Approximate controllability of fractional differential equations via resolvent operators

where D is the Caputo fractional derivative of order α with  < α < , A :D(A)⊂ X → X is the infinitesimal generator of a resolvent Sα(t), t ≥ , B : U → X is a bounded linear operator, u ∈ L([,b],U), X and U are two real Hilbert spaces, J–α t h denotes the  – α order fractional integral of h ∈ L([,b],X). The controllability problem has attracted a lot of mathematicians and engineers’ attention since it plays a key role in control theory and engineering and has very important applications in these fields. Many contributions on exact and approximate controllability have been made in recent years. We refer the reader to the recent papers [–] and the references therein. However, there are few articles to study fractional control system (.) governed by a linear closed operator which generates a resolvent. The main difficulty is that the resolvent does not have the semigroup property, even the continuity in the uniform operator topology. Fortunately, we can prove the continuity of a resolvent in the uniform operator topology and the compactness of the solution operator in the case of an analytic resolvent. For more details, we refer the reader to the papers [, ] by Fan and Mophou. A similar idea on the uniform continuity of operators can be found in [] by Liang, Liu and Xiao. In the present paper, we study approximate controllability of fractional control system (.) by using the analytic resolvent method and the uniform continuity of the resolvent.