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

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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.

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Notes

  1. Archibald Vivian Hill (1886–1977) was the co-recipient of 1922 Nobel prize in Medicine for this equation.

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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.

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Authors and Affiliations

  1. Univ. Bourgogne Franche-Comté, Le2i Laboratory EA 7508, Dijon, France

    Toufik Bakir

  2. Univ. Bourgogne Franche-Comté, IMB Laboratory UMR CNRS 5584, Dijon, France

    Bernard Bonnard

  3. Inria Sophia Antipolis Méditerranée, team McTAO, Valbonne, France

    Bernard Bonnard

  4. XLIM Research Institute, UMR CNRS 7252, University of Limoges, Limoges, France

    Loïc Bourdin

  5. ISEN Brest, 20 Rue Cuirassé Bretagne, 29200, Brest, France

    Jérémy Rouot

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  1. Toufik Bakir
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  2. Bernard Bonnard
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  3. Loïc Bourdin
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  4. Jérémy Rouot
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Correspondence to Jérémy Rouot.

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Appendix: Physical Descriptions of the Force–Fatigue Model

Appendix: Physical Descriptions of the Force–Fatigue Model

Table 3 provides the values of the constants and the physical descriptions of the variables for the force–fatigue model presented in Sect. 3.

Table 3 Values of the constant parameters in the Ding et al. model
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Bakir, T., Bonnard, B., Bourdin, L. et al. Pontryagin-Type Conditions for Optimal Muscular Force Response to Functional Electrical Stimulations. J Optim Theory Appl 184, 581–602 (2020). https://doi.org/10.1007/s10957-019-01599-4

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  • DOI: https://doi.org/10.1007/s10957-019-01599-4

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