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Impulsive semilinear heat equation with delay in control and in state

In this paper, we prove the interior approximate controllability of the impulsive semilinear heat equation with delay in control and in state by proving first that the linear heat equation with delay in control is approximately controllable. After that, we add impulses and a nonlinear perturbation with delay in state, and using Rothe's fixed point theorem, we prove that the interior approximate controllability of the impulsive semilinear system. Finally, we present some open problems and a possible general framework to study the controllability of impulsive semilinear diffusion process in Hilbert spaces with delay in control and in state.