Approximate controllability of the wave equation with mixed boundary conditions
We consider initial boundary-value problem for acoustic equation in the time space cylinder Ω×(0, 2T) with unknown variable speed of sound, zero initial data, and mixed boundary conditions. We assume that (Neumann) controls are located at some part Σ × [0, T], Σ ⊂ ∂Ω of the lateral surface of the cy...
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| Опубліковано в: : | Український математичний вісник |
|---|---|
| Дата: | 2018 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2018
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/169401 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Approximate controllability of the wave equation with mixed boundary conditions / L. Pestov, D. Strelnikov // Український математичний вісник. — 2018. — Т. 15, № 2. — С. 251-263. — Бібліогр.: 14 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862546800908435456 |
|---|---|
| author | Pestov, L. Strelnikov, D. |
| author_facet | Pestov, L. Strelnikov, D. |
| citation_txt | Approximate controllability of the wave equation with mixed boundary conditions / L. Pestov, D. Strelnikov // Український математичний вісник. — 2018. — Т. 15, № 2. — С. 251-263. — Бібліогр.: 14 назв. — англ. |
| collection | DSpace DC |
| container_title | Український математичний вісник |
| description | We consider initial boundary-value problem for acoustic equation in the time space cylinder Ω×(0, 2T) with unknown variable speed of sound, zero initial data, and mixed boundary conditions. We assume that (Neumann) controls are located at some part Σ × [0, T], Σ ⊂ ∂Ω of the lateral surface of the cylinder Ω × (0, T). The domain of observation is Σ × [0, 2T], and the pressure on another part (∂Ω\Σ) × [0, 2T]) is assumed to be zero for any control. We prove the approximate boundary controllability for functions from the subspace V ⊂ H¹(Ω) whose traces have vanished on Σ provided that the observation time is 2T more than two acoustic radii of the domain Ω. We give an explicit procedure for solving Boundary Control Problem (BCP) for smooth harmonic functions from V (i.e., we are looking for a boundary control f which generates a wave uf such that uf (., T) approximates any prescribed harmonic function from V ). Moreover, using the Friedrichs–Poincar´e inequality, we obtain a conditional estimate for this BCP. Note that, for solving BCP for these harmonic functions, we do not need the knowledge of the speed of sound.
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| first_indexed | 2025-11-25T13:06:47Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-169401 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1810-3200 |
| language | English |
| last_indexed | 2025-11-25T13:06:47Z |
| publishDate | 2018 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Pestov, L. Strelnikov, D. 2020-06-12T15:38:13Z 2020-06-12T15:38:13Z 2018 Approximate controllability of the wave equation with mixed boundary conditions / L. Pestov, D. Strelnikov // Український математичний вісник. — 2018. — Т. 15, № 2. — С. 251-263. — Бібліогр.: 14 назв. — англ. 1810-3200 2010 MSC. Primary 35R30; Secondary 35M33, 46E35 https://nasplib.isofts.kiev.ua/handle/123456789/169401 We consider initial boundary-value problem for acoustic equation in the time space cylinder Ω×(0, 2T) with unknown variable speed of sound, zero initial data, and mixed boundary conditions. We assume that (Neumann) controls are located at some part Σ × [0, T], Σ ⊂ ∂Ω of the lateral surface of the cylinder Ω × (0, T). The domain of observation is Σ × [0, 2T], and the pressure on another part (∂Ω\Σ) × [0, 2T]) is assumed to be zero for any control. We prove the approximate boundary controllability for functions from the subspace V ⊂ H¹(Ω) whose traces have vanished on Σ provided that the observation time is 2T more than two acoustic radii of the domain Ω. We give an explicit procedure for solving Boundary Control Problem (BCP) for smooth harmonic functions from V (i.e., we are looking for a boundary control f which generates a wave uf such that uf (., T) approximates any prescribed harmonic function from V ). Moreover, using the Friedrichs–Poincar´e inequality, we obtain a conditional estimate for this BCP. Note that, for solving BCP for these harmonic functions, we do not need the knowledge of the speed of sound. This work was supported by the Volkswagen Foundation project “Modeling, Analysis and Approximation Theory toward Applications in Tomography and Inverse Problem”. The authors would like also to thank M. Belishev, V. Derkach and T. Fastovska for useful discussions and valuable remarks. en Інститут прикладної математики і механіки НАН України Український математичний вісник Approximate controllability of the wave equation with mixed boundary conditions Article published earlier |
| spellingShingle | Approximate controllability of the wave equation with mixed boundary conditions Pestov, L. Strelnikov, D. |
| title | Approximate controllability of the wave equation with mixed boundary conditions |
| title_full | Approximate controllability of the wave equation with mixed boundary conditions |
| title_fullStr | Approximate controllability of the wave equation with mixed boundary conditions |
| title_full_unstemmed | Approximate controllability of the wave equation with mixed boundary conditions |
| title_short | Approximate controllability of the wave equation with mixed boundary conditions |
| title_sort | approximate controllability of the wave equation with mixed boundary conditions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/169401 |
| work_keys_str_mv | AT pestovl approximatecontrollabilityofthewaveequationwithmixedboundaryconditions AT strelnikovd approximatecontrollabilityofthewaveequationwithmixedboundaryconditions |