From ControlTheoryPro.com

1 Introduction to BIBO Stability
A system is defined to be BIBO Stable if every bounded input to the system results in a bounded output over the time interval . This must hold for all initial times t_{o}. So long as we don't input infinity to our system, we won't get infinity output.
A system is defined to be uniformly BIBO Stable if there exists a positive constant k that is independent of t_{0} such that for all t_{0} the following conditions:
implies that
2 Determining BIBO Stability
Mathematically a system f is BIBO stable if an arbitrary input x is bounded by two finite but large arbitrary constants M and M:
Apply the input x, and the arbitrary boundries M and M to the system to produce three outputs:
Now, all three outputs should be finite for all possible values of M and x, and they should satisfy the following relationship:
If this condition is satisfied, then the system is BIBO stable.