The damping problem for a non-stationary system with delay of neutral type

We consider the damping problem for a non-stationary control system described by a system of differential-difference equations of neutral type with smooth matrix coefficients and several delays, and establish the relationship of the variational problem for the nonlocal functionals and the corresponding boundary value problem for differential-difference equations. It is proved the existence and uniqueness of generalized solution to the boundary value problem for this system of differential-difference equations. It is proved that the smoothness of this solution can be violated on the considered interval and is preserved only on some subintervals.