Some Topics Research On Stabilization And Synchronization Of Boolean Networks Via STP Method

Logic systems are often used to simulate gene regulatory networks when study-ing system biology.In 1969,Kauffman first introduced logical relation into Boolean function,that is Boolean networks(BNs),which are proposed to simulate and analyze gene regulatory networks.According to different research environment,the variables in BNs can be abstracted into different nodes interactions in gene regulatory network-s.In BNs,there are two ways to represent each gene node at each moment:1(or 0)corresponds to the on(or off),and the value of each node at the current time is determined by the logical relationship of the node adjacent to it in the system at the previous moment,so the BN is a discrete of the time logic system."When the control variables are added into the BNs,it becomes Boolean control networks(BCNs).At present,the research on BNs by experts and scholars have made great achievements,which is a great improvement on the study of gene regulation networks.As the math-ematics tool for BNs is more suitable for semi-tensor product(STP),So we use STP to transform the logical relation in BNs into algebraic form,and then conduct further research.To be concrete,the contributions of this dissertation are as follows:Chapter 1 describes the research background of this paper.First,the development of BNs and probabilistic Boolean networks(PBNs),and the definition and properties of semi-tensor products are introduced.Then there are some preparatory knowledge,including the matrix representation of the logical function in BNs and its algebraic expression.Chapter 2 studies the complete synchronization problem of two coupled BNs.Synchronization is a very important and basic topic in control theory,which can explain and solve some biological phenomena,such as Escherichia coli lac operon,coupled vibration problem in the cell cycle and apoptosis.We use the sampled-data state feedback controller to synchronize the two coupled BNs in finite time.First,based on the analysis of the drive system in the coupled BNs,the synchronization path of two coupled BNs is determined(i.e.,the number and size of the limit cycles),so as to find out the sampling points and sampling periods,and design the sampled-data state feedback controller.Finally,the important conclusions are applied to the lac operon problem of Escherichia coli.Chapter 3 studies the robust sampled-data control invariance of BCNs.Firstly,based on the sampled-data state feedback controller,we obtain some necessary and sufficient conditions for checking whether or not a given non-empty state set is a robust s mpled-data control invariant sat.Then,for agiven non-empty state set,a sampled-data state feedback controller is designed through analyzing the sampling points,so that the set is a robust sampled-data control invariant set.Finally,a biological example is provided to illustrate the effectiveness of the results.In Chapter 4,the robust control invariance of probabilistic Boolean control net-works(PBCNs)based on event-triggered control is studied.Firstly,the transition probability matrix of PBCNs is analyzed,the event-triggered controller is introduced and the triggering conditions are given.Secondly,using the transition probability ma-trix,we obtain some necessary and sufficient conditions for checking whether or not a given non-empty state set is a robust event-triggered control invariant set.Thirdly,for a given non-empty state set,the event-triggered controller is designed by altering the columns of the initial time-variant event-triggered matrix,so that the set is a robust event-triggered control invariant set.In Chapter 5,we investigate the output feedback set stabilization of context-sensitive probabilistic Boolean control networks(CS-PBCNs).First,we analyze the CS-PBCNs based on PBCNs and give the concept of set stabilization.Then,We design an algorithm for largest control invariant set with probability 1.Based on the analysis of set stabilization,we obtain the necessary and sufficient condition for s-stabilization.Then we design the output feedback controller to make the CS-PBCNs to be s-Stabilization with Probability 1.Finally,we present an example of studying metastatic melanoma to show the validity of our main results.