Eventual stability

Abstract

In many real situations one wishes to consider the stability of states which are not equilibrium states. This immediately rules out Liapunov stability since his stability definitions imply that a stable state is an equilibrium state. A particularly good example is found in the study of stability of adaptive control systems. The desired state may not be an equilibrium state but may tend in time to act more and more like a stable equilibrium state. Such a stability, which here is called an eventual stability, is given a precise definition, and some basic properties of such stabilities are obtained. For example, if the system is autonomous or the state is an equilibrium state, then the eventual stabilities are the same as Liapunov stabilities, and the theorems can be viewed as generalizations of Liapunov's theorems. However, when the system is non-autonomous and the state is not an equilibrium state, then it is something new. Theorems are then given which show how Liapunov's direct method can be extended to study eventual stabilities and how qualitative estimates of the extent of the stability can be obtained. The authors believe this new concept of stability may play an important role in the theory and design of adaptive control systems and an example is given which illustrates how these ideas can be used in the design of an adaptive control system.