Finite-dimensional filters with nonlinear drift .12. Mitter conjecture and structure of eta

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. Tn 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 derivative f(j)/partial derivative x(i) - partial derivative f(i)/partial derivative 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, Math. Control Signals Systems, 9 (1996), to appear]. We also give a structure theorem for eta = Sigma(i=1)(n) partial derivative f(i)/partial derivative x(i) + Sigma(i=1)(n) f(i)(2) + Sigma(i=1)(m) h(i)(2).