Stabilization of Boolean control networks

The paper deals with problems on the stability and stabilization of Boolean networks with respect to a given part of variables characterizing the Boolean network. Using the semi-tensor product of matrices and the matrix expression of logic, the dynamics of a Boolean (control) network can be converted to discrete time linear (bilinear) dynamics called the algebraic form of the Boolean (control) network. The two main results are as follows: (i) From the algebraic form of a Boolean (control) network, two partial stability definitions are introduced. One is partial stability with respect to another variable set, and the other is partial stability with respect to arbitrary initial states. (ii) Based on the algebraic form, necessary and sufficient conditions for the stability of global partial states with respect to the variable set and arbitrary initial states are obtained. Examples are provided to illustrate the effectiveness of our results.

Global robust stability and stabilization of Boolean network with disturbances

The ring structure and Boolean variables are investigated and they are used to define the product of Boolean matrices. The logical operators have also been extended to the operators on Boolean matrices. Based on these, the vector distance between Boolean matrices is defined. Some properties are revealed. Finally, the vector distance is used to the analysis of the stability of Boolean networks and the stabilization of Boolean control networks.

Ministry in 2005, and the National Prize of Natural Science of China in 2008. Currently, he is Associate Editor IMA Journal of Math Control and Inform., and a Technical Committee member of IFAC (TC2.3).This paper investigates the state feedback stabilization of Boolean control networks (BCNs) with state and input constraints by using the semi‐tensor product of matrices. Firstly, a kind of constrained input‐state incidence matrix is proposed for constrained BCNs, which contains all the reachability information of the constrained systems. Secondly, based on the constrained input‐state incidence matrix, a necessary and sufficient condition is presented for the stabilization of constrained BCNs via state feedback. Thirdly, a general procedure is proposed to design all possible minimum‐time state feedback stabilizers. The study of an illustrative example shows that the new results obtained are effective in analyzing the stabilization of constrained BCNs.

The traditional set stability of Boolean networks (BNs) refers to whether all the states can converge to a given state subset. Different from the existing results, the set stability investigated in this paper is whether all states in a given initial set can converge to a given destination set. This paper studies the set stability and set stabilization avoiding undesirable sets of BNs and Boolean control networks (BCNs), respectively. First, by virtue of the semi-tensor product (STP) of matrices, the dynamics of BNs avoiding a given undesirable set are established. Then, the set reachability and set stability of BNs from the initial set to destination set avoiding an undesirable set are investigated, respectively. Furthermore, the set stabilization of BCNs from the initial set to destination set avoiding a given undesirable set are investigated. Finally, a design method for finding the time optimal set stabilizer is proposed, and an example is provided to illustrate the effectiveness of the results.

Partial controllability of system is a special definition of controllability with respect to only a pre-specified subset of a system's state variables. The paper considers problems on the controllability of Boolean control networks with respect to a given part of variables that characterize the Boolean network. Using a semi-tensor product of matrices and the matrix expression of logic, the dynamics of a Boolean control network are converted to discrete time bilinear dynamics. First, the definition of partial controllability of Boolean control networks is introduced. Then, based on the algebraic form, necessary and sufficient conditions for partial state variable controllability with respect to a variables set are obtained. In addition, an illustrative example is provided to construct a proper control sequence with which to steer the partial variables' states to those desired.

This paper investigates pinning control for stabilizing Boolean networks with time delays. The purpose of this paper is to design a control algorithm to select a fraction of key nodes for a Boolean network with time delays such that the considered Boolean network can achieve stability. First, by changing the columns of the transition matrix of a Boolean network, the stabilization of a Boolean network is investigated. By doing this, we can obtain a new transition matrix for which we can select the key nodes. Furthermore, an algorithm is then proposed to design a pinning controller. Finally, an example of pinning control of coupled oscillators in the cell cycle is presented to show the effectiveness of the proposed results.

Nominal Boolean network is a kind of Boolean network, which is obtained from a Boolean control network with all controls disconnected. In this paper, the relationship between the transition matrix of the original Boolean control network and the transition matrix of the corresponding nominal Boolean control network is investigated. Due to the fact that transition matrices can be derived from structure matrices, the relationship between structure matrices of the original Boolean control network and structure matrices of the nominal Boolean network is studied. Via constant values corresponding to disconnected controls, structure matrices of the nominal Boolean network can be derived from structure matrices of the original Boolean control network. Finally, a concrete algorithm is presented to derive transition matrices of nominal Boolean networks.

Minimum energy control and optimal-satisfactory control of Boolean control network

Further results on feedback stabilization control design of Boolean control networks

Controllability and observability of singular Boolean control networks are proposed in this paper from its algebraic form via semi-tensor product. First, the input-state incidence matrix of singular Boolean control networks is presented based on which the generalized input-state incidence matrix is also defined. Second, a necessary and sufficient condition for the controllability of singular Boolean control networks is given, as well as an algorithm to design controls which drive a state to a reachable state. At last, similar to Boolean control networks, a sufficient condition of observability of singular Boolean control networks is obtained.

On the ensemble controllability of Boolean control networks using STP method

State feedback controller design for synchronization of master–slave Boolean networks based on core input-state cycles

Note that state transition space of Boolean networks is determined by both network structure (node’s coupling relationships) and logical functions (update rules for nodes). This paper studies structural controllability of Boolean control networks (BCNs) under partial information, where only part of information about connections among nodes are known. Then, using semi-tensor product of matrices and algebraic forms of BCNs, two types of structural controllability are presented according to different cases of logical functions. Subsequently, certain sufficient and necessary criteria are established for structurally controllablility of BCNs with partial information. Finally, a numerical example is provided to illustrate the effectiveness of the obtained theoretical 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.