Infinite-dimensional input-to-state stability

In this talk we discuss infinite-dimensional versions of well-known stability notions relating the external input $u$ and the state $x$ of a linear system governed by the equation $$\dot{x}=Ax+Bu, \quad x(0)=x_{0}.$$ Here, $A$ and $B$ are unbounded operators. For instance, the system is called \textit{$L^{p}$-input-to-state stable} if $$u(\cdot)\mapsto x(t)$$ is bounded as a mapping from $L^{p}(0,t)$ to the state space $X$ for all $t\u3e0$. In particular, we elaborate on the relation of this notion to \textit{integral input-to-state} stability and \textit{(zero-class) admissibility} with a special focus on the case $p=\infty$.\\ This is joint work with B.~Jacob, R.~Nabiullin and J.R.~Partington