A Graph-theoretical Approach Of Optimal Control Problem Of Boolean Control Networks And Nominal Boolean Networks

Motivated by the development of systems biology,the investigation of Boolean networks has become an important topic.This paper investigates three kinds of problems including the relationship between transition matrices of Boolean control networks and nominal Boolean networks,and the relation-ship between topological structures of Boolean control networks and nominal Boolean networks,and the graph-theoretical approach of optimal control prob-lem of Boolean control networks.Chapter 1 is the introduction of the thesis.Backgrounds about Boolean networks,Boolean control networks,nominal Boolean networks and the semi-tensor product are introduced firstly.Then motivations of our research are presented.Next,main work of the thesis is summarized.Finally,organization of this paper is given.Chapter 2 is the preparation of the thesis.The definition and correspond-ing properties of the semi-tensor product are specified.Then some related results about Boolean control networks and nominal Boolean networks are presented.Finally,the Boolean algebra and the Dijkstra's algorithm are in-troduced.In Chapter 3,the relationship between transition matrices of Boolean control networks and transition matrices of nominal Boolean networks is in-vestigated.Via constants corresponding to disconnected controls,structure matrices of nominal Boolean networks are derived from structure matrices of Boolean control networks.Then transition matrices of nominal Boolean networks are calculated from structure matrices of nominal Boolean network-s.Finally,a concrete algorithm is presented to derive transition matrices of nominal Boolean networks.In Chapter 4,the comparison of topological structures between Boolean control networks and nominal Boolean networks is investigated.First,tran-sient periods and attractor basins of Boolean control networks are discussed,and three necessary and sufficient conditions are derived for determining tran-sient periods and attractor basins.From two ways of attractors and tran-sient periods,topological structures of Boolean control networks and nominal Boolean networks are compared,and the existences of different relations are illustrated via some examples.Finally,four necessary and sufficient conditions are proposed for judging relations of topological structures between Boolean control networks and nominal Boolean networks.Chapter 5 studies two kinds of optimal control problems of Boolean control networks.For these two optimal control problems,optimal control sequences are illustrated to exist.Via the investigation of controllability of Boolean con-trol networks,corresponding weighted directed graphs are built.Then based on the Dijkstra's algorithm for the shortest path problem,graph-theoretical approaches are proposed to find the optimal control sequences.Chapter 6 is a concise summary of the thesis.