TY - GEN
T1 - Control interpretations of products in the Hopf algebra
AU - Kawski, Matthias
PY - 2009
Y1 - 2009
N2 - 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 powerseries 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.
AB - 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 powerseries 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.
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U2 - 10.1109/CDC.2009.5400441
DO - 10.1109/CDC.2009.5400441
M3 - Conference contribution
AN - SCOPUS:77950873197
SN - 9781424438716
T3 - Proceedings of the IEEE Conference on Decision and Control
SP - 7503
EP - 7508
BT - Proceedings of the 48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009
PB - Institute of Electrical and Electronics Engineers Inc.
T2 - 48th IEEE Conference on Decision and Control held jointly with 2009 28th Chinese Control Conference, CDC/CCC 2009
Y2 - 15 December 2009 through 18 December 2009
ER -