It is a common belief that the order of a Boolean network is mainly determined by its attractors, including fixed points and cycles. Using the semi-tensor product (STP) of matrices and the algebraic state-space representation (ASSR) of the Boolean networks, this article reveals that in addition to this explicit order, there is a certain implicit or hidden order, which is determined by the fixed points and limit cycles of their dual networks. The structure and certain properties of dual networks are investigated. Instead of a trajectory, which describes the evolution of a state, the hidden order provides a global horizon to describe the evolution of the overall network. We conjecture that the order of networks is mainly determined by the dual attractors via their corresponding hidden orders. Then these results about the Boolean networks are further extended to the k -valued case.

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.

In this article, a novel integral reinforcement learning (IRL) algorithm is proposed to solve the optimal control problem for continuous-time nonlinear systems with unknown dynamics. The main challenging issue in learning is how to reject the oscillation caused by the externally added probing noise. This article challenges the issue by embedding an auxiliary trajectory that is designed as an exciting signal to learn the optimal solution. First, the auxiliary trajectory is used to decompose the state trajectory of the controlled system. Then, by using the decoupled trajectories, a model-free policy iteration (PI) algorithm is developed, where the policy evaluation step and the policy improvement step are alternated until convergence to the optimal solution. It is noted that an appropriate external input is introduced at the policy improvement step to eliminate the requirement of the input-to-state dynamics. Finally, the algorithm is implemented on the actor-critic structure. The output weights of the critic neural network (NN) and the actor NN are updated sequentially by the least-squares methods. The convergence of the algorithm and the stability of the closed-loop system are guaranteed. Two examples are given to show the effectiveness of the proposed algorithm.

In this article, the set stabilization problem of Boolean control networks (BCNs) using output-feedback controllers is investigated comprehensively. First, a novel method for stabilizing BCNs, namely, the state dynamics matrix-based method, is proposed. Then, based on the above method, the time-invariant output-feedback (TIOF) and the time-variant output-feedback (TVOF) laws are derived. The necessary and sufficient conditions of both TIOF and TVOF stabilization of BCNs are also given. In addition, corresponding algorithms are provided such that all possible TIOF stabilizers and TVOF stabilizers for given BCNs can be designed. Finally, some illustrative examples are demonstrated, where the results are discussed and the effectiveness of the presented method is validated.

In this article, a general approach for directed graph clustering and two new density-based clustering objectives are presented. First, using an equivalence between the clustering objective functions and a trace maximization expression, the directed graph clustering objectives are converted into the corresponding weighted kernel$k$-means problems. Then, a nonspectral algorithm, which covers both the direction and weight information of the directed graphs, is thus proposed. Next, with Rayleigh’s quotient, the upper and lower bounds of clustering objectives are obtained. After that, we introduce a new definition of weak links to characterize the effectiveness of clustering. Finally, illustrative examples are given to demonstrate effectiveness of the results. This article provides a glance at the potential connection between density-based and pattern-based clustering. Compared with other approaches for directed graph clustering, the method proposed in this article naturally avoids the loss of the nonsymmetric edge data because there is no need for any additional symmetrization.

Given a static logical network, partition-based solutions are investigated. Easily verifiable necessary and sufficient conditions are obtained, and the corresponding formulas are presented to provide all types of the partition-based solutions. Then, the results are extended to mix-valued logical networks. Finally, two applications are presented: 1) an implicit function (IF) theorem of logical equations, which provides necessary and sufficient condition for the existence of IF and 2) converting the difference-algebraic network into a standard difference network.

A major challenge to solve problems in control of Boolean networks is that the computational cost increases exponentially when the number of nodes in the network increases. We consider the problem of controllability and stabilizability of Boolean control networks, address the increasing cost problem by partitioning the network graph into several subnetworks, and analyze the subnetworks separately. Easily verifiable necessary conditions for controllability and stabilizability are proposed for a general aggregation structure. For acyclic aggregation, we develop a sufficient condition for stabilizability. It dramatically reduces the computational complexity if the number of nodes in each block of the acyclic aggregation is small enough compared with the number of nodes in the entire Boolean network.

The evolutionarily stable strategy (ESS) of networked evolutionary games (NEGs) is studied. Analyzing the ESS of infinite popular evolutionary games and comparing it with networked games, a new verifiable definition of ESS for NEGs is proposed. Then, the fundamental evolutionary equation (FEE) is investigated and used to construct the strategy profile dynamics (SPDs) of homogeneous NEGs. Two ways for verifying the ESS are proposed: 1) using the SPDs to verify it directly. The SPDs provides complete information about the NEGs, and then necessary and sufficient conditions are revealed. It can be used for NEGs with small size and 2) some sufficient conditions are proposed to verify the ESS of NEGs via their FEEs. This method is particularly suitable for large scale networks. Some illustrative examples are included to demonstrate the theoretical results.

A class of control systems, which are emerged from dynamic games, are considered. Using semi-tensor product of matrices, the set of strategies can be described as a set of matrices. Then the dynamics of such systems can be converted from logical type dynamics into standard discrete-time dynamic systems. Hence, the classical techniques for control systems are applicable to such systems. Semi-tensor product formulation of such systems is investigated. Some related optimal control problems are also investigated.

The problem of solving max-min fuzzy relational equations is investigated. First, we show that if there is a solution, then there is a corresponding solution within the set of parameters [briefly, the parameter set solution (PSS)]. Then, the semitensor product of matrices is used to convert the logical equations into algebraic equations via the vector expression of logical variables. Under this form, every PSS can be obtained. It is proved that all the solutions can be revealed from their corresponding PSS. Some examples are presented to demonstrate the algorithm to solve fuzzy relational equations.

In this paper, a set of data is assumed to be obtained from an experiment that satisfies a Boolean dynamic process. For instance, the dataset can be obtained from the diagnosis of describing the diffusion process of cancer cells. With the observed datasets, several methods to construct the dynamic models for such Boolean networks are proposed. Instead of building the logical dynamics of a Boolean network directly, its algebraic form is constructed first and then is converted back to the logical form. Firstly, a general construction technique is proposed. To reduce the size of required data, the model with the known network graph is considered. Motivated by this, the least in-degree model is constructed that can reduce the size of required data set tremendously. Next, the uniform network is investigated. The number of required data points for identification of such networks is independent of the size of the network. Finally, some principles are proposed for dealing with data with errors.

This paper provides a comprehensive framework for the state-space approach to Boolean networks. First, it surveys the authors' recent work on the topic: Using semitensor product of matrices and the matrix expression of logic, the logical dynamic equations of Boolean (control) networks can be converted into standard discrete-time dynamics. To use the state-space approach, the state space and its subspaces of a Boolean network have been carefully defined. The basis of a subspace has been constructed. Particularly, the regular subspace, Y-friendly subspace, and invariant subspace are precisely defined, and the verifying algorithms are presented. As an application, the indistinct rolling gear structure of a Boolean network is revealed.

This paper investigates the structure of Boolean networks via input-state structure. Using the algebraic form proposed by the author, the logic-based input-state dynamics of Boolean networks, called the Boolean control networks, is converted into an algebraic discrete-time dynamic system. Then the structure of cycles of Boolean control systems is obtained as compounded cycles. Using the obtained input-state description, the structure of Boolean networks is investigated, and their attractors are revealed as nested compounded cycles, called rolling gears. This structure explains why small cycles mainly decide the behaviors of cellular networks. Some illustrative examples are presented.

10.1360/03yf0601

This paper deals with chained form systems with strongly nonlinear disturbances and drift terms. The objective is to design robust nonlinear output feedback laws such that the closed-loop systems are globally exponentially stable. The systematic strategy combines the input-state-scaling technique with the so-called backstepping procedure.