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Pontryagin-Type Conditions for Optimal Muscular Force Response to Functional Electrical Stimulations

  • Toufik Bakir
  • Bernard Bonnard
  • Loïc Bourdin
  • Jérémy RouotEmail author
Article

Abstract

In biomechanics, recent mathematical models allow one to predict the muscular force response to functional electrical stimulations. The main concern of the present paper is to deal with the computation of optimized electrical pulses trains (for example in view of maximizing the final force response). Using the fact that functional electrical stimulations are modeled as Dirac pulses, our problem is rewritten as an optimal sampled-data control problem, where the control parameters are the pulses amplitudes and the pulses times. We establish the corresponding Pontryagin first-order necessary optimality conditions and we show how they can be used in view of numerical simulations.

Keywords

Functional electrical stimulation Muscle mechanics Optimal control problems Sampled-data controls Pontryagin-type necessary optimality conditions 

Mathematics Subject Classification

49K15 93B07 92B05 

Notes

Acknowledgements

This research paper benefited from the support of the FMJH Program PGMO and from the support of EDF, Thales, Orange. T. Bakir, B. Bonnard and J. Rouot are partially supported by the Labex AMIES.

References

  1. 1.
    Law, L.F., Shields, R.: Mathematical models of human paralyzed muscle after long-term training. J. Biomech. 40, 2587–2595 (2007)CrossRefGoogle Scholar
  2. 2.
    Ding, J., Binder-Macleod, S.A., Wexler, A.S.: Two-step, predictive, isometric force model tested on data from human and rat muscles. J. Appl. Physiol. 85, 2176–2189 (1998)CrossRefGoogle Scholar
  3. 3.
    Ding, J., Wexler, A.S., Binder-Macleod, S.A.: Development of a mathematical model that predicts optimal muscle activation patterns by using brief trains. J. Appl. Physiol. 88, 917–925 (2000)CrossRefGoogle Scholar
  4. 4.
    Gesztelyi, R., Zsuga, J., Kemeny-Beke, A., Varga, B., Juhasz, B., Tosaki, A.: The Hill equation and the origin of quantitative pharmacology. Arch. Hist. Exact Sci. 66(4), 427–438 (2012)CrossRefGoogle Scholar
  5. 5.
    Ding, J., Wexler, A.S., Binder-Macleod, S.A.: A predictive model of fatigue in human skeletal muscles. J. Appl. Physiol. 89, 1322–1332 (2000)CrossRefGoogle Scholar
  6. 6.
    Ding, J., Wexler, A.S., Binder-Macleod, S.A.: Mathematical models for fatigue minimization during functional electrical stimulation. J. Electromyogr. Kinesiol. 13, 575–588 (2003)CrossRefGoogle Scholar
  7. 7.
    Wilson, E.: Force response of locust skeletal muscle. Southampton University, Ph.D. thesis (2011)Google Scholar
  8. 8.
    Bourdin, L., Trélat, E.: Optimal sampled-data control, and generalizations on time scales. Math. Cont. Relat. Fields 6, 53–94 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bourdin, L., Trélat, E.: Pontryagin maximum principle for optimal sampled-data control problems. In: Proceedings of 16th IFAC Workshop on Control Applications of Optimization CAO’2015 (2015)CrossRefGoogle Scholar
  10. 10.
    Bourdin, L., Dhar, G.: Continuity/constancy of the Hamiltonian function in a Pontryagin maximum principle for optimal sampled-data control problems with free sampling times. Math. Control Signals Syst. (2019).  https://doi.org/10.1007/s00498-019-00247-6 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bonnans, F., Giorgi, D., Maindrault, S., Martinon, P., Grélard, V.: Bocop—a collection of examples. Inria Research Report, Project-Team Commands, 8053 (2014)Google Scholar
  12. 12.
    Cots, O.: Contrôle optimal géométrique : méthodes homotopiques et applications. Ph.D. thesis, Université de Bourgogne, Dijon (2012)Google Scholar
  13. 13.
    Bakir, T., Bonnard, B., Rouot, J.: A case study of optimal input–output system with sampled-data control: Ding et al. force and fatigue muscular control model. Netw. Heterog. Med. 14(1), 79–100 (2019)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Yochum, M.: Contribution à la conception d’un électromyostimulateur intelligent. Thèse de doctorat, Instrumentation et informatique de l’image Dijon (2013)Google Scholar
  15. 15.
    Bakir, T.: Contribution à la modélisation, l’estimation et la commande de systèmes non linéaires dans les domaines de la cristallisation et de l’électrostimulation musculaire. HDR Université de Bourgogne (2018)Google Scholar
  16. 16.
    Gelfand, I.M., Fomin, S.V.: Calculus of Variations. Prentice Hall Inc., Englewood Cliffs (1963)zbMATHGoogle Scholar
  17. 17.
    Schättler, H., Ledzewicz, U.: Geometric Optimal Control. Theory, Methods and Examples. Interdisciplinary Applied Mathematics, vol. 38. Springer, New York (2012)CrossRefGoogle Scholar
  18. 18.
    Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962)zbMATHGoogle Scholar
  19. 19.
    Trélat, E.: Contrôle Optimal: théorie et Applications. Vuibert, Paris (2005)zbMATHGoogle Scholar
  20. 20.
    Vinter, R.: Optimal control. Systems & Control: Foundations and Applications. Birkhäuser, Boston (2000)zbMATHGoogle Scholar
  21. 21.
    Rackauckas, C., Nie, Q.: Differentialequations.jl—a performant and feature-rich ecosystem for solving differential equations in julia. J. Open Res. Softw 5, 15 (2017)CrossRefGoogle Scholar
  22. 22.
    Hermes, H.: Lie algebras of vector fields and local approximation of attainable sets. SIAM J. Control Optim. 16(5), 715–727 (1978)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Bourdin, L., Dhar, G.: Optimal sampled-data controls with running inequality state constraints—Pontryagin maximum principle and bouncing trajectory phenomenon (2019). Submitted—available on HAL (hal id: hal-02160231)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Univ. Bourgogne Franche-Comté, Le2i Laboratory EA 7508DijonFrance
  2. 2.Univ. Bourgogne Franche-Comté, IMB Laboratory UMR CNRS 5584DijonFrance
  3. 3.Inria Sophia Antipolis Méditerranée, team McTAOValbonneFrance
  4. 4.XLIM Research Institute, UMR CNRS 7252University of LimogesLimogesFrance
  5. 5.ISEN BrestBrestFrance

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