A first order system is given by

$\backslash$pm $($t$)=$ x$($t$)+$ u$($t$).$

$($a$)$ Find the unconstrained optimal control law which minimizes the

performance index

such that the final time t f is fixed and the final state x$($t f$)$ is free.

$($b$)$ Find the optimal control law as t f $-*00.$