The maximum response ratio of linear systems

EXCEPT in cases where pure undamped resonances occur, stable linear transmission or filter systems are characterized by bounded response to every bounded input. What is the maximum response amplitude that can be obtained in a linear, time-invariant system subjected to inputs having limited amplitudes? This question may be stated in another way. To what level must the input to a system be limited to insure that the output (or signal level at any given point in the system) does not exceed a given level? This question is of importance in many physical problems.