
Optimal control problem
Given that
$\displaystyle F\{y(x)\}= \int\begin{array}{cc}1\\0\end{array} (y'^2y^2) dx $
with the constraint on y(x) such that
$\displaystyle \int \begin{array}{cc}1\\0\end{array} \sqrt{1+y'^2}dx = \sqrt{2} $
and the end condition y(0)=0 and y(1)=1,
prove that F{y(x)} achieves its minimum value for y = x

$\displaystyle
\int \begin{array}{cc}1\\0\end{array} \sqrt{1+y'^2}dx = \sqrt{2}
$
this is length of a curve from (0,0) to (1,1). Its length is sqrt(2) only if it is a straight line.
y=x.
May be it may be solved to get this solution.
Without length restriction F min is sinx/sin1.