I've seen the relative questions. I'm studying a textbook and I'm confused as to how it connects bibo stability with asymptotic stability for an LTI system.
For asymptotic stability we check if the system's response goes to zero for zero input. For bibo stability if the input is finite the output must be finite for zero initial conditions.
The way I see it the output consists of two terms, the zero state response and the zero input response. Each of the stability checks above has to do with one of the terms.
If a system is asymptotically stable the zero input response tends to zero as time goes to infinity. How can this imply that the zero state response is finite (bibo stability) if it is not taken into account (initial conditions are set to 0) when we check asymptotic stability?