  New trajectory-based small-gain results are obtained for nonlinear feedback
systems under relaxed assumptions. Specifically, during a transient period, the
solutions of the feedback system may not satisfy some key inequalities that
previous small-gain results usually utilize to prove stability properties. The
results allow the application of the small-gain perspective to various systems
which satisfy less demanding stability notions than the Input-to-Output
Stability property. The robust global feedback stabilization problem of an
uncertain time-delayed chemostat model is solved by means of the
trajectory-based small-gain results.
