### 统计代写|最优控制作业代写optimal control代考|MA409

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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|Basic Concepts and Definitions

We will use the word system as a primitive term in this book. The only property that we require of a system is that it is capable of existing in various states. Let the (real) variable $x(t)$ be the state variable of the system at time $t \in[0, T]$, where $T>0$ is a specified time horizon for the system under consideration. For example, $x(t)$ could measure the inventory level at time $t$, the amount of advertising goodwill at time $t$, or the amount of unconsumed wealth or natural resources at time $t$.

We assume that there is a way of controlling the state of the system. Let the (real) variable $u(t)$ be the control variable of the system at time $t$. For example, $u(t)$ could be the production rate at time $t$, the advertising rate at time $t$, etc.

Given the values of the state variable $x(t)$ and the control variable $u(t)$ at time $t$, the state equation, a differential equation,
$$\dot{x}(t)=f(x(t), u(t), t), \quad x(0)=x_{0},$$
specifies the instantaneous rate of change in the state variable, where $\dot{x}(t)$ is a commonly used notation for $d x(t) / d t, f$ is a given function of $x, u$, and $t$, and $x_{0}$ is the initial value of the state variable. If we know the initial value $x_{0}$ and the control trajectory, i.e., the values of $u(t)$ over the whole time interval $0 \leq t \leq T$, then we can integrate (1.1) to get the state trajectory, i.e., the values of $x(t)$ over the same time interval. We want to choose the control trajectory so that the state and control trajectories maximize the objective functional, or simply the objective function,
$$J=\int_{0}^{T} F(x(t), u(t), t) d t+S[x(T), T]$$
In (1.2), $F$ is a given function of $x, u$, and $t$, which could measure the benefit minus the cost of advertising, the utility of consumption, the negative of the cost of inventory and production, etc. Also in (1.2), the function $S$ gives the salvage value of the ending state $x(T)$ at time $T$. The salvage value is needed so that the solution will make “good sense” at the end of the horizon.

## 统计代写|最优控制作业代写optimal control代考|Formulation of Simple Control Models

We now formulate three simple models chosen from the areas of production, advertising, and economics. Our only objective here is to identify and interpret in these models each of the variables and functions described in the previous section. The solutions for each of these models will be given in detail in later chapters.
Example 1.1 A Production-Inventory Model. The various quantities that define this model are summarized in Table $1.1$ for easy comparison with the other models that rulluw.

We consider the production and inventory storage of a given good, such as steel, in order to meet an exogenous demand. The state variable $I(t)$ measures the number of tons of steel that we have on hand at time $t \in[0, T]$. There is an exogenous demand rate $S(t)$ tons of steel per day at time $t \in[0, T]$, and we must choose the production rate $P(t)$ tons of steel per day at time $t \in[0, T]$. Given the initial inventory of $I_{0}$ tons of steel on hand at $t=0$, the state equation
$$\dot{I}(t)=P(t)-S(t)$$
describes how the steel inventory changes over time. Since $h(I)$ is the cost of holding inventory $I$ in dollars per day, and $c(P)$ is the cost of producing steel at rate $P$, also in dollars per day, the objective function is to maximize the negative of the sum of the total holding and production costs over the period of $T$ days. Of course, maximizing the negative sum is the same as minimizing the sum of holding and production costs. The state variable constraint, $I(t) \geq 0$, is imposed so that the demand is satisfied for all $t$. In other words, backlogging of demand is not permitted. (An alternative formulation is to make $h(I)$ become very large when $I$ becomes negative, i.e., to impose a stockout penalty cost.) The control constraints keep the production rate $P(t)$ between a specified lower bound $P_{\min }$ and a specified upper bound $P_{\max }$. Finally, the terminal constraint $I(T) \geq I_{\min }$ is imposed so that the terminal inventory is at least $I_{\min }$.

The statement of the problem is lengthy because of the number of variables, functions, and parameters which are involved. However, with the production and inventory interpretations as given, it is not difficult to see the reasons for each condition. In Chap. 6, various versions of this model will be solved in detail. In Sect. $12.2$, we will deal with a stochastic version of this model.

## 统计代写|最优控制作业代写optimal control代考|Basic Concepts and Definitions

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

$$J=\int_{0}^{T} F(x(t), u(t), t) d t+S[x(T), T]$$

## 统计代写|最优控制作业代写optimal control代考|Formulation of Simple Control Models

$$\dot{I}(t)=P(t)-S(t)$$

## 有限元方法代写

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## MATLAB代写

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