zbMATH — the first resource for mathematics

Finite-dimensional filters with nonlinear drift. XII: Linear and constant structure of Wong-matrix. (English) Zbl 1038.93085
Pasik-Duncan, Bozenna (ed.), Stochastic theory and control. Proceedings of the workshop, Lawrence, KS, USA, October 18–20, 2001. Berlin: Springer (ISBN 3-540-43777-0/pbk). Lect. Notes Control Inf. Sci. 280, 507-518 (2002).
Summary: This is the first of two final papers in the series [for Part XI see Adv. Math. 140, 156–189 (1998; Zbl 0935.93059)] which will give a complete classification of the finite-dimensional estimation algebra of maximal rank, a problem posed by R. Brockett in his invited lecture at the International Congress of Mathematicians in 1983. 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 \(\Omega= (\omega_{ij}) =(\partial f_j/ \partial x_i-\partial f_i/ \partial x_j)\), where \(f\) denotes a drift term. He solved Brockett’s problem when \(\Omega\) has only constant entries. Yau’s program is to show that \(\Omega\) must have constant entries for a 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 \((\omega_{ij})\), \(1\leq i,j\leq k\), of \(\Omega\) is a matrix with constant coefficients. In this paper, we show that the lower right corner \((\omega_{ij})\), \(k+1\leq i,j\leq n\), of \(\Omega\) is also a constant matrix.
For the entire collection see [Zbl 1005.00048].

93E11 Filtering in stochastic control theory
93C10 Nonlinear systems in control theory
93B29 Differential-geometric methods in systems theory (MSC2000)