Abstract. In this work, the relation between input-to-state stability and integral input-to-5 state stability is studied for linear infinite-dimensional systems with an unbounded control operator. 6Although a special focus is laid on the case L ∞ , general function spaces are considered for the inputs. 7We show that integral input-to-state stability can be characterized in terms of input-to-state stability 8 with respect to Orlicz spaces. Since we consider linear systems, the results can also be formulated 9 in terms of admissibility. For parabolic diagonal systems with scalar inputs, both stability notions 10 with respect to L ∞ are equivalent. 11Key words. Input-to-state stability, integral input-to-state stability, C 0 -semigroup, admissibil-12 ity, Orlicz spaces 13 AMS subject classifications. 93D20, 93C05, 93C20, 37C75 14

We study minimal conditions under which mild solutions of linear evolutionary control systems are continuous for arbitrary bounded input functions. This question naturally appears when working with boundary controlled, linear partial differential equations. Here, we focus on parabolic equations which allow for operator-theoretic methods such as the holomorphic functional calculus. Moreover, we investigate stronger conditions than continuity leading to input-to-state stability with respect to Orlicz spaces. This also implies that the notions of input-to-state stability and integral-input-to-state stability coincide if additionally the uncontrolled equation is dissipative and the input space is finite-dimensional.

Input-to-state stability (ISS) for systems described by partial differential equations has seen intensified research activity recently, and in particular the class of boundary control systems, for which truly infinite-dimensional effects enter the situation. This note reviews input-to-state stability for parabolic equations with respect to general L p -input-norms in the linear case and includes extensions of recent results on semilinear equations.

We investigate the boundedness of the H ∞ -calculus by estimating the bound b(ε) of the mapping H ∞ → B(X): f → f (A)T (ε) for ε near zero. Here, −A generates the analytic semigroup T and H ∞ is the space of bounded analytic functions on a domain strictly containing the spectrum of A. We show that b(ε) = O(| log ε|) in general, whereas b(ε) = O(1) for bounded calculi. This generalizes a result by Vitse and complements work by Haase and Rozendaal for non-analytic semigroups. We discuss the sharpness of our bounds and show that single square function estimates yield b(ε) = O( | log ε|).

This is a repository copy of On input-to-state-stability and integral input-to-state-stability for parabolic boundary control systems.