Let $\dot{x} = A x + B u$, $y = x$ be a BIBO (bounded input, bounded output) stable system. Given an output bound $y_l \leq y(t) \leq y_h$, how can we determine the maximum input bound $u_l \leq u(t) \leq u_h$ so that any such bounded input $u(t)$ yields an output bounded by $y_l$, $y_t$?
Conversely, what is the minimum output bound $y_l \leq y(t) \leq y_h$ for a given input bound $u_l \leq u(t) \leq u_h$?
Pointers to literature are welcome.
Based on the system response you can calculate the general bound $$\|x(t)\|\leq \|e^{At}x(0)\|+\left[\int_0^t{\|e^{As}B\|ds}\right]\sup_{t}\|u(t)\|$$ Assuming that $A$ is stable and $x(0)=0$ then $$ \sup_t\|x(t)\|\leq \left[\int_0^{\infty}{\|e^{As}B\|ds}\right]\sup_{t}\|u(t)\|$$ Note that $g(t):=e^{At}B$ is the impulse response matrix and the above condition results in the standard textbook condition $$\sup_t\|y(t)\|\leq \left[\int_0^{\infty}{\|g(s)\|ds}\right]\sup_{t}\|u(t)\|$$