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Classification of finite-dimensional estimation algebras of maximal rank with arbitrary state – space dimension and Mitter conjecture. (English) Zbl 1121.93068
Summary: In the late seventies, the concept of the estimation algebra of a filtering system was introduced. It was proven to be an invaluable tool in the study of non-linear filtering problems. In the early eighties, Brockett proposed to classify finite dimensional estimation algebras and Mitter conjectured that all functions in finite dimensional estimation algebras are necessarily polynomials of total degree at most one. Despite the massive effort in understanding the finite dimensional estimation algebras, the 20 year old problem of Brockett and Mitter conjecture remains open. In this paper, we give a classification of finite dimensional estimation algebras of maximal rank and solve the Mitter conjecture affirmatively for finite dimensional estimation algebras of maximal rank. In particular, for an estimation algebra \(E\) of maximal rank, we give a necessary and sufficient conditions for \(E\) to be finite dimensional in terms of the drift \(f_i(x)\) and observation \(h_j(x)\). As an important corollary, we show that the number of statistics needed to compute the conditional density of the state given the observation \(\{y(s):0=s=t\}\) by the algebraic method is \(n\) where \(n\) is the dimension of the state.

MSC:
93E10 Estimation and detection in stochastic control theory
93E11 Filtering in stochastic control theory
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