摘要
在我们生活中，特别是在力学和工程技术领域，经常会遇到非光滑现象：比如勺子在碗里刮擦时发出刺耳的噪声，或者机器的振动、刹车的摩擦、打孔机的冲击、自动控制器的转换以及晶体的振荡等等。从自然规律来讲，这些现象通常都是因为相互接触的两个刚体之间发生相对移动而造成的，即由摩擦引起的。另外，系统不同部分所受冲击不均匀也会产生非光滑现象。从数学的观点来看，这一类系统称为非光滑动力系统，即这些系统的右端是不连续或不可微的。由于许多来自于经典动力系统的概念都依赖于光滑性，现有的经典微分动力系统理论尚不能直接用于解决非光滑动力系统的问题，这就需要将相关概念及理论推广到非光滑动力系统中。同时，由于这类系统的特殊性，自然也需要发展一些新理论来解决实际问题。而事实上大部分推广及相关新理论都不是平凡的。
继电反馈系统是一类重要的非光滑动力系统，被广泛的应用于自动控制领域。本文主要研究继电反馈系统周期轨的问题。在绪论中我们将对相关的背景知识进行介绍，包括前人对继电反馈系统周期轨存在性所取得的进展。为以后深入研究周期轨稳定性，我们还介绍积分不等式的一些进展。绪论的最后部分介绍了本文的主要工作。
在第二章我们研究二维继电反馈系统单峰周期轨的存在性。之前Astrom、Johansson等人给出了周期轨存在的必要条件。Varigonda和Georgiou在确定的切换条件下给出了周期轨存在的充要条件。一个有趣的问题是：在待定的切换条件下周期轨存在的充分条件是什么?这个问题的重要性在于只有在待定的切换条件下寻找的条件才能对控制参数作出回答。我们针对二维系统在待定的切换条件下研究这个问题，得到了周期轨存在的充分条件。这个条件使我们可以设定控制参数使系统具有周期轨，甚至构造出具有一定对称性的周期轨。同前人遇到的困难一样，我们的困难也来自超越函数，即要求解一个联立的超越方程和超越不等式。前人对此只做了数值计算，而本文从这个超越问题中给出了系统参数和输入、输出参数的直接关系，使获得的充分条件更易于验证。
研究周期轨的渐近稳定性的一个重要工具是积分不等式。在本文第三章中，我们研究了一个带有时滞的、未知函数项带有方幂的积分不等式组。和前人如Greene和Pachpatte等的工作相比，我们的困难是未知函数更具有隐含性。我们沿着Greene和Pachpatte的思想，结合Bernoulli不等式和推广的Gronwall不等式，给出了未知函数的估计。这个结果还被用于证明一个泛函微分方程组解的有界性。
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.
引文
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