Finite-dimensional filters with nonlinear drift - XII: linear and constant structure of Wong-matrix

摘要

This is the first of final two papers in this series which will give complete classification of the finite dimensional estimation algebra of maximal rank (cf. Definition 2 in Sec. 2), a problem proposed by R. Brockett in his invited lecture at the International Congress of Mathematics in 1983. The concept of estimation algebra (Lie algebra) was first introduced by Brockett and Mitter independently. This concept plays a crucial role in the investigation of finite-dimensional nonlinear filters. Since 1990, Yau has launched a program to study Brockett's problem. He first considered Wong's anti-symmetric matrix Ω = (w{sub}(ij)) = ((partial deriv)f{sub}j/(partial deriv)x{sub}i -(partial deriv)f{sub}i/(partial deriv)x{sub}j), where f denotes the drift term in equation (1). He solved the Brockett's problem when P matrix has only constant entries. Yau's program is to show that P matrix must have constant entries for finite dimensional estimation algebra. Recently Chen and Yau studied the structure of quadratic forms in a finite-dimensional estimation algebra. Let k be the quadratic rank of the estimation algebra and n be the dimension of the state space. They showed that the left upper corner (w{sub}(ij)), 1≤i, j≤k, of Ω matrix is a matrix with constant coefficients. In this paper, we shall show that the lower right corner (w{sub}(ij)), k + 1 ≤ i,j ≤ n, of Ω matrix is also a constant matrix.