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

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


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.


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

Mathematics Subject Classification

49K15 93B07 92B05 



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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© 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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