Finite dimensional filters with nonlinear drift, XI - Explicit solution ofthe generalized Kolmogorov equation in Brockett-Mitter program

Ever since the technique of the Kalman-Bucy filter was popularized, there h as been an intense interest in finding new classes of finite-dimensional re cursive filters. In the late 1970s the concept of the estimation algebra of a filtering system was introduced. Brockett, Clark, and Mitter proposed to use the Wei-Norman approach to solve the nonlinear filtering problem. In 1 990, Tam, Wong, and Yau presented a rigorous proof of the Brocket-Mitter pr ogram which allows one to construct finite-dimensional recursive filters fr om finite-dimensional estimation algebras. Later Yau wrote down explicitly a system of ordinary differential equations and generalized Kolmogorov equa tion to which the robust Duncan-Mortenser-Zakai equation can be reduced. Th us there remains three fundamental problems in Brockett-Mitter program. The first is the problem of finding explicit solution to the generalized Kolmo gorov equation. The second is the problem of finding real-time solution of a system of ODEs. The third is the Brockett's problem of classification of finite-dimensional estimation algebras. In this paper, we solve the first p roblem. (C) 1998 Academic Press.