Rothe's fixed point theorem and the approximate controllability of semilinear heat equation with impulses, delays, and nonlocal conditions

Under certain conditions, impulses, delays, and nonlocal conditions are known to be negligible in comparison with the duration of the process. From the practical (engineering) point of view, delays and nonlocal conditions are intrinsic phenomena of a system that do not violate certain properties of it, such as the controllability. In other words, as a rule, the controllability is robust under the influence of impulses, delays and nonlocal conditions. In this paper, we apply Rothe's fixed-point theorem to prove the interior approximate controllability of the semilinear heat equation with impulses, delays and nonlocal conditions. Also we obtain conditions under which the system considered is approximately controllable.