Discretized LKF Method for Coupled Functional-Differential Equations with Continuous Distributed Delays

This article discusses the stability of a class of differential-difference systems with continuous distributed delays. Based on the discretized Lyapunov-Krasovskii functional (LKF) method, a sufficient condition for ensuring system stability is given by three LMIs. Firstly, the system is reconstructed by decomposing the distributed time-delay term into the nominal part and the disturbance part. Secondly, in order to smoothly implement the discretization of LKF, the perturbation matrix and related lagged state variables under the integral sign are treated as a whole variable, and the loss of system information caused by this is compensated by introducing a quadratic functional constructed from the disturbance matrix. Therefore, the problem is successfully solved by the discretized LKF method. Meanwhile, the obtained results can not only be used to test the timedelay bound of stable systems using numerical methods, but also to estimate the disturbance bound when the time delay is fixed. Finally, the two examples are provided to demonstrate the effectiveness and superiority of the results.