The realization of input-output maps using bialgebras

Abstract

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 $p\in H^{\ast }$ is differentially produced by the algebra $R$ with the augmentation $\epsilon$ if there is right $H$-module algebra structure on $R$ and there exists $f\in R$ satisfying $p(h)=\epsilon (f\cdot h)$. We characterize those $p\in H^{\ast }$ which are differentially produced.

16 pages