Search

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.

For a Boolean network with disturbances and outputs, a necessary and sufficient graphic condition for the original disturbance decoupling is proposed, and it reveals that the outputs are unaffected by the disturbances if and only if the vertex-colored state transition graph has a concolorous perfect vertex partition (CP-VP). In addition, if the CP-VP is an equal partition, a corresponding system decomposition is implemented. Furthermore, an algorithm is designed to check the existence of a CP-VP.

This paper devotes to establishing a bridge between asymptotical stability of a probabilistic Boolean network (PBN) and a solution to its induced equations, which are induced from the PBN's transition matrix. By utilizing the semitensor product technique routinely, the dynamics of a PBN with coincident state delays can be equivalently converted into that of a higher dimensional PBN without delays. Subsequently, several novel stability criteria are derived from the standpoint of equations’ solution. The most significant finding is that a PBN is globally asymptotically stable at a predesignated one-point distribution if and only if a vector, obtained by adding 1 at the bottom of this distribution, is the unique nonnegative solution to PBN's induced equations. Moreover, the influence of coincident state delays on PBN's asymptotical stability is explicitly analyzed without consideration of the convergence rate. Consequently, such bounded state delays are verified to have no impact on PBN's stability, albeit delays are time-varying. Based on this worthwhile observation, the time computational complexity of the aforementioned approach can be reduced by removing delays directly. Furthermore, this universal procedure is summarized to reduce the time complexity of some previous results in the literature to a certain extent. Two examples are employed to demonstrate the feasibility and effectiveness of the obtained theoretical results.

This paper presents some results on stability and $l_1$ gain analysis of Boolean networks with Markovian jump parameters. A necessary and sufficient condition for global stability of the concerned Boolean networks is given in terms of linear programming by utilizing the semi-tensor product of matrices and some properties of linear positive systems. Then, the definition of Lyapunov function for stochastic Boolean networks is presented and Lyapunov theorem is derived. Moreover, an $l_1$ gain problem for stochastic Boolean networks with external disturbances is formulated and solved by a sufficient condition. Examples are shown to illustrate the effectiveness of the obtained results.

This note investigates the output regulation of Boolean control networks (BCNs) by using the semi-tensor product of matrices, and presents a number of new results. Firstly, based on an algebraic state-space representation of BCNs, a necessary and sufficient condition is presented for the solvability of the output regulation problem. Secondly, an effective method is proposed for the control design of the output regulation problem by using an augmented system approach. The study of an illustrative example shows that the obtained new results are effective in dealing with the output regulation of BCNs.

This note investigates the logical matrix factorization with application to the topological structure analysis of Boolean networks. First, the concepts of both factorization and rank are defined for logical matrices, and two factorization problems are then studied. Using nonsingular logical matrix transformations, several necessary and sufficient conditions for the factorization of a given logical matrix are presented. Second, the logical matrix factorization is applied to the topological structure analysis of a given Boolean network, and a size-reduced structure-equivalent logical network is constructed for the given system. It is shown that the topological structure (including all the fixed points and cycles) of the resulting size-reduced logical network is the same as that of the original Boolean network. The study of an illustrative example shows that the new results presented in this note are effective in analyzing the topological structure of Boolean networks.

This paper considers the infinite horizon optimal control of logical control networks, including Boolean control networks as a special case. Using the framework of game theory, the optimal control problem is formulated. In the sight of the algebraic form of a logical control network, its cycles can be calculated algebraically. Then the optimal control is revealed over a certain cycle. When the games, using memory μ >; 1 (which means the players only consider previous μ steps' action at each step), are considered, the higher order logical control network is introduced and its algebraic form is also presented, which corresponds to a conventional logical control network (i.e., μ = 1 ). Then it is proved that the optimization technique developed for conventional logical control networks is also applicable to this μ-memory case.

Boolean networks have recently been attracting considerable interest as computational models for genetic and cellular networks. We consider a Mayer-type optimal control problem for a single-input Boolean network, and derive a necessary condition for a control to be optimal. This provides an analog of Pontryagin's maximum principle for single-input Boolean networks.

Disturbance decoupling problem (DDP) of Boolean control networks is considered. Using semi-tensor product of matrices and the matrix expression of logical functions, a working procedure is proposed to solve the problem. This procedure consists of two key design steps. First, how to convert a system into an output-friendly coordinate frame. An algorithm is provided to calculate the output-friendly subspaces. Secondly, it was shown how to find proper controllers to solve the problem if it is solvable. A state variable separation form is introduced to guide the design of controllers. Based on the design technique, necessary and sufficient conditions are obtained for the solvability of DDP.

A new matrix product, called semi-tensor product of matrices, is reviewed. Using it, a matrix expression of logic is proposed, where a logical variable is expressed as a vector, a logical function is expressed as a multiple linear mapping. Under this framework, a Boolean network equation is converted into an equivalent algebraic form as a conventional discrete-time linear system. Analyzing the transition matrix of the linear system, formulas are obtained to show a) the number of fixed points; b) the numbers of cycles of different lengths; c) transient period, for all points to enter the set of attractors; and d) basin of each attractor. The corresponding algorithms are developed and used to some examples.

We propose a new concept, stability in distribution (SD) of a probabilistic Boolean network (PBN), which determines whether the probability distribution converges to the distribution of the target state (namely, a one-point distributed random variable). In a PBN, stability with probability one, stability in the stochastic sense, and SD are equivalent. The SD is easily generalized to subset stability, i.e., to set stability in distribution (SSD). We prove that the transition probability from any state to an invariant subset (or to a fixed point) is nondecreasing in time. This monotonicity is an important property in establishing stability criteria and in calculating or estimating the transient period. We also obtain a verifiable, necessary, and sufficient condition for SD of PBNs with independently and identically distributed switching. We then show that SD problems of PBNs with Markovian switching and PBN synchronizations can be recast as SSD problems of Markov chains. After calculating the largest invariant subset of a Markov chain in a given set by the newly proposed algorithm, we propose a necessary and sufficient condition for SSDs of Markov chains. The proposed method and results are supported by examples.

In this paper, stability and stabilization of Boolean networks with stochastic delays are studied via semi-tensor product of matrices. The stochastic delays, randomly attaining finite values, are modeled by Markov chains. By utilizing an augmented method, the considered Boolean network is first converted into two coupled Markovian switching systems without delays. Then, some stochastic stability results are obtained based on stability results of positive systems. Subsequently, the stabilization of Boolean networks with stochastic delays is further investigated, and an equivalent condition for the existence of feedback controllers is provided in terms of a convex programming problem, which can be easily solved and also conveniently applied to design controller gains. Finally, numerical examples are given to illustrate feasibility of the obtained results.