Q : Consider the system:

$x^{}=\frac{1}{2}x+u$

it is required to find an unconstrained optimal control law u$(t)$ for $0t1$ so$,$ that the performance index

$J={\displaystyle \underset{0}{\overset{1}{}}}(2e^{-t}{u}^2+\frac{1}{2}{e}^{-t}{x}^2)dt$

is minimized The initial state is $x(0)=1$ and the final state at $t=1$ is completely free.

$tf$fixed.

X$($tf$)$ free