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The Liapunov's second method for continuous time difference equations

Among many other cases such as economic and lossless propagation models, continuous time difference equations are encountered as the internal dynamics in a class of non‐linear time delay systems, when controlled by a suitable state feedback which drives the output exponentially to zero. The Liapunov's second method for these infinite dimensional systems has not been extensively investigated in the literature. This paper has the aim of filling this gap. Liapunov's second method theorems for checking the stability and the asymptotic stability of this class of infinite dimensional systems are built up, in both a finite and an infinite dimensional setting. In the finite dimensional setting, the Liapunov function is defined on finite dimensional sets. The conditions for stability are given as inequalities on continuous time. No derivatives are involved, as in the dynamics of the studied systems. In the infinite dimensional setting, the continuous time difference equation is transformed into a discrete time system evolving on an infinite dimensional space, and then the classical Liapunov theorem for the system in the new form is written. In this paper the very general case is considered, that is non‐linear continuous time difference equations with multiple non commensurate delays are considered, and moreover the functions involved in the dynamics are allowed to be discontinuous, as well as the initial state. In order to study the stability of the internal dynamics in non‐linear time delay feedback systems, an exogenous disturbance is added, which goes to zero exponentially as the time goes to infinity. An example is considered, from non‐linear time delay feedback theory. While the results available in the literature are inconclusive as far as the stability of that example is concerned, such stability is proved to hold by the theorems developed in this paper, and is validated by simulation results. Copyright © 2003 John Wiley & Sons, Ltd.