Finite-Dimensional Filters with Nonlinear Drift VII: Mitter Conjecture and Structure of $\eta$

Abstract

The concept of estimation algebra introduced independently by Brockett and Mitter has been playing a fundamental role in the investigation of finite-dimensional nonlinear filters. Mitter conjectured that the observation terms $h_i(x)$ are polynomials of degree one if the corresponding estimation algebra is finite dimensional. Chiou, Leung, and the present authors classify all finite-dimensional estimation algebra of maximal rank with dimension of the state space less than or equal to three. In this paper, we prove the Mitter conjecture for finite-dimensional estimation algebra of maximal rank with arbitrary state space dimension. In the course of our proof, we show that the $\Omega= ({\partial f_j\over\partial x_i} - {\partial f_i\over\partial x_j})$ matrix, where $f$ denotes the drift term, has special linear structure which generalizes our previous result in [J. Chen and S. S.-T. Yau, \textit{Math. Control Signals Systems}, 9 (1996), to appear]. We also give a structure theorem for $\eta= \sum^n_{i=1} {\partial f_i\over\partial x_i} + \sum^n_{i=1} f_i^2 + \sum^m_{i=1} h_i^2$.