泛在双线性随机微分系统的非常规估计定理：控制观点
Non-conventional Estimation Theorems Concerning a Ubiquitous Bilinear Stochastic Differential System: a Control Perspective
摘要：
随机微分方程及其相关的偏微分方程是随机控制问题的基础形式。双线性随机系统的普遍性可以在自治系统、非线性动态电路和数学金融学中找到。Ito与Stratonovich的共识建议嵌入Stratonovich微分的随机系统来描述真实物理系统的状态向量的随机演化。标量时变双线性Stratonovich随机微分方程的数学理论在当前的教科书中是可用的。标量Stratonovich系统的理论是通过推导其封闭形式的解决方案，然后条件矩。服从向量时变双线性Stratonovich随机微分方程的实际问题是普遍存在的。然而，他们的正式和系统的估计理论是不可用的。本文推广了向量Stratonovich双线性随机微分方程的非齐次马氏过程的结果。然后，将本文的理论应用于随机三相整流电路。利用Euler-Lagrange理论和嵌入Stratonovich微分构造了整流器状态向量的随机演化。由于本文的理论既考虑了双线性系统的多维性，又考虑了其Stratonovich随机性，因此估计定理对于非线性实际物理系统的估计和控制具有重要意义。最后，本文将感兴趣的控制和计算从业者渴望的正式理论的双线性随机系统所产生的应用程序以及应用数学家寻找正式的双线性随机估计理论的应用。
英文摘要：
Stochastic differential equations and the associated partial differential equations are the cornerstone formalism in stochastic control problems. The universality of bilinear stochastic systems can be found in autonomous systems, non-linear dynamic circuits, and mathematical finance. Consensus on the Ito versus Stratonovich suggests stochastic systems embedded with Stratonovich differential to describe the stochastic evolution of the state vector of real physical systems. The mathematical theory of a scalar time-varying bilinear Stratonovich stochastic differential equation is available in current texts. The theory of scalar Stratonovich systems was developed by deriving their closed-form solutions and then conditional moments. Practical problems obeying vector time-varying bilinear Stratonovich stochastic differential equations are ubiquitous. However, their formal and systematic estimation theory is not available. In this paper, we develop the results for non-homogeneous Markov processes obeying the vector Stratonovich bilinear stochastic differential equation. Then, the theory of the paper is applied to a stochastic three-phase rectifier circuit. The stochastic evolution of the rectifier state vector is constructed by utilizing the Euler-Lagrange theory and embedding Stratonovich differential. Since the theory of the paper accounts for the multidimensionality as well as respects Stratonovich stochasticity of the ubiquitous bilinear system, the estimation Theorems will be remarkably useful to estimation and control of nonlinear real physical systems. Finally, this paper will be of interest to control and computational practitioners aspiring for the formal theory of bilinear stochastic systems arising from their application as well as applied mathematicians looking for applications of formal bilinear stochastic estimation theory.
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