Is a transfer function of a hole system BIBO and asymptotically stable, if the poles of the two sub systems shorten each other out?
Bibo stability is all about systems external stability which is determined by applying the external input with zero initial condition (transfer function in other words) so if you check bibo stability of G(s) ,it would be bibo stable
But
Asymptotic stability is all about systems internal stability which can be determined by applying the non zero initial condition and no external input ,and if two system as in your example are cascaded if any one of them is unstable(or both) then its modes will not decay and reaches up to infinity (in your example) after infinite time ,so G(s) would be asymptotic unstable