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PreprintarXiv licencearXiv

Quantitative linear approximation for controllability of quasi-linear parabolic equations

  • Manish Kumar
  • Xu Liu
  • Xu Zhang
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Abstract

This paper studies the relationship between the controllability of a quasi-linear parabolic equation and that of its linear counterpart. Under suitable assumptions, we prove that for sufficiently small initial data both systems are null controllable, and that the differences between their corresponding controls and states satisfy a higher-order error estimate. Unlike standard controllability problems, the controls constructed here not only steer the system to a prescribed target at a given terminal time, but are also designed to ensure that the deviation between the nonlinear and linear dynamics obeys the aforementioned estimate. This method is applicable to general nonlinear partial differential equations, and the approximation order for controllability is optimal.

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Cite

BibTeXRIS
BibTeX
@misc{kumar2026quantitative,
  title = {{Quantitative linear approximation for controllability of quasi-linear parabolic equations}},
  author = {Manish Kumar and Xu Liu and Xu Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04969},
  eprint = {2609.04969},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04969},
}
RIS
TY  - UNPB
TI  - Quantitative linear approximation for controllability of quasi-linear parabolic equations
AU  - Manish Kumar
AU  - Xu Liu
AU  - Xu Zhang
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.04969
UR  - https://arxiv.org/abs/2609.04969
AB  - This paper studies the relationship between the controllability of a quasi-linear parabolic equation and that of its linear counterpart. Under suitable assumptions, we prove that for sufficiently small initial data both systems are null controllable, and that the differences between their corresponding controls and states satisfy a higher-order error estimate. Unlike standard controllability problems, the controls constructed here not only steer the system to a prescribed target at a given terminal time, but are also designed to ensure that the deviation between the nonlinear and linear dynamics obeys the aforementioned estimate. This method is applicable to general nonlinear partial differential equations, and the approximation order for controllability is optimal.
ER  -