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# 数学代写|优化理论代写Optimization Theory代考|TWO-POINT BOUNDARY-VALUE PROBLEMS

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## 数学代写|优化理论代写Optimization Theory代考|TWO-POINT BOUNDARY-VALUE PROBLEMS

Assuming that the state and control variables are not constrained by any boundaries, that the final time $t_f$ is fixed, and that $\mathbf{x}\left(t_f\right)$ is free, we can summarize the two-point boundary-value problem that results from the variational approach by the equations
\begin{aligned} \dot{\mathbf{x}}^(t)= & \frac{\partial \mathscr{H}}{\partial \mathbf{p}}=\mathbf{a}\left(\mathbf{x}^(t), \mathbf{u}^(t), t\right) \ \dot{\mathbf{p}}^(t)= & -\frac{\partial \mathscr{H}}{\partial \mathbf{x}}=-\left[\frac{\partial \mathbf{a}}{\partial \mathbf{x}}\left(\mathbf{x}^(t), \mathbf{u}^(t), t\right)\right]^T \mathbf{p}^(t) \ & -\frac{\partial g}{\partial \mathbf{x}}\left(\mathbf{x}^(t), \mathbf{u}^(t), t\right) \ \mathbf{0}= & \frac{\partial \mathscr{H}}{\partial \mathbf{u}}=\left[\frac{\partial \mathbf{a}}{\partial \mathbf{u}}\left(\mathbf{x}^(t), \mathbf{u}^(t), t\right)\right]^T \mathbf{p}^(t) \ & +\frac{\partial g}{\partial \mathbf{u}}\left(\mathbf{x}^(t), \mathbf{u}^(t), t\right) \ \mathbf{x}^\left(t_0\right)= & \mathbf{x}_0 \ \mathbf{p}^\left(t_f\right)= & \frac{\partial h}{\partial \mathbf{x}}\left(\mathbf{x}^\left(t_f\right)\right) \cdot \dagger \end{aligned} From these five sets of conditions it is desired to obtain an explicit relationship for $\mathbf{x}^(t)$ and $\mathbf{u}^(t), t \in\left[t_0, t_f\right]$. Notice that the expressions for $\mathbf{x}^(t)$ and $\mathbf{u}^*(t)$ will be implicitly dependent on the initial state $x_0$.
Let us assume that Eq. (6.1-3) can be solved to obtain an expression for $\mathbf{u}^(t)$ in terms of $\mathbf{x}^(t), \mathbf{p}^(t)$, and $t$; that is, $$\mathbf{u}^(t)=\mathbf{f}\left(\mathbf{x}^(t), \mathbf{p}^(t), t\right) .$$
If this expression is substituted into Eqs. (6.1-1) and (6.1-2), we have a set of $2 n$ first-order ordinary differential equations (called the reduced differential equations) involving only $\mathbf{x}^(t), \mathbf{p}^(t)$, and $t$. The boundary conditions for these differential equations (which are generally nonlinear) are given by Eq. (6.1-4).

## 数学代写|优化理论代写Optimization Theory代考|THE METHOD OF STEEPEST DESCENT

Let us begin our discussion of the method of steepest descent (or gradients) by considering an analogous calculus problem. Let $f$ be a function of two independent variables $y_1$ and $y_2$; the value of the function at the point $y_1$, $y_2$ is denoted by $f\left(y_1, y_2\right)$. It is desired to find the point $y_1^, y_2^$, where $f$ assumes its minimum value, $f\left(y_1^, y_2^\right)$.
If it is assumed that the variables $y_1$ and $y_2$ are not constrained by any boundaries, a necessary condition for $y_1^, y_2^$ to be a point where $f$ has a (relative) minimum is that the differential of $f$ vanish at $y_1^, y_2^$, that is,
\begin{aligned} d f\left(y_1^, y_2^\right)= & {\left[\frac{\partial f}{\partial y_1}\left(y_1^, y_2^\right)\right] \Delta y_1+\left[\frac{\partial f}{\partial y_2}\left(y_1^, y_2^\right)\right] \Delta y_2 } \ & \triangleq\left[\frac{\partial f}{\partial \mathbf{y}}\left(\mathbf{y}^*\right)\right]^T \Delta \mathbf{y}=0 . \end{aligned}
$\partial f / \partial \mathbf{y}$ is called the gradient of $f$ with respect to $\mathbf{y}$. Since $y_1$ and $y_2$ are independent, the components of $\Delta \mathbf{y}$ are independently arbitrary and (6.2-1)
implies
$\frac{\partial f}{\partial \mathbf{y}}\left(\mathbf{y}^*\right)=\mathbf{0}$.

## 数学代写|优化理论代写Optimization Theory代考|TWO-POINT BOUNDARY-VALUE PROBLEMS

\begin{aligned} \dot{\mathbf{x}}^(t)= & \frac{\partial \mathscr{H}}{\partial \mathbf{p}}=\mathbf{a}\left(\mathbf{x}^(t), \mathbf{u}^(t), t\right) \ \dot{\mathbf{p}}^(t)= & -\frac{\partial \mathscr{H}}{\partial \mathbf{x}}=-\left[\frac{\partial \mathbf{a}}{\partial \mathbf{x}}\left(\mathbf{x}^(t), \mathbf{u}^(t), t\right)\right]^T \mathbf{p}^(t) \ & -\frac{\partial g}{\partial \mathbf{x}}\left(\mathbf{x}^(t), \mathbf{u}^(t), t\right) \ \mathbf{0}= & \frac{\partial \mathscr{H}}{\partial \mathbf{u}}=\left[\frac{\partial \mathbf{a}}{\partial \mathbf{u}}\left(\mathbf{x}^(t), \mathbf{u}^(t), t\right)\right]^T \mathbf{p}^(t) \ & +\frac{\partial g}{\partial \mathbf{u}}\left(\mathbf{x}^(t), \mathbf{u}^(t), t\right) \ \mathbf{x}^\left(t_0\right)= & \mathbf{x}_0 \ \mathbf{p}^\left(t_f\right)= & \frac{\partial h}{\partial \mathbf{x}}\left(\mathbf{x}^\left(t_f\right)\right) \cdot \dagger \end{aligned} 从这五组条件中，我们希望得到一个显式关系 $\mathbf{x}^(t)$ 和 $\mathbf{u}^(t), t \in\left[t_0, t_f\right]$． 注意，for的表达式 $\mathbf{x}^(t)$ 和 $\mathbf{u}^*(t)$ 会隐式依赖于初始状态吗 $x_0$．

## 数学代写|优化理论代写Optimization Theory代考|THE METHOD OF STEEPEST DESCENT

\begin{aligned} d f\left(y_1^, y_2^\right)= & {\left[\frac{\partial f}{\partial y_1}\left(y_1^, y_2^\right)\right] \Delta y_1+\left[\frac{\partial f}{\partial y_2}\left(y_1^, y_2^\right)\right] \Delta y_2 } \ & \triangleq\left[\frac{\partial f}{\partial \mathbf{y}}\left(\mathbf{y}^\right)\right]^T \Delta \mathbf{y}=0 . \end{aligned} $\partial f / \partial \mathbf{y}$称为$f$相对于$\mathbf{y}$的梯度。由于$y_1$和$y_2$是独立的，$\Delta \mathbf{y}$的分量是独立任意的，并且(6.2-1) 暗示 $\frac{\partial f}{\partial \mathbf{y}}\left(\mathbf{y}^\right)=\mathbf{0}$。

## Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。