Anke Kalauch

Stability radii for positive linear time-invariant systems on time scales

We deal with dynamic equations on time scales, where we characterize the positivity of a system. Uniform exponential stability of a system is determined by the spectrum of its matrix. We investigate the corresponding stability radii with respect to structured perturbations and show that, for positive systems, the complex and the real stability radius coincide.

A Constructive Approach to Linear Lyapunov Functions for Positive Switched Systems Using Collatz-Wielandt Sets

We establish the link between linear Lyapunov functions for positive switched systems and corresponding Collatz-Wielandt sets. This leads to an algorithm to compute a linear Lyapunov function whenever a Lyapunov function exists.

Stability of positive linear switched systems on ordered Banach spaces

Abstract We provide sufficient criteria for the stability of positive linear switched systems on ordered Banach spaces. The switched systems can be generated by finitely many bounded operators in infinite-dimensional spaces with a general class of order-inducing cones. In the discrete-time case, we assume an appropriate interior point of the cone, whereas in the continuous-time case an appropriate interior point of the dual cone is sufficient for stability. This is an extension of the concept of linear Lyapunov functions for positive systems to the setting of infinite-dimensional partially ordered spaces. We illustrate our results with examples.

On Positive-off-Diagonal Operators on Ordered Normed Spaces

On a normed space X ordered by a cone K we consider a continuous linear operator A: X → X of the following kind: If a positive continuous functional f attains 0 on some positive element x, then f(Ax) ≥ 0. If X is a vector lattice, then such operators can be represented as sI + B, where B is a positive operator, I is the identity and s ∈ R. We generalize this assertion for weaker assumptions on X, using the Riesz decomposition property.

On maximum principles for M-operators

Abstract For M -matrices a condition to satisfy the “maximum principle for inverse column entries” is known. We generalize this result (concerning a more general maximum principle) for M -operators on R n , ordered by some cone, as well as, to a certain extent, for M -operators on infinite-dimensional ordered normed spaces.

Vector lattice covers of ideals and bands in Pre-Riesz spaces

Abstract Pre-Riesz spaces are ordered vector spaces which can be order densely embedded into vector lattices, their so-called vector lattice covers. Given a vector lattice cover Y for a pre-Riesz space X, we address the question how to find vector lattice covers for subspaces of X, such as ideals and bands. We provide conditions such that for a directed ideal I in X its smallest extension ideal in Y is a vector lattice cover. We show a criterion for bands in X and their extension bands in Y as well. Moreover, we state properties of ideals and bands in X which are generated by sets, and of their extensions in Y.

Hyperbolicity Radius of Time-Invariant Linear Systems

Hyperbolicity of linear systems of difference and differential equations is a robust property. We provide a quantity to measure the maximal size of perturbations under which hyperbolicity is preserved. This so-called hyperbolicity radius is calculated by two methods, using the transfer operator and the input–output operator.