A system with inputs (or controls) has the form

where the (generally time-dependent) input u(t) may be viewed as a "control", "external input",
"stimulus", "disturbance", or "forcing function". The study of such systems is the subject
of control theory and applied in control engineering. For systems with inputs, one must
quantify the effect of inputs on the stability of the system. The main two approaches to this
analysis are BIBO stability (for linear systems) and input-to-state (ISS) stability (for nonlinear systems)

Stability for systems with inputs
Lyapunov's second method for stability
Barbalat's lemma and stability of time-varying systems
Example