In the realm of LTI systems (perhaps even in general systems, I am not sure) it is so that if a system is asymptotically stable then it is also BIBO stable. Is that the case when a system is marginally stable too? Meaning that if it is, then it too is BIBO stable.
For an LTI system to be BIBO stable we simply need that for any bounded input, the response does not exceed some finite bound. According to my understanding of marginal stability it is so that there does exist inputs that lead to both bounded and unbounded outputs. As such is it so in general that marginally stable systems ARE NOT BIBO stable?
I would like to say that marginally stable systems are NOT BIBO stable. As an example I can give you a system with transfer function $H(s)=1/s$ and give you as input $x(t)=u(t)$ where $u(t)$ is the heaviside function a.k.a the unit step function. That would yield unbounded outputs no?
Marginally stable refers to the internal stability of system, BIBO refers to the input output properties.
A system can be marginally stable but not BIBO stable.
For example, $\hat{h}(s) = {s \over 1+s^2}$ is marginally stable, but the bounded input $u(t) = \sin t$ will result in an unbounded response.
A system can BIBO stable but internally unstable (if the unstable modes are unobservable).