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
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show all references

References:
[1]

Encyclopedia of Math. and Its Appl., 115, Cambridge Univ. Press, 2007.  Google Scholar

[2]

Israel J. Math., 116 (2000), 29-70. doi: 10.1007/BF02773211.  Google Scholar

[3]

J. Dynam. Differential Equations, 19 (2007), 215-241. doi: 10.1007/s10884-006-9026-1.  Google Scholar

[4]

J. Differential Equations, 244 (2008), 2407-2447. doi: 10.1016/j.jde.2008.02.028.  Google Scholar

[5]

Lect. Notes in Math., 1926, Springer, 2008. doi: 10.1007/978-3-540-74775-8.  Google Scholar

[6]

Grundlehren der mathematischen Wissenschaften, 161, Springer, 1970.  Google Scholar

[7]

Translations of Mathematical Monographs, 43, Amer. Math. Soc., 1974.  Google Scholar

[8]

Grundlehren der mathematischen Wissenschaften, 138, Springer, 1967.  Google Scholar

[9]

Mathematics in Science and Engineering, 4, Academic Press, New York-London, 1961.  Google Scholar

[10]

Taylor and Francis, 1992.  Google Scholar

[11]

Pure and Applied Mathematics, 21, Academic Press, 1966.  Google Scholar

[12]

Trans. Moscow Math. Soc., 19 (1968), 197-221. Google Scholar

[13]

Ergodic Theory Dynam. Systems, 5 (1985), 145-161. doi: 10.1017/S0143385700002807.  Google Scholar

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