A weak invariance principle and asymptotic stability for evolution equations with bounded generators
If V is a Lyapunov function of an equation du/dt=u′=Zu in a Banach space then asymptotic stability of an equilibrium point may be easily proved if it is known that sup(V′)<0 on sufficiently small spheres centered at the equilibrium point. In this paper weak asymptotic stability is proved for a bo...
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| Format: | Article |
| Language: | English |
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Wiley
1995-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171295000317 |
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| author | E. N. Chukwu P. Smoczynski |
| author_facet | E. N. Chukwu P. Smoczynski |
| author_sort | E. N. Chukwu |
| collection | DOAJ |
| description | If V is a Lyapunov function of an equation du/dt=u′=Zu in a Banach space then
asymptotic stability of an equilibrium point may be easily proved if it is known that sup(V′)<0 on
sufficiently small spheres centered at the equilibrium point. In this paper weak asymptotic stability is
proved for a bounded infinitesimal generator Z under a weaker assumption V′≤0 (which alone
implies ordinary stability only) if some observability condition, involving Z and the Frechet derivative
of V′, is satisfied. The proof is based on an extension of LaSalle's invariance principle, which yields
convergence in a weak topology and uses a strongly continuous Lyapunov function. The theory is
illustrated with an example of an integro-differential equation of interest in the theory of chemical
processes. In this case strong asymptotic stability is deduced from the weak one and explicit sufficient
conditions for stability are given. In the case of a normal infinitesimal generator Z in a Hilbert
space, strong asymptotic stability is proved under the following assumptions Z*+Z is weakly
negative definite and Ker Z={0}. The proof is based on spectral theory. |
| format | Article |
| id | doaj-art-c55666743fd5428a9725b2f1cbc79fa9 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1995-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-c55666743fd5428a9725b2f1cbc79fa92025-02-03T01:03:39ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251995-01-0118225526410.1155/S0161171295000317A weak invariance principle and asymptotic stability for evolution equations with bounded generatorsE. N. Chukwu0P. Smoczynski1Department of Mathematics, Box 8205, North Carolina State University, Raleigh 27695-8205, N. C., USADepartment of Mathematics and Statistics, Simon Fraser University, B.C., Burnaby V5A 1S6, CanadaIf V is a Lyapunov function of an equation du/dt=u′=Zu in a Banach space then asymptotic stability of an equilibrium point may be easily proved if it is known that sup(V′)<0 on sufficiently small spheres centered at the equilibrium point. In this paper weak asymptotic stability is proved for a bounded infinitesimal generator Z under a weaker assumption V′≤0 (which alone implies ordinary stability only) if some observability condition, involving Z and the Frechet derivative of V′, is satisfied. The proof is based on an extension of LaSalle's invariance principle, which yields convergence in a weak topology and uses a strongly continuous Lyapunov function. The theory is illustrated with an example of an integro-differential equation of interest in the theory of chemical processes. In this case strong asymptotic stability is deduced from the weak one and explicit sufficient conditions for stability are given. In the case of a normal infinitesimal generator Z in a Hilbert space, strong asymptotic stability is proved under the following assumptions Z*+Z is weakly negative definite and Ker Z={0}. The proof is based on spectral theory.http://dx.doi.org/10.1155/S0161171295000317asymptotic stabilityinvariance principleLyapunov functions. |
| spellingShingle | E. N. Chukwu P. Smoczynski A weak invariance principle and asymptotic stability for evolution equations with bounded generators International Journal of Mathematics and Mathematical Sciences asymptotic stability invariance principle Lyapunov functions. |
| title | A weak invariance principle and asymptotic stability for evolution equations with bounded generators |
| title_full | A weak invariance principle and asymptotic stability for evolution equations with bounded generators |
| title_fullStr | A weak invariance principle and asymptotic stability for evolution equations with bounded generators |
| title_full_unstemmed | A weak invariance principle and asymptotic stability for evolution equations with bounded generators |
| title_short | A weak invariance principle and asymptotic stability for evolution equations with bounded generators |
| title_sort | weak invariance principle and asymptotic stability for evolution equations with bounded generators |
| topic | asymptotic stability invariance principle Lyapunov functions. |
| url | http://dx.doi.org/10.1155/S0161171295000317 |
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