This paper is mainly concerned with controlled time fractional differential equations of Sobolev type in Caputo and Riemann-Liouville fractional derivatives with the order in $(1,2)$ respectively. By properties on some corresponding fractional resolvent operators family, we first establish sufficient conditions for the existence of mild solutions to these controlled time fractional differential equations of Sobolev type. Then, we present the existence of optimal controls of systems governed by corresponding time fractional differential equations of Sobolev type via setting up approximating minimizing sequences of suitable functions twice.

中文翻译：

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Sobolev型时间分数阶微分方程和具有$（1,2）$阶的最优控制

本文主要涉及Caputo和Riemann-Liouville分数阶导数中Sobolev型受控时间分数阶微分方程，其阶次分别为$（1,2）$。通过一些相应的分数阶分解算子族的性质，我们首先为这些Sobolev型受控时间分数阶微分方程的温和解的存在建立了充分的条件。然后，我们通过设置两次近似最小化适当函数的序列，给出了由相应的Sobolev型时间分数微分方程控制的系统的最优控制的存在。