脉冲微分控制系统的稳定性研究

发布时间：2018-06-29 18:36

本文选题：非线性脉冲微分控制系统 + (h_0　；参考：《山东师范大学》2015年硕士论文

【摘要】：本文考虑具固定时刻脉冲的微分控制系统和具依赖状态脉冲的微分控制系统分别讨论它们关于两个测度的稳定性性质和有界性性质,其中脉冲微分控制系统是从数学的角度对各种控制系统的动力学模型进行阐释,在描述现实世界的各种控制现象中具有非常重要的作用.它使人们更加科学的认识到系统的内部规律,从而可以更好的对系统进行有目的的控制.近年来,随着现代科学技术的不断发展,脉冲控制问题已在工业、生物、经济等领域中有着大量的实际应用.例如,为了维持金融市场的稳定性,中央银行不可能每天都改变存款利率,而是让其在一段时间内保持一致,这类问题就可以归纳为脉冲控制系统的稳定性.因此,脉冲微分控制系统在动力学研究方面具有广泛的应用前景.许多情况下,脉冲控制和连续控制需要相辅相成才能对系统产生较好的控制效果.在控制理论中,连续控制体现在系统的表达式右端函数含有一个满足一定条件的控制向量,且在脉冲函数中也含有控制向量,而脉冲微分控制系统就描述了这类脉冲控制问题.具依赖状态脉冲的微分控制系统包含具固定时刻脉冲的微分控制系统这一特殊情况,因此,对于具依赖状态脉冲的微分控制系统的研究具有更广泛的应用背景.目前关于脉冲微分控制系统的稳定性研究引起了很多研究者的兴趣，但所研究的控制系统的控制向量大多是定义在控制集合Ω={u∈Rm:U(t,U)≤ r(t),t≥to}中的,而对于控制集合E={u∈Rm:U(t,u)≤λ0(t),t≥t0),研究结果还比较少,Lakshmikantam等人在文献[1]，，[3]中研究了无脉冲作用下的微分控制系统在控制集合E上的实际稳定性.文献[21]基于Lyapunov第二方法得到一些稳定性结果.在此基础上,本文利用锥值Lyapunov函数方法及锥值变分Lyapunov函数方法研究脉冲微分控制系统在控制集合E上的稳定性问题,得到了若干新结果,全文分两章.本文第一章重点研究如何采用锥值变分函数方法研究系统(1)的Lyapunov稳定性和有界性.在第一章中,首先借助锥值变分函数方法的基本思Lyapunov想,建立一个新的比较原理,从而克服了右端函数在整个R+N拟单调的条件.在这个比较定理的基础上,研究系统(1)的(h0,h)-稳定、渐近稳定、一致稳定、实际稳定、最终稳定、有界、一致有界等性质,最后给出一个例子说明定理的实用性.在第二章中,我们主要研究具依赖状态脉冲微分控制系统的稳定性.目前对具依赖状态脉冲的微分控制系统(2)稳定性的研究主要是借鉴文献[22]中的转化思想,将具依赖状态脉冲转化为不依赖状态脉冲,用向量函数和微分Lyapunov不等式,通过与不带脉冲的非扰动系统作比较建立一个比较原理,讨论系统(2)关于两个测度的稳定性及有界性,而具脉动的脉冲微分控制系统的稳定性结果尚不多见.第二章第三节采用锥值函数比较方法,建立了一个新的比较Lyapunov原理,在允许依赖状态脉冲的微分控制系统(2)的解曲线与同一脉冲面碰撞有限次的条件下,讨论了微分系统(2)的稳定性性质并给出系统(2)的比较结果.在以上比较结果的研究中,我们总是允许具依赖状态脉冲的微分控制系统(2)的解曲线与同一脉冲面碰撞有限次.
[Abstract]:In this paper, we consider the differential control systems with fixed time pulses and differential control systems with dependent state pulses, respectively, to discuss their stability and boundedness of two measures, respectively, in which the impulsive differential control system interprets the dynamic models of various control systems from a mathematical point of view and describes the real world. All kinds of control phenomena have a very important role. It makes people more scientific to recognize the internal rules of the system, and thus can better control the system. In recent years, with the continuous development of modern science and technology, the problem of pulse control has been applied in a large number of practical applications in industrial, biological, economic and other fields. For example, in order to maintain the stability of the financial market, the central bank can not change the deposit interest rate every day, but keep it consistent for a period of time. This kind of problem can be summed up as the stability of the pulse control system. Therefore, the impulsive differential control system has a wide application prospect in the field of dynamics research. In many cases, the pulse is used. Control and continuous control need to complement each other in order to produce a better control effect on the system. In the control theory, the continuous control is embodied in the right end function of the system's expression, which contains a control vector that satisfies certain conditions, and also contains a control vector in the pulse function, and the pulse impulse differential control system describes this kind of pulse control. The differential control system with dependent state pulses contains a special case of a differential control system with fixed time pulses. Therefore, the study of a differential control system with dependent state pulses has a wider application background. However, the control vector of the control system is mostly defined in the control set Omega ={u Rm:U (T, U) < R (T), t > to}, but for the control set E={u Rm:U (T, U) less than lambda 0, the results are relatively few. The actual stability of the [21] is based on the Lyapunov second method. On this basis, this paper uses the cone value Lyapunov function method and the cone value variational Lyapunov function method to study the stability of the impulsive differential control system on the control set E. Some new results are obtained in this paper. The first chapter of this paper is the first chapter of this paper. This paper studies how to use the cone value variational function method to study the Lyapunov stability and boundedness of the system (1). In the first chapter, a new comparison principle is established with the help of the basic thought Lyapunov of the cone value variational function method, so as to overcome the condition of the right end function in the whole R+N quasi single modulation. The system (1) (H0, H) - stable, asymptotically stable, uniformly stable, stable, final stable, bounded, uniformly bounded and so on. Finally, an example is given to illustrate the practicability of the theorem. In the second chapter, we mainly study the stability of the impulsive differential control system with dependence state. At present, the differential control system with dependent state pulses (2) is stable. The qualitative research is mainly based on the transformation thought in the literature [22], transforming the dependent state pulse into non dependent state pulse. Using vector function and differential Lyapunov inequality, a comparison principle is established by comparing with the non disturbing system without pulse, and the stability and boundedness of the two measures are discussed (2), and the pulsation has a pulsation. The stability results of the impulsive differential control system are still rare. In the second chapter and third section, a new comparison Lyapunov principle is established by using the conical function comparison method. The stability of the differential system (2) is discussed under the condition that the solution curve of the differential control system (2) which allows the dependent state pulse (2) is limited to the same pulse plane. The comparison results of the system (2) are given. In the study of the above comparison results, we always allow the solution of the differential control system (2) with dependent state pulses (2) to collide with the same pulse plane for a finite time.
【学位授予单位】：山东师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175;O231

【参考文献】