Conventional input/output approaches, to nonlinear system stability, often ignore the transient behavior of systems by employing the assumption that the initial state is at the steady state of interest. Furthermore, these same input/output approaches allow inputs to span the whole space often assuming values which are not physically realizable. Some of these limitations can be alleviated by considering an input-state-output system description and sets of allowable initial conditions and input signals. To this end, a measure for bounded input/initial state bounded output (BIISBO) stability over ball is introduced. This notion is then employed to develop a theorem which allows one to deduce the closed-loop system's BIISBO stability over ball from those of the plant and the controller. Computation of the measure for BIISBO stability over ball is shown to be possible through optimal control theory. A physical system is employed to demonstrate the computation and various features of the proposed analysis.