Time optimal control problems governed by Riemann-Liouville fractional differential system are considered in this paper. Firstly, the existence results are obtained by using the theory of semigroup and Schauder’s fixed point. Secondly, the new approach of establishing time minimizing sequences twice is applied to acquire the time optimal pairs without the Lipschitz continuity of nonlinear function. Moreover, the reflexivity of state space is removed with the help of compact method. Finally, an example is given to illustrate the main conclusions. Our work essentially improves and generalizes the corresponding results in the existing literature.

Under a compactness assumption on the resolvent, some properties on relevant operators generated by resolvent are given. Existence results of fractional control systems are obtained by Schauder?s fixed point theorem and approximation techniques. Furthermore, the approximately controllable result is acquired under the assumption that the corresponding linear system is approximately controllable, which improves and extends some results on this topic.

This article is concerned with the existence of mild solutions for fractional differential inclusions with nonlocal conditions in Banach spaces. The results are obtained by using fractional calculus, Hausdorff measure of noncompactness, and the multivalued fixed point theorem. The results obtained in the present paper extend some related results on this topic.