Euler Lagrange Equation

描述

求解有界泛函的极值：$J=F[y(x)]$, $y(x)$是一个可变的函数，例如可能是任意一条连接两点的曲线

例子

1. Lagrange mechanics motion：$s[q(t)]={\int }_{t0}^{t1}L(t,q,\dot{q})dt$
2. Path planning：$s[y(x)]={\int }_{x0}^{x1}F(t,y,\dot{y})dx$

公式推导

1. 借助扰动函数$y=y^{\ast }+\epsilon \eta (x)$，推导:
   $J^{\prime }(\epsilon )=\int \frac{\partial F}{\partial y}\frac{\partial y}{\partial \epsilon }+\frac{\partial F}{\partial \dot{y}}\frac{\partial \dot{y}}{\partial \epsilon }dx=\int \frac{\partial F}{\partial y}\eta (x)+\frac{\partial F}{\partial \dot{y}}\dot{\eta (x)}dx=0$

2. 起终点的时刻和状态都确定：
   $\int \frac{\partial F}{\partial \dot{y}}\dot{\eta (x)}dx=(\frac{\partial F}{\partial \dot{y}}\eta (x)){\mid }_{x0}^{x1}-\int \frac{\mathrm{d}}{\mathrm{d}\mathrm{x}}\eta (x)\frac{\partial F}{\partial \dot{y}}dx=-\int \frac{\mathrm{d}}{\mathrm{d}\mathrm{x}}\eta (x)\frac{\partial F}{\partial \dot{y}}dx$
   $\eta (x0)=\eta (x1)=0$ ==> $\frac{\partial F}{\partial y}-\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\partial F}{\partial y^{\prime }})=0$

3. 如果终点时刻缺点状态不确定，需要额外满足横截条件
   $\eta (x0)=0,\eta (x1)\ne 0$ ==> $\frac{\partial F}{\partial \dot{y}}{|}_{t1}=0$

参考

[1]: https://zhuanlan.zhihu.com/p/148949128
[2]: https://www.youtube.com/watch?v=V0wx0JBEgZc
[3]: 最优控制理论基础-吕显瑞，黄庆道