We study linear control systems in infinite--dimensional Banach spacesgoverned by analytic semigroups. For $p\in[1,\infty]$ and $\alpha\in\RR$ weintroduce the notion of $L^p$--admissibility of type $\alpha$ for unboundedobservation and control operators. Generalising earlier work by Le Merdy andthe first named author and Le Merdy we give conditions under which$L^p$--admissibility of type $\alpha$ is characterised by boundednessconditions which are similar to those in the well--known Weiss conjecture. Wealso study $L^p$--wellposedness of type $\alpha$ for the full system. Here weuse recent ideas due to Pruess and Simonett. Our results are illustrated by acontrolled heat equation with boundary control and boundary observation wherewe take Lebesgue and Besov spaces as state space. This extends theconsiderations from Byrnes, Gilliam, Shubov and Weiss to non--Hilbertiansettings and to $p\neq 2$.
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机译：我们研究由解析半群控制的无限维Banach空间中的线性控制系统。对于$ p \ in [1，\ infty] $和$ \ alpha \ in \ RR $，我们引入了$ L ^ p $的概念-无界观察和控制算子的类型为$ \ alpha $的可允许性。归纳一下Le Merdy和第一位作者及Le Merdy的早期工作，我们给出了条件$ L ^ p $（类型$ \ alpha $的可容许性）的有界性条件，与众所周知的魏斯猜想中的相似。我们还研究了$ L ^ p $-整个系统$ \ alpha $类型的健康状况。在这里，我们使用由于Pruess和Simonett而产生的最新想法。我们的结果通过带有边界控制和边界观测的受控热方程式进行了说明，其中我们以Lebesgue空间和Besov空间为状态空间。这将考虑范围从Byrnes，Gilliam，Shubov和Weiss扩展到非希尔伯特式设置和$ p \ neq 2 $。
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