Analysis and control of general logical networks – An algebraic approach

Since Boolean network is a powerful tool in describing the genetic regulatory networks, accompanying the development of systems biology, the analysis and control of Boolean networks have attracted much attention from biologists, physicists, and systems scientists. From mathematical point of view, the dynamics of a Boolean (control) network is a discrete-time logical dynamic process. This paper surveys a recently developed technique, called the algebraic approach, based on semi-tensor product. The new technique can deal with not only Boolean networks, which allow each node to take two values, but also k-valued networks, which allow each node to take k different values, and mix-valued networks, which allow nodes to take different numbers of values.The paper provides a comprehensive introduction to the new technique, including (1) mathematical background of this new technique – semi-tensor product of matrices and the matrix expression of logic; (2) dynamic models of Boolean networks, and general (multi- or mix-valued) logical networks; (3) the topological structure of Boolean networks and general networks; (4) the basic control problems of Boolean/general control networks, which include the controllability, observability, realization, stability and stabilization, disturbance decoupling, identification and optimization, etc.; (5) some other related applications.