The Application Of Monotone Control Systems Theory To Some Biology Systems

D.Angeli, together with Professor Eduardo D. Sontag, published an article named Monotone Control Systems in IEEE Transactions On Automatic Control in 2003. This article has laid the foundation for the development of the Monotone Control Systems theory. At present, Monotone control system is one of the hot research subjects in the control field and it plays an important role in biology systems research. This theory provides a method for studying the stability which is of great importance to a system. The application of monotone control system is to make use of the achievements, ideas and methods of monotone control theory to deal with biology systems problems, especially the globally asymptotical stability. Based on some biology systems models, this paper is developed from the following aspects:The first part of this paper presents a brief review about the development of monotone control theory and its application background in the recent years. This paper also introduces a few new research directions in biology system, and the related basic theories, researching methods and some mathematic tools.Second, because nonmonotone systems, existing in large number in nature, are usually decomposed into monotone systems with negative feedback interconnection to be analyzed, the decomposition of some nonlinear system is discussed to gain the sufficient condition of decomposition and global stability. Based on the incidence graph theory of the monotone control systems, through discussing the I/O monotone systems, competitive and cooperative systems, the author puts forward a method, that is, the incidence graph theory can be applied to study the global stability of double-nutrient bacteria growth model, thus to gain the sufficient condition of asymptotical stability of Chemostat model and the multi-stability of the MAPK cascade model. Practically, the author simplifies the method of analyzing the global stability and convergence of a system.In the last part, this paper provides a brief summary about the research in monotone control systems and presents a future development prospect in this field.