Linearized stability in the context of an example by Rodrigues and Sola-Morales

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Garab, Abel¹  Pituk, Mihaly²  Poetzsche, Christian¹ 

[1] Univ Klagenfurt, Inst Math, A-9020 Klagenfurt, Austria

[2] Pannon Egyet, Matemat Tanszek, H-8201 Veszprem, Hungary

关键词: Exponential asymptotic stability;    Exponential stability;    Linearized stability;    Spectral gap condition;   

DOI  :  10.1016/j.jde.2020.07.011

来源: Elsevier

【 摘 要 】

In a recent paper, Rodrigues and Sola-Morales construct an example of a continuously Frechet differentiable discrete dynamical system in a separable Hilbert space for which the origin is an exponentially asymptotically stable fixed point, although its derivative at 0 has spectral radius greater than one. For maps on general Banach spaces we demonstrate that the slightly stronger, but also widely used concept of exponential stability allows a complete characterization in terms of the spectral radius. Moreover, under a spectral gap condition valid for compact and finite-dimensional linearizations these two stability notions are shown to be equivalent. (C) 2020 The Author(s). Published by Elsevier Inc.

【 授权许可】

Free