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In practice one is not only interested in the qualitative type of information obtainable from the stability (in the Lyapunov sense) of a dynamic system, but also in quantitative data, such as specific trajectory bounds and specific transient behavior. A system could, for example, be stable and still be completely useless because it may exhibit undesirable transient characteristics (e.g., it may exceed certain limits imposed on the trajectory bounds). In order to develop a meaningful quantitative theory for the analysis of dynamic systems, stability is defined here in terms of subsets of the state space that are prespecified in a given problem and, in general, may be time varying. The properties of these subsets yield not only information about the stability of a system, but they also yield estimates of trajectory bounds and of trajectory behavior. The theory developed here is general enough to include autonomous and nonautonomous systems, linear and nonlinear systems, simple systems and interconnected systems. The composite, or interconnected systems considered are analyzed and treated in terms of their subsystems. After stating various definitions of stability and instability, theorems that yield sufficient conditions for stability and instability are stated and proved. These theorems involve the existence of Lyapunov-like functions which in general do not possess the usual definiteness requirements on V and V. In order to demonstrate the developed theory, several examples are considered.