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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 Differential inclusion ： ウィキペディア英語版 Differential inclusion In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form :$\backslash frac(t)\backslash in\; F(t,x(t)),$ where ''F'' is a multivalued map, i.e. ''F''(''t'', ''x'') is a ''set'' rather than a single point in $\backslash scriptstyle^d$. Differential inclusions arise in many situations including differential variational inequalities, projected dynamical systems, dynamic Coulomb friction problems and fuzzy set arithmetic. For example, the basic rule for Coulomb friction is that the friction force has magnitude ''μN'' in the direction opposite to the direction of slip, where ''N'' is the normal force and ''μ'' is a constant (the friction coefficient). However, if the slip is zero, the friction force can be ''any'' force in the correct plane with magnitude smaller than or equal to ''μN'' Thus, writing the friction force as a function of position and velocity leads to a set-valued function. ==Theory== Existence theory usually assumes that ''F''(''t'', ''x'') is an upper hemicontinuous function of ''x'', measurable in ''t'', and that ''F''(''t'', ''x'') is a closed, convex set for all ''t'' and ''x''. Existence of solutions for the initial value problem :$\backslash frac(t)\backslash in\; F(t,x(t)),\; \backslash quad\; x(t\_0)=x\_0$ for a sufficiently small time interval [''t''0, ''t''0 + ''ε''), ''ε'' > 0 then follows. Global existence can be shown provided ''F'' does not allow "blow-up" ($\backslash scriptstyle\; \backslash Vert\; x(t)\backslash Vert\backslash ,\backslash to\backslash ,\backslash infty$ as $\backslash scriptstyle\; t\backslash ,\backslash to\backslash ,\; t^$ * for a finite $\backslash scriptstyle\; t^$ *). Existence theory for differential inclusions with non-convex ''F''(''t'', ''x'') is an active area of research. Uniqueness of solutions usually requires other conditions. For example, suppose $F(t,x)$ satisfies a one-sided Lipschitz condition: :$(x\_1-x\_2)^T(F(t,x\_1)-F(t,x\_2))\backslash leq\; C\backslash Vert\; x\_1-x\_2\backslash Vert^2$ for some ''C'' for all ''x''1 and ''x''2. Then the initial value problem :$\backslash frac(t)\backslash in\; F(t,x(t)),\; \backslash quad\; x(t\_0)=x\_0$ has a unique solution. This is closely related to the theory of maximal monotone operators, as developed by Minty and Haïm Brezis. Filippov's theory only allows for disconituities in the derivative $\backslash frac(t)$, but allows no discontinuities in the state, i.e. $x(t)$ need be continuous. Schatzman and later Moreau (who gave it the currently accepted name) extended the notion to ''measure differential inclusion'' (MDI) in which the inclusion is evaluated by taking the limit from above for $x(t)$. 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「Differential inclusion」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.