几种随机系统的控制问题

发布时间：2018-02-28 15:28

本文关键词： 倒向随机微分时滞方程最优控制随机最大值原理对偶方程线性二次最优控制　出处：《电子科技大学》2017年博士论文　论文类型：学位论文

【摘要】：众所周知,在自然现象和社会生活中,很多事物的动态演化过程可以用一个随机微分方程(SDE)来描述。其中的某些过程不仅和它当前所处的状态有关,而且和它的历史状态有关。这样,这类事物的动态演化过程可以用一个含有时间延迟项的随机微分方程来刻画,通常称之为随机微分时滞方程(SDDE)。如今,SDDE广泛应用于生物、金融、物理等领域并且关于它的理论研究也得到很大发展。近几年,一种新的SDDE被提出并且引起很多研究者的关注,这里我们称作倒向随机微分时滞方程(BSDDE)。这种方程可以看做是一种含有时间延迟的倒向随机微分方程(BSDE),关于它的一些理论和应用也有一定的发展。其中,探讨这种方程最优控制问题是一个值得关注的方向。基于此,本文研究了几种正向、倒向随机时滞系统的最优控制问题。研究的内容主要有以下几个方面:1.研究了BSDDE的非零和的微分对策问题。通过引入一种三对偶的方程来作为对偶过程,并且结合凸变分技术,得到了纳什均衡点满足的必要条件,并证明在附加某些凹性的条件下,它也是充分的。所得结果应用于一个最优消费选择的问题并得到精确的纳什均衡点。此外,在应用当中给出了关于BSDDE解的一些特性。2.研究了一种一般的BSDE的最优控制问题。假定控制域非凸,并且控制进入扩散项中,通过引入二阶的变分方程和对偶方程,并结合针状变分和BSDE的估计技术,建立了最优控制满足的必要条件,即最大值原理。3.把经典变分、对偶技术和滤波结果结合,通过引入时间超前的随机微分方程,建立了局部信息下的最大值原理。理论的结果应用于线性二次最优控制问题(LQ问题),在获得最优控制的过程中,提出了一种新的正、倒向耦合的随机微分方程,因其含有时间超前、滞后以及方程的滤波项,称之为一般的随机微分滤波方程,并讨论了这种方程的解的存在性。以此为基础,获得了最优控制的一个表示。4.考虑到某些时候人们不能完全观测到系统状态,因此研究了局部可观测的SDDE的最优控制问题,通过哥萨诺夫变换,这个问题可以被转换成一个类似于完全信息下的最优控制问题。在利用SDDE的估计技术处理延迟项并引入时间超前的SDE后,得到了这种问题下的最大值原理。5.研究了局部可观测的SDDE的LQ问题,利用倒向分离技术得到观测延迟的情况下的最优控制的反馈形式。此外,在一般情形下表示最优控制时对时间超前的BSDE的滤波问题做了初步的探讨,所得结果可以看做是对经典滤波理论的有益的补充。
[Abstract]:As we all know, in natural phenomena and social life, the dynamic evolution of many things can be described by a stochastic differential equation (SDE). And it has something to do with its historical state. In this way, the dynamic evolution of this kind of thing can be described by a stochastic differential equation with a time delay term, commonly known as the stochastic differential delay equation. Now SDDE is widely used in biology and finance. In recent years, a new kind of SDDE has been proposed and attracted the attention of many researchers. Here we call the backward stochastic differential delay equation BSDDE.This equation can be regarded as a kind of backward stochastic differential equation with time delay, and some theories and applications about it have also been developed. It is an interesting direction to study the optimal control problem of this kind of equation. Based on this, several kinds of forward control problems are studied. The main contents of this paper are as follows: 1. The differential game problem of BSDDE's non-zero sum is studied. By introducing a three-duality equation as duality process, and combining convex variational technique, we introduce a three-duality equation to solve the problem of optimal control for backward stochastic time-delay systems. The necessary conditions for Nash equilibrium point to be satisfied are obtained, and it is proved that it is also sufficient under some concave conditions. The results obtained are applied to a problem of optimal consumption choice and exact Nash equilibrium points are obtained. In application, some properties of BSDDE solution are given. 2. A general optimal control problem of BSDE is studied. Assuming that the control domain is nonconvex, and the control enters the diffusion term, by introducing the second order variational equation and dual equation, Combined with needle-shaped variation and BSDE estimation technique, the necessary condition of optimal control is established, I. E. maximum principle .3.Uniting classical variation, duality technique and filtering results, the stochastic differential equation with time advance is introduced. The maximum principle under local information is established. The theoretical results are applied to the LQ problem of linear quadratic optimal control. In the process of obtaining the optimal control, a new positive and backward coupled stochastic differential equation is proposed. Because it contains the filtering terms of time leading, delay and equation, it is called the general stochastic differential filtering equation, and the existence of the solution of the equation is discussed. A representation of optimal control is obtained. Considering that the state of the system can not be observed completely at some time, the optimal control problem of locally observable SDDE is studied. This problem can be transformed into an optimal control problem similar to the one under complete information. After using the SDDE estimation technique to deal with the delay term and introduce the time-advanced SDE, The maximum principle of this problem is obtained. 5. The LQ problem of locally observable SDDE is studied, and the feedback form of optimal control in the case of observation delay is obtained by using backward separation technique. In the case of optimal control, the filtering problem of BSDE in advance of time is preliminarily discussed, and the results obtained can be regarded as a useful supplement to the classical filtering theory.
【学位授予单位】：电子科技大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O231

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