Theory And Application Of Initial Valued Problems And Stability Of Delayed Differential Inclusions

Differential inclusions are an important branch of nonlinear analysis theory, which originated from the need of development of control theory and differential equations with discontinuous right hand sides. They have been extensively applied not only to self-adaptive control systems, but also to dynamic systems derived from economics, sociology and biological science. In recent years much attention has been attracted in nonlinear delayed systems with discontinuous right hand sides, since delays are usually unavoidable in many practical problems even though in systems with speed of light. However, a great number of theories and methods for dynamical systems and functional differential equations (FDEs) couldn't be applied straightforward to delayed systems with discontinuous right hand sides, so it is necessary to establish corresponding theories and methods. Reviewing of recent literature, delayed differential inclusions are a useful tool for delayed systems with discontinuous right hand sides. Hence, it is important to develop the theory of delayed differential inclusions. In addition, the need from the time delay control theory is another reason for the development.Based on the theory of set-valued and nonsmooth analysis, we study the initial valued problems (IVP) and stability theory of delayed differential inclusions. As applications of the theory, we investigate Hopfield neural networks with discontinuous activations and piecewise linear biological networks with autoregulation.In the aspect of the IVP of delayed differential inclusions, a definition of solutions is introduced in the sense of Filippov by the concept of "locally absolutely continuous", which is a generalization of the definitions in ordinary differential equations (ODEs) and FDEs. By applying Kakutani fixed point theorem, we obtain a result for existence of solutions. Furthermore, some results are given on uniqueness and continuous dependence of solutions. As to the continuation of solutions, we get several result, especially the result of entire existence of solutions.The problem of stability of delayed differential inclusions is another key research topic in this thesis. Based on ODEs and FDEs, some definitions about the stability of delayed differential inclusions are introduced. By the generalized chain rule, we obtain some local and global results about stability, asymptotical stability, uniform stability, uniform asymptotical stability and exponential stability. These results are the improvement or generalization of the stability theory of ODEs, FDEs and ordinary differential inclusions, and fill some gaps in stability theory of delayed differential inclusions.As the application of the theory of delayed differential inclusions to neural networks, we give an insight into dynamical behavior of a class of Hopfield neural networks with discontinuous activations. Without the assumption of monotonicity and boundedness of the activation functions, we obtain some results of existence and global asymptotical stability for equilibria. Furthermore, the existence of periodic solutions is derived by Leray-Schauder alternative theorem without monotonicity of the activation functions and the stability of periodic solutions is investigated by generalized Lyapunov method. Moreover, we study the dynamical behavior of a class of delayed Hopfield neural networks with discontinuous activations and obtain some results about the existence and stability of equilibria for the autonomous systems as well as the existence and stability of periodic solutions for time-varying periodic systems.Applying the the theory of delayed differential inclusions to biological science, we study dynamical behavior of a class of piecewise linear biological networks with autoregu-lation and several thresholds per variable. A state transition graph, in which vertexes can be represented by coordinates of a space with finite dimension, is introduced to show the message of transition among threshold hyperplanes of any dimensions. Based on the method of Filippov, we discuss the existence and geometrical properties of different types of equilibria. Moreover, the analysis of closed trajectories is carried out, especially close trajectories with sliding and homoclinic trajectories with sliding.