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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].

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