Center problem, Abel equation and the Faa di Bruno Hopf algebra for output feedback
Center problem, Abel equation and the Faa di Bruno Hopf algebra for output feedback
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DOI：
10.1093/imrn/rnw167
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发表时间：
2015-07
期刊：
影响因子：
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通讯作者：
K. Ebrahimi-Fard;W. S. Gray
中科院分区：
文献类型：
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作者：
K. Ebrahimi-Fard;W. S. Gray
给出了非自治微分方程经典Poincare中心焦点问题背后的Devlin字问题的组合解释。从连通分次Hopf代数的角度出发，证明了Devlin的正则多项式与应用于Ferfera形式幂函数级数的Hopf代数对极的分次分支密切相关。这种联系是通过控制理论建立的，因为描述中心的阿贝尔方程等价于输出反馈方程，并且输出反馈的霍普夫代数是由迭代积分的合成而不只是迭代积分的乘积得出的，这就产生了混洗代数。这意味着在Devlin的方法中起作用的主要代数结构实际上不是Shuffle代数，而是FAA di Bruno型Hopf代数，它是根据Shuffle乘积定义的，但却是一种独特的代数结构。
A combinatorial interpretation is given of Devlin's word problem underlying the classical center-focus problem of Poincare for non-autonomous differential equations. It turns out that the canonical polynomials of Devlin are from the point of view of connected graded Hopf algebras intimately related to the graded components of a Hopf algebra antipode applied to the formal power series of Ferfera. The link is made by passing through control theory since the Abel equation, which describes a center, is equivalent to an output feedback equation, and the Hopf algebra of output feedback is derived from the composition of iterated integrals rather than just the products of iterated integrals, which yields the shuffle algebra. This means that the primary algebraic structure at play in Devlin's approach is actually not the shuffle algebra, but a Faa di Bruno type Hopf algebra, which is defined in terms of the shuffle product but is a distinct algebraic structure.