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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

Hildebrando M. Rodrigues 1,, , J. Solà-Morales 2, and  G. K. Nakassima 1,

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]).

Keywords: Robustness of stability, generalised almost periodic functions, integrally small perturbation.
Mathematics Subject Classification: 34G20, 35B15, 34D30, 34E10.
Citation: Hildebrando M. Rodrigues, J. Solà-Morales, G. K. Nakassima. Stability problems in nonautonomous linear differential equations in infinite dimensions. Communications on Pure and 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.

[2]

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

[3]

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

[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.

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[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.

[11]

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

[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.

[13]

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

[14]

H. M. Rodrigues, T. 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.

[15]

H. M. Rodrigues, M. 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.

[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.

[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.

[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.

[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.

[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.

[21]

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

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.

[2]

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

[3]

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

[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.

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[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.

[11]

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

[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.

[13]

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

[14]

H. M. Rodrigues, T. 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.

[15]

H. M. Rodrigues, M. 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.

[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.

[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.

[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.

[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.

[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.

[21]

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

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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