On the stability of solutions of neutral differential equations with distributed delay

We consider one class of systems of linear nonautonomous differential equations of neutral type with distributed delay. The matrix in front of the derivative of the unknown vector-function with delay is constant, the matrix in front of the unknown vector-function has continuous T-periodic elements, the kernel of the integral operator consists of continuous functions, which are T-periodic with respect to argument t. The aim of the work is to study the asymptotic stability of the zero solution using the method of modified Lyapunov–Krasovskii functionals. The method of Lyapunov–Krasovskii functionals is the development of the Lyapunov second method. The advantage of this method is the simplicity of formulations and the reduction of the study of asymptotic stability to solving well-conditioned problems. In addition, the method of modified Lyapunov–Krasovskii functionals allows to obtain estimates of solutions to linear systems of delay differential equations. Note that the use of modified Lyapunov–Krasovskii functionals also allows to obtain estimates of solutions to nonlinear differential equations and estimates of the attraction set. Previously, a system of periodic differential equations of neutral type was considered in the works by G.V. Demidenko and I.I. Matveeva, where sufficient conditions of the asymptotic stability of the zero solution were obtained and estimates of solutions to this system were established. A system of linear periodic differential equations with distributed delay was considered by the author of this paper. For this system it was also obtained sufficient conditions of the asymptotic stability of the zero solution and established estimates of solutions. In the present paper, we obtain sufficient conditions of the asymptotic stability of the zero solution to the neutral type system with distributed delay in terms of matrix inequalities and establish estimates of solutions to the system characterizing the exponential decay at infinity.