Algebraic Aspects of Control Systems and Realizations Original Research Article

Actions of formal groups on formal schemes correspond to Hopf algebra actions. They provide a natural framework for the discussion of local aspects of control theory. The existence of a unique minimal realization of an input-output map or its generating function is a consequence of general properties of such actions. Moreover, the syntactic Lie algebra is related to the largest Hopf algebra quotient acting faithfully on the minimal realization. The generating functions which have linear realizations, i.e., correspond to linear systems, are exactly the representative functions (rational, in the free case).