Delay-dependent methods and the first delay interval

This paper deals with the solution bounds for time-delay systems via delay-dependent Lyapunov-Krasovskii methods. Solution bounds are widely used for systems with input saturation caused by actuator saturation or by the quantizers with saturation. We show that an additional bound for solutions is needed for the first time-interval, where t<τ(t), both in the continuous and in the discrete time. This first time-interval does not influence on the stability and the exponential decay rate analysis. The analysis of the first time-interval is important for nonlinear systems, e.g., for finding the domain of attraction. Regional stabilization of a linear (probably, uncertain) system with unknown and bounded input delay under actuator saturation is revisited, where the saturation avoidance approach is used.