Estimating unobservable state

We consider the scalar system

$\frac{d}{dt}x=-x+u$

We do not observe the system, but we would like to estimate the state $x.$ For that purpose, we run the state estimator

$\frac{d}{dt}$hat$(x)=-$hat$(x)+u$

$($a$)$ Show that regardless of $u,$hat$(x)$ converges towards $x.$ Thus the estimator works, even though it reads no information from the system.

$($b$)$ Would this estimator approach work if the system was

$\frac{d}{dt}x=x+u?$

Explain your reasoning.

$($c$)$ We go back to the system

$\frac{d}{dt}x=-ax+u$

where $a=1.$ Unfortunately, the control engineer has the wrong estimate of $a,$ and she thinks $a=-0\mathrm{.}5.$ Considering the state estimator

$\frac{d}{dt}$hat$(x)=-0\mathrm{.}5$hat$(x)+u$

compute $x$ and hat$(x)$ when $x(0)=0,$hat$(x)(0)=0,u=\sqrt[\phantom{2}]{t}sin(t),$ and $0t50.$ Would you consider your estimator to be good?

$($d$)$ Using the same information as in the previous question, we now assume $y=x$ is