## 统计代写|最优控制作业代写optimal control代考|MA30061

2022年7月7日

couryes-lab™ 为您的留学生涯保驾护航 在代写最优控制optimal control方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写最优控制optimal control代写方面经验极为丰富，各种代写最优控制Soptimal control相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
couryes™为您提供可以保分的包课服务

## 统计代写|最优控制作业代写optimal control代考|Sufficiency Conditions

So far, we have shown the necessity of the maximum principle conditions for optimality. Next we prove a theorem that gives qualifications under which the maximum principle conditions are also sufficient for optimality. This theorem is important from our point of view since the models derived from many management science applications will satisfy conditions required for the sufficiency result. As remarked earlier, our technique for proving existence will be to display for any given model, a solution that satisfies both necessary and sufficient conditions. A good reference for sufficiency conditions is Seierstad and Sydsæter (1987).

We first define a function $H^{0}: E^{n} \times E^{m} \times E^{1} \rightarrow E^{1}$ called the derived Hamiltonian as follows:
$$H^{0}(x, \lambda, t)=\max _{u \in \Omega(t)} H(x, u, \lambda, t)$$
We assume that by this equation a function $u^{}(x, \lambda, t)$ is implicitly and uniquely defined. Given these assumptions we have by definition, $$H^{0}(x, \lambda, t)=H\left(x, u^{}, \lambda, t\right)$$

For our proof of the sufficiency of the maximum principle, we also need the derivative $H_{x}^{0}(x, \lambda, t)$, which by use of the Envelope Theorem can be given as
$$H_{x}^{0}(x, \lambda, t)=H_{x}\left(x, u^{}, \lambda, t\right):=\left.H_{x}(x, u, \lambda, t)\right|{u=u^{}}$$
To see this in the case when $u^{}(x, \lambda, t)$ is differentiable in $x$, let us differentiate (2.62) with respéct to $x$ : $$H{x}^{0}(x, \lambda, t)=H_{x}\left(x, u^{}, \lambda, t\right)+H_{u}\left(x, u^{}, \lambda, t\right) \frac{\partial u^{}}{\partial x}$$
To obtain (2.63) from (2.64), we need to show that the second term on the right-hand side of (2.64) vanishes, i.e.,
$$H_{u}\left(x, u^{}, \lambda, t\right) \frac{\partial u^{}}{\partial x}=0$$ for each $x$. There are two cases to consider. If $u^{}$ is in the interior of $\Omega(t)$, then it satisfies the first-order condition $H_{u}\left(x, u^{}, \lambda, t\right)=0$, thereby implying (2.65). Otherwise, $u^{}$ is on the boundary of $\Omega(t)$. Then, for each $i, j$, either $H_{u_{i}}=0$ or $\partial u_{i}^{} / \partial x_{j}=0$ or both. Once again, (2.65) holds. Exercise $2.25$ gives a specific instance of this case.

## 统计代写|最优控制作业代写optimal control代考|Solving a TPBVP by Using Excel

A number of examples and exercises found throughout this book involve finding a numerical solution to a two-point boundary value problem (TPBVP). In this section we will show how the GOAL SEEK function in Excel can be used for this purpose. We will solve the following example.
Example $2.8$ Consider the problem:
$$\max \left{J=\int_{0}^{1}-\frac{1}{2}\left(x^{2}+u^{2}\right) d t\right}$$

subject to
$$\dot{x}=-x^{3}+u, x(0)=5 .$$
Solution We form the Hamiltonian
$$H=-\frac{1}{2}\left(x^{2}+u^{2}\right)+\lambda\left(-x^{3}+u\right)$$
where the adjoint variable $\lambda$ satisfies the equation
$$\dot{\lambda}=x+3 x^{2} \lambda, \lambda(1)=0$$
Since $u$ is unconstrained, wee set $H_{u}=0$ to obtaiin $u^{*}=\lambda$. With this, the stateé equation (2.76) becomes
$$\dot{x}=-x^{3}+\lambda, x(0)=5$$
Thus, the TPBVP is given by the system of Eqs. (2.77) and (2.78).
A simple method to solve the TPBVP uses what is known as the shooting method, explained in the flowchart in Fig. 2.6.

## 统计代写|最优控制作业代写optimal control代考|Sufficiency Conditions

$$H^{0}(x, \lambda, t)=\max {u \in \Omega(t)} H(x, u, \lambda, t)$$ 我们假设通过这个方程一个函数 $u(x, \lambda, t)$ 是隐含且唯一定义的。鉴于这些假设，我们根据 定义, $$H^{0}(x, \lambda, t)=H(x, u, \lambda, t)$$ 为了证明最大原理的充分性，我们还需要导数 $H{x}^{0}(x, \lambda, t)$, 通过使用包絡定理可以给出
$$H_{x}^{0}(x, \lambda, t)=H_{x}(x, u, \lambda, t):=H_{x}(x, u, \lambda, t) \mid u=u$$

$$H x^{0}(x, \lambda, t)=H_{x}(x, u, \lambda, t)+H_{u}(x, u, \lambda, t) \frac{\partial u}{\partial x}$$

$$H_{u}(x, u, \lambda, t) \frac{\partial u}{\partial x}=0$$

## 统计代写|最优控制作业代写optimal control代考|Solving a TPBVP by Using Excel

\max \eft $\left{J=\backslash\right.$ int_{ ${0} \wedge{1}-\backslash \operatorname{frac}{1}{2} \backslash \operatorname{lft}\left(\mathrm{x}^{\wedge}{2}+\mathrm{u}^{\wedge}{2} \backslash\right.$ right) $\mathrm{d} t \backslash$ right $}$

$$\dot{x}=-x^{3}+u, x(0)=5 .$$

$$H=-\frac{1}{2}\left(x^{2}+u^{2}\right)+\lambda\left(-x^{3}+u\right)$$

$$\dot{\lambda}=x+3 x^{2} \lambda, \lambda(1)=0$$

$$\dot{x}=-x^{3}+\lambda, x(0)=5$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。