Control interpretations of products in the Hopf algebra

Families of differential operators, like those defining affine, generally nonlinear, control systems are known to have natural Hopf algebra structures. These provide deeper insight into relationships and properties of such objects as the Chen-Fliess series and state space realizations of systems defined by input-output operators. A starting point for this work are representations of control objects by formal power-series in, generally noncommuting, indeterminates. Such series are elements of a free associative algebra which contains several rich substructure that are equipped with various products. This article summarizes correspondences between objects from control theory and algebraic combinatorics, with special focus on elements of the two Hopf algebra structures on this algebra of formal power series, including the antipode and two convolution products. Newly found relationships are discussed, and open questions are posed. Recent work has demonstrated that these abstract combinatorial algebra structures help obtain effective solution formulas for various control problems including questions about controllability and algorithms for path planning. They also find applications in the field of geometric numerical integration algorithms.