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The realization of input-output maps using bialgebras

We use the theory of bialgebras to provide the algebraic background 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 systems and more recent results involving analysis related to families of trees. If $H$ is a bialgebra, we say that $p \in H^*$ is differentially produced by the algebra $R$ with the augmentation $ε$ if there is right $H$-module algebra structure on $R$ and there exists $f \in R$ satisfying $p(h) = ε(f \cdot h)$. We characterize those $p \in H^*$ which are differentially produced.