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# 数学代写|优化理论代写Optimization Theory代考|FUNDAMENTAL CONCEPTS

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## 数学代写|优化理论代写Optimization Theory代考|FUNDAMENTAL CONCEPTS

In optimal control problems the objective is to determine a function that minimizes a specified functional – the performance measure. The analogous problem in calculus is to determine a point that yields the minimum value of a function. In this section we shall introduce some new concepts concerning functionals by appealing to some familiar results from the theory of functions. $\dagger$
Functionals
To begin, let us review the definition of a function.
DEFINITION 4-1
A function $f$ is a rule of correspondence that assigns to each element $q$ in a certain set $\mathscr{D}$ a unique element in a set $\mathscr{R} . \mathscr{D}$ is called the domain of $f$ and $\mathscr{R}$ is the range.
We shall be considering functions that assign a real number to each point (or vector) in $n$-dimensional Euclidean space. $\ddagger$
$$f(\mathbf{q})=\sqrt{q_1^2+q_2^2+\cdots+q_n^2}$$
The real number assigned by $f$ is the distance of the point $q$ from the origin.
The definition of a functional parallels that of a function.
DEFINITION 4-2
A functional $J$ is a rule of correspondence that assigns to each function $\mathbf{x}$ in a certain class $\Omega$ a unique real number. $\Omega$ is called the domain of the functional, and the set of real numbers associated with the functions in $\Omega$ is called the range of the functional.

## 数学代写|优化理论代写Optimization Theory代考|Linearity of Functiona/s

Let us review the concept of linearity, which will be useful to us later, by considering a function $f$ of $\mathbf{q}$, defined for $\mathbf{q} \in \mathscr{D}$.
DEFINITION 4-3
$f$ is a linear function of $\mathbf{q}$ if and only if it satisfies the principle of homogeneity
$$f(\alpha \mathbf{q})=\alpha f(\mathbf{q})$$
for all $\mathbf{q} \in \mathscr{D}$ and for all real numbers $\alpha$ such that $\alpha \mathbf{q} \in \mathscr{D}$, and the principle of additivity
$$f\left(\mathbf{q}^{(1)}+\mathbf{q}^{(2)}\right)=f\left(\mathbf{q}^{(1)}\right)+f\left(\mathbf{q}^{(2)}\right)$$
for all $\mathbf{q}^{(1)}, \mathbf{q}^{(2)}$, and $\mathbf{q}^{(1)}+\mathbf{q}^{(2)}$ in $\mathscr{D} . \dagger$
Example 4.1-3. If $f(t)=5 t$ for all $t$, then
$$f(\alpha t)=5[\alpha t]$$
and
$$\alpha f(t)=\alpha[5 t]$$
therefore, since
$$5[\alpha t]=\alpha[5 t]$$
for all $t$, the principle of homogeneity is satisfied. Now, let us test to see if the property of additivity is satisfied.
$$f\left(t^{(1)}+t^{(2)}\right)=5\left[t^{(1)}+t^{(2)}\right]$$
and
$$f\left(t^{(1)}\right)+f\left(t^{(2)}\right)=5 t^{(1)}+5 t^{(2)}$$
thus, since
$$5\left[t^{(1)}+t^{(2)}\right]=5 t^{(1)}+5 t^{(2)}$$
for all $t^{(1)}, t^{(2)}$, the principle of additivity is satisfied. Since the principle of homogeneity and the principle of additivity are both satisfied, $f$ is a linear function.

## 数学代写|优化理论代写Optimization Theory代考|FUNDAMENTAL CONCEPTS

Functionals

$$f(\mathbf{q})=\sqrt{q_1^2+q_2^2+\cdots+q_n^2}$$
$f$分配的实数是点$q$到原点的距离。

## 数学代写|优化理论代写Optimization Theory代考|Linearity of Functiona/s

$f$ 一个线性函数是 $\mathbf{q}$ 当且仅当满足同质性原则
$$f(\alpha \mathbf{q})=\alpha f(\mathbf{q})$$

$$f\left(\mathbf{q}^{(1)}+\mathbf{q}^{(2)}\right)=f\left(\mathbf{q}^{(1)}\right)+f\left(\mathbf{q}^{(2)}\right)$$

$$f(\alpha t)=5[\alpha t]$$

$$\alpha f(t)=\alpha[5 t]$$

$$5[\alpha t]=\alpha[5 t]$$

$$f\left(t^{(1)}+t^{(2)}\right)=5\left[t^{(1)}+t^{(2)}\right]$$

$$f\left(t^{(1)}\right)+f\left(t^{(2)}\right)=5 t^{(1)}+5 t^{(2)}$$

$$5\left[t^{(1)}+t^{(2)}\right]=5 t^{(1)}+5 t^{(2)}$$

## 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。