Razumikhin and Krasovskii stability theorems for time-varying time-delay systems

The main results of the paper are generalizations of the Razumikhin and of the Krasovskii classical stability theorems for stability analysis of time-varying time-delay systems. The condition of negativity of the time-derivative of Razumikhin functions and Krasovskii functionals is weakened. This is achieved by using the notion and properties of uniformly stable functions. We also show how to apply the results to the stability analysis of linear time-varying time-delay systems of retarded type. Both the system matrices and time-delays are allowed to be time-varying. Some constructive sufficient stability conditions are obtained and their effectiveness is demonstrated by some examples.