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# 数学代写|优化理论代写Optimization Theory代考|Constrained Minimization of Functionals

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## 数学代写|优化理论代写Optimization Theory代考|Constrained Minimization of Functionals

We are now ready to consider the presence of constraints in variational problems. To simplify the variational equations, it will be assumed that the admissible curves are smooth.

Point Constraints. Let us determine a set of necessary conditions for a function $w^*$ to be an extremal for a functional of the form
$$J(w)=\int_{t_0}^{t t} g(w(t), \dot{w}(t), t) d t$$
$\mathbf{w}$ is an $(n+m) \times 1$ vector of functions $(n, m \geq 1)$ that is required to satisfy $n$ relationships of the form
$$f_i(\mathrm{w}(t), t)=0, \quad i=1,2, \ldots, n,$$
which are called point constraints. Constraints of this type would be present if, for example, the admissible trajectories were required to lie on a specified surface in the $n+m+1$-dimensional $w(t)-t$ space. The presence of these $n$ constraining relations means that only $m$ of the $n+m$ components of $w$ are independent.

We have previously found that the Euler equations must be satisfied regardless of the boundary conditions, so we will ignore, temporarily, terms that enter only into the determination of boundary conditions.

One way to attack this problem might be to solve Eqs. (4.5-25) for $n$ of the components of $\mathbf{w}(t)$ in terms of the remaining $m$ components-which can then be regarded as $m$ independent functions-and use these equations to eliminate the $n$ dependent components of $w(t)$ and $\mathbf{w}(t)$ from $J$. If this can be done, then the equations of Sections 4.2 and 4.3 apply. Unfortunately, the constraining equations (4.5-25) are generally nonlinear algebraic equations, which may be quite difficult to solve.

As an alternative approach we can use Lagrange multipliers. The first step is to form the augmented functional by adjoining the constraining relations to $J$, which yields
\begin{aligned} J_a(\mathbf{w}, \mathbf{p})= & \int_{t_0}^{t_t}\left{g(\mathbf{w}(t), \dot{\mathbf{w}}(t), t)+p_1(t)\left[f_1(\mathbf{w}(t), t)\right]\right. \ & \left.+p_2(t)\left[f_2(\mathbf{w}(t), t)\right]+\cdots+p_n(t)\left[f_n(\mathbf{w}(t), t)\right]\right} d t \ = & \int_{t_0}^{t f}\left{g(\mathbf{w}(t), \dot{w}(t), t)+\mathbf{p}^T(t)[\mathbf{f}(\mathbf{w}(t), t)]\right} d t . \end{aligned}

## 数学代写|优化理论代写Optimization Theory代考|NECESSARY CONDITIONS FOR OPTIMAL CONTROL

Let us now employ the techniques introduced in Chapter 4 to determine necessary conditions for optimal control. As stated in Chapter 1, the problem is to find an admissible control $\mathbf{u}^$ that causes the system $$\dot{\mathbf{x}}(t)=\mathbf{a}(\mathbf{x}(t), \mathbf{u}(t), t)$$ to follow an admissible trajectory $\mathbf{x}^$ that minimizes the performance measure
$$J(\mathbf{u})=h\left(\mathbf{x}\left(t_f\right), t_f\right)+\int_{t_0}^{t_f} g(\mathbf{x}(t), \mathbf{u}(t), t) d t . \dagger$$
We shall initially assume that the admissible state and control regions are not bounded, and that the initial conditions $\mathbf{x}\left(t_0\right)=\mathbf{x}_0$ and the initial time $t_0$ are specified. As usual, $\mathbf{x}$ is the $n \times 1$ state vector and $\mathbf{u}$ is the $m \times 1$ vector of control inputs.

In the terminology of Chapter 4 , we have a problem involving $n+m$ functions which must satisfy the $n$ differential equation constraints (5.1-1). The $m$ control inputs are the independent functions.

The only difference between Eq. (5.1-2) and the functionals considered in Chapter 4 is the term involving the final states and final time. However, assuming that $h$ is a differentiable function, we can write
$$h\left(\mathbf{x}\left(t_f\right), t_f\right)=\int_{s_0}^{t_f} \frac{d}{d t}[h(\mathbf{x}(t), t)] d t+h\left(\mathbf{x}\left(t_0\right), t_0\right)$$
so that the performance measure can be expressed as
$$J(\mathbf{u})=\int_{t_0}^{t_s}\left{g(\mathbf{x}(t), \mathbf{u}(t), t)+\frac{d}{d t}[h(\mathbf{x}(t), t)]\right} d t+h\left(\mathbf{x}\left(t_0\right), t_0\right)$$
Since $\mathbf{x}\left(t_0\right)$ and $t_0$ are fixed, the minimization does not affect the $h\left(\mathbf{x}\left(t_0\right), t_0\right)$ term, so we need consider only the functional
$$J(\mathbf{u})=\int_{t_0}^{t_s}\left{g(\mathbf{x}(t), \mathbf{u}(t), t)+\frac{d}{d t}[h(\mathbf{x}(t), t)]\right} d t$$

## 数学代写|优化理论代写Optimization Theory代考|Constrained Minimization of Functionals

$$J(w)=\int_{t_0}^{t t} g(w(t), \dot{w}(t), t) d t$$
$\mathbf{w}$是满足表单$n$关系所需的函数$(n, m \geq 1)$的$(n+m) \times 1$向量
$$f_i(\mathrm{w}(t), t)=0, \quad i=1,2, \ldots, n,$$

\begin{aligned} J_a(\mathbf{w}, \mathbf{p})= & \int_{t_0}^{t_t}\left{g(\mathbf{w}(t), \dot{\mathbf{w}}(t), t)+p_1(t)\left[f_1(\mathbf{w}(t), t)\right]\right. \ & \left.+p_2(t)\left[f_2(\mathbf{w}(t), t)\right]+\cdots+p_n(t)\left[f_n(\mathbf{w}(t), t)\right]\right} d t \ = & \int_{t_0}^{t f}\left{g(\mathbf{w}(t), \dot{w}(t), t)+\mathbf{p}^T(t)[\mathbf{f}(\mathbf{w}(t), t)]\right} d t . \end{aligned}

## 数学代写|优化理论代写Optimization Theory代考|NECESSARY CONDITIONS FOR OPTIMAL CONTROL

$$J(\mathbf{u})=h\left(\mathbf{x}\left(t_f\right), t_f\right)+\int_{t_0}^{t_f} g(\mathbf{x}(t), \mathbf{u}(t), t) d t . \dagger$$

Eq.(5.1-2)和第4章中考虑的泛函之间的唯一区别是涉及最终状态和最终时间的术语。然而，假设$h$是一个可微函数，我们可以写
$$h\left(\mathbf{x}\left(t_f\right), t_f\right)=\int_{s_0}^{t_f} \frac{d}{d t}[h(\mathbf{x}(t), t)] d t+h\left(\mathbf{x}\left(t_0\right), t_0\right)$$

$$J(\mathbf{u})=\int_{t_0}^{t_s}\left{g(\mathbf{x}(t), \mathbf{u}(t), t)+\frac{d}{d t}[h(\mathbf{x}(t), t)]\right} d t+h\left(\mathbf{x}\left(t_0\right), t_0\right)$$

$$J(\mathbf{u})=\int_{t_0}^{t_s}\left{g(\mathbf{x}(t), \mathbf{u}(t), t)+\frac{d}{d t}[h(\mathbf{x}(t), t)]\right} d t$$

## MATLAB代写

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