Information states in stochastic control and filtering: A Lie algebraic theoretic approach

The purpose of this paper is twofold: i) to introduce the sufficient statis tic algebra which is responsible for propagating the sufficient statistics, or information state, in the optimal control of stochastic systems and ii) to apply certain Lie algebraic methods and gauge transformations, widely u sed in nonlinear control theory and quantum physics, to derive new results concerning finite-dimensional controllers. This enhances our understanding of the role played by the sufficient statistics. The sufficient statistic a lgebra enables us to determine a priori whether there exist finite-dimensio nal controllers; it also enables us to classify all finite-dimensional cont rollers. Relations to minimax dynamic games are delineated.