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June  2020, 19(6): 3189-3207. doi: 10.3934/cpaa.2020138

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

Received  June 2019 Revised  November 2019 Published  March 2020

Fund Project: 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

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]).

Citation: Hildebrando M. Rodrigues, J. Solà-Morales, G. K. Nakassima. Stability problems in nonautonomous linear differential equations in infinite dimensions. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3189-3207. doi: 10.3934/cpaa.2020138
References:
[1]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors of Infinite Dimensional Nonautonomous Dynamical Syustems, Springer-Verlag Berlin, 2011. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[2]

W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath & Co., Boston, 1965.  Google Scholar

[3]

W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag Berlin Heidelberg New York, 1970.  Google Scholar

[4]

Ju. L. Dalekĭi and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Translation of Mathematical Monographs, Vol. 43, American Mathematical Society, Providence, RI, 1974.  Google Scholar

[5]

A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathmatics, Vol. 377, Springer-Verlag, Berlin-Heidelberg-New York, 1974.  Google Scholar

[6]

P. R. Halmos, A Hilbert Space Problem Book, Graduate texts in Mathematics, Vol. 19, Springer-Verlag, New York, Heidelberg, Berlin, 1974.  Google Scholar

[7]

J. K. Hale, Ordinary Differential Equations, 2$^nd$ edition, Krieger Publishing Co., Huntington, New York, 1980.  Google Scholar

[8]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes Math., Vol 840, Springer-Verlag, Berlin, 1981.  Google Scholar

[9]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1980.  Google Scholar

[10]

P. E. Kloeden and H. M. Rodrigues, Dynamics of a Class of ODEs more general than almost periodic, Nonlinear Anal., 74 (2011), 2695-2719.  doi: 10.1016/j.na.2010.12.025.  Google Scholar

[11]

H. M. Rodrigues, Invariância para sistemas de equações diferenciais com retardamento e aplicações, Tese de Mestrado, Universidade de São Paulo (São Carlos), 1970. Google Scholar

[12]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York Berlin Heidelberg Tokyo, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[13]

Ch. E. Rickart, General Theory of Banach Algebras, Princeton, D. van Nostrand, 1960.  Google Scholar

[14]

H. M. RodriguesT. Caraballo and M. Gameiro, Dynamics of a class of ODEs via wavelets, Commun. Pure Appl. Anal., 16 (2017), 2337-2355.  doi: 10.3934/cpaa.2017115.  Google Scholar

[15]

H. M. RodriguesM. A. Teixeira and Ma. Gameiro, On exponential decay and the Markus-Yamabe conjecture in infinite dimensions with applications to the Cima system, J. Dyn. Differ. Equ., 30 (2018), 1199-1219.  doi: 10.1007/s10884-017-9598-y.  Google Scholar

[16]

H. M. Rodrigues and J. G. Ruas-Filho, Evolution equations: dichotomies and the Fredholm alternative for bounded solutions, J. Differ. Equ., 119 (1995), 263-283.  doi: 10.1006/jdeq.1995.1091.  Google Scholar

[17]

H. M. Rodrigues and J. Solà-Morales, Linearization of Class $C^1$ for Contractions on Banach Spaces, J. Differ. Equ., 201 (2004), 351-382.  doi: 10.1016/j.jde.2004.02.013.  Google Scholar

[18]

H. M. Rodrigues and J. Solà-Morales, On the Hartman-Grobman Theorem with Parameters, J. Dyn. Differ. Equ., 22 (2010), 473-489.  doi: 10.1007/s10884-010-9160-7.  Google Scholar

[19]

H. M. Rodrigues and J. Solà-Morales, Invertible Contractions and Asymptotically Stable ODE's that are not $C^1$-Linearizable, J. Dyn. Differ. Equ., 18 (2006), 961-973.  doi: 10.1007/s10884-006-9050-1.  Google Scholar

[20]

H. M. Rodrigues and J. Solà-Morales, Smooth Linearization for a Saddle on Banach Spaces, J. Dyn. Differ. Equ., 16 (2004), 767-793.  doi: 10.1007/s10884-004-6116-9.  Google Scholar

[21]

H. M. Rodrigues, J. Solà-Morales, An example on Lyapunov stability and linearization, preprint, arXiv: 1902.02111, (2019). Google Scholar

show all references

References:
[1]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors of Infinite Dimensional Nonautonomous Dynamical Syustems, Springer-Verlag Berlin, 2011. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[2]

W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D. C. Heath & Co., Boston, 1965.  Google Scholar

[3]

W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag Berlin Heidelberg New York, 1970.  Google Scholar

[4]

Ju. L. Dalekĭi and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Translation of Mathematical Monographs, Vol. 43, American Mathematical Society, Providence, RI, 1974.  Google Scholar

[5]

A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathmatics, Vol. 377, Springer-Verlag, Berlin-Heidelberg-New York, 1974.  Google Scholar

[6]

P. R. Halmos, A Hilbert Space Problem Book, Graduate texts in Mathematics, Vol. 19, Springer-Verlag, New York, Heidelberg, Berlin, 1974.  Google Scholar

[7]

J. K. Hale, Ordinary Differential Equations, 2$^nd$ edition, Krieger Publishing Co., Huntington, New York, 1980.  Google Scholar

[8]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes Math., Vol 840, Springer-Verlag, Berlin, 1981.  Google Scholar

[9]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1980.  Google Scholar

[10]

P. E. Kloeden and H. M. Rodrigues, Dynamics of a Class of ODEs more general than almost periodic, Nonlinear Anal., 74 (2011), 2695-2719.  doi: 10.1016/j.na.2010.12.025.  Google Scholar

[11]

H. M. Rodrigues, Invariância para sistemas de equações diferenciais com retardamento e aplicações, Tese de Mestrado, Universidade de São Paulo (São Carlos), 1970. Google Scholar

[12]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York Berlin Heidelberg Tokyo, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[13]

Ch. E. Rickart, General Theory of Banach Algebras, Princeton, D. van Nostrand, 1960.  Google Scholar

[14]

H. M. RodriguesT. Caraballo and M. Gameiro, Dynamics of a class of ODEs via wavelets, Commun. Pure Appl. Anal., 16 (2017), 2337-2355.  doi: 10.3934/cpaa.2017115.  Google Scholar

[15]

H. M. RodriguesM. A. Teixeira and Ma. Gameiro, On exponential decay and the Markus-Yamabe conjecture in infinite dimensions with applications to the Cima system, J. Dyn. Differ. Equ., 30 (2018), 1199-1219.  doi: 10.1007/s10884-017-9598-y.  Google Scholar

[16]

H. M. Rodrigues and J. G. Ruas-Filho, Evolution equations: dichotomies and the Fredholm alternative for bounded solutions, J. Differ. Equ., 119 (1995), 263-283.  doi: 10.1006/jdeq.1995.1091.  Google Scholar

[17]

H. M. Rodrigues and J. Solà-Morales, Linearization of Class $C^1$ for Contractions on Banach Spaces, J. Differ. Equ., 201 (2004), 351-382.  doi: 10.1016/j.jde.2004.02.013.  Google Scholar

[18]

H. M. Rodrigues and J. Solà-Morales, On the Hartman-Grobman Theorem with Parameters, J. Dyn. Differ. Equ., 22 (2010), 473-489.  doi: 10.1007/s10884-010-9160-7.  Google Scholar

[19]

H. M. Rodrigues and J. Solà-Morales, Invertible Contractions and Asymptotically Stable ODE's that are not $C^1$-Linearizable, J. Dyn. Differ. Equ., 18 (2006), 961-973.  doi: 10.1007/s10884-006-9050-1.  Google Scholar

[20]

H. M. Rodrigues and J. Solà-Morales, Smooth Linearization for a Saddle on Banach Spaces, J. Dyn. Differ. Equ., 16 (2004), 767-793.  doi: 10.1007/s10884-004-6116-9.  Google Scholar

[21]

H. M. Rodrigues, J. Solà-Morales, An example on Lyapunov stability and linearization, preprint, arXiv: 1902.02111, (2019). Google Scholar

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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