American Institute of Mathematical Sciences

Advanced
July  2013, 33(7): 2631-2650. doi: 10.3934/dcds.2013.33.2631

Stability of nonautonomous equations and Lyapunov functions

Luis Barreira 1, and  Claudia Valls 2,

1. 

Departamento de Matemática, Instituto Superior Técnico, UTL, 1049-001 Lisboa
2. 

Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa

Received  May 2011 Revised  November 2012 Published  January 2013

We consider nonautonomous linear equations $x'=A(t)x$ in a Banach space, and we give a complete characterization of those admitting nonuniform exponential contractions in terms of strict Lyapunov functions. The uniform contractions are a very particular case of nonuniform exponential contractions. In addition, we establish ``inverse theorems'' that give explicitly a strict Lyapunov function for each nonuniform contraction. These functions are constructed in terms of Lyapunov norms, which transform the nonuniform behavior of the contraction into a uniform exponential behavior. Moreover, we use the characterization of nonuniform exponential contractions in terms of strict Lyapunov functions to establish in a very simple manner, in comparison with former works, the persistence of the asymptotic stability under sufficiently small linear and nonlinear perturbations.
Keywords: nonuniform exponential contractions., Lyapunov functions.
Mathematics Subject Classification: Primary: 37D99, 93D9.
Citation: Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631
References:
[1]

L. Barreira and Ya. Pesin, "Nonuniform Hyperbolicity," Encyclopedia of Math. and Its Appl., 115, Cambridge Univ. Press, 2007.  Google Scholar

[2]

L. Barreira and J. Schmeling, Sets of "non-typical" points have full topological entropy and full Hausdorff dimension, Israel J. Math., 116 (2000), 29-70. doi: 10.1007/BF02773211.  Google Scholar

[3]

L. Barreira and C. Valls, Nonuniform exponential dichotomies and Lyapunov regularity, J. Dynam. Differential Equations, 19 (2007), 215-241. doi: 10.1007/s10884-006-9026-1.  Google Scholar

[4]

L. Barreira and C. Valls, Robustness of nonuniform exponential dichotomies in Banach spaces, J. Differential Equations, 244 (2008), 2407-2447. doi: 10.1016/j.jde.2008.02.028.  Google Scholar

[5]

L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations," Lect. Notes in Math., 1926, Springer, 2008. doi: 10.1007/978-3-540-74775-8.  Google Scholar

[6]

N. Bhatia and G. Szegö, "Stability Theory of Dynamical Systems," Grundlehren der mathematischen Wissenschaften, 161, Springer, 1970.  Google Scholar

[7]

Ju. Dalec$'$kiĭ and M. Kreĭn, "Stability of Solutions of Differential Equations in Banach Space," Translations of Mathematical Monographs, 43, Amer. Math. Soc., 1974.  Google Scholar

[8]

W. Hahn, "Stability of Motion," Grundlehren der mathematischen Wissenschaften, 138, Springer, 1967.  Google Scholar

[9]

J. LaSalle and S. Lefschetz, "Stability by Liapunov's Direct Method, with Applications," Mathematics in Science and Engineering, 4, Academic Press, New York-London, 1961.  Google Scholar

[10]

A. Lyapunov, "The General Problem of the Stability of Motion," Taylor and Francis, 1992.  Google Scholar

[11]

J. Massera and J. Schäffer, "Linear Differential Equations and Function Spaces," Pure and Applied Mathematics, 21, Academic Press, 1966.  Google Scholar

[12]

V. Oseledets, A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-221. Google Scholar

[13]

M. Wojtkowski, Invariant families of cones and Lyapunov exponents, Ergodic Theory Dynam. Systems, 5 (1985), 145-161. doi: 10.1017/S0143385700002807.  Google Scholar

show all references

References:
[1]

L. Barreira and Ya. Pesin, "Nonuniform Hyperbolicity," Encyclopedia of Math. and Its Appl., 115, Cambridge Univ. Press, 2007.  Google Scholar

[2]

L. Barreira and J. Schmeling, Sets of "non-typical" points have full topological entropy and full Hausdorff dimension, Israel J. Math., 116 (2000), 29-70. doi: 10.1007/BF02773211.  Google Scholar

[3]

L. Barreira and C. Valls, Nonuniform exponential dichotomies and Lyapunov regularity, J. Dynam. Differential Equations, 19 (2007), 215-241. doi: 10.1007/s10884-006-9026-1.  Google Scholar

[4]

L. Barreira and C. Valls, Robustness of nonuniform exponential dichotomies in Banach spaces, J. Differential Equations, 244 (2008), 2407-2447. doi: 10.1016/j.jde.2008.02.028.  Google Scholar

[5]

L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations," Lect. Notes in Math., 1926, Springer, 2008. doi: 10.1007/978-3-540-74775-8.  Google Scholar

[6]

N. Bhatia and G. Szegö, "Stability Theory of Dynamical Systems," Grundlehren der mathematischen Wissenschaften, 161, Springer, 1970.  Google Scholar

[7]

Ju. Dalec$'$kiĭ and M. Kreĭn, "Stability of Solutions of Differential Equations in Banach Space," Translations of Mathematical Monographs, 43, Amer. Math. Soc., 1974.  Google Scholar

[8]

W. Hahn, "Stability of Motion," Grundlehren der mathematischen Wissenschaften, 138, Springer, 1967.  Google Scholar

[9]

J. LaSalle and S. Lefschetz, "Stability by Liapunov's Direct Method, with Applications," Mathematics in Science and Engineering, 4, Academic Press, New York-London, 1961.  Google Scholar

[10]

A. Lyapunov, "The General Problem of the Stability of Motion," Taylor and Francis, 1992.  Google Scholar

[11]

J. Massera and J. Schäffer, "Linear Differential Equations and Function Spaces," Pure and Applied Mathematics, 21, Academic Press, 1966.  Google Scholar

[12]

V. Oseledets, A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-221. Google Scholar

[13]

M. Wojtkowski, Invariant families of cones and Lyapunov exponents, Ergodic Theory Dynam. Systems, 5 (1985), 145-161. doi: 10.1017/S0143385700002807.  Google Scholar

[1]

Luis Barreira, Claudia Valls. Nonuniform exponential dichotomies and admissibility. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 39-53. doi: 10.3934/dcds.2011.30.39
[2]

Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219
[3]

Luis Barreira, Claudia Valls. Admissibility versus nonuniform exponential behavior for noninvertible cocycles. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1297-1311. doi: 10.3934/dcds.2013.33.1297
[4]

Luis Barreira, Claudia Valls. Characterization of stable manifolds for nonuniform exponential dichotomies. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1025-1046. doi: 10.3934/dcds.2008.21.1025
[5]

Peter Giesl, Sigurdur Hafstein. Computational methods for Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : i-ii. doi: 10.3934/dcdsb.2015.20.8i
[6]

Peter Giesl, Sigurdur Hafstein. Review on computational methods for Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2291-2331. doi: 10.3934/dcdsb.2015.20.2291
[7]

Sigurdur Freyr Hafstein. A constructive converse Lyapunov theorem on exponential stability. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 657-678. doi: 10.3934/dcds.2004.10.657
[8]

Ezzeddine Zahrouni. On the Lyapunov functions for the solutions of the generalized Burgers equation. Communications on Pure & Applied Analysis, 2003, 2 (3) : 391-410. doi: 10.3934/cpaa.2003.2.391
[9]

Gunther Dirr, Hiroshi Ito, Anders Rantzer, Björn S. Rüffer. Separable Lyapunov functions for monotone systems: Constructions and limitations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2497-2526. doi: 10.3934/dcdsb.2015.20.2497
[10]

C. Connell Mccluskey. Lyapunov functions for tuberculosis models with fast and slow progression. Mathematical Biosciences & Engineering, 2006, 3 (4) : 603-614. doi: 10.3934/mbe.2006.3.603
[11]

Tomoharu Suda. Construction of Lyapunov functions using Helmholtz–Hodge decomposition. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2437-2454. doi: 10.3934/dcds.2019103
[12]

Peter Giesl, Boumediene Hamzi, Martin Rasmussen, Kevin Webster. Approximation of Lyapunov functions from noisy data. Journal of Computational Dynamics, 2020, 7 (1) : 57-81. doi: 10.3934/jcd.2020003
[13]

Arnaud Goullet, Shaun Harker, Konstantin Mischaikow, William D. Kalies, Dinesh Kasti. Efficient computation of Lyapunov functions for Morse decompositions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2419-2451. doi: 10.3934/dcdsb.2015.20.2419
[14]

Jóhann Björnsson, Peter Giesl, Sigurdur F. Hafstein, Christopher M. Kellett. Computation of Lyapunov functions for systems with multiple local attractors. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4019-4039. doi: 10.3934/dcds.2015.35.4019
[15]

Peter Giesl, Sigurdur Hafstein. Existence of piecewise linear Lyapunov functions in arbitrary dimensions. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3539-3565. doi: 10.3934/dcds.2012.32.3539
[16]

Frédéric Grognard, Frédéric Mazenc, Alain Rapaport. Polytopic Lyapunov functions for persistence analysis of competing species. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 73-93. doi: 10.3934/dcdsb.2007.8.73
[17]

Volodymyr Pichkur. On practical stability of differential inclusions using Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1977-1986. doi: 10.3934/dcdsb.2017116
[18]

Connell McCluskey. Lyapunov functions for disease models with immigration of infected hosts. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4479-4491. doi: 10.3934/dcdsb.2020296
[19]

Lars Grüne. Computing Lyapunov functions using deep neural networks. Journal of Computational Dynamics, 2021, 8 (2) : 131-152. doi: 10.3934/jcd.2021006
[20]

Peter Giesl, Sigurdur Hafstein. System specific triangulations for the construction of CPA Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6027-6046. doi: 10.3934/dcdsb.2020378

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy