Some Topics Research Of Probabilistic Boolean Networks Via Semi-tensor Product

Logic is the science of exploration,elaboration and establishment of effective rea-soning,the origin of which can be traced back to the ancient Greek scholar Aristotle.It was in 1847 that British mathematician Boolean published the article Mathematic Analysis of Logic,which initially lay the foundation of mathematical logic.Since it has constructed Boolean algebra and created the corresponding signal system.The algebra idea began to be applied to solving logic problem.A network can be defined as a graphin mathematic since it is constructed by some nodes and edges,where the nodes represent the states and the edge between two nodes implies some relationship.Boolean networks,a model which has been used to characterize genetic regulatory networks were firstly introduced by Kauffman in 1969.The states of all nodes belong to a Boolean set,which contains two elements 1 and 0 and they update their states based on a Boolean function,whose region is also a Boolean set.Boolean Networks,as one of a discrete time dynamic systems,the most early method used in it is the graph method.As a new subject,graph theory has its advantages in the application of Boolean networks,and the most important one is its low computational complexity.However,it also has some limit in solving some issues.Probabilistic Boolean networks is a kind of more complex but more efficient model compared with Boolean networks.Probabilistic Boolean networks can be viewed as a combination of multiple Boolean networks that are subject to some probability distributions.Meanwhile,the update of the states obey the Markov process.It was in 2009 that professor Cheng Daizhan from Chinese academy of sciences provided a novel method called semi-tensor product(STP)of matrices,which offers an algebraic method to investigate Boolean systems,as well as discrete-time logical systems.Since they can be converted into discrete-time linear systems,which facilitates the analysis of Boolean networks and probabilistic Boolean networks.Concretely,the contributions of this dissertation are as follows:Chapter one contains some signal representations,the definition and properties of STP.Meanwhile,it introduce how to convert the logical expressions into algebra?ic expressions,as well as the algebraic dynamic expression of probabilistic Boolean networks.Chapter two is a section investigating how to use sampled-data state feedback controllers to stabilize a probabilistic Boolean control network.A sufficient condi-tion has been obtained to stabilize the system considered.What's more,it has been compared with sampled-data state feedback controllers for Boolean control networks and state feedback controllers for probabilistic Boolean control networks to show its advantages.At last,the results have been applied in a biology system.Chapter three is about another controllers called pinning control for probabilistic Boolean networks.This system is a bit different since it does not have controllers first.The main purpose is to find the pinned nodes and add the controllers,which has been designed to these nodes so as to stabilize the system considered.Pinning stabilization of probabilistic Boolean networks is different from Boolean networks.This chapter obtains a sufficient condition to pinning stabilize a probabilistic Boolean network and a theorem to find the pinned nodes,design the controllers and add the controllers.Moreover,the minimal number of pinned nodes has been considered,which is the most important and creative part of the thesis.Chapter four is about a farther results of state feedback stabilizers of probabilistic Boolean control networks and a corresponding global optimization problem.In this chapter,we construct an algorithm to find the whole state feedback stabilizers of a probabilistic Boolean control networks,which degenerate the results of Boolean control networks.And then another algorithm is given to find the optimal value of the problem considered.At last,the results have been applied in game theory.Chapter five focus on the controllability and reachability with probability p for a probabilistic Boolean control network.It is the unique property of probabilistic Boolean control network and degenerate the results of Boolean control networks.Chapter six is an application on game theory,which is also an exploration of how to find the minimum pinned nodes.What's more,the replicator dynamic equation has been converted to an algebraic form.