本文選題：非線性脈沖微分控制系統(tǒng) + (h_0��； 參考：《山東師范大學》2015年碩士論文

【摘要】：本文考慮具固定時刻脈沖的微分控制系統(tǒng)和具依賴狀態(tài)脈沖的微分控制系統(tǒng)分別討論它們關于兩個測度的穩(wěn)定性性質(zhì)和有界性性質(zhì),其中脈沖微分控制系統(tǒng)是從數(shù)學的角度對各種控制系統(tǒng)的動力學模型進行闡釋,在描述現(xiàn)實世界的各種控制現(xiàn)象中具有非常重要的作用.它使人們更加科學的認識到系統(tǒng)的內(nèi)部規(guī)律,從而可以更好的對系統(tǒng)進行有目的的控制.近年來,隨著現(xiàn)代科學技術的不斷發(fā)展,脈沖控制問題已在工業(yè)、生物、經(jīng)濟等領域中有著大量的實際應用.例如,為了維持金融市場的穩(wěn)定性,中央銀行不可能每天都改變存款利率,而是讓其在一段時間內(nèi)保持一致,這類問題就可以歸納為脈沖控制系統(tǒng)的穩(wěn)定性.因此,脈沖微分控制系統(tǒng)在動力學研究方面具有廣泛的應用前景.許多情況下,脈沖控制和連續(xù)控制需要相輔相成才能對系統(tǒng)產(chǎn)生較好的控制效果.在控制理論中,連續(xù)控制體現(xiàn)在系統(tǒng)的表達式右端函數(shù)含有一個滿足一定條件的控制向量,且在脈沖函數(shù)中也含有控制向量,而脈沖微分控制系統(tǒng)就描述了這類脈沖控制問題.具依賴狀態(tài)脈沖的微分控制系統(tǒng)包含具固定時刻脈沖的微分控制系統(tǒng)這一特殊情況,因此,對于具依賴狀態(tài)脈沖的微分控制系統(tǒng)的研究具有更廣泛的應用背景.目前關于脈沖微分控制系統(tǒng)的穩(wěn)定性研究引起了很多研究者的興趣，但所研究的控制系統(tǒng)的控制向量大多是定義在控制集合Ω={u∈Rm:U(t,U)≤ r(t),t≥to}中的,而對于控制集合E={u∈Rm:U(t,u)≤λ0(t),t≥t0),研究結(jié)果還比較少,Lakshmikantam等人在文獻[1]，，[3]中研究了無脈沖作用下的微分控制系統(tǒng)在控制集合E上的實際穩(wěn)定性.文獻[21]基于Lyapunov第二方法得到一些穩(wěn)定性結(jié)果.在此基礎上,本文利用錐值Lyapunov函數(shù)方法及錐值變分Lyapunov函數(shù)方法研究脈沖微分控制系統(tǒng)在控制集合E上的穩(wěn)定性問題,得到了若干新結(jié)果,全文分兩章.本文第一章重點研究如何采用錐值變分函數(shù)方法研究系統(tǒng)(1)的Lyapunov穩(wěn)定性和有界性.在第一章中,首先借助錐值變分函數(shù)方法的基本思Lyapunov想,建立一個新的比較原理,從而克服了右端函數(shù)在整個R+N擬單調(diào)的條件.在這個比較定理的基礎上,研究系統(tǒng)(1)的(h0,h)-穩(wěn)定、漸近穩(wěn)定、一致穩(wěn)定、實際穩(wěn)定、最終穩(wěn)定、有界、一致有界等性質(zhì),最后給出一個例子說明定理的實用性.在第二章中,我們主要研究具依賴狀態(tài)脈沖微分控制系統(tǒng)的穩(wěn)定性.目前對具依賴狀態(tài)脈沖的微分控制系統(tǒng)(2)穩(wěn)定性的研究主要是借鑒文獻[22]中的轉(zhuǎn)化思想,將具依賴狀態(tài)脈沖轉(zhuǎn)化為不依賴狀態(tài)脈沖,用向量函數(shù)和微分Lyapunov不等式,通過與不帶脈沖的非擾動系統(tǒng)作比較建立一個比較原理,討論系統(tǒng)(2)關于兩個測度的穩(wěn)定性及有界性,而具脈動的脈沖微分控制系統(tǒng)的穩(wěn)定性結(jié)果尚不多見.第二章第三節(jié)采用錐值函數(shù)比較方法,建立了一個新的比較Lyapunov原理,在允許依賴狀態(tài)脈沖的微分控制系統(tǒng)(2)的解曲線與同一脈沖面碰撞有限次的條件下,討論了微分系統(tǒng)(2)的穩(wěn)定性性質(zhì)并給出系統(tǒng)(2)的比較結(jié)果.在以上比較結(jié)果的研究中,我們總是允許具依賴狀態(tài)脈沖的微分控制系統(tǒng)(2)的解曲線與同一脈沖面碰撞有限次.
[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.
【學位授予單位】：山東師范大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：O175;O231

【參考文獻】

相關博士學位論文 前1條

1 趙海清;脈沖微分系統(tǒng)的穩(wěn)定性和可控性[D];大連理工大學;2006年

本文編號：2082991