# 统计代写|最优控制作业代写optimal control代考|SF2852

#### Doug I. Jones

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## 统计代写|最优控制作业代写optimal control代考|Economic Interpretations of the Maximum Principle

Recall from Sect. 2.1.3 that the objective function $(2.3)$ is
$$J=\int_{0}^{T} F(x, u, t) d t+S(x(T), T),$$
where $F$ is considered to be the instantaneous profit rate measured in dollars per unit of time, and $S(x, T)$ is the salvage value, in dollars, of the system at time $T$ when the terminal state is $x$. For purposes of discussion it will be convenient to consider the system as a firm and the state $x(t)$ as the stock of capital at time $t$.

In (2.17), we interpreted $\lambda(t)$ to be the per unit change in the value function $V(x, t)$ for small changès in cāpitall stock $x$. In othèr words, $\lambda(t) \mathrm{~ i s ~ t h e ~ m a r g i n a}$ value perr uñit of capital at time $t$, añ it is also refeerrè to as the price or shadow price of a unit of capital at time $t$. In particular, the value of $\lambda(0)$ is the marginal rate of change of the maximum value of $J$ (the objective function) with respect to the change in the initial capital stock, $x_{0}$.

Remark 2.2 As mentioned in Appendix C, where we prove a maximum principle without any smoothness assumption on the value function, there arise cases in which the value function may not be differentiable with respect to the state variables. In such cases, when $V_{x}\left(x^{*}(t), t\right)$ does not exist, then (2.17) has no meaning. See Bettiol and Vinter (2010), Yong and Zhou (1999), and Cernea and Frankowska (2005) for interpretations of the adjoint variables or extensions of (2.17) in such cases.

Next we interpret the Hamiltonian function in (2.18). Multiplying (2.18) formally by $d t$ and using the state equation (2.1) gives
$$H d t=F d t+\lambda f d t=F d t+\lambda \dot{x} d t=F d t+\lambda d x$$

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

In order to absorb the maximum principle, the reader should study very carefully the examples in this section, all of which are problems having only one state and one control variable. Some or all of the exercises at the end of the chapter should also be worked.

In the following examples and others in this book, we will at times omit the superscript $*$ on the optimal values of the state variables as long as no confusion arises from doing so.
Example 2.2 Consider the problem:
$$\max \left{J=\int_{0}^{1}-x d t\right}$$
subject to the state equation
$$\dot{x}=u, x(0)=1$$
and the control constraint
$$u \in \Omega=[-1,1]$$
Note that $T=1, F=-x, S=0$, and $f=u$. Because $F=-x$, we can interpret the problem as one of minimizing the (signed) area under the curve $x(t)$ for $0 \leq t \leq 1$.

## 统计代写|最优控制作业代写optimal control代考|Economic Interpretations of the Maximum Principle

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

$$H d t=F d t+\lambda f d t=F d t+\lambda \dot{x} d t=F d t+\lambda d x$$

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

\max \eft ${1=\backslash \operatorname{lnt}{0} \wedge{1}-x d$ t \right } }

$$\dot{x}=u, x(0)=1$$

$$u \in \Omega=[-1,1]$$

## 有限元方法代写

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## MATLAB代写

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