Observability And Stabilization Of Probabilistic Boolean Networks Based On Semi-tensor Product Of Matrices

With the rapid development of biological systems and medicine,the analysis and application of Boolean networks have become a research hot topic in the field of control.By using semi-tensor product of matrices,the Boolean network and its generalized forms,such as Boolean control networks and probabilistic Boolean networks can be equivalently transformed into linear forms.Under this setting,many fundamental control problems of Boolean networks can be solved.In this thesis,the observability problem and stabilization problem of probabilistic Boolean networks and their generalizations,Markovian jump Boolean networks,are discussed.The main results are presented in the following seven chapters.In chapter 1,the research background and status of Boolean networks and probabilistic Boolean networks are introduced.The origin and the application of semi-tensor product,which is the main mathematical tool for the research of Boolean networks in this thesis.In chapter 2,the definition and basic properties of semi-tensor of matrix are introduced.Meanwhile,the processes of equivalent transformations of Boolean networks and probabilistic Boolean networks into algebraic forms are deduced in detail.In chapter 3,the observability of probabilistic Boolean networks is inves-tigated.By using semi-tensor product of matrices,a matrix sequence is firstly constructed for calculating the distinguishable state pairs,based on which,a nov-el criterion for examining the observability of probabilistic Boolean networks is given.Then,an algorithm has been proposed using the above criterion.Com-pared with the existing results,the proposed method in this paper reduces the computation complexity.Chapter 4 is devoted to the observability of probabilistic Boolean control networks.Firstly,four types of observability are extended from Boolean control networks to probabilistic Boolean control networks are introduced.Then,the parallel expanded system are proposed.By reconstructing the structure matrices of this expanded system,we obtained a reconstructed system.Finally,the observ-ability of probabilistic Boolean control networks is solved by this reconstructed system.In chapter 5,the set stabilization problem of Markovian jump Boolean con-trol networks is investigated via linear programming approach.First,the concep-tions of set stabilization and control invariant subset are extended to Markovian jump Boolean control networks.Then based on the algebraic expression of Marko-vian jump Boolean control network,the necessary and sufficient condition for set stabilization problem is proposed by a linear programming problem,which is sim-ple to solve.Moreover,by solving this linear programming problem,an algorithm for designing a state feedback controller is developed.In chapter 6,the set stabilization problem for Markovian jump Boolean con-trol networks by state feedback control is addressed via optimal approach.Firstly,the set stabilization of a Markovian jump Boolean control network is converted into the set stabilization of a augmented system with input constraints.Then,by utilizing the relation between stabilization problem and average cost optimal problem of Boolean control networks,the necessary and sufficient condition for the set stabilization of this augmented system is put forward.Moreover,a policy iterative algorithm is deduced for checking the set stabilization of a Markovian jump Boolean control network and designing the state feedback control.Chapter 7 concludes the results on probabilistic Boolean networks and lists some prospects of study about probabilistic Boolean networks.