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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2026
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| Accès en ligne: | https://arxiv.org/abs/2605.26993 |
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| _version_ | 1866911720261287936 |
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| author | Cao, Xiao-Dong Xu, Chao-Jiang Xu, Yan |
| author_facet | Cao, Xiao-Dong Xu, Chao-Jiang Xu, Yan |
| contents | We prove backward uniqueness for a class of ultraparabolic operators with coupled linear drift. The main difficulty is that the Fourier transform in the degenerate variables turns the coupled drift into a transport operator in the dual frequency variables, so the classical Littlewood--Paley Carleman argument does not apply directly. We overcome this by introducing an invariant frequency variable and establishing a frequency-localized Carleman estimate adapted to the transport structure. The result gives a partial answer to the question of W. Wang and L. Zhang $\left[ \emph {Methods Appl. Anal.}, \ 20 \ (1) \ (2013) \ 79-88 \right]$ for constant coupled drift, with diffusion and lower-order coefficients depending on time and the diffusive variables. As an application, for a jerk-driven control model, we prove backward uniqueness for the equation describing the position, velocity, acceleration, or jerk error: under bounded lower-order coefficients, zero final error in $L^2$ implies zero error at all earlier times. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26993 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Backward Uniqueness for Coupled Ultraparabolic Operators and an Application to Jerk-Driven Control Models Cao, Xiao-Dong Xu, Chao-Jiang Xu, Yan Analysis of PDEs 35K70, 35A02 We prove backward uniqueness for a class of ultraparabolic operators with coupled linear drift. The main difficulty is that the Fourier transform in the degenerate variables turns the coupled drift into a transport operator in the dual frequency variables, so the classical Littlewood--Paley Carleman argument does not apply directly. We overcome this by introducing an invariant frequency variable and establishing a frequency-localized Carleman estimate adapted to the transport structure. The result gives a partial answer to the question of W. Wang and L. Zhang $\left[ \emph {Methods Appl. Anal.}, \ 20 \ (1) \ (2013) \ 79-88 \right]$ for constant coupled drift, with diffusion and lower-order coefficients depending on time and the diffusive variables. As an application, for a jerk-driven control model, we prove backward uniqueness for the equation describing the position, velocity, acceleration, or jerk error: under bounded lower-order coefficients, zero final error in $L^2$ implies zero error at all earlier times. |
| title | Backward Uniqueness for Coupled Ultraparabolic Operators and an Application to Jerk-Driven Control Models |
| topic | Analysis of PDEs 35K70, 35A02 |
| url | https://arxiv.org/abs/2605.26993 |