Controllability of retarded semilinear systems with control delay

This paper presents the controllability of a family of linear and semilinear systems with control delay in Hilbert spaces. Firstly, the approximate controllability of the linear control delay system is proved by assuming that the linear system without control delay is completely controllable. Then, Nemytskii operators have been constructed associated to control operators and the nonlinear function. The approximate controllability of the retarded semilinear system is established by using the Rothe type fixed point theorem. The applications of results are explained through examples of parabolic and hyperbolic partial differential equations.