Advertisement

Journal of Dynamics and Differential Equations

, Volume 31, Issue 3, pp 1079–1106 | Cite as

Feedback Stabilization of Stem Growth

  • Fabio Ancona
  • Alberto BressanEmail author
  • Olivier Glass
  • Wen Shen
Article

Abstract

The paper studies a PDE model describing the elongation of a plant stem and its bending as a response to gravity. For a suitable range of parameters in the defining equations, it is proved that a feedback response produces stabilization of growth, in the vertical direction.

Keywords

Feedback stabilization First order PDE Stable linear semigroup Stem growth 

Notes

Acknowledgements

The first author was partially supported by the Istituto Nazionale di Alta Matematica “F.Severi”, through GNAMPA. The research of the second author was partially supported by NSF, with Grant DMS-1714237 “Models of controlled biological growth”. The third author thanks the Agence Nationale de la Recherche for its financial support, with Projects DYFICOLTI (Grant ANR-13-BS01-0003-01) and IFSMACS (Grant ANR-15-CE40-0010). This work was initiated while the first and third authors were visiting the Department of Mathematics at Penn State University, which they thank for the kind hospitality.

References

  1. 1.
    Bressan, A., Palladino, M., Shen, W.: Growth models for tree stems and vines. J. Differ. Equ. 263, 2280–2316 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bressan, A., Palladino, M.: Well-posedness of a model for the growth of tree stems and vines. Discrete Contin. Dyn. Syst., to appearGoogle Scholar
  3. 3.
    Evans, L.C.: Partial Differential Equations, 2nd edn. American Mathematical Society, Providence (2010)zbMATHGoogle Scholar
  4. 4.
    Hale, J.: Theory of Functional Differential Equations, 2nd edn. Springer, New York (1977)CrossRefzbMATHGoogle Scholar
  5. 5.
    Leyser, O., Day, S.: Mechanisms in Plant Development. Blackwell Publishing, Hoboken (2003)Google Scholar
  6. 6.
    Martin Jr., R.: Functional Analysis and Differential Equations in Banach Spaces. Wiley, Hoboken (1975)Google Scholar
  7. 7.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  8. 8.
    Sell, G., You, Y.: Dynamics of Evolutionary Equations. Springer, New York (2002)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  • Fabio Ancona
    • 1
  • Alberto Bressan
    • 2
    Email author
  • Olivier Glass
    • 3
  • Wen Shen
    • 2
  1. 1.Dipartimento di MatematicaUniversità di PadovaPaduaItaly
  2. 2.Department of MathematicsPenn State UniversityUniversity ParkUSA
  3. 3.CEREMADEUniversité Paris-DauphineParisFrance

Personalised recommendations