【作者】 郭飞；

【导师】 傅希林；

【作者基本信息】 山东师范大学 ， 应用数学， 2003， 硕士

【摘要】 本文，我们主要讨论了脉冲混合微分系统，即 其中f∈C[R_+×R~n×R~m，R~n]，I_k∈C[R~n，R~n]，λ_k∈C[R~n，R~m)，k=0，1，2，…，少量内容涉及其摄动系统．脉冲混合微分系统来源于实际生产研究中的物理模型，是脉冲微分系统的推广。 随着工业的发展，出现了许多新的物理模型，其中有一类仅用脉冲微分系统无法恰当地描述，这时，就需要考虑具有瞬时脉冲摄动性质的一族新的脉冲微分方程，对这样的系统我们一般称为具有可变结构的脉冲微分系统。本文所讨论的脉冲混合微分系统是一类特殊的但很重要的具有可变结构的脉冲微分系统，它的特点就是不同时间段内微分系统可以不同，并且后一时间段的系统依赖于前一时间段。比如给厂房配电，不同时间段内电流所满足的微分方程是不同的，甚至依赖于前一时间段最后一时刻电流的值。脉冲混合微分系统是脉冲微分系统的推广，当不同时间段内微分系统相同时就简化成为脉冲微分系统，因而本文所得到的结果也适用于脉冲微分系统。 到目前为止，有关该系统的稳定性和有界性结果并不多见，本文把Lyapunov直接方法和比较方法应用到脉冲混合系统上得到了该系统关于两个测度的实际稳定性、积分稳定性和有界性结果。 由于右端出现了λ_k(x_k)一项，故与以往不同，需要先假设系统 的解对任意固定的y∈R~n和k=0，1，2…都在[t_k，t_k+1]上存在。 我们主要借助Lyapunov直接方法和比较方法的思想讨论了脉冲混合微分系统的关于两个测度的稳定性及有界性问题。本文所用的Lyapunov直接方法不同于以 往的常规方法，主要表现在所找的LyaPunov函数不再局限于沿系统轨线单调递减， 也不再局限于对离散或连续部分分别设置条件，而是对其离散和连续部分设置混合 条件以保证其不能增长太快．基于这些思想，本文给出了一系列用这种扩展的rya Punov直接方法得到的充分条件以判别脉冲混合微分系统的关于两个测度的实际稳 定性和有界性．有时，我们不能选取到一个恰当的Lyapunov函数满足定理的条件 来直接判断系统的稳定性，但是，我们可以考虑其摄动系统以及采用多个Lyapunov 函数来解决问题．本文又得到了脉冲混合微分系统的摄动系统的关于两个测度的实 际稳定性定理，给出了一个例子，并利用多个tapanunov函数得到了关于该系统的 实际稳定性结果．同时，比较原理可以建立复杂的向量系统与简单纯量系统之间的 关系，这使得在较少条件下得到较强结果成为可能．本文结合脉冲混合系统的比较 原理给出了该系统关于两个测度的实际稳定性、积分稳定性和有界性的比较结果． 第一章，给出引言和预备知识． 第H章，主要利用Lyspunov函数来研究脉冲混合微分系统的关于两个测度的 稳定性．52J 利用Ly＆punov函数直接方法、比较方法和多个Ly3punov函数研究 了脉冲混合微分系统关于两个测度的实际稳定性，还得到了一个关于其摄动系统的 实际稳定性结果；52．2利用比较方法给出了该系统关于两个测度的积分稳定性结 果；52．3讨论了该系统关于两个测度的不稳定性． 第三章，主要利用Lyapunov函数来研究脉冲混合微分系统的关于两个测度的 有界性．53．1 主要利用LyaPunov直接方法得到脉冲混合系统的关于两个测度的 有界性结果；53．2主要利用比较方法得到脉冲混合系统的关于两个测度的有界性 的比较结果．更多还原

【Abstract】 In this paper, we manily study the impulsive hybrid differential system, that is,and a little of content is about its perturbed system. The impulsive hybrid differential system derives from physical models in practical investigation and is further extension of impulsive differential system.With development of industry, there are a lot of new kinds of physcial models, of these there is one that can not be described only by impulsive differential system. In this case, we should switch to a new set of differential equations taking into consideration momentary perturbations of impulsive nature. A general description of such systems was called impulsive systems with variable structure. The impulsive hybrid differential system studied in this paper is a special but important case of impulsive differential systems with variable structure, its characteristic is that its equations in different time periods may be different and the equation in the latter depends on the former. It is further extension of impulsive system. When the system in different time periods is the same, it becomes impulsive differential system. So the results in this paper also hold for impulsive differential systems.So far, the results about its stability and boundedness are few. In this paper, we apply Lyapunov’s direct method and comparison principle in impulsive hybrid differential systms to get some new results about practical stability, integral stability and boundedness in terms of two measures.Since there exists λk(xk) in the right of the systems, different from former investi-gation, we need do some convertion. So, we assume existence of solutions of the systemsWe mainly study stability properties and boundedness in terms of two measures for impulsive hybrid differential systems using Lyapunov’s direct method and comparison principle. Here these stability results do not require a Lyapunov function to decrease along trajectories of the system, and we also do not give conditions on continuous portion or discrete portion of the systems respectively, but we can give mixing conditions on them to ensure that it doesn’t increase quickly. Based on this idea, we give a series of sufficient conditions to determine practical stability properties and boundedness in terms of two measures for impulsive hybrid systems. Sometimes , we can not choose a proper Lyapunov function that satifies theory’s conditions, but we can consider its perturbed systems and can use sevsral Lyapunov functions to study the systems. This paper get a result about practical stability of its perturbed systems and a result about its practical stability using several Lyapunov functions. In the same time, comparison principle establishs the relation between complicated vector systems and simple scalar systems, which makes it possible to get stronger results with fewer conditions. The paper gives some comparison results about practical stability and boundedness in terms of two measures using comparison principle about impulsive hybrid differential systems. In chapter one, we give intruduction and preliminaries.In chapter two, we study the stability of impulsive hybrid differential systems in terms of two measures. In section one, we study its practical stability using Lyapunov’s direct method and comparison method, we also get a result about practical stability of its perturbed systems and a result about practical stability using several Lyapunov functions. In section two, we get some results about integral stability using comparison method. In section three, we study its unstability.In chapter three, we study the boundedness in terms of two measures for impulsive hybrid systems. In section one, we get the boundedness in terms of two measures for impulsive hybrid systems using Lyapunov’s direct method. In section two, we get the boundedness in terms of two measures for impulsive hybrid systems using comparison principle.更多还原