turkmath.org

 Time-delay systems are dynamic systems in which a delay between the input and output significantly affects system performance. These delays, whether constant or variable, discrete or distributed, often arise in engineering applications such as networked control systems, biological systems, and process control. Managing these delays is crucial for ensuring system stability and achieving robust control. One of the most common approaches for 

analyzing the stability of time-delay systems is the Lyapunov-Krasovskii functional method. This involves constructing a functional that decreases over time, ensuring system stability. The LKF method is powerful for both constant and time-varying delays. The Lyapunov-Razumikhin approach uses a Lyapunov function that considers the delayed state explicitly,providing conditions under which the system remains stable despite the delay. It is often simpler to apply than the LKF method but might be less conservative. Various stability criteria have been developed, including the delay-independent and delay-dependent criteria. 
Delay-independent criteria provide stability conditions that do not depend on the magnitude of the delay, while delay-dependent criteria take the specific value of the delay into account for a more precise stability assessment.