# 数学代写|凸分析和最优控制代写Convex Analysis and Optimal Control代考|MATH4071 The variational limit

my-assignmentexpert™提供最专业的一站式服务：Essay代写，Dissertation代写，Assignment代写，Paper代写，Proposal代写，Proposal代写，Literature Review代写，Online Course，Exam代考等等。my-assignmentexpert™专注为留学生提供Essay代写服务，拥有各个专业的博硕教师团队帮您代写，免费修改及辅导，保证成果完成的效率和质量。同时有多家检测平台帐号，包括Turnitin高级账户，检测论文不会留痕，写好后检测修改，放心可靠，经得起任何考验！

## 数学代写|凸分析和最优控制代写Convex Analysis and Optimal Control代考|The variational limit

With the aid of the narrow limit of the parameter $\rho_{\varepsilon}(\cdot)$ in $(1.1)$, we exhibit in this section a candidate for a variational limit for (1.1). We $\mathcal{C}\left(R^n\right)$. It is true (see Lemma 4.2) that the the narrow convergence of $\rho_{\varepsilon}(\cdot)$ to $\mu(\cdot)$ implies the narrow convergence of $F\left(\cdot, \rho_{\varepsilon}(\cdot)\right)$ to $F(\cdot, \mu(\cdot))$.
Let $\nu$ be a probability measure on $\mathcal{C}\left(R^n\right)$, and let $p$ be a probability measure on $R^n$. The measure $p$ is selectionable with respect to $\nu$ if there are a probability measure space $(\Omega, \Sigma, \lambda)$, a measurable set-valued map $H$ from $\Omega$ into $\mathcal{C}\left(R^n\right)$ and a measurable point-valued map $h$ from $\Omega$ into $R^n$, such that $h(\omega) \in H(\omega)$ for all $\omega, D h=p$ and $D H=\nu$, where $D h$ and $D H$ are the probability distributions induced by $h$ and $H$, namely
$$D h(A)=\lambda({\omega: h(\omega) \in A}),$$
and likewise for $H$.

## 数学代写|凸分析和最优控制代写Convex Analysis and Optimal Control代考|The convergence

This section verifies that the optimization problem (3.1) is a variational limit of (1.1) as $\varepsilon$ tends to 0 , in the sense expressed in the following theorem. In the next section we show how a solution of (3.1) gives rise to near optimal solutions of (1.1) for $\varepsilon$ small. Recall that we work under Assumptions 2.1, $2.2$ and 3.1.

Theorem 4.1. Suppose that a fixed open neighborhood of the resource vector $\hat{a}$ is in the feasible set of (1.1) for $\varepsilon$ small enough. Then $\hat{a}$ is in the feasible set of (3.1), and the optimal values $\operatorname{val}(\varepsilon)$ of (1.1) converge as $\varepsilon$ tends to 0 , to the optimal value $\operatorname{val}(0)$ of (3.1). Furthermore, let $x_{\varepsilon}^(\cdot)$ for $\varepsilon$ fixed be an optimal solution of (1.1). Then for any sequence $\varepsilon_i$ converging to 0 , there exists a subsequence $\varepsilon_j$ such that $x_{\varepsilon_j}^(\cdot)$ converge narrowly to a solution $\xi(\cdot)$ of (3.1).

A term which is self explanatory, yet may need a formal definition, is the feasible set. To this end recall the integral of a set-valued function, say $H$ from $[0,1]$ into $\mathcal{C}\left(R_{+}^n\right)$, given by (see Aumann [7] and Klein and Thompson [21]):
\begin{aligned} &\int_0^1 H(t) d t= \ &\quad\left{\int_0^1 h(t) d t: h(t) \in H(t) \text { for almost every } t\right} . \end{aligned}

## 数学代写凸分析和最优控制代写CONVEX ANALYSIS AND OPTIMAL CONTROL代考|THE VARIATIONAL LIMIT

$F\left(\cdot, \rho_{\varepsilon}(\cdot)\right)$ 至 $F(\cdot, \mu(\cdot))$.

$$\operatorname{Dh}(A)=\lambda(\omega: h(\omega) \in A),$$

## 数学代写凸分析和最优控制代写CONVEX ANALYSIS AND OPTIMAL CONTROL代考|THE CONVERGENCE

seeAumann [7] andKleinandThompson [21]:

## 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。