# 庞特里亚金最大化原理

${\displaystyle {\mathcal {U}}}$是所有可能控制值的集合，則此原理指出，最優控制${\displaystyle u^{*}}$必須滿足以下條件：

${\displaystyle H(x^{*}(t),u^{*}(t),\lambda ^{*}(t),t)\leq H(x^{*}(t),u,\lambda ^{*}(t),t),\quad \forall u\in {\mathcal {U}},\quad t\in [t_{0},t_{f}]}$

${\displaystyle H(x^{*}(t),u^{*}(t),\lambda ^{*}(t))\equiv \mathrm {constant} \,}$

${\displaystyle H(x^{*}(t),u^{*}(t),\lambda ^{*}(t))\equiv 0.\,}$

## 符號

${\displaystyle \Psi _{T}(x(T))={\frac {\partial \Psi (x)}{\partial T}}|_{x=x(T)}\,}$
${\displaystyle \Psi _{x}(x(T))={\begin{bmatrix}{\frac {\partial \Psi (x)}{\partial x_{1}}}|_{x=x(T)}&\cdots &{\frac {\partial \Psi (x)}{\partial x_{n}}}|_{x=x(T)}\end{bmatrix}}}$
${\displaystyle H_{x}(x^{*},u^{*},\lambda ^{*},t)={\begin{bmatrix}{\frac {\partial H}{\partial x_{1}}}|_{x=x^{*},u=u^{*},\lambda =\lambda ^{*}}&\cdots &{\frac {\partial H}{\partial x_{n}}}|_{x=x^{*},u=u^{*},\lambda =\lambda ^{*}}\end{bmatrix}}}$
${\displaystyle L_{x}(x^{*},u^{*})={\begin{bmatrix}{\frac {\partial L}{\partial x_{1}}}|_{x=x^{*},u=u^{*}}&\cdots &{\frac {\partial L}{\partial x_{n}}}|_{x=x^{*},u=u^{*}}\end{bmatrix}}}$
${\displaystyle f_{x}(x^{*},u^{*})={\begin{bmatrix}{\frac {\partial f_{1}}{\partial x_{1}}}|_{x=x^{*},u=u^{*}}&\cdots &{\frac {\partial f_{1}}{\partial x_{n}}}|_{x=x^{*},u=u^{*}}\\\vdots &\ddots &\vdots \\{\frac {\partial f_{n}}{\partial x_{1}}}|_{x=x^{*},u=u^{*}}&\ldots &{\frac {\partial f_{n}}{\partial x_{n}}}|_{x=x^{*},u=u^{*}}\end{bmatrix}}}$

## 最小化問題必要條件的正式敘述

${\displaystyle {\dot {x}}=f(x,u),\quad x(0)=x_{0},\quad u(t)\in {\mathcal {U}},\quad t\in [0,T]}$

${\displaystyle {\mathcal {U}}}$ 為可行控制的集合
${\displaystyle T}$ 為系統的結束時間。

${\displaystyle J=\Psi (x(T))+\int _{0}^{T}L(x(t),u(t))\,dt}$

${\displaystyle H(x(t),u(t),\lambda (t),t)=\lambda ^{\rm {T}}(t)f(x(t),u(t))+L(x(t),u(t))\,}$

${\displaystyle (1)\qquad H(x^{*}(t),u^{*}(t),\lambda ^{*}(t),t)\leq H(x^{*}(t),u,\lambda ^{*}(t),t)\,}$

${\displaystyle (2)\qquad \Psi _{T}(x(T))+H(T)=0\,}$

${\displaystyle (3)\qquad -{\dot {\lambda }}^{\rm {T}}(t)=H_{x}(x^{*}(t),u^{*}(t),\lambda (t),t)=\lambda ^{\rm {T}}(t)f_{x}(x^{*}(t),u^{*}(t))+L_{x}(x^{*}(t),u^{*}(t))}$

${\displaystyle (4)\qquad \lambda ^{\rm {T}}(T)=\Psi _{x}(x(T))\,}$

## 相關條目

• ，變分法下中的拉格朗日法
• 奇異控制：無法利用龐特里亞金最小化原理求出完整解的最優控制問題。

## 腳註

1. ^ 參考資料中有最早發表的論文
2. ^ C1BV空間條目中有更多的資訊
3. ^ 參照 Pontryagin 1962年的書，第13頁
4. ^ Haggag, S.; Desokey, F.; Ramadan, M.,. A cosmological inflationary model using optimal control. Gravitation and Cosmology (Pleiades Publishing). 2017, 23 (3): 236–239. ISSN 1995-0721. doi:10.1134/S0202289317030069.

## 參考資料

• Boltyanskii, V. G.; Gamkrelidze, R. V.; Pontryagin, L. S. К теории оптимальных процессов [Towards a Theory of Optimal Processes]. Dokl. Akad. Nauk SSSR. 1956, 110 (1): 7–10. MR 0084444 （俄语）.
• Pontryagin, L. S.; Boltyanskii, V. G.; Gamkrelidze, R. V.; Mishchenko, E. F. The Mathematical Theory of Optimal Processes. English translation. Interscience. 1962. ISBN 2-88124-077-1.
• Fuller, A. T. Bibliography of Pontryagin's maximum principle. J. Electronics & Control. 1963, 15 (5): 513–517.
• Kirk, D. E. Optimal Control Theory: An Introduction. Prentice Hall. 1970. ISBN 0-486-43484-2.
• Sethi, S. P.; Thompson, G. L. Optimal Control Theory: Applications to Management Science and Economics 2nd. Springer. 2000. ISBN 0-387-28092-8. Slides are available at [1]
• Geering, H. P. Optimal Control with Engineering Applications. Springer. 2007. ISBN 978-3-540-69437-3.
• Ross, I. M. A Primer on Pontryagin's Principle in Optimal Control. Collegiate. 2009. ISBN 978-0-9843571-0-9.
• Cassel, Kevin W. Variational Methods with Applications in Science and Engineering. Cambridge University Press. 2013.