Posted on Categories:Ordinary Differential Equations, 常微分方程, 数学代写

# 数学代写|常微分方程代考Ordinary Differential Equations代写|Equality constraints

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## 数学代写|常微分方程代考Ordinary Differential Equations代写|Equality constraints

Consider the problem of minimising the following functional
$$J(y)=\int_a^b \ell\left(x, y(x), y^{\prime}(x)\right) d x$$
in the set given by
$$V=\left{v \in C^1([a, b]): v(a)=y_a, v(b)=y_b ; K(v)=c\right}$$
where $K: C^1([a, b]) \rightarrow \mathbb{R}$ is given by
$$K(y)=\int_a^b \varphi\left(x, y(x), y^{\prime}(x)\right) d x$$
The equation $K(y)=c$, with $c$ a given constant, defines a global equality constraint for $y$, and (9.33)-(9.35) defines a constrained optimisation problem.
Now, assume that $\ell, \varphi \in C^2$ in all their arguments. If $y \in V$ is a weak local minimiser of the constrained optimisation problem, and the Fréchet derivative $\partial K$ has a continuous inverse in a neighbourhood of $y$; in the present case, $\partial K(y) \neq 0$, then there exists a $\lambda \in \mathbb{R}$ such that the following EL equations are satisfied. We have
$$\frac{\partial}{\partial y}(\ell+\lambda \varphi)\left(x, y, y^{\prime}\right)-\frac{d}{d x} \frac{\partial}{\partial y^{\prime}}(\ell+\lambda \varphi)=0, \quad x \in[a, b]$$
The condition $\partial K(y) \neq 0$ means that $y$ is not an extremal of (9.35). In fact, consider a variation $y+\alpha \eta$, where $\eta \in W$. This variation should preserve the constraint, and hence considering the first variation of $K$ we require
$$\delta K(y ; \eta)=0$$
This means that
$$\int_a^b\left(\frac{\partial \varphi}{\partial y}-\frac{d}{d x} \frac{\partial \varphi}{\partial y^{\prime}}\right) \eta d x=0$$

## 数学代写|常微分方程代考Ordinary Differential Equations代写|The Legendre condition

The EL equation, with the appropriate boundary conditions, represents the first-order optimality condition for a minimiser of $J$ in $V$. However, in order to characterise an extremal solution as the minimum sought, we need to consider second-order optimality conditions. For this purpose, we focus on the problem of the calculus of variation given by (9.15)-(9.16), that is, the following optimisation problem
$$\min {y \in V} J(y):=\int_a^b \ell\left(x, y(x), y^{\prime}(x)\right) d x$$ where $$V=\left{v \in C^1([a, b]): v(a)=y_a, v(b)=y_b\right}$$ Now, we compute the second variation of $J$ in $V$, assuming that $\ell \in C^2$. Recall that the second variation is given by $\delta^2 J(y ; h)=\left.\frac{d^2}{d t^2} J(y+t h)\right|{t=0}$. We have
\begin{aligned} \delta^2 J(y ; h) & =\int_a^b\left{\frac{\partial^2 \ell}{\partial y^{\prime 2}}\left(x, y, y^{\prime}\right)\left(h^{\prime}\right)^2+2 \frac{\partial^2 \ell}{\partial y \partial y^{\prime}}\left(x, y, y^{\prime}\right) h h^{\prime}\right. \ & \left.+\frac{\partial^2 \ell}{\partial y^2}\left(x, y, y^{\prime}\right) h^2\right} d x \end{aligned}
As discussed above, the necessary optimality conditions for $y \in V$ to be a weak local minimiser are given by $(\nabla J(y), h)=0, h \in W$, and the following second-order condition:
$$\left(\nabla^2 J(y) h, h\right) \geq 0, \quad h \in W$$
Next, we discuss this condition for our specific problem (9.39)-(9.40). We have 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|Equality constraints

$$J(y)=\int_a^b \ell\left(x, y(x), y^{\prime}(x)\right) d x$$

$$V=\left{v \in C^1([a, b]): v(a)=y_a, v(b)=y_b ; K(v)=c\right}$$
$K: C^1([a, b]) \rightarrow \mathbb{R}$是由谁给出的
$$K(y)=\int_a^b \varphi\left(x, y(x), y^{\prime}(x)\right) d x$$

$$\frac{\partial}{\partial y}(\ell+\lambda \varphi)\left(x, y, y^{\prime}\right)-\frac{d}{d x} \frac{\partial}{\partial y^{\prime}}(\ell+\lambda \varphi)=0, \quad x \in[a, b]$$

$$\delta K(y ; \eta)=0$$

$$\int_a^b\left(\frac{\partial \varphi}{\partial y}-\frac{d}{d x} \frac{\partial \varphi}{\partial y^{\prime}}\right) \eta d x=0$$

## 数学代写|常微分方程代考Ordinary Differential Equations代写|The Legendre condition

$$\min {y \in V} J(y):=\int_a^b \ell\left(x, y(x), y^{\prime}(x)\right) d x$$其中$$V=\left{v \in C^1([a, b]): v(a)=y_a, v(b)=y_b\right}$$现在，我们计算$V$中$J$的第二种变化，假设$\ell \in C^2$。回想一下，第二个变体由$\delta^2 J(y ; h)=\left.\frac{d^2}{d t^2} J(y+t h)\right|{t=0}$给出。我们有
\begin{aligned} \delta^2 J(y ; h) & =\int_a^b\left{\frac{\partial^2 \ell}{\partial y^{\prime 2}}\left(x, y, y^{\prime}\right)\left(h^{\prime}\right)^2+2 \frac{\partial^2 \ell}{\partial y \partial y^{\prime}}\left(x, y, y^{\prime}\right) h h^{\prime}\right. \ & \left.+\frac{\partial^2 \ell}{\partial y^2}\left(x, y, y^{\prime}\right) h^2\right} d x \end{aligned}

$$\left(\nabla^2 J(y) h, h\right) \geq 0, \quad h \in W$$

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