This manuscript is mainly focusing on approximate controllability for fractional differential evolution equations of order 1 < r < 2 in Hilbert spaces. We consider a class of control systems governed by the fractional differential evolution equations. By using the results on fractional calculus, cosine and sine functions of operators, and Schauder’s fixed point theorem, a new set of sufficient conditions are formulated which guarantees the approximate controllability of fractional differential evolution systems. The results are established under the assumption that the associated linear system is approximately controllable. Then, we develop our conclusions to the ideas of nonlocal conditions. Lastly, we present theoretical and practical applications to support the validity of the study.

该手稿主要关注 希尔伯特空间中1 < r <2的分数阶微分演化方程的近似可控性 。我们考虑一类由分数阶微分演化方程控制的控制系统。通过使用分数微积分，算子的余弦和正弦函数以及Schauder不动点定理的结果，制定了一组新的充分条件，保证了分数微分演化系统的近似可控性。在相关线性系统近似可控的假设下建立结果。然后，我们对非本地条件的想法得出结论。最后，我们提出理论和实际应用以支持研究的有效性。