BIBO stability from impulse response

Question

I've seen two versions of checking BIBO stability from the impulse response $g(t)$.

1. $\int_{-\infty}^{+\infty}|g(t)|dt < \infty$
2. $\int_{0}^{+\infty}|g(t)|dt < \infty$

These two integrals can lead to different results because of different integral limits. So I consider these are different conditions. I wonder what is the difference between them and how should I understand them, mathematically and intuitively?

Thank you!

Answer 1

The former condition is the most general one. The latter applies whenever the signal is $g(t)=0$ for $t<0$ that, without loss of generality, is a common assumption in control. Thus, it is a special case of the former and there is no physical difference between them.