Problem:
I have to give the minima, subject to $x'(t) = u(t)$, $x(0) = 0$ and $x(1) = 1$ of the following function:
$\int_0^1 u(t) dt$
I have to find this minima using the Pontryagin's Maximum Principle.
What we have so far:
$H = pu- \lambda_0 u$
$x' = \hat{H_p}=u$
$p' =0$
$p-\lambda_0=0 \iff p=\lambda_0$
Exclude bad case: if $\lambda_0=0 \iff p=0$, and this cannot happen, since they cannot both be zero. So we choose $\lambda_0=1$
$H_u=p-1$, Since $p=\lambda_0=1, H_u=0$
Can anyone please help us!
