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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|Derivation of the Adjoint Equation

The derivation of the adjoint equation proceeds from the HJB equation (2.19), and is similar to those in Fel’dbaum (1965) and Kirk (1970). Note that, given the optimal path $x^{}$, the optimal control $u^{}$ maximizes the left-hand side of $(2.19)$, and its maximum value is zero. We now consider small perturbations of the values of the state variables in a neighborhood of the optimal path $x^{}$. Thus, let $$x(t)=x^{}(t)+\delta x(t)$$
where $|\delta x(t)|<\varepsilon$ for a small positive $\varepsilon$.

We now consider a ‘fixed’ time instant $t$. We can then write $(2.19)$ as
\begin{aligned} 0 &=H\left[x^{}(t), u^{}(t), V_{x}\left(x^{}(t), t\right), t\right]+V_{t}\left(x^{}(t), t\right) \ & \geq H\left[x(t), u^{}(t), V_{x}(x(t), t), t\right]+V_{t}(x(t), t) . \end{aligned} To explain, we note from (2.19) that the left-hand side of $\geq$ in (2.23) equals zero. The right-hand side can attain the value zero only if $u^{}(t)$ is also an optimal control for $x(t)$. In general, for $x(t) \neq x^{}(t)$, this will not be so. From this observation, it follows that the expression on the right-hand side of (2.23) attains its maximum (of zero) at $x(t)=x^{}(t)$. Furthermore, $x(t)$ is not explicitly constrained. In other words, $x^{}(t)$ is an unconstrained local maximum of the right-hand side of (2.23), so that the derivative of this expression with respect to $x$ must vanish at $x^{}(t)$, i.e.,
$$H_{x}\left[x^{}(t), u^{}(t), V_{x}\left(x^{}(t), t\right), t\right]+V_{t x}\left(x^{}(t), t\right)=0,$$
provided the derivative exists, and for which, we must further assume that $V$ is a twice continuously differentiable function of its arguments. With $H=F+V_{x} f$ from (2.17) and (2.18), we obtain
$$H_{x}=F_{x}+V_{x} f_{x}+f^{T} V_{x x}=F_{x}+V_{x} f_{x}+\left(V_{x x} f\right)^{T}$$
by using $g=V_{x}$ in the identity (1.15). Substituting this in (2.24) and recognizing the fact that $V_{x x}=\left(V_{x x}\right)^{T}$, we obtain
$$F_{x}+V_{x} f_{x}+f^{T} V_{x x}+V_{t x}=F_{x}+V_{x} f_{x}+\left(V_{x x} f\right)^{T}+V_{t x}=0,$$
where the superscript ${ }^{T}$ denotes the transpose operation. See (1.16) or Exercise $1.10$ for further explanation.

## 统计代写|最优控制作业代写optimal control代考|The Maximum Principle

The necessary conditions for $u^{}(t), t \in[0, T]$, to be an optimal control are: $$\left{\begin{array}{l} \dot{x}^{}=f\left(x^{}, u^{}, t\right), x^{}(0)=x_{0}, \ \dot{\lambda}=-H_{x}\left[x^{}, u^{}, \lambda, t\right], \lambda(T)=S_{x}\left(x^{}(T), T\right), \ H\left[x^{}, u^{}, \lambda, t\right] \geq H\left[x^{*}, u, \lambda, t\right], \forall u \in \Omega(t), t \in[0, T] . \end{array}\right.$$

It should be emphasized that the state and the adjoint arguments of the Hamiltonian are $x^{}(t)$ and $\lambda(t)$ on both sides of the Hamiltonian maximizing condition in (2.31), respectively. Furthermore, $u^{}(t)$ must provide a global maximum of the Hamiltonian $H\left[x^{*}(t), u, \lambda(t), t\right]$ over $u \in \Omega(t)$. For this reason the necessary conditions in (2.31) are called the maximum principle.

Note that in order to apply the maximum principle, we must simultaneously solve two sets of differential equations with $u^{}$ obtained from the Hamiltonian maximizing condition in $(2.31)$. With the control variable $u^{}$ so obtained, the state equation for $x^{}$ is given with the initial value $x_{0}$, and the adjoint equation for $\lambda$ is specified with a condition on the terminal value $\lambda(T)$. Such a system of equations, where initial values of some variables and final values of other variables are specified, is called a two-point boundary value problem (TPBVP). The general solution of such problems can be very difficult; see Bryson and Ho (1975), Roberts and Shipman (1972), and Feichtinger and Hartl (1986). However, there are certain special cases which are easy. One such is the case in which the adjoint equation is independent of the state and the control variables; here we can solve the adjoint equation first, then get the optimal control $u^{}$, and then solve for $x^{*}$.

Note also that if we can solve the Hamiltonian maximizing condition for an optimal control function in closed form $u^{}(x, \lambda, t)$ so that $$u^{}(t)=u^{}\left[x^{}(t), \lambda(t), t\right]$$
then we can substitute this into the state and adjoint equations to get the TPBVP just in terms of a set of differential equations, i.e.,
$$\left{\begin{array}{l} \dot{x}^{}=f\left(x^{}, u^{}\left(x^{}, \lambda, t\right), t\right), x^{}(0)=x_{0}, \ \dot{\lambda}=-H_{x}\left(x^{}, u^{}\left(x^{}, \lambda, t\right), \lambda, t\right), \lambda(T)=S_{x}\left(x^{*}(T), T\right) . \end{array}\right.$$

## 统计代写|最优控制作业代写optimal control代考|Derivation of the Adjoint Equation

$$x(t)=x(t)+\delta x(t)$$

$$0=H\left[x(t), u(t), V_{x}(x(t), t), t\right]+V_{t}(x(t), t) \quad \geq H\left[x(t), u(t), V_{x}(x(t), t), t\right]+V_{t}(x(t), t) .$$

$$H_{x}\left[x(t), u(t), V_{x}(x(t), t), t\right]+V_{t x}(x(t), t)=0,$$

$$H_{x}=F_{x}+V_{x} f_{x}+f^{T} V_{x x}=F_{x}+V_{x} f_{x}+\left(V_{x x} f\right)^{T}$$

$$F_{x}+V_{x} f_{x}+f^{T} V_{x x}+V_{t x}=F_{x}+V_{x} f_{x}+\left(V_{x x} f\right)^{T}+V_{t x}=0$$

## 统计代写|最优控制作业代写optimal control代考|The Maximum Principle

$$\dot{x}=f(x, u, t), x(0)=x_{0}, \dot{\lambda}=-H_{x}[x, u, \lambda, t], \lambda(T)=S_{x}(x(T), T), H[x, u, \lambda, t] \geq H\left[x^{}, u, \lambda, t\right], \forall u \in \Omega(t), t \in[0, T] .$$ 正确的。 $\$ \$$应该强调的是，哈密顿量的状态和伴随论证是 x(t) 和 \lambda(t) 分别在 (2.31) 中的哈密顿最大化条件的两侧。此外， u(t) 必须提供哈密顿量的 全局最大值 H\left[x^{}(t), u, \lambda(t), t\right] 超过 u \in \Omega(t). 因此，（2.31) 中的必要条件被称为最大原理。 请注意，为了应用极大值原理，我们必须同时求解两组微分方程 u 从哈密顿量最大化条件中获得 (2.31). 与控制变量 u 如此得到，状态方程 为 x 以初始值给出 x_{0} ，以及伴随方程 \lambda 用终端值的条件指定 \lambda(T). 这样一个方程组，其中指定了一些变量的初始值和其他变量的最终值， 称为两点边值问题 (TPBVP)。此类问题的一般解决方案可能非常困难；参见 Bryson 和 Ho (1975)、Roberts 和 Shipman (1972) 以及 Feichtinger 和 Hartl (1986)。但是，有些特殊情况很容易。其中一种情况是伴随方程独立于状态和控制变量。这里我们可以先求解伴随方 程，然后得到最优控制 u ，然后求解 x^{}. 还要注意，如果我们可以解决封闭形式的最优控制函数的哈密顿最大化条件 u(x, \lambda, t) 以便$$ u(t)=u[x(t), \lambda(t), t] $$然后我们可以将其代入状态和伴随方程，以仅根据一组微分方程得到 TPBVP，即 \ \$$ Veft { $$\dot{x}=f(x, u(x, \lambda, t), t), x(0)=x_{0}, \dot{\lambda}=-H_{x}(x, u(x, \lambda, t), \lambda, t), \lambda(T)=S_{x}\left(x^{}(T), T\right)$$
、正确的。
$\$ \

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assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师