Necessary and Sufficient Razumikhin-Type Conditions for Stability of Delay Difference Equations

This technical note considers stability analysis of time-delay systems described by delay difference equations (DDEs). All existing analysis methods for DDEs that rely on the Razumikhin approach provide sufficient, but not necessary conditions for asymptotic stability. Nevertheless, Lyapunov-Razumikhin functions are of interest because they induce invariant sets in the underlying state space of the dynamics. Therefore, we propose a relaxation of the Razumikhin conditions and prove that the relaxed conditions are necessary and sufficient for asymptotic stability of DDEs. For linear DDEs, it is shown that the developed conditions can be verified by solving a linear matrix inequality. Moreover, it is indicated that the proposed relaxation of Lyapunov-Razumikhin functions has an important implication for the construction of invariant sets for linear DDEs.