Consider the system

$x^{}=Ax+Bu,$

$y=Cx.$

The system is called strongly observable if for any input function $u$ and

$x_0$inx the following holds: $y_u(t,x_0)=0$ for all $t0$ implies $x_0=0.$

In other words: the output of the system can only be zero if the initial

condition is zero. Prove that the following three statements are equiva$-$

lent:

$(1)$ The system is strongly observable,

$(2)V^{**}(kerC)=\{0\},$

$(3)$ For all $F$ the pair $(C,A+BF)$ is observable.