Strong stability of bounded evolution families and semigroups
We prove several characterizations of strong stability of uniformly bounded evolution families (U(t, s))t≥s≥0 of bounded operators on a Banach space X, i.e. we characterize the property limt→∞U(t, s)x = 0 for all s ≥ 0 and all x ε X. These results are connected to the asymptotic stability of the wel...
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| Format: | Journal article |
| Language: | English |
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Elsevier
2002
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| _version_ | 1797078974872944640 |
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| author | Batty, C Chill, R Tomilov, Y |
| author_facet | Batty, C Chill, R Tomilov, Y |
| author_sort | Batty, C |
| collection | OXFORD |
| description | We prove several characterizations of strong stability of uniformly bounded evolution families (U(t, s))t≥s≥0 of bounded operators on a Banach space X, i.e. we characterize the property limt→∞U(t, s)x = 0 for all s ≥ 0 and all x ε X. These results are connected to the asymptotic stability of the well-posed linear nonautonomous Cauchy problem In the autonomous case, i.e. when U(t, s) = T(t - s) for some C0-semigroup (T(t))t≥0, we present, in addition, a range condition on the generator A of (T(t))t≥0 which is sufficient for strong stability. This condition is more general than the condition in the ABLV-Theorem involving countability of the imaginary part of the spectrum of A. © 2002 Elsevier Science (USA). |
| first_indexed | 2024-03-07T00:39:02Z |
| format | Journal article |
| id | oxford-uuid:826884f6-11c2-41f4-a616-8804ff95a2ab |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T00:39:02Z |
| publishDate | 2002 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:826884f6-11c2-41f4-a616-8804ff95a2ab2022-03-26T21:37:09ZStrong stability of bounded evolution families and semigroupsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:826884f6-11c2-41f4-a616-8804ff95a2abEnglishSymplectic Elements at OxfordElsevier2002Batty, CChill, RTomilov, YWe prove several characterizations of strong stability of uniformly bounded evolution families (U(t, s))t≥s≥0 of bounded operators on a Banach space X, i.e. we characterize the property limt→∞U(t, s)x = 0 for all s ≥ 0 and all x ε X. These results are connected to the asymptotic stability of the well-posed linear nonautonomous Cauchy problem In the autonomous case, i.e. when U(t, s) = T(t - s) for some C0-semigroup (T(t))t≥0, we present, in addition, a range condition on the generator A of (T(t))t≥0 which is sufficient for strong stability. This condition is more general than the condition in the ABLV-Theorem involving countability of the imaginary part of the spectrum of A. © 2002 Elsevier Science (USA). |
| spellingShingle | Batty, C Chill, R Tomilov, Y Strong stability of bounded evolution families and semigroups |
| title | Strong stability of bounded evolution families and semigroups |
| title_full | Strong stability of bounded evolution families and semigroups |
| title_fullStr | Strong stability of bounded evolution families and semigroups |
| title_full_unstemmed | Strong stability of bounded evolution families and semigroups |
| title_short | Strong stability of bounded evolution families and semigroups |
| title_sort | strong stability of bounded evolution families and semigroups |
| work_keys_str_mv | AT battyc strongstabilityofboundedevolutionfamiliesandsemigroups AT chillr strongstabilityofboundedevolutionfamiliesandsemigroups AT tomilovy strongstabilityofboundedevolutionfamiliesandsemigroups |