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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 singular control ： ウィキペディア英語版 singular control In optimal control, problems of singular control are problems that are difficult to solve because a straightforward application of Pontryagin's minimum principle fails to yield a complete solution. Only a few such problems have been solved, such as Merton's portfolio problem in financial economics or trajectory optimization in aeronautics. A more technical explanation follows. The most common difficulty in applying Pontryagin's principle arises when the Hamiltonian depends linearly on the control $u$, i.e., is of the form: $H(u)=\backslash phi(x,\backslash lambda,t)u+\backslash cdots$ and the control is restricted to being between an upper and a lower bound: $a\backslash le\; u(t)\backslash le\; b$. To minimize $H(u)$, we need to make $u$ as big or as small as possible, depending on the sign of $\backslash phi(x,\backslash lambda,t)$, specifically: : If $\backslash phi$ is positive at some times, negative at others and is only zero instantaneously, then the solution is straightforward and is a bang-bang control that switches from $b$ to $a$ at times when $\backslash phi$ switches from negative to positive. The case when $\backslash phi$ remains at zero for a finite length of time $t\_1\backslash le\; t\backslash le\; t\_2$ is called the singular control case. Between $t\_1$ and $t\_2$ the maximization of the Hamiltonian with respect to u gives us no useful information and the solution in that time interval is going to have to be found from other considerations. (One approach would be to repeatedly differentiate $\backslash partial\; H/\backslash partial\; u$ with respect to time until the control u again explicitly appears, which is guaranteed to happen eventually. One can then set that expression to zero and solve for u. This amounts to saying that between $t\_1$ and $t\_2$ the control $u$ is determined by the requirement that the singularity condition continues to hold. The resulting so-called singular arc will be optimal if it satisfies the Kelley condition: :$(-1)^k\; \backslash frac\; \backslash left(\backslash right)\}^\; H\_u\; \backslash right)\; \backslash ge\; 0\; ,\backslash ,\; k=0,1,\backslash cdots$ .〔Bryson, Ho: Applied Optimal Control, Page 246〕 This condition is also called the generalized Legendre-Clebsch condition). The term bang-singular control refers to a control that has a bang-bang portion as well as a singular portion. ==References== 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「singular control」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.