Function Perturbation Of Mix-valued Logical Networks And Pinning Stabilization Of Boolean Networks

With the development of biological and genetic systems in recent years,logical networks have played more and more important roles in both theoretical and practical fields.This paper includes two parts,the first part generalizes results of one-bit perturbation in Boolean networks to that of mix-valued log-ical networks and investigates the impacts of function perturbation on limit sets.Specifically,motivated by the concept of one-bit perturbation in Boolean networks,the definition of general perturbation in mix-valued logical networks is presented and the algebraic expression of the perturbed networks is given by applying semi-tensor product(STP)of matrices.Also,necessary and suf-ficient conditions are proposed for function perturbations’ impacts on limit sets of mix-valued logical networks.Three kinds of function perturbations are listed and their impacts on fixed points and limited circles are discussed by analyzing the changes of transition matrix in the algebraic form.In addition to identifying one perturbation in mix-valued logical networks,a new way to identify multi-perturbation is given.The results of perturbation identification are applied to the WNT5A gene network,which shows broad prospects of ap-plication.In the second part,pinning stabilization of Boolean networks is investi-gated based on semi-tensor product of matrices.By studying the structure of Boolean networks,a necessary and sufficient condition is established to en-sure global stability of Boolean networks.An algorithm is proposed to modify the transition matrix through as few function perturbations as possible.The method of pinning control design is improved,which is embodied in system-atizing and simplifying the pinning control design procedure.Finally,one illustrative example is displayed to show the feasibility of the theoretical re-sults.The thesis consists of five Chapters.In Chapter 1,the background of Boolean networks and mix-valued logical networks is discussed.Present results concerning function perturbations in Boolean networks and mix-valued logical networks,stability and the pinning control are provided.Also,semi-tensor product of matrices and its applica-tions are introduced.At last,the main research issues and organization of this thesis are given.Chapter 2 introduces the theory of semi-tensor product of matrices and some of its useful and important properties.Then,mix-valued logical net-works are introduced.In the framework of semi-tensor product of matrices and the definition of structure matrix,the algebraic form of mix-valued logical networks is presented.In Chapter 3,the definition of general perturbation in mix-valued logical networks is presented and the algebraic expression of the perturbed networks is given by STP.The impacts of function perturbation on fixed points and limit cycles are discussed by analyzing the changes of transition matrix in algebra-ic form.Three kinds of function perturbations are discussed.Perturbation identification is analyzed and related results are applied to the WNT5A gene network.Chapter 4 is mainly about pinning stabilization of Boolean networks.A new necessary and sufficient condition of global stability of Boolean networks is derived and one algorithm is proposed to ensure the network globally stable through as few function perturbations as possible.Also,improved method of pinning control design is proposed,which achieves global stabilization and simplifies pinning control design procedure.Finally,Chapter 5 concludes this thesis.