Relationship between BIBO, Marginal and Asymptotic stability: statements I know that asymptotic stability implies BIBO stability.
But then, which of the following can be held true:
  1. If a system is BIBO stable would it definitely be asymptotic stable?

  2. If a system is marginally stable, it will definitely be   
asymptotically stable? 
  3. If a system is asymptotic stable, it will   
definitely be marginally stable?

Kepeing in view the poles real part values in mind, what could be held true?
 A: A linear system is said to be asymptotically stable if $\lim_{t \to \infty} x(t) = 0$, where $x$ is the states of the system. A linear system is asymptotically stable if and only if real parts of all poles (or eigenvalues of the system matrix) are negative.
A linear system is said to be marginally stable if $\lim_{t \to \infty} x(t) \neq 0$ but $x$ is bounded. A linear system is marginally stable if and only if it has at least one simple pole (not repeated) with real part zero, and all other poles have negative real parts. Therefore, a system cannot be both asymptotically stable and marginally stable.
A linear system is said to be BIBO stable if the output is bounded for an arbitrary bounded input. If a linear system is asymptotically stable, then it is BIBO stable. If a linear system is BIBO stable and the state space representation is minimal, i.e. both controllable and observable, then the system is asymptotically stable. This also implies that a marginally stable system with minimal realization is not BIBO stable.
