Existence and controllability of second-order nonlinear retarded integro-differential systems with multiple delays in control

This work analyzes the existence and exact controllability of nonlinear second-order retarded integro-differential equations involving delays in control. Making use of fixed point principle and cosine family we determine the existence of solutions. Then, under some assumptions, we show that the controllability of the associated linear system without delay implies the controllability of the associated linear delay system and the actual system by applying an iterative technique. To illustrate the results, we introduce an example.