<p style='text-indent:20px;'>Our investigations are motivated by the well - posedness problem of some dynamical models with anomalous diffusion described by the Caputo spatial fractional derivative of order <inline-formula><tex-math id="M1">\begin{document}$ \alpha \in (1, 2) $\end{document}</tex-math></inline-formula>. We propose a characterization of an exponentially stable analytic semigroup generator using the inverse operator. This characterization enables us to establish the form of a generator involving the Caputo fractional derivative, under various boundary conditions. In particular, the results simplify those known from literature obtained by means of the fractional Sobolev spaces and some perturbation results. Going further, we show how to construct a control system in factor form, having such a generator as the state operator.</p>

中文翻译：

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关于涉及 Caputo 分数导数的解析半群生成器

<p style='text-indent:20px;'>我们的研究是由一些动力学模型的适定性问题引起的"M1">\begin{document}$ \alpha \in (1, 2) $\end{document}</tex-math></inline-formula>。我们提出了使用逆算子来表征指数稳定的解析半群生成器。这种表征使我们能够在各种边界条件下建立涉及 Caputo 分数导数的生成器的形式。特别是，这些结果简化了通过分数 Sobolev 空间和一些扰动结果获得的文献中已知的结果。更进一步，我们展示了如何以因子形式构建控制系统，