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

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. 

References: 

[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.