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# 数学代写|常微分方程代考Ordinary Differential Equations代写|Optimality conditions

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## 数学代写|常微分方程代考Ordinary Differential Equations代写|Optimality conditions

Once existence of a minimiser in $V$ is established, one can consider its characterisation based on derivatives of the functional $J: V \rightarrow \mathbb{R}, V$ subset of $B$.
In the following, we illustrate different concepts of derivatives in functional spaces. For this purpose, let us denote with $\mathscr{L}(B, \mathbb{R})$ the Banach space of all bounded linear functionals $\mathcal{A}: B \rightarrow \mathbb{R}$.
Definition 9.3 The functional $J$ is said to be Fréchet differentiable at $y \in V$ if there exists an operator $\mathcal{A}y \in \mathscr{L}(B, \mathbb{R})$ such that $$\lim {z \rightarrow y} \frac{J(z)-J(y)-\mathcal{A}_y(z-y)}{|z-y|}=0 .$$
This generalisation of the concept of the derivative is named after Maurice René Fréchet.
We have the following equivalent properties [4].
( $\alpha$ ) The functional $J$ is Fréchet differentiable with respect to $y$.
( $\beta$ ) There exists $\mathcal{A}_y \in \mathscr{L}(B, \mathbb{R})$, and the real-valued functional $r_y: V \rightarrow \mathbb{R}$, where $r_y$ is continuous in $y$ and $r_y(y)=0$, such that the following holds:
$$J(z)=J(y)+\mathcal{A}_y(z-y)+r_y(y)|z-y|, \quad z \in V .$$
$(\gamma)$ There exists $\mathcal{A}_y \in \mathscr{L}(B, \mathbb{R})$ with
$$J(z)=J(y)+\mathcal{A}_y(z-y)+o(|y-z|),$$
where $o$ represents the “small” Landau symbol.
Notice that the operator $\mathcal{A}_y$ is uniquely determined. It is called the Fréchet derivative of $J$, and we denote it with $\partial J(y)$. Further, if $J$ is Fréchet differentiable in $y$, so it is continuous in $y$.
The Fréchet derivative allows to construct an approximation to $J$ at $\bar{y} \in V$ in the following sense. Define
$$g: B \rightarrow \mathbb{R}, \quad y \mapsto J(\bar{y})+\partial J(\bar{y})(y-\bar{y})$$
Then it holds
$$\lim {y \rightarrow \bar{y}} \frac{|J(y)-g(y)|}{|y-\bar{y}|}=0 .$$ Next, we consider the following one-sided limit: $$\delta J(y ; h):=\lim {\alpha \rightarrow 0^{+}} \frac{J(y+\alpha h)-J(y)}{\alpha},$$
for $y \in V, h \in B \backslash{0}$. If this limit exists, it is called the directional derivative or first (right) variation of $J$ in $y$ in the direction $h$. This concept of directional derivative in functional spaces was made mathematically rigorous thanks to the work of René Eugène Gâteaux.

## 数学代写|常微分方程代考Ordinary Differential Equations代写|First-order optimality conditions

We discuss first-order necessary optimality conditions for a minimum.
Theorem 9.4 Let $V \subset B$ be a non-empty convex subset and suppose that $y \in V$ is a weak local minimiser of $J$ in $V$, and assume that $(9.9)$ holds. Then
$$(\nabla J(y), v) \geq 0, \quad v \in T(V, y)$$
Proof. Let $t_n, y_n$ be the sequence associated to $v \in T(V, y)$. Then, for $n$ sufficient large it holds
\begin{aligned} J(y) & \leq J\left(y_n\right)=J\left(y+\left(y_n-y\right)\right) \ & =J(y)+\left(\nabla J(y), y_n-y\right)+\eta\left(\left|y_n-y\right|_B\right)\left|y_n-y\right|_H \end{aligned}
Thus for $n$ sufficient large such that $t_n>1$, we have
$$0 \leq\left(\nabla J(y), t_n\left(y_n-y\right)\right)+\eta\left(\left|y_n-y\right|_B\right)\left|t_n\left(y_n-y\right)\right|_H$$
Taking the limit $n \rightarrow \infty,(9.10)$ is proved.
Notice that, if $V$ is an open subset of $B$, and $y \in V$ is a weak local minimiser of $J$ in $V$, then $(9.10)$ becomes
$$\nabla J(y)=0$$
A first-order sufficient condition for a minimum is given by the following theorem.

# 常微分方程代写

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Optimality conditions

($\alpha$)函数$J$对于$y$是fr可微的。
($\beta$)存在$\mathcal{A}_y \in \mathscr{L}(B, \mathbb{R})$和实值泛函$r_y: V \rightarrow \mathbb{R}$，其中$r_y$在$y$和$r_y(y)=0$中连续，从而成立:
$$J(z)=J(y)+\mathcal{A}_y(z-y)+r_y(y)|z-y|, \quad z \in V .$$
$(\gamma)$存在$\mathcal{A}_y \in \mathscr{L}(B, \mathbb{R})$ with
$$J(z)=J(y)+\mathcal{A}_y(z-y)+o(|y-z|),$$

$$g: B \rightarrow \mathbb{R}, \quad y \mapsto J(\bar{y})+\partial J(\bar{y})(y-\bar{y})$$

$$\lim {y \rightarrow \bar{y}} \frac{|J(y)-g(y)|}{|y-\bar{y}|}=0 .$$接下来，我们考虑以下单侧极限:$$\delta J(y ; h):=\lim {\alpha \rightarrow 0^{+}} \frac{J(y+\alpha h)-J(y)}{\alpha},$$

## 数学代写|常微分方程代考Ordinary Differential Equations代写|First-order optimality conditions

$$(\nabla J(y), v) \geq 0, \quad v \in T(V, y)$$

\begin{aligned} J(y) & \leq J\left(y_n\right)=J\left(y+\left(y_n-y\right)\right) \ & =J(y)+\left(\nabla J(y), y_n-y\right)+\eta\left(\left|y_n-y\right|_B\right)\left|y_n-y\right|_H \end{aligned}

$$0 \leq\left(\nabla J(y), t_n\left(y_n-y\right)\right)+\eta\left(\left|y_n-y\right|_B\right)\left|t_n\left(y_n-y\right)\right|_H$$

$$\nabla J(y)=0$$

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