Periodic Orbits For Non-smooth Dynamical Systems And Integral Inequalities Systems

In our life, especially in mechanics and engineering sciences, there are many nonsmooth phenomena. For example, the noise of a squeaking spoon in a bowl, or the chattering of machines, the grating of brakes, the percussion of drilling machines, the switches of auto-controller, the oscillation of crystal, etc. Physically speaking, these phenomena are often due to the fact that there are rigid bodies which attach each other moving relative to another, i.e., this kind of non-smooth effects are caused by friction. In addition, uneven impacts for different parts of the systems also generate non-smooth phenomena. In mathematical viewpoint, this kind of systems are called non-smooth dynamical systems, i.e., right-hand sides of the systems are not continuous or differentiable. Because many concepts from classical dynamical systems theory depend on the smoothness, this type of problems can not be solved by visible classical differential dynamical systems theory, thus it is necessary to generalize these concepts and theory to non-smooth dynamical systems. At the same time, because of the particularity of these systems, we need to construct some new theory to solve these problems. In fact, most of these generalizations and new theory are non-trivial.An important kind of non-smooth dynamical systems are relay feedback systems which are widely applied in auto-control. In this paper, we mainly discuss the periodic orbits. In the first chapter, we introduce some relative background knowledge, including the latest results of existence of periodic orbits for the relay feedback systems. Moreover, some integral inequalities and these latest results are stated in order to discuss the stability of periodic orbits. At last, we give our results.In the second chapter, we discuss the existence of unimodal periodic orbits for 2-D SISO relay feedback systems. Astrom, Johansson and so on gave necessary conditions of existence of periodic orbits in 1995. Varigonda and Georgiou obtained the necessary and sufficient condition of existence of periodic orbits under determined relay. An interesting problem is to get the sufficient condition of existence of periodic orbits under unknown relay. It is a very important problem because the control parameter problem can only be solved by using these conditions. We discuss this problem for 2-D systems, and obtain the sufficient condition of existence of periodic orbits. From these conditions we can set control parameter so that systems have periodic orbits, even construct the periodic orbits with certain symmetric. Our difficulty is the same as the others, derived from transcendental functions, in details, some transcendental equations and transcendental inequalities have to be solved. There are only some numerical solutions, in this paper, we obtain the direct relationship of input, output parameter and system parameters by solving the transcendental problem. Our results can be used to verify the system conveniently.Integral inequality is an useful tool of researching asymptotic stability of the closed orbit. In the third chapter, we discuss an integral inequalities system with retard and unknown functions of power. The unknown functions in our paper are more implicit than those in Greene and Pachpatte's results. Based on the idea of Greene and Pachpatte, we get the estimations of unknown functions by combining the Bernoulli inequality with generalized Gronwall inequalities. In the end, we prove the boundedness of the solutions for a fnnctional differential equations system by using our theorem.