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# 数学代写|优化理论代写Optimization Theory代考|PIECEWISE-SMOOTH EXTREMALS

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## 数学代写|优化理论代写Optimization Theory代考|PIECEWISE-SMOOTH EXTREMALS

In the preceding sections we have derived necessary conditions that must be satisfied by extremal curves. The admissible curves were assumed to be continuous and to have continuous first derivatives; that is, the admissible curves were smooth. This is a very restrictive requirement for many practical problems. For example, if a control signal is the output of a relay, we know that this signal will contain discontinuities and that when such a control discontinuity occurs, one or more of the components of $\dot{\mathbf{x}}(t)$ will be discontinuous. Thus, we wish to enlarge the class of admissible curves to include functions that have only piecewise-continuous first derivatives; that is, $\dot{\mathbf{x}}$ will be continuous except at a finite number of times in the interval $\left(t_0, t_f\right)$. At a time when $\dot{x}$ is discontinuous, $\mathbf{x}$ is said to have a corner. Let us begin by considering functionals involving only a single function.
The problem is to find a necessary condition that must be satisfied by extrema of the functional
$$J(x)=\int_{t_0}^{t s} g(x(t), \dot{x}(t), t) d t$$
It is assumed that $g$ has continuous first and second partial derivatives with respect to all of its arguments, and that $t_0, t_f, x\left(t_0\right)$, and $x\left(t_f\right)$ are specified. $\dot{x}$ is a piecewise-continuous function (or we say that $x$ is a piecewise-smooth curve). Assume that $\dot{x}$ has a discontinuity at some point $t_1 \in\left(t_0, t_f\right) ; t_1$ is not fixed, nor is it usually known in advance.
Let us first express the functional $J$ as
\begin{aligned} J(x) & =\int_{t_0}^{t_1} g(x(t), \dot{x}(t), t) d t+\int_{t_1}^{t_s} g(x(t), \dot{x}(t), t) d t \ & \triangleq J_1(x)+J_2(x) \end{aligned}
functions that have only piecewise-continuous first derivatives; that is, $\dot{\mathbf{x}}$ will be continuous except at a finite number of times in the interval $\left(t_0, t_f\right)$. At a time when $\dot{x}$ is discontinuous, $\mathbf{x}$ is said to have a corner. Let us begin by considering functionals involving only a single function.
The problem is to find a necessary condition that must be satisfied by extrema of the functional
$$J(x)=\int_{t_0}^{t s} g(x(t), \dot{x}(t), t) d t$$
It is assumed that $g$ has continuous first and second partial derivatives with respect to all of its arguments, and that $t_0, t_f, x\left(t_0\right)$, and $x\left(t_f\right)$ are specified. $\dot{x}$ is a piecewise-continuous function (or we say that $x$ is a piecewise-smooth curve). Assume that $\dot{x}$ has a discontinuity at some point $t_1 \in\left(t_0, t_f\right) ; t_1$ is not fixed, nor is it usually known in advance.
Let us first express the functional $J$ as
\begin{aligned} J(x) & =\int_{t_0}^{t_1} g(x(t), \dot{x}(t), t) d t+\int_{t_1}^{t_s} g(x(t), \dot{x}(t), t) d t \ & \triangleq J_1(x)+J_2(x) \end{aligned}

## 数学代写|优化理论代写Optimization Theory代考|CONSTRAINED EXTREMA

So far, we have discussed functionals involving $\mathbf{x}$ and $\dot{\mathbf{x}}$, and we have derived necessary conditions for extremals assuming that the components of $\mathbf{x}$ are independent. In control problems the situation is more complicated, because the state trajectory is determined by the control $\mathbf{u}$; thus, we wish to consider functionals of $n+m$ functions, $\mathbf{x}$ and $\mathbf{u}$, but only $m$ of the functions are independent-the controls. Let us now extend the necessary conditions we have derived to include problems with constraints.
To begin, we shall review the analogous problem from the calculus, and introduce some new variables-the Lagrange multipliers-that will be required for our subsequent discussion.
Constrained Minimization of Functions
Example 4.5-1. Find the point on the line $y_1+y_2=5$ that is nearest the origin.
To solve this problem we need only apply elementary plane geometry to Fig. 4-19 to obtain the result that the minimum distance is $5 / \sqrt{2}$, and the extreme point is $y_1^=2.5, y_2^=2.5$.
Most problems cannot be solved by inspection, so let us consider alternative methods of solving this simple example.
The Elimination Method. If $\mathbf{y}^$ is an extreme point of a function, it is necessary that the differential of the function, evaluated at $\mathbf{y}^$, be zero. $\dagger$ In our example, the function
$$f\left(y_1, y_2\right)=y_1^2+y_2^2 \quad \text { (the square of the distance) }$$
is to be minimized subject to the constraint
$$y_1+y_2=5$$
The differential is
$$d f\left(y_1, y_2\right)=\left[\frac{\partial f}{\partial y_1}\left(y_1, y_2\right)\right] \Delta y_1+\left[\frac{\partial f}{\partial y_2}\left(y_1, y_2\right)\right] \Delta y_2,$$
and if $\left(y_1^, y_2^\right)$ is an extreme point,
$$d f\left(y_1^, y_2^\right)=\left[\frac{\partial f}{\partial y_1}\left(y_1^, y_2^\right)\right] \Delta y_1+\left[\frac{\partial f}{\partial y_2}\left(y_1^, y_2^\right)\right] \Delta y_2=0$$

## 数学代写|优化理论代写Optimization Theory代考|PIECEWISE-SMOOTH EXTREMALS

$$J(x)=\int_{t_0}^{t s} g(x(t), \dot{x}(t), t) d t$$

\begin{aligned} J(x) & =\int_{t_0}^{t_1} g(x(t), \dot{x}(t), t) d t+\int_{t_1}^{t_s} g(x(t), \dot{x}(t), t) d t \ & \triangleq J_1(x)+J_2(x) \end{aligned}

$$J(x)=\int_{t_0}^{t s} g(x(t), \dot{x}(t), t) d t$$

\begin{aligned} J(x) & =\int_{t_0}^{t_1} g(x(t), \dot{x}(t), t) d t+\int_{t_1}^{t_s} g(x(t), \dot{x}(t), t) d t \ & \triangleq J_1(x)+J_2(x) \end{aligned}

## 数学代写|优化理论代写Optimization Theory代考|CONSTRAINED EXTREMA

$$f\left(y_1, y_2\right)=y_1^2+y_2^2 \quad \text { (the square of the distance) }$$

$$y_1+y_2=5$$

$$d f\left(y_1, y_2\right)=\left[\frac{\partial f}{\partial y_1}\left(y_1, y_2\right)\right] \Delta y_1+\left[\frac{\partial f}{\partial y_2}\left(y_1, y_2\right)\right] \Delta y_2,$$

$$d f\left(y_1^, y_2^\right)=\left[\frac{\partial f}{\partial y_1}\left(y_1^, y_2^\right)\right] \Delta y_1+\left[\frac{\partial f}{\partial y_2}\left(y_1^, y_2^\right)\right] \Delta y_2=0$$

## 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。