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

The purpose of the paper is two-fold: (i) to introduce the sufficient statistic 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 used 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 algebra enables us to determine a priori whether there exist finite-dimensional controllers; it also enables us to classify all finite-dimensional controllers. Relations to minimax dynamic games are delineated.