On Lyapunov theory for delay difference inclusions

This paper provides a complete collection of Lyapunov methods for delay difference inclusions. We discuss the Lyapunov-Krasovskii (LK) approach, which uses a Lyapunov function that depends on both the current state and the entire delayed state trajectory. It is shown that such a function exists if and only if the delay difference inclusion is globally asymptotically stable (GAS). We also study the Lyapunov-Razumikhin (LR) method, which employs a Lyapunov function that is required to decrease only if the state trajectory satisfies a certain condition. It is proven that the LR method provides a sufficient condition for GAS. Moreover, an example of a linear system which is globally exponentially stable but does not admit a Lyapunov-Razumikhin function (LRF) is provided. Then, we show that the existence of a LRF is a sufficient condition for the existence of a Lyapunov-Krasovskii function and that only under certain additional assumptions the converse is true. For both methods, we establish what type of invariant/contractive sets can be obtained from the respective functions.