On Lipschitz conditions of infinite dimensional systems

Abstract

This paper addresses Lipschitz conditions of infinite dimensional systems. Three types of Lipschitz conditions are discussed, namely, the compact set Lipschitz condition, the local Lipschitz condition and the bounded set Lipschitz condition. These three conditions are equivalent in finite dimensional systems but not necessarily equivalent in infinite dimensional systems on the basis of the Riesz’s lemma. It is first proved that in infinite dimensional systems, the compact set Lipschitz condition is equivalent to the local Lipschitz condition. Then by providing a counterexample which is locally Lipschitz but is not bounded set Lipschitz, it is shown that the local Lipschitz condition is strictly weaker than the bounded set Lipschitz condition. In addition, the right hand side Lipschitz condition, which is frequently used in the study of time delay systems, is considered. It is shown that the right hand side Lipschitz condition and the local Lipschitz condition cannot imply each other. Furthermore, impacts of the obtained results on time delay systems, which are special cases of infinite dimensional systems, are discussed.