Direct and indirect methods to optimize the muscular force response to a pulse train of electrical stimulation

Abstract

Recent force-fatigue mathematical models in biomechanics [7] allow to predict the muscular force response to functional electrical stimulation (FES) and leads to the optimal control problem of maximizing the force. The stimulations are Dirac pulses and the control parameters are the pulses amplitudes and times of application, the number of pulses is physically limited and the model leads to a sampled data control problem. The aim of this article is to present and compare two methods. The first method is a direct optimization scheme where a further refined numerical discretization is applied on the dynamics. The second method is an indirect scheme: first-order Pontryagin type necessary conditions are derived and used to compute the optimal sampling times.