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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 Pontryagin's maximum principle ： ウィキペディア英語版 Pontryagin's maximum principle Pontryagin's maximum (or minimum) principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. It was formulated in 1956 by the Russian mathematician Lev Pontryagin and his students.〔See ref. below for first published work.〕 It has as a special case the Euler–Lagrange equation of the calculus of variations. The principle states, informally, that the ''control Hamiltonian'' must take an extreme value over controls in the set of all permissible controls. Whether the extreme value is maximum or minimum depends both on the problem and on the sign convention used for defining the Hamiltonian. The normal convention, which is the one used in Hamiltonian, leads to a maximum hence ''maximum principle'' but the sign convention used in this article, which apparently comes from, makes the extreme value a minimum, hence the unusual name ''minimum principle''. If $\backslash mathcal$ is the set of values of permissible controls then the principle states that the optimal control u * must satisfy: :$H(x^$ *(t),u^ *(t),\lambda^ *(t),t) \leq H(x^ *(t),u,\lambda^ *(t),t), \quad \forall u \in \mathcal, \quad t \in (t_f ) where $x^$ *\in C^1() is the optimal state trajectory and $\backslash lambda^$ * \in BV() is the optimal costate trajectory.〔More info on C1 and BV spaces〕 The result was first successfully applied to minimum time problems where the input control is constrained, but it can also be useful in studying state-constrained problems. Special conditions for the Hamiltonian can also be derived. When the final time $t\_f$ is fixed and the Hamiltonian does not depend explicitly on time $\backslash left(\backslash tfrac\; \backslash equiv\; 0\backslash right)$, then: :$H(x^$ *(t),u^ *(t),\lambda^ *(t)) \equiv \mathrm\, and if the final time is free, then: :$H(x^$ *(t),u^ *(t),\lambda^ *(t)) \equiv 0.\, More general conditions on the optimal control are given below. When satisfied along a trajectory, Pontryagin's minimum principle is a necessary condition for an optimum. The Hamilton–Jacobi–Bellman equation provides a necessary and sufficient condition for an optimum, but this condition must be satisfied over the whole of the state space. ==Maximization and minimization== The principle was first known as ''Pontryagin's maximum principle'' and its proof is historically based on maximizing the Hamiltonian. The initial application of this principle was to the maximization of the terminal speed of a rocket. However as it was subsequently mostly used for minimization of a performance index it has here been referred to as the ''minimum principle''. Pontryagin's book solved the problem of minimizing a performance index.〔See p.13 of the 1962 book of Pontryagin et al. referenced below.〕 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「Pontryagin's maximum principle」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.