# American Institute of Mathematical Sciences

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

## Stability of nonautonomous equations and Lyapunov functions

 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.
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

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##### 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
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