In this paper, we mainly consider a time-varying semi-linear integro-differential inclusion with Clarke sub-differential and a non-local initial condition. By a suitable Green function combined with a resolvent operator, we firstly formulate its mild solutions and show that it admits at least one mild solution which can exist in a well-defined ball with a radius big enough. Through constructing a proper functional, we then derive a useful characterization of the approximate controllability for its related linear system in Green function terms, and establish a sufficient condition for the approximate controllability of the time-varying semi-linear integro-differential inclusion. Lastly, we also consider the finite approximate controllability of the time-varying semi-linear integro-differential inclusion via variational method.

中文翻译：

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具有Clarke次微分和非局部初始条件的时变积分微分包含物：存在和近似可控性

在本文中，我们主要考虑具有Clarke次微分和非局部初始条件的时变半线性积分微分包含。通过将适当的格林函数与可分解算符结合，我们首先制定了它的温和解，并表明它接受了至少一个温和解，该温和解可以存在于半径足够大的轮廓分明的球中。通过构造适当的泛函，我们然后用格林函数项推导了其相关线性系统的近似可控性的有用表征，并为时变半线性积分-微分包含的近似可控性建立了充分条件。最后，我们还通过变分方法考虑了时变半线性积分微分包含的有限近似可控性。