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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Krener's theorem ： ウィキペディア英語版 Krener's theorem In mathematics, Krener's theorem is a result attributed to Arthur J. Krener in geometric control theory about the topological properties of attainable sets of finite-dimensional control systems. It states that any attainable set of a bracket-generating system has nonempty interior or, equivalently, that any attainable set has nonempty interior in the topology of the corresponding orbit. Heuristically, Krener's theorem prohibits attainable sets from being hairy. ==Theorem== Let $\backslash dot\; q=f(q,u)$ be a smooth control system, where $$ belongs to a finite-dimensional manifold $\backslash \; M$ and $\backslash \; u$ belongs to a control set $\backslash \; U$. Consider the family of vector fields $=\backslash$. Let $\backslash \; \backslash mathrm\backslash ,\backslash mathcal$ be the Lie algebra generated by $$ with respect to the Lie bracket of vector fields. Given $\backslash \; q\backslash in\; M$, if the vector space $\backslash \; \backslash mathrm\_q\backslash ,\backslash mathcal=\backslash \backslash \}$ is equal to $\backslash \; T\_q\; M$, then $\backslash \; q$ belongs to the closure of the interior of the attainable set from $\backslash \; q$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Krener's theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.