On the controllability of nonlinear systems with time-variable delays

This note deals with the controllability, of a certain class of nonlinear time-varying systems with time-variable delays in state and implicit derivative. The notion of measure of noncompactness of a set and methods for its determination in the spaces of continuous and differentiable functions are considered. By using the measure of noncompactness of a set and the fixed-point theorem of Darbo, a theorem is proved on the existence of solutions of a certain type of nonlinear time-varying systems with multiple time-variable delays in state with derivative of the state-vector on the right-hand side. Sufficient conditions are given for global and local controllability of such systems.