Mutual spectrum of commutating conjugate operators and correctness and stability criteria for differential-operator equations
Tests are established for the propriety of the Cauchy problem, the stability, stabilization, asymptotic stability, and exponential stability for the equation $y'' + By' + By = 0, t \in [0, +\infty)$, where $A$ and $B$ are arbitrary commuting self-adjoint operators on a sep...
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| Date: | 1991 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian |
| Published: |
Institute of Mathematics, NAS of Ukraine
1991
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/9623 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860513543291404288 |
|---|---|
| author | Shklyar , A. Ya. Шкляр , А. Я. |
| author_facet | Shklyar , A. Ya. Шкляр , А. Я. |
| author_sort | Shklyar , A. Ya. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-10-09T12:05:28Z |
| description | Tests are established for the propriety of the Cauchy problem, the stability, stabilization, asymptotic stability, and exponential stability for the equation $y'' + By' + By = 0, t \in [0, +\infty)$, where $A$ and $B$ are arbitrary commuting self-adjoint operators on a separable Hilbert space. For this, in terms of the arrangement in $R^2$ of the joint spectrum of $A$ and $B$ tests are obtained for $B$ to be subordinate (strongly subordinate, equivalent) to $A$. The results on propriety and stability are illustrated by the example of model partial differential equations. |
| first_indexed | 2026-03-24T03:46:21Z |
| format | Article |
| fulltext |
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| id | umjimathkievua-article-9623 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T03:46:21Z |
| publishDate | 1991 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b4/a23bd1aedd03ead82cd1223e61372ab4 |
| spelling | umjimathkievua-article-96232025-10-09T12:05:28Z Mutual spectrum of commutating conjugate operators and correctness and stability criteria for differential-operator equations Совместный спектр коммутирующих самосопряженных операторов и критерии корректности и устойчивости для дифференциально-операторных уравнений Shklyar , A. Ya. Шкляр , А. Я. - Tests are established for the propriety of the Cauchy problem, the stability, stabilization, asymptotic stability, and exponential stability for the equation $y'' + By' + By = 0, t \in [0, +\infty)$, where $A$ and $B$ are arbitrary commuting self-adjoint operators on a separable Hilbert space. For this, in terms of the arrangement in $R^2$ of the joint spectrum of $A$ and $B$ tests are obtained for $B$ to be subordinate (strongly subordinate, equivalent) to $A$. The results on propriety and stability are illustrated by the example of model partial differential equations. Установлены критерии корректности задачи Коши, устойчивости, стабилизации, асимптотической устойчивости, экспоненциальной устойчивости для уравнения $y'' + By' + By = 0, t \in [0, +\infty)$, где $A, B$ — произвольные коммутирующие самосопряженные операторы в сепарабельном гильбертовом пространстве. Для этого в терминах размещения в $R^2$ совместного спектра $A, B$ получены критерии того, что $B$ подчинен (сильно подчинен, эквивалентен) $A$. Результаты о корректности и устойчивости проиллюстрированы на примере модельных уравнений с частными производными. Установлено критерії коректності задачі Коші, стійкості, стабілізації, асимптотично стійкості, експоненціальної стійкості для рівняння $y'' + By' + By = 0, t \in [0, +\infty)$, де $A, B$ — довільні самоспряжені оператори в гільбертовому просторі, які комутують. Для цього в термінах розміщення в $R^2$ спільного спектра $A, B$ одержано критерії того, що $B$ є підпорядкованим (сильно підпорядкованим, еквівалентним) $A$. Результати про коректність стійкість проілюстровано прикладом модельних рівнянь з частинними похідними. Institute of Mathematics, NAS of Ukraine 1991-02-28 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/9623 Ukrains’kyi Matematychnyi Zhurnal; Vol. 43 No. 3 (1991); 406-414 Український математичний журнал; Том 43 № 3 (1991); 406-414 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/9623/10591 Copyright (c) 1991 А. Я. Шкляр |
| spellingShingle | Shklyar , A. Ya. Шкляр , А. Я. Mutual spectrum of commutating conjugate operators and correctness and stability criteria for differential-operator equations |
| title | Mutual spectrum of commutating conjugate operators and correctness and stability criteria for differential-operator equations |
| title_alt | Совместный спектр коммутирующих самосопряженных операторов и критерии корректности и устойчивости для дифференциально-операторных уравнений |
| title_full | Mutual spectrum of commutating conjugate operators and correctness and stability criteria for differential-operator equations |
| title_fullStr | Mutual spectrum of commutating conjugate operators and correctness and stability criteria for differential-operator equations |
| title_full_unstemmed | Mutual spectrum of commutating conjugate operators and correctness and stability criteria for differential-operator equations |
| title_short | Mutual spectrum of commutating conjugate operators and correctness and stability criteria for differential-operator equations |
| title_sort | mutual spectrum of commutating conjugate operators and correctness and stability criteria for differential-operator equations |
| topic_facet | - |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/9623 |
| work_keys_str_mv | AT shklyaraya mutualspectrumofcommutatingconjugateoperatorsandcorrectnessandstabilitycriteriafordifferentialoperatorequations AT šklâraâ mutualspectrumofcommutatingconjugateoperatorsandcorrectnessandstabilitycriteriafordifferentialoperatorequations AT shklyaraya sovmestnyjspektrkommutiruûŝihsamosoprâžennyhoperatorovikriteriikorrektnostiiustojčivostidlâdifferencialʹnooperatornyhuravnenij AT šklâraâ sovmestnyjspektrkommutiruûŝihsamosoprâžennyhoperatorovikriteriikorrektnostiiustojčivostidlâdifferencialʹnooperatornyhuravnenij |