A new method to enclose the pseudospectrum via the numerical range of the inverse of a matrix or linear operator is presented. The method is applied to finite-dimensional discretizations of an operator on an infinite-dimensional Hilbert space, and convergence results for different approximation schemes are obtained, including finite element methods. We show that the pseudospectrum of the full operator is contained in an intersection of sets which are expressed in terms of the numerical ranges of shifted inverses of the approximating matrices. The results are illustrated by means of two examples: the advection–diffusion operator and the Hain–Lüst operator.

We investigate input-to-state stability (ISS) of infinite-dimensional collocated control systems subject to saturated feedback. Here, the unsaturated closed loop is dissipative and uniformly globally asymptotically stable. Under an additional assumption on the linear system, we show ISS for the saturated one. We discuss the sharpness of the conditions in light of existing results in the literature.

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The following corrections should be made to this article.In Theorem 9 and Theorem 11, one needs to add the assumption that the admissible feedback function σ is globally Lipschitz continuous. If we denote the globally Lipschitz constant by k, then in the proofs of Theorems 9 and 11 the definition of r should be omitted and the constant k r replaced by k.Otherwise, the K ∞ -function candidate for the functions ρ in ( 7) would depend on the solution x and the disturbance d in both theorems.Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.