A new exploration on the existence and approximate controllability for fractional semilinear impulsive control systems of order r∈(1,2)

In this article, we mainly focus on the existence and approximate controllability results for the fractional semilinear impulsive control system of order r∈(1,2). We consider two different sets of sufficient conditions. In the first set, we derive the results by using the theories on the fractional calculus, compactness of the cosine family, and Schauder's fixed point theorem. In the second set, we prove the main results by using Gronwall's inequality, avoids the usage of the compactness of cosine family and fixed point theorems. By introducing the suitable assumptions, we discuss the existence and uniqueness of mild solutions for the fractional semilinear impulsive system. Finally, we provide theoretical and practical applications to assist in the effectiveness of the discussion.