sampling instants of aperiodic sampled-data control (ASDC) are uncertain and only the activation frequencies of the sampling interval are known. Using the semitensor product of matrices, a BCN under ASDC can be transformed into a Boolean network (BN) with stochastic delays. Specifically, the ASDC is represented as a delayed control. Here, the time-varying delay is a random variable generated by a Markov chain and its transition probability matrix can be obtained by the activation frequencies of the sampling interval. Notably, the value of the time-varying delay is less than the upper bound of the sampling interval and when its present value is given, there are only two possible values that can be taken at the next moment. Subsequently, by using the Lyapunov function and augmented method, a sufficient condition for the global stochastic stability of BCNs under ASDC is provided. In particular, the aforementioned results are applicable to the sampled-data control with constant sampling interval. Finally, a numerical example is presented to demonstrate our results.

Stabilization of Delayed Boolean Control Networks With State Constraints: A Barrier Lyapunov Function Method

The stabilization for delayed Boolean control networks (DBCNs) with state constraints is studied. The definition of barrier Lyapunov function (BLF) is put forward for DBCNs with state constraints. The stabilization of DBCNs with state constraints is proved to equal the existence of BLF. Furthermore, the delayed state feedback gain can be constructed via BLF. Finally, the obtained results are verified by networked evolutionary games with memories.

Reduction and Analysis of Boolean Control Networks by Bisimulation

The concept of bisimulation has been recently introduced in the study of Boolean control networks (BCNs). The notion provides a perspective that explains how a state trajectory of one BCN can be paired with a state trajectory of another BCN, and vice versa. The present paper further pursues the notion of bisimulation in BCNs. It first gives necessary and sufficient conditions for checking this notion. The conditions are more general and neater than those previously published; more importantly, they allow us to go further to consider reducing a BCN via bisimulation. A bisimulation-based methodology is then proposed to reduce the size of the state space of a BCN. Applying the reduction methodology allows one to infer control properties of a BCN from those of the corresponding reduced system. Possibilities of applications of the results are illustrated on several Boolean models of real biomolecular systems.

Structural Controllability of Boolean Control Networks under Partial Information

It is worth noting that both nodes’ coupling connections and logical updating functions play a vital role in state evolutions of Boolean networks (BNs). In this paper, a new concept named structural controllability (SC) about Boolean control networks (BCNs) with known partial information on nodes’ connections is studied. Then, by referring to semi‐tensor product (STP) techniques, several types of SC are presented according to different issues of Boolean functions. Thereafter, several necessary and sufficient conditions are derived for SC of BCNs. Finally, a biological model of the lactose operon in Escherichia coli is simulated to show the effectiveness of the main theoretical results.

Optimal Control of Probabilistic Boolean Networks: An Information-Theoretic Approach

The primary challenge with biological sciences is to control gene regulatory networks (GRNs), thereby creating therapeutic intervention methods that alter network dynamics in the desired manner. The optimal control of GRNs with probabilistic Boolean control networks (PBCNs) as the underlying structure is a solution to this challenge. Owing to the exponential growth in network size with the increase in the number of genes, we need an optimal control approach that scales to large systems without imposing any limitations on network dynamics. Furthermore, we are encouraged to use the graphics processing unit (GPU) to reduce time complexity utilizing the easily available and enhanced computational resources. The optimal control of PBCNs in the Markovian framework is developed in this paper employing an information-theoretic approach which includes Kullback-Leibler (KL) divergence. We convert the nonlinear optimal control problem of PBCN to a linear problem by using the exponential transformation of the cost function, also known as the desirability function. The linear formulation enables us to compute an optimal control using the path integral (PI) method. Furthermore, we offer sampling-based methodologies for approximating PI and therefore optimizing PBCN control. The sampling-based method can be implemented in parallel, which solves the optimal control problem for large PBCNs.

Detectability of Delayed Boolean Control Networks Based on Full-Order Observer

This paper investigates the detectability of delayed Boolean control networks (DBCNs) via the Boolean semi-tensor product (BSTP) method. Firstly, three concepts of weak detectability, strong detectability, and detectability for DBCNs are proposed and the dynamics of DBCN are converted into an equivalent algebraic form. Secondly, a full-order observer of DBCNs is constructed to judge the detectabilities of DBCNs by using BSTP. Thirdly, based on the observer, some necessary and sufficient conditions of weak detectability, strong detectability, and detectability for DBCNs are presented. Finally, two examples are illustrated to verify the obtained results.

On quotients of Boolean control networks

In this paper, we focus on the study of quotients of Boolean control networks (BCNs) with the motivation that they might serve as smaller models that still carry enough information about the original network. Given a BCN and an equivalence relation on the state set, we consider a labeled transition system that is generated by the BCN. The resulting quotient transition system then naturally captures the quotient dynamics of the BCN concerned. We therefore develop a method for constructing a Boolean system that behaves equivalently to the resulting quotient transition system. The use of the obtained quotient system for control design is discussed and we show that for BCNs, controller synthesis can be done by first designing a controller for a quotient and subsequently lifting it to the original model. We finally demonstrate the applicability of the proposed techniques on a biological example.

Pinning Stabilization of Boolean Control Networks via a Minimum Number of Controllers.

The stabilization problem of Boolean control networks (BCNs) under pinning control is investigated in this article, and the set of pinned nodes is minimized. A BCN is a Boolean network with Boolean control inputs in it. When the given BCNs cannot realize stabilization under existing Boolean control inputs, pinning control strategy is introduced to make the BCNs achieve stabilization. The Warshall algorithm is introduced to verify the stabilizability of BCNs, then novel computational feasible algorithms are developed to design the minimum number pinning controller for the system. By using our method, the minimum set of pinned nodes can be found with relatively low computational complexity. Finally, the theoretical result is validated using a biological example.