For the following inequalities refer
- Zhou, Kemin; Doyle, John C.; Glover, Keith, Robust and optimal control, Upper Saddle River, NJ: Prentice Hall. xx, 596 p. (1996). ZBL0999.49500.
Consider a general LTI system
For $u(t) \in \mathcal{L}_2$ the input signal and $y(t) \in \mathcal{L}_2$ the output signal, the induced system gain is given by $\lVert G \rVert_\infty$ as shown below (Ref. Page 107 Table 4.2)
$$ \lVert z \rVert_2 \leq \lVert G \rVert_\infty \lVert u \rVert_2 $$
similarly, for $u(t) \in \mathcal{L}_\infty$ the input signal and $y(t) \in \mathcal{L}_\infty$ the output signal, the induced system gain is given by $\leq \int_0^\infty \lVert g(t \rVert dt$ as shown below (Ref. Page 107 Table 4.2)
$$ \lVert z \rVert_\infty \leq \int_0^\infty \left\lVert g(\tau) \right\rVert d\tau \lVert u \rVert_\infty $$
Also we have the inequality (Ref. Page 111 Theorem 4.5)
$$ \lVert G \rVert_\infty \leq \int_0^\infty \lVert g(t) \rVert dt $$
The question that I want to ask is under what conditions does equality holds for the above stated inequality?
There is no known necessary conditions for that and it is unlikely that there are any as it can just be a coincidence.
In any way, there is a small class of systems for which this is always true and this is the case of SISO internally positive systems. A system is said to be internally positive if for any nonnegative initial condition and any nonnegative input, the state remains nonnegative and the output is nonnegative. A necessary and sufficient for a system $(A,B,C,D)$ to be internally positive is that $A$ be Metzler, and that the other matrices be nonnegative (i.e. componentwise).
In the case of a SISO stable internally positive system, we have that all the $L_p$-gains are the same and are equal to $G(0)$ where $G(s):=C(sI-A)^{-1}B+D$.