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• 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

## 统计代写|最优控制作业代写optimal control代考|The Mathematical Model

An important part of any control problem is the process of modeling the dynamic system under consideration, be it physical, business, or otherwise. The aim is to arrive at a mathematical description which is simple enough to deal with, and realistic enough to be able to predict the response of the system to any given input. Our model is restricted to systems that can be characterized by a set of ordinary differential equations (or, ordinary difference equations in the discrete-time case treated in Chap. 8). Thus, given the initial state $x_{0}$ of the system and control history $u(t), t \in[0, T]$, of the process, the evolution of the system may be described by the first-order differential equation, known also as the state equation,
$$\dot{x}(t)=f(x(t), u(t), t), \quad x(0)=x_{0}$$
where the vector of state variables, $x(t) \in E^{n}$, the vector of control variables, $u(t) \in E^{m}$, and $f: E^{n} \times E^{m} \times E^{1} \rightarrow E^{n}$. Furthermore, the function $f$ is assumed to be continuously differentiable. Here we assume $x$ to be a column vector and $f$ to be a column vector of functions. The path $x(t), t \in[0, T]$, is called a state trajectory and $u(t), t \in[0, T]$, is called a control trajectory or simply, a control. The terms vector of state variables, state vector, and state will be used interchangeably; similarly for the terms vector of control variables, control vector, and control. As mentioned earlier, when no confusion arises, we will usually suppress the time notation $(t)$; thus, e.g., $x(t)$ will be written simply as $x$. Furthermore, it should be inferred from the context whether $x$ denotes the state at time $t$ or the entire state trajectory. A similar statement holds for $u$.

## 统计代写|最优控制作业代写optimal control代考|The Optimal Control Problem

Given the preceding definitions we can state the optimal control problem, which we will be concerned with in this chapter. The problem is to find an admissible control $u^{}$, which maximizes the objective function (2.3) subject to the state equation (2.1) and the control constraints (2.2). We now restate the optimal control problem as: $$\left{\begin{array}{l} \max {u(t) \in \Omega(t)}\left{J=\int{0}^{T} F(x, u, t) d t+S(x(T), T)\right} \ \text { subject to } \ \dot{x}=f(x, u, t), x(0)=x_{0} . \end{array}\right.$$
The control $u^{}$ is called an optimal control and $x^{}$, determined by means of the state equation with $u=u^{}$, is called the optimal trajectory or an optimal path. The optimal value $J\left(u^{}\right)$ of the objective function will be denoted as $J^{}$, and occasionally as $J_{\left(x_{0}\right)}^{*}$ when we need to emphasize its dependence on the initial state $x_{0}$.

The optimal control problem (2.4) is said to be in Bolza form because of the form of the objective function in (2.3). It is said to be in Lagrange form when $S \equiv 0$. We say the problem is in Mayer form when $F \equiv 0$. Furthermore, it is in linear Mayer form when $F \equiv 0$ and $S$ is linear, i.e.,
$$\left{\begin{array}{l} \max {u(t) \in \Omega(t)}{J=c x(T)} \ \text { subject to } \ \dot{x}=f(x, u, t), x(0)=x{0} \end{array}\right.$$
where $c=\left(c_{1}, c_{2}, \cdots, c_{n}\right)$ is an $n$-dimensional row vector of given constants. In the next paragraph and in Exercise $2.5$, it will be demonstrated that all of these forms can be converted into the linear Mayer form.

## 统计代写|最优控制作业代写optimal control代考|The Mathematical Model

$$\dot{x}(t)=f(x(t), u(t), t), \quad x(0)=x_{0}$$

## 统计代写|最优控制作业代写optimal control代考|The Optimal Control Problem

$\backslash$ left 的分隔符缺失或无法识别

$\$ \$$控件 u 称为最优控制，并且 x ，通过状态方程确定 u=u ，称为最优轨迹或最优路径。最优值 J(u) 目标函数的值将表示为 J ，有时作为 J_{\left(x_{0}\right)}^{*} 当我们需要强调它对初始状态的依赖时 x_{0} . 由于 (2.3) 中目标函数的形式，最优控制问题 (2.4) 被称为 Bolza 形式。据说当它是拉格朗日形式时 S \equiv 0. 我们说问题是 Mayer 形式的， 当 F \equiv 0. 此外，它是线性迈耶形式，当 F \equiv 0 和 S 是线性的，即 \ \$$
Veft{
$$\max u(t) \in \Omega(t) J=c x(T) \text { subject to } \dot{x}=f(x, u, t), x(0)=x 0$$

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## MATLAB代写

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assignmentutor™您的专属作业导师
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