Best regards,
I am asked to minimize the next functional
$T(v)=\int_{A}^{B}\frac{dx}{v(x)}$,
with $v\neq 0\text{ and } v\in V$, where $V=\{v\in C([A,B]):v(A)=v(B)=0\}$.
If we assume that there exists $v^*$ that minimizes the functional, then $v^*$ satisfies the Euler-Lagrange equation.
Euler-Lagrange Equation: $\frac{\partial L}{\partial y}-\frac{d}{dx}(\frac{\partial L}{\partial y'})=0$.
If $L=\frac{1}{v},v\neq 0$
$y=v\Rightarrow \frac{\partial L}{\partial y}=-\frac{1}{v^2},$
$\frac{\partial L}{\partial y'}=0\Rightarrow\frac{d}{dx}(\frac{\partial L}{\partial y'})=0,$
then $\frac{\partial L}{\partial y}-\frac{d}{dx}(\frac{\partial L}{\partial y'})=-\frac{1}{v^2}-0=0\Rightarrow \frac{1}{v^2}=0,$ which is impossible, then must resort to the Principle of Maximum of Pontryagin.
How can I use the Pontryagin maximum principle in this case and I have also been told that the optimal $v$ is reached at point $\frac{A+B}{2}$?