Controllability of a semilinear neutral dynamic equation on time scales with impulses and nonlocal conditions

Abstract

In this paper we consider a control system governed by a neutral differential equation on time scales with impulses and nonlocal conditions. We obtain conditions under which the system is approximately controllable, on one hand, and on the other hand, the exactly controllable is also proved. Concretely, first of all, we prove the existence of solutions. After that, we prove approximate controllability assuming that the associated linear system on time scales is exactly controllable, and applying a technique developed by Bashirov et al. [8, 9, 10] where we can avoid fixed point theorems. Next, assuming certain conditions on the nonlinear term, we can apply Banach Fixed Point Theorem to prove exact controllability. Finally, we propose an example to illustrate the applicability of our results.