Ventilation optimale de type bang-bang sous contrainte de fermeture du cycle mécanique : un modèle réduit des échanges gazeux pulmonaires

Many reduced and control-oriented respiratory models prescribe a ventilatory waveform. Here we ask a narrower inverse question: once cycle duration and mechanical closure are fixed, what ventilation pattern maximizes a reduced oxygen-transfer criterion? We study a minimal 0D model coupling global lung deformation to alveolar oxygen concentration. Under explicit quantitative and nondegeneracy assumptions, every optimal control is bang-bang with exactly two switchings, producing one inflation phase followed by one deflation phase. Thus optimality selects the internal two phase inspiration-expiration organization. The concentration of oxygen is not impose at the beginning and the end of a cycle. We prove that repetition of a fixed mechanical cycle induces a unique stable periodic concentration regime, where the concentration at the beginning and the end remains the same. Numerical simulations recover the two-switch structure for the nominal parameter set and indicate its persistence in a linear viscoelastic extension.