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# 数学代写|优化理论代写Optimization Theory代考|CONSTRAINED EXTREMA

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## 数学代写|优化理论代写Optimization Theory代考|CONSTRAINED EXTREMA

So far, we have discussed functionals involving $\mathbf{x}$ and $\dot{\mathbf{x}}$, and we have derived necessary conditions for extremals assuming that the components of $\mathbf{x}$ are independent. In control problems the situation is more complicated, because the state trajectory is determined by the control u; thus, we wish to consider functionals of $n+m$ functions, $\mathbf{x}$ and $\mathbf{u}$, but only $m$ of the functions are independent – the controls. Let us now extend the necessary conditions we have derived to include problems with constraints.

To begin, we shall review the analogous problem from the calculus, and introduce some new variables-the Lagrange multipliers-that will be required for our subsequent discussion.
Constrained Minimization of Functions
Example 4.5-1. Find the point on the line $y_1+y_2=5$ that is nearest the origin.

To solve this problem we need only apply elementary plane geometry to Fig. 4-19 to obtain the result that the minimum distance is $5 / \sqrt{2}$, and the extreme point is $y_1^=2.5, y_2^=2.5$.

Most problems cannot be solved by inspection, so let us consider alternative methods of solving this simple example.

The Elimination Method. If $\mathbf{y}^$ is an extreme point of a function, it is necessary that the differential of the function, evaluated at $\mathbf{y}^$, be zero. $\dagger$ In our example, the function
$$f\left(y_1, y_2\right)=y_1^2+y_2^2 \quad \text { (the square of the distance) }$$
is to be minimized subject to the constraint
$$y_1+y_2=5$$
The differential is
$$d f\left(y_1, y_2\right)=\left[\frac{\partial f}{\partial y_1}\left(y_1, y_2\right)\right] \Delta y_1+\left[\frac{\partial f}{\partial y_2}\left(y_1, y_2\right)\right] \Delta y_2,$$
and if $\left(y_1^, y_2^\right)$ is an extreme point,
$$d f\left(y_1^, y_2^\right)=\left[\frac{\partial f}{\partial y_1}\left(y_1^, y_2^\right)\right] \Delta y_1+\left[\frac{\partial f}{\partial y_2}\left(y_1^, y_2^\right)\right] \Delta y_2=0$$

## 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
$$J(\mathbf{w})=\int_{t_0}^{t s} g(\mathbf{w}(t), \dot{w}(t), t) d t$$
$\mathbf{w}$ is an $(n+m) \times 1$ vector of functions $(n, m \geq 1)$ that is required to satisfy $n$ relationships of the form
$$f_i(\mathbf{w}(t), t)=0, \quad i=1,2, \ldots, n$$
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.

## 数学代写|优化理论代写Optimization Theory代考|CONSTRAINED EXTREMA

$$f\left(y_1, y_2\right)=y_1^2+y_2^2 \quad \text { (the square of the distance) }$$

$$y_1+y_2=5$$

$$d f\left(y_1, y_2\right)=\left[\frac{\partial f}{\partial y_1}\left(y_1, y_2\right)\right] \Delta y_1+\left[\frac{\partial f}{\partial y_2}\left(y_1, y_2\right)\right] \Delta y_2,$$

$$d f\left(y_1^, y_2^\right)=\left[\frac{\partial f}{\partial y_1}\left(y_1^, y_2^\right)\right] \Delta y_1+\left[\frac{\partial f}{\partial y_2}\left(y_1^, y_2^\right)\right] \Delta y_2=0$$

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

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

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