本文關(guān)鍵詞： 倒向隨機(jī)微分時滯方程 最優(yōu)控制 隨機(jī)最大值原理 對偶方程 線性二次最優(yōu)控制　出處：《電子科技大學(xué)》2017年博士論文　論文類型：學(xué)位論文

【摘要】：眾所周知,在自然現(xiàn)象和社會生活中,很多事物的動態(tài)演化過程可以用一個隨機(jī)微分方程(SDE)來描述。其中的某些過程不僅和它當(dāng)前所處的狀態(tài)有關(guān),而且和它的歷史狀態(tài)有關(guān)。這樣,這類事物的動態(tài)演化過程可以用一個含有時間延遲項的隨機(jī)微分方程來刻畫,通常稱之為隨機(jī)微分時滯方程(SDDE)。如今,SDDE廣泛應(yīng)用于生物、金融、物理等領(lǐng)域并且關(guān)于它的理論研究也得到很大發(fā)展。近幾年,一種新的SDDE被提出并且引起很多研究者的關(guān)注,這里我們稱作倒向隨機(jī)微分時滯方程(BSDDE)。這種方程可以看做是一種含有時間延遲的倒向隨機(jī)微分方程(BSDE),關(guān)于它的一些理論和應(yīng)用也有一定的發(fā)展。其中,探討這種方程最優(yōu)控制問題是一個值得關(guān)注的方向�；诖�,本文研究了幾種正向、倒向隨機(jī)時滯系統(tǒng)的最優(yōu)控制問題。研究的內(nèi)容主要有以下幾個方面:1.研究了BSDDE的非零和的微分對策問題。通過引入一種三對偶的方程來作為對偶過程,并且結(jié)合凸變分技術(shù),得到了納什均衡點(diǎn)滿足的必要條件,并證明在附加某些凹性的條件下,它也是充分的。所得結(jié)果應(yīng)用于一個最優(yōu)消費(fèi)選擇的問題并得到精確的納什均衡點(diǎn)。此外,在應(yīng)用當(dāng)中給出了關(guān)于BSDDE解的一些特性。2.研究了一種一般的BSDE的最優(yōu)控制問題。假定控制域非凸,并且控制進(jìn)入擴(kuò)散項中,通過引入二階的變分方程和對偶方程,并結(jié)合針狀變分和BSDE的估計技術(shù),建立了最優(yōu)控制滿足的必要條件,即最大值原理。3.把經(jīng)典變分、對偶技術(shù)和濾波結(jié)果結(jié)合,通過引入時間超前的隨機(jī)微分方程,建立了局部信息下的最大值原理。理論的結(jié)果應(yīng)用于線性二次最優(yōu)控制問題(LQ問題),在獲得最優(yōu)控制的過程中,提出了一種新的正、倒向耦合的隨機(jī)微分方程,因其含有時間超前、滯后以及方程的濾波項,稱之為一般的隨機(jī)微分濾波方程,并討論了這種方程的解的存在性。以此為基礎(chǔ),獲得了最優(yōu)控制的一個表示。4.考慮到某些時候人們不能完全觀測到系統(tǒng)狀態(tài),因此研究了局部可觀測的SDDE的最優(yōu)控制問題,通過哥薩諾夫變換,這個問題可以被轉(zhuǎn)換成一個類似于完全信息下的最優(yōu)控制問題。在利用SDDE的估計技術(shù)處理延遲項并引入時間超前的SDE后,得到了這種問題下的最大值原理。5.研究了局部可觀測的SDDE的LQ問題,利用倒向分離技術(shù)得到觀測延遲的情況下的最優(yōu)控制的反饋形式。此外,在一般情形下表示最優(yōu)控制時對時間超前的BSDE的濾波問題做了初步的探討,所得結(jié)果可以看做是對經(jīng)典濾波理論的有益的補(bǔ)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.
【學(xué)位授予單位】：電子科技大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O231

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本文編號：1547895