Deutsch Intern
Dynamics and Control

Oberseminar "Dynamische Systeme und Kontrolltheorie" - Prof. Dr. Christian Pötzsche

Exponential Stability: From Time Scales to Infinite Dimensions
Date: 11/19/2021, 4:00 PM
Category: Veranstaltung
Location: Hubland Nord, Geb. 40, 01.003
Speaker: Prof. Dr. Christian Pötzsche, Alpen-Adria Universität Klagenfurt, Institut für Mathematik

In this talk we consider classical notions of exponential stability from two perspectives: 


First, we review a characterization of exponential stability for linear, finite-dimensional and autonomous dynamic equations on time scales in terms of the set of exponential stability. This is based on a joint work [2] with Stefan Siegmund and Fabian Wirth in the early 2000s. 


Second, it came as a surprise when Rodrigues and Solà-Morales constructed a nonlinear autonomous difference equation in a separable Hilbert space, whose trivial solution is exponentially asymptotically stable, while its linearization has a spectral radius greater than 1 (see [3]). This means that the principle of linearized stability provides a sufficient, but not a necessary condition for exponential asymptotic stability. In a recent collaboration [1] with Ábel Garab and Mihály Pituk we prove that the principle of linearized stability indeed yields a necessary and sufficient condition for the slightly stronger notion of exponential stability. This characterization remains true when dealing with nonautonomous difference or differential equations in terms of the dichotomy spectrum. 




[1] Á. Garab, M. Pituk, C. Pötzsche, Linearized stability in the context of an example by Rodrigues and Solà-Morales, J. Differ. Equations 269 (2020), no. 11, 9838–9845.


[2] C. Pötzsche, S. Siegmund, F. Wirth, A spectral characterization of exponential stability for linear time-invariant systems on time scales, Discrete Contin. Dyn. Syst. 9 (2003), no. 5, 1223–1241.


[3] H.M. Rodrigues, J. Solà-Morales, An example on Lyapunov stability and linearization, J. Differ. Equations 269 (2020), no. 2, 1349–1359.