On the stability of coupled delay differential and continuous time difference equations

A Lyapunov-type stability analysis for coupled delay differential and continuous time difference equations is given. Such systems are encountered as the internal dynamics of input output linearized nonlinear time delay systems. They appear also in lossless propagation models, economics and gas dynamics. The main contribution of this note is a two-step algorithm for the stability analysis of such systems. At the first step, only the L₂ stability is checked; while at the second step, the Lyapunov stability of the difference part of the system is analyzed. In these two steps, the stability of the whole system can be checked. Using this method, we show the stability of a system (dependent on the delay), for which the known methods in literature are inconclusive.