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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 Hamilton-Jacobi-Bellman Equation

Suppose $V(x, t): E^{n} \times E^{1} \rightarrow E^{1}$ is a function whose value is the maximum value of the objective function of the control problem for the system, given that we start at time $t$ in state $x$. That is,
$$V(x, t)=\max {u(s) \in \Omega(s)}\left[\int{t}^{T} F(x(s), u(s), s) d s+S(x(T), T)\right]$$
where for $s \geq t$,
$$\frac{d x(s)}{d s}=f(x(s), u(s), s), x(t)=x$$
We initially assume that the value function $V(x, t)$ exists for all $x$ and $t$ in the relevant ranges. Later we will make additional assumptions about the function $V(x, t)$.

Bellman (1957) in his book on dynamic programming states the principle of optimality as follows:
An optimal policy has the property that, whatever the initial state and initial decision are, the remaining decision must constitute an optimal policy with regard to the outcome resulting from the initial decision.
Intuitively this principle is obvious, for if we were to start in state $x$ at time $t$ and did not follow an optimal path from then on, there would then exist (by assumption) a better path from $t$ to $T$, hence, we could improve the proposed solution by following this better path. We will use the principle of optimality to derive conditions on the value function $V(x, t)$.

Figure $2.1$ is a schematic picture of the optimal path $x^{*}(t)$ in the state-time space, and two nearby points $(x, t)$ and $(x+\delta x, t+\delta t)$, where $\delta t$ is a small increment of time and $x+\delta x=x(t+\delta t)$. The value function changes from $V(x, t)$ to $V(x+$ $\delta x, t+\delta t)$ between these two points. By the principle of optimality, the change in the objective function is made up of two parts: first, the incremental change in $J$ from $t$ to $t+\delta t$, which is given by the integral of $F(x, u, t)$ from $t$ to $t+\delta t$; second, the value function $V(x+\delta x, t+\delta t)$ at time $t+\delta t$. The control actions $u(\tau)$ should be chosen to lie in $\Omega(\tau), \tau \in[t, t+\delta t]$, and to maximize the sum of these two terms. In equation form this is
$$V(x, t)=\max {\substack{u(\tau) \in \Omega(\tau) \ \tau \in[t, t+\delta t]}}\left{\int{t}^{t+\delta t} F[x(\tau), u(\tau), \tau] d \tau+V[x(t+\delta t), t+\delta t]\right}$$
where $\delta t$ represents a small increment in $t$. It is instructive to compare this equation to definition (2.9).

## 统计代写|最优控制作业代写optimal control代考|

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 Hamilton-Jacobi-Bellman Equation

$$V(x, t)=\max u(s) \in \Omega(s)\left[\int t^{T} F(x(s), u(s), s) d s+S(x(T), T)\right]$$

$$\frac{d x(s)}{d s}=f(x(s), u(s), s), x(t)=x$$

Bellman (1957) 在他的关于动态规划的书中陈述了最优性原则如下:

$\backslash$ left 的分隔符缺失或无法识别

## 统计代写|最优控制作业代写optimal control代考|

$\backslash$ 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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assignmentutor™您的专属作业导师
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