Controllability Of Boolean Control Networks

Boolean networks are a kind of particular discrete dynamical systems expressed by bi-valued logical variables and logical functions. The concepts of Boolean network was proposed by Kauffman, an American theoretical biologist, to describe the dynam-ical behavior of genetic regulation networks. A Boolean network with external inputs is called a logical control network. In a long time, the research work focuses on simu-lations and computation for some practical Boolean networks for pursuing, modelling or predicting some new biological phenomena. However, the theoretical analysis for Boolean networks had been relatively weak due to lacking efficient mathematical tools to deal with logical expressions. In recent years, an important contribution to Boolean network theory lies in successfully introducing semi-tensor product of matrices into the analysis of Boolean control networks, which has provided a general theoretical frame. In this frame, lots of traditional control problems are extended to Boolean control systems, such as controllability, observability, stability and stabilization. Con-trollability problem is one of the foundational problems of control systems. It is very meaningful to research controllability criteria for some Boolean control networks.Based on the existing methods and results of Boolean control networks, this the-sis further researches the controllability problem for Boolean control networks. First, for Boolean control networks without delay, a new necessary and sufficient condition is obtained. Second, for Boolean control networks with one-step input-delays, a test-ing condition for controllability is proposed. Third, an array product is proposed to construct a criterion for the controllability of Boolean control networks with multi-step input-delays. Finally, for the case that both state-delays and input-delays exist, a connection is established between the controllability of the delayed Boolean net-works and that of the Boolean networks without delays, which subsequently solves the controllability problem of the delayed Boolean networks.