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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 Riccati differential equation ： ウィキペディア英語版 Riccati equation In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form :$y\text{'}(x)\; =\; q\_0(x)\; +\; q\_1(x)\; \backslash ,\; y(x)\; +\; q\_2(x)\; \backslash ,\; y^2(x)$ where $q\_0(x)\; \backslash neq\; 0$ and $q\_2(x)\; \backslash neq\; 0$. If $q\_0(x)\; =\; 0$ the equation reduces to a Bernoulli equation, while if $q\_2(x)\; =\; 0$ the equation becomes a first order linear ordinary differential equation. The equation is named after Jacopo Riccati (1676–1754).〔Riccati, Jacopo (1724) ("Animadversiones in aequationes differentiales secundi gradus" ) (Observations regarding differential equations of the second order), ''Actorum Eruditorum, quae Lipsiae publicantur, Supplementa'', 8 : 66-73. (Translation of the original Latin into English ) by Ian Bruce.〕 More generally, the term Riccati equation is used to refer to matrix equations with an analogous quadratic term, which occur in both continuous-time and discrete-time linear-quadratic-Gaussian control. The steady-state (non-dynamic) version of these is referred to as the algebraic Riccati equation. == Reduction to a second order linear equation == The non-linear Riccati equation can always be reduced to a second order linear ordinary differential equation (ODE): If :$y\text{'}=q\_0(x)\; +\; q\_1(x)y\; +\; q\_2(x)y^2\backslash !$ then, wherever $q\_2$ is non-zero and differentiable, $v=yq\_2$ satisfies a Riccati equation of the form :$v\text{'}=v^2\; +\; R(x)v\; +S(x),\backslash !$ where $S=q\_2q\_0$ and $R=q\_1+\backslash left(\backslash frac\backslash right)$, because :$v\text{'}=(yq\_2)\text{'}=\; y\text{'}q\_2\; +yq\_2\text{'}=(q\_0+q\_1\; y\; +\; q\_2\; y^2)q\_2\; +\; v\; \backslash frac=q\_0q\_2\; +\backslash left(q\_1+\backslash frac\backslash right)\; v\; +\; v^2.\backslash !$ Substituting $v=-u\text{'}/u$, it follows that $u$ satisfies the linear 2nd order ODE :$u\text{'}\text{'}-R(x)u\text{'}\; +S(x)u=0\; \backslash !$ since :$v\text{'}=-(u\text{'}/u)\text{'}=-(u\text{'}\text{'}/u)\; +(u\text{'}/u)^2=-(u\text{'}\text{'}/u)+v^2\backslash !$ so that :$u\text{'}\text{'}/u=\; v^2\; -v\text{'}=-S\; -Rv=-S\; +Ru\text{'}/u\backslash !$ and hence :$u\text{'}\text{'}\; -Ru\text{'}\; +Su=0.\backslash !$ A solution of this equation will lead to a solution $y=-u\text{'}/(q\_2u)$ of the original Riccati equation. 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「Riccati equation」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.