发表机构
Beijing Institute of Mathematical Sciences and Applications; Qiuzhen College, Tsinghua University; Department of Mathematical Sciences, Tsinghua University（北京数学科学与应用研究院; 清华大学求真书院; 清华大学数学科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在统一缓冲加权框架下，引入指数型权函数及加权Sobolev空间，证明三类滤波方程的解存在且唯一，为非线性滤波系统提供了统一处理方法。
AI 中文摘要
非线性滤波问题是现代控制理论的核心课题之一。本文在统一的缓冲加权框架内，研究连续时间非线性滤波中产生的三个基本演化方程——鲁棒Duncan-Mortensen-Zakai（DMZ）方程、随机DMZ方程以及Kushner-Stratonovich方程的适定性。引入指数型权函数及对应的加权Sobolev空间，以在更一般的框架下对滤波方程进行变分处理，该框架中滤波系统的系数可具有多项式增长的无界性。在温和且易于验证的假设下，我们首先建立鲁棒DMZ方程在这些加权空间中弱解的适定性。利用规范（指数）变换及其逆变换，将上述结果推广至随机DMZ方程和Kushner-Stratonovich方程，证明这两类方程的解在缓冲加权Sobolev空间中存在且唯一，从而实现对三个滤波方程的统一处理。本文还总结了适定性的充分条件，表明所提出的框架对一般非线性滤波系统具有广泛适用性。
英文摘要
Nonlinear filtering problem is one of the core subjects in modern control theory. In this paper, we will study the well-posedness of the three fundamental evolution equations arising in continuous-time nonlinear filtering--the robust Duncan-Mortensen-Zakai (DMZ) equation, the stochastic DMZ equation, and the Kushner-Stratonovich equation--within a unified buffered weighted formulation. An exponential-type weight function and the corresponding weighted Sobolev spaces are introduced to enable a variational treatment of the filtering equations in a more general setting, in which the coefficients of the filtering system may be unbounded with polynomial growth. Under mild and easily verifiable assumptions, we first establish the well-posedness of the weak solution to the robust DMZ equation in these weighted spaces. Using the gauge (exponential) transformation and its inverse, these results are then transferred to the stochastic DMZ equation and the Kushner-Stratonovich equation, whose solutions are shown to exist and be unique in buffered weighted Sobolev spaces, yielding a unified treatment of all three filtering equations. Sufficient conditions for the well-posedness are also summarized, which illustrate the wide applicability of the proposed framework to general nonlinear filtering systems.
