On the Asymptotic Stability and Ultimate Boundedness of Solutions of a Class of Nonlinear Systems with Delay

For a certain class of nonlinear systems of differential equations with constant delay, we study the conditions for the asymptotic stability of the zero solution and the ultimate boundedness of the solutions. To obtain such conditions, we propose special constructions of Lyapunov–Krasovskii full-type functionals. Estimates of the transient time are found, and an analysis of the influence of perturbations on the dynamics of systems is carried out. In addition, we study the case in which the systems have switching operation modes and determine conditions under which the asymptotic stability or ultimate boundedness is preserved for any admissible switching laws.