Sharp estimates for biorthogonal families to exponential functions associated to complex sequences without gap conditions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914659557179392 |
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| author | González-Burgos, Manuel Ouaili, Lydia |
| author_facet | González-Burgos, Manuel Ouaili, Lydia |
| contents | The general goal of this work is to obtain upper and lower bounds for the $L^2$-norm of biorthogonal families to complex exponential functions associated to sequences $\{ Λ_k \}_{k \ge 1} \subset \mathbb C$ which satisfy appropriate assumptions but without imposing a gap condition on the elements of the sequence. As a consequence, we also present new results on the cost of the boundary null controllability of two parabolic systems at time $T > 0$: a phase-field system and a parabolic system whose generator has eigenvalues that accumulate. In the latter case, the behavior of the control cost when $T$ goes to zero depends strongly on the accumulation parameter of the eigenvalue sequence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17128 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp estimates for biorthogonal families to exponential functions associated to complex sequences without gap conditions González-Burgos, Manuel Ouaili, Lydia Optimization and Control Functional Analysis 35K40, 35Q93, 93B05, 93B60, 93C20 The general goal of this work is to obtain upper and lower bounds for the $L^2$-norm of biorthogonal families to complex exponential functions associated to sequences $\{ Λ_k \}_{k \ge 1} \subset \mathbb C$ which satisfy appropriate assumptions but without imposing a gap condition on the elements of the sequence. As a consequence, we also present new results on the cost of the boundary null controllability of two parabolic systems at time $T > 0$: a phase-field system and a parabolic system whose generator has eigenvalues that accumulate. In the latter case, the behavior of the control cost when $T$ goes to zero depends strongly on the accumulation parameter of the eigenvalue sequence. |
| title | Sharp estimates for biorthogonal families to exponential functions associated to complex sequences without gap conditions |
| topic | Optimization and Control Functional Analysis 35K40, 35Q93, 93B05, 93B60, 93C20 |
| url | https://arxiv.org/abs/2401.17128 |