Research On Functional Differential Equations With Discontinuous Right-hand Sides

As far as we know, there is a lot of functional differential equations with discontinuous right-hand sides in many fields such as mechanical engineering, mechanics,automatic control, neural network and biology. Generally, the classical theory of functional differential equation has been shown to be invalid for functional differential equations with discontinuous right-hand sides due to the discontinuous of right-hand side function. In order to analysis and investigate the basic properties and some dynamic behaviors of functional differential equations with discontinuous right-hand sides, we consider the corresponding functional differential inclusions in view of Filippov differential regularization method. Based on theory of functional differential inclusions, we give definition of solutions in the sense of Filippov and the initial value problem for the discontinuous functional differential equation. On this basis, we further investigate a large number of basic properties and complex dynamic behaviors of discontinuous functional differential equations with time-varying and distributed delays via theory of functional differential inclusions.The main topics include the local and global existence of solutions(the extension of solution), periodicity and almost periodicity dynamic behaviors of trajectory solutions in sense of Filippov, different kinds of stability and convergence behaviors for solutions in sense of Filippov(e.g., global exponential stability, synchronization and global dissipativity) and many others.This paper mainly discusses the following two problems. On the one hand, according to some discontinuous phenomenon arising from the actual production and scientific practice, we establish some mathematical models which can be described by discontinuous functional differential equations. And then, based on Filippov differential regularization method, these functional differential equations with discontinuous right-hand sides could be transformed into functional differential inclusions. On the other hand, under the framework of Filippov functional differential inclusions, we discuss various dynamic behaviors of solutions in sense of Filippov.We introduce some new methods involving fixed point theory of set-valued maps,the topological degree theory in set-valued analysis, non-smooth analysis theory,matrix analysis, the matrix measure approach, generalized Lyapunov functional approach etc to study various dynamic behaviors. The main dynamic behaviors include the existence of periodic solution and multiple periodic solutions; the existence and uniqueness of periodic solutions and almost periodic solutions; various kinds of stability and convergence behaviors for solutions in sense of Filippov. This thesis is composed of six chapters.In the first chapter, the historical background and development of the theory of functional differential inclusions and functional differential equations with discontinuous right-hand sides are briefly stated. Meanwhile, the history and research status of the discontinuous neural networks and discontinuous biological 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, which is necessary in the next discussion.In the third chapter, a class of Cohen-Grossberg neural networks with timevarying and distributed delays is investigated, in which the neuron activation functions are modeled by discontinuous functions with single variable(piecewise continuous). The used methods and tools include: functional differential inclusions,fixed point theory of set-valued maps, non-smooth analysis theory, generalized Lyapunov functional approach, etc. Firstly, without assumption the discontinuous neuron activation functions are bounded or satisfy linear growth condition, we investigate the existence of periodic solutions and multiple periodic solutions for delayed Cohen-Grossberg neural networks with discontinuous neuron activations.Secondly, without assumption the discontinuous neuron activation functions are monotonically non-decreasing in R, we discuss the existence, uniqueness and global stability of periodic solutions for delayed Cohen-Grossberg neural networks with discontinuous neuron activations. Meanwhile, we also discuss the global convergence in measure of output solution. Lastly, without assumption the discontinuous neuron activation functions are bounded and monotonically non-decreasing in R,we study the dynamic behaviors of almost periodicity for delayed Cohen-Grossberg neural networks with discontinuous neuron activations. These results on discontinuous neural networks with time-varying and distributed delays are generalize and improve some known results.In the fourth chapter, the synchronization of discontinuous neural networks with time-varying delays is investigated. Based on functional differential inclusions, non-smooth analysis theory, generalized Lyapunov functional approach and some technique of inequalities, the drive-response system of neural network with discontinuous neuron activations can realized global synchronization. It is worthy to point out that the discontinuous neuron activation functions are allowed to be non-monotonically in R, to be super linear growth, even to be exponential growth.And the results are generalize and improve some known results.In the fifth chapter, a class of delayed BAM neural networks is investigated,in which the neuron activation functions are modeled by discontinuous functions with two variables. Firstly, we give the definition of solutions in sense of Filipov for BAM neural networks with discontinuous binary functions via appropriate Filippov differential inclusions. And then, we study the local and global existence of solutions in sense of Filippov, and also study the global dissipativity of solution via functional differential inclusions. Secondly, by designing appropriate discontinuous state feedback controllers, the drive-response system of neural network with discontinuous binary functions can realized global synchronization. Lastly, by applying the topological degree theory in set-valued analysis, we study the existence of periodic solutions for BAM neural networks with discontinuous binary functions.In the sixth chapter, in order to exploit and manage renewable resources efficiency, we consider the harvesting management policies are modeled by general discontinuous functions. And we study Lotka-Volterra competition system with discontinuous harvesting strategy. Based on the theory of functional differential inclusions, fixed point theorem in set-valued analysis and some analysis techniques,we investigate the local and global existence of solutions in sense of Filippov, the existence of periodic solutions for discontinuous biological systems. Finally, some numerical examples are given to illustrate our main results.These research not only enriches and develops some basic theory of functional differential inclusions and functional differential equation with the discontinuous right-hand sides, but also provides effective method and theoretical basis to solve many practical problems with discontinuous character in the areas such as neural networks and biological systems.