Applied Mathematics and Optimization

, Volume 57, Issue 1, pp 98–124 | Cite as

Geometric Aspects of Force Controllability for a Swimming Model

  • A. Y. Khapalov


We study controllability properties (swimming capabilities) of a mathematical model of an abstract object which “swims” in the 2-D Stokes fluid. Our goal is to investigate how the geometric shape of this object affects the forces acting upon it. Such problems are of interest in biology and engineering applications dealing with propulsion systems in fluids.


Stokes equation Swimming model Coupled systems Multiplicative control Controllability 


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  1. 1.
    Ball, J.M., Mardsen, J.E., Slemrod, M.: Controllability for distributed bilinear systems. SIAM J. Control Opt. 20, 575–597 (1982) MATHCrossRefGoogle Scholar
  2. 2.
    Childress, S.: Mechanics of Swimming and Flying. Cambridge University Press, Cambridge (1981) MATHGoogle Scholar
  3. 3.
    Fauci, L.J.: Computational modeling of the swimming of biflagellated algal cells. Contemp. Math. 141, 91–102 (1993) MathSciNetGoogle Scholar
  4. 4.
    Fauci, L.J., Peskin, C.S.: A computational model of aquatic animal locomotion. J. Comp. Phys. 77, 85–108 (1988) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hirose, S.: Biologically Inspired Robots: Snake-like Locomotors and Manipulators. Oxford University Press, Oxford (1993) Google Scholar
  6. 6.
    Khapalov, A.Y.: Bilinear control for global controllability of the semilinear parabolic equations with superlinear terms, the Special volume “Control of Nonlinear Distributed Parameter Systems,” dedicated to D. Russell, G. Chen, I. Lasiecka, J. Zhou (eds.) pp. 139–155. Dekker (2001) Google Scholar
  7. 7.
    Khapalov, A.Y.: Global non-negative controllability of the semilinear parabolic equation governed by bilinear control. ESAIM: Contrôl. Optim. Calc. Var. 7, 269–283 (2002) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Khapalov, A.Y.: Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: a qualitative approach. SIAM J. Control Optim. 41, 1886–1900 (2003) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Khapalov, A.Y.: Controllability properties of a vibrating string with variable axial load. Discret. Cont. Dyn. Syst. 11, 311–324 (2004) MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Khapalov, A.Y.: The well-posedness of a model of an apparatus swimming in the 2-D Stokes fluid, preprint. Available as Techn. Rep. 2005-5, Washington State University, Department of Mathematics,
  11. 11.
    Khapalov, A.Y.: Local controllability for a “swimming” model. SIAM J. Control Opt. 46, 655–682 (2007) CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Ladyzhenskaya, O.H.: The Mathematical Theory of Viscous Incompressible Flow. Cordon and Breach, New York (1963) MATHGoogle Scholar
  13. 13.
    Lighthill, M.J.: Mathematics of Biofluiddynamics. Society for Industrial and Applied Mathematics, Philadelphia (1975) Google Scholar
  14. 14.
    McIsaac, K.A., Ostrowski, J.P.: Motion planning for dynamic eel-like robots. In Proc. IEEE Int. Conf. Robotics and Automation, San Francisco, pp. 1695–1700 (2000) Google Scholar
  15. 15.
    Martinez, S., Cortés, J.: Geometric control of robotic locomotion systems, In Proc. X Fall Workshop on Geometry and Physics, Madrid, 2001, Publ. de la RSME, vol. 4, pp. 183–198 (2001) Google Scholar
  16. 16.
    Peskin, C.S.: Numerical analysis of blood flow in the heart. J. Comp. Phys. 25, 220–252 (1977) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Peskin, C.S., McQueen, D.M.: A general method for the computer simulation of biological systems interacting with fluids. SEB Symposium on biological fluid dynamics, Leeds, England, 5–8 July 1994 Google Scholar
  18. 18.
    Symon, K.R.: Mechanics. Addison–Wesley, Reading (1971) Google Scholar
  19. 19.
    Temam, R.: Navier-Stokes Equations. North-Holland, Amsterdam (1984) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of MathematicsWashington State UniversityPullmanUSA

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