Stability problems in nonautonomous linear differential equations in infinite dimensions

Hildebrando M. Rodrigues; J. Solà-Morales; G. K. Nakassima

doi:10.3934/cpaa.2020138

Article Contents

2020, Volume 19, Issue 6: 3189-3207. Doi: 10.3934/cpaa.2020138

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Stability problems in nonautonomous linear differential equations in infinite dimensions

1.
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, 13560-970, São Carlos, SP, Brazil
2.
Departament de Matemàtiques, Universitat Politècnica de Catalunya, ETSEIB, Av. Diagonal 647, 08028 Barcelona, Spain

^*Corresponding author
^*Corresponding author

Received: May 31, 2019

Revised: October 31, 2019

Published: March 2020

The first author was partially supported by FAPESP grant Processo 2018/05218-8 and CNPq grant Processo 304767/2018-2, Brazil. The second author was partially supported by MINECO grant MTM2017-84214-C2-1-P, is a faculty member of the Barcelona Graduate School of Mathematics (BGSMath) and part of the Catalan research group 2017 SGR 01392

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Abstract

In this paper we study the robustness of the stability in nonautonomous linear ordinary differential equations under integrally small perturbations in infinite dimensional Banach spaces. Some applications are obtained to the case of rapidly oscillating perturbations, with arbitrarily small periods, showing that even in this case the stability is robust. These results extend to infinite dimensions some results given in Coppel [3]. Based in Rodrigues [11] and in Kloeden & Rodrigues [10] we introduce a class of functions that we call Generalized Almost Periodic Functions that extend the usual class of almost periodic functions and are suitable to model these oscillating perturbations. We also present an infinite dimensional example of the previous results.

As counterparts, e show first in another example that it is possible to stabilize an unstable system by using a perturbation with a large period and a small mean value, and finally we give an example where we stabilize an unstable linear ODE with a small perturbation in infinite dimensions by using some ideas developed in Rodrigues & Solà-Morales [21] after an example due to Kakutani (see [13]).

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Mathematics Subject Classification: 34G20, 35B15, 34D30, 34E10.

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Figure 1. Left: The spectrum of $ L $ given by $ \sigma (L) = B_{\nu}(a) $. Right: The spectrum of $ A $ given by $ \sigma (A) = \log (\sigma (L)) $, with $ a = 1/2 $ and $ \nu = 1/4 $

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