The realization of input-output maps using bialgebras

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Abstract
We use the theory of bialgebras to provide the algebraic back- ground for state space realization theorems for input-output maps of control systems. This allows us to consider from a common viewpoint classical results about formal state space realizations of nonlinear sys- tems and more recent results involving analysis related to families of trees. If H is a bialgebra, we say that p 2 H is dierentially produced by the algebra R with the augmentation if there is right H-module al- gebra structure on R and there exists f 2 R satisfying p(h) = (f ·h). We characterize those p 2 H which are dierentially produced.

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