On problems of partial stability for delay systems

The problem of stability with respect to a part of variables of both zero equilibrium state and “partial” equilibrium state is considered for nonlinear nonstationary delay systems of differential equations. As compared to known assumptions, more general assumptions are made for the values of the supremum norm of components of the initial vector function that correspond to the “uncontrolled” variables. Within the method of Lyapunov–Krasovskii functionals, new conditions for stability and asymptotic stability of this type are obtained, which generalize a number of existing results.