Calculate first/second variation of the functional $f : V \rightarrow \mathbb{R}$, $y \rightarrow \sin(y(1))$

Question

Hey I have a problem with this exercise.

I have to calculate the first and second variation of the functional $f : V \rightarrow \mathbb{R}$

$y \rightarrow \sin(y(1))$ for $V = C^0([a, b])$ with $a < 1 < b$

I have used the definition and I have done the following:

$δf |_y(a)=\lim_{a\rightarrow 0} \frac{f(y+ay)-f(y)}{a}=\lim_{a\rightarrow 0} \frac{\sin(y(1)+ay(1))-\sin(y(1))}{a}$

But now I don't know how to proceed. Can someone help me?

Answer 1

The first and second variations of $f$ are given by $$\begin{align}\delta f(y,h)&=\left.\frac d{da}f(y+ah)\right|_{a=0}\\&=\left.\frac d{da}\sin(y(1)+ah(1))\right|_{a=0}\\&=\left.h(1)\cos(y(1)+ah(1))\right|_{a=0}\\&=h(1)\cos y(1)\end{align}$$ and $$\begin{align}\delta^2f(y,h)&=\left.\frac{d^2}{da^2}f(y+ah)\right|_{a=0}\\&=\left.\frac d{da}h(1)\cos(y(1)+a h(1))\right|_{a=0}\\&=\left.-h^2(1)\sin(y(1)+ah(1))\right|_{a=0}\\&=-h^2(1)\sin y(1).\end{align}$$