Dynamics Research For Several Classes Of Discontinuous Systems Based On Differential Inclusion

As far as we know, functional differential inclusion (FDI) system is considered as a generalization of the system described by functional differential equation. Since given vector field is no longer smooth or globally Lipschitz, the classical solutions are not guaranteed when discontinuities occur in a dynamical system defined by some differential equation. In this thesis, in order to deal with and investigate the fundamental questions of solutions and dynamical behaviors for delayed differential equations with discontinuous right-hand sides, we give a new definition of solutions in the sense of Filippov and the initial value problem for the discontinuous delayed differential equation by Filippov regularization. On this basis, we further investigate a large number of basic questions about the properties of Filippov solutions and stability behaviors for time-varying delayed differential equations with discontinuous right-hand sides or functional differential inclusions via the Filippov framework. The topics include the local existence of solutions, the extension of solution, different kinds of stability and convergence behaviors for solutions (e.g., global asymptotic or exponential stability, synchronization and quasi-synchronization, global dissipativity and robust stability) and many others. By doing so, we extend and generalize the theory of delayed differential equa-tions with discontinuous right-hand sides or functional differential inclusions to a certain extent. Then, we apply the theoretical results obtained to various fields of science and engineering including neural networks, automatic control and en-gineering, population biology and so on. On the one hand, according to the discontinuous phenomena in reality, we formulate and study different kinds of mathematical models described by differential equations with discontinuous right-hand sides. Moreover, by constructing the Filippov set-valued maps, these dif-ferential equations with discontinuous right-hand sides could be transformed into differential inclusions, which are also called as the Filippov regularization. On the other hand, under the framework of Filippov differential inclusions, we introduce some new methods involving fixed point theory of set-valued maps, generalized Lyapunov approach, the topological degree theory in set-valued analysis, matrix analysis, the matrix measure approach and generalized inequalities, non-smooth analysis etc to investigate various dynamic behaviors. These dynamic behaviors include (positive) equilibrium points,(positive) periodic solutions, the stability of solution, etc. This thesis is composed of five chapters. In the first chapter, the historical background and development of the theory of differential inclusions and discontinuous differential dynamical systems are briefly stated. Meanwhile, the history and research status of the discontinuous biological dynamical systems and discontinuous neural networks dynamical systems are also introduced. Finally, the main content and structure arrangements of this thesis are summarized.In the second chapter, we give some basic theoretical knowledge. The rest of this thesis presents our main research results and is organized as follows.In the third chapter, we first generalize a class of inequalities:generalized Halanay’s inequalities. Such a class of new inequalities will be very effective to deal with the stability for time-varying delayed differential equations with discontinuous right-hand sides or functional differential inclusions. Second, we give the standard definitions of Filippov solutions for functional differential inclusions with single and multiple time-varying delay(s), respectively. Then, we mainly discuss the extension problem of the Filippov solution with given initial condition and obtain two important theorems on viability. Finally, by virtue of the generalized Halanay’s inequalities, we study the stability of Filippov solutions in the sense of perturbation for functional differential inclusions with time-varying delays. We not only analyze the robust dissipativity and global robust stability for FDI with perturbation by constructing radially unbounded auxiliary functions, but also discuss the robust quasi-synchronization for FDI with perturbation by introducing appropriate state-feedback controller.In the fourth chapter, a general class of delayed Lotka-Volterra competition systems is considered, where the harvesting management policies are modeled by non-Lipschitz functions or by discontinuous functions. Based on differential inclu-sions theory introduced by Filippov, cone expansion and compression fixed point theorem of set-valued maps and nonsmooth analysis theory with generalized Lya-punov approach, we establish a series of useful criteria on existence, uniqueness and global asymptotic stability of the positive periodic solution for the delayed Lotka-Volterra competition systems with discontinuous right-hand sides. Moreover, we also discuss the global convergence in measure of harvesting solution. Finally, we provide some corollaries and numerical examples to show the applicability and effectiveness of the main results.In the fifth chapter, we investigate the dynamical behaviors of three classes of discontinuous delayed neural networks. Our methods to be used involve the application of set-valued version of fixed point theorem, the generalized Lyapunov theory, the topological degree theory in set-valued analysis, the M-matrix theory, the matrix measure approach, some generalized inequalities techniques, the frame-work of Filippov differential inclusions theory and so on. Firstly, without assuming the boundedness or monotonicity of the discontinuous neuron activation functions, we perform a thorough analysis of the existence, uniqueness and global exponen-tial stability of the periodic solution for a general class of time-varying delayed cellular neural networks with discontinuous right-hand sides. Furthermore, we also study the global convergence of the output and the convergence in finite time of the state for the autonomous systems corresponding to the non-autonomous neural networks. Next, we study the periodic dynamics of a class of neural net-works with discontinuous activation and general mixed time-delays involving both time-varying delays and distributed delays. Finally, we formulate and investigate a class of memristor-based BAM neural networks with time-varying delays. Some sufficient conditions are given to guarantee the global dissipativity and the exis-tence of positive periodic solutions for the memristive BAM neural network system. The obtained results extend and improve some previous works on discontinuous or continuous neural network dynamical systems.