We provide an introduction to infinite-dimensional port-Hamiltonian systems. As this research field is quite rich, we restrict ourselves to the class of infinite-dimensional linear port-Hamiltonian systems on a one-dimensional spatial domainand we will focus on topics such as Dirac structures, well-posedness, stability and stabilizability, Riesz-bases and dissipativity. We combine the abstract operator theoretic approach with the more physical approach based on Hamiltonians. This enables us to derive easy verifiable conditions for well-posedness and stability.

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 the non-autonomous version of an infinite-dimensional linear port-Hamiltonian system on an interval [a, b]. Employing abstract results on evolution families, we show C 1 -well-posedness of the corresponding Cauchy problem, and thereby existence and uniqueness of classical solutions for sufficiently regular initial data. Further, we demonstrate that a dissipation condition in the style of the dissipation condition sufficient for uniform exponential stability in the autonomous case also leads to a uniform exponential decay rate of the energy in this non-autonomous setting.

This paper gives embedding theorems for a very general class of weighted Bergman spaces: the results include a number of classical Carleson embedding theorems as special cases. The little Hankel operators on these Bergman spaces are also considered. Next, a study is made of Carleson embeddings in the right half-plane induced by taking the Laplace transform of functions defined on the positive half-line (these embeddings have applications in control theory): particular attention is given to the case of a sectorial measure or a measure supported on a strip, and complete necessary and sufficient conditions for a bounded embedding are given in many cases.

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