  A variety of optimal control, estimation, system identification and design
problems can be formulated as functional optimization problems with
differential equality and inequality constraints. Since these problems are
infinite-dimensional and often do not have a known analytical solution, one has
to resort to numerical methods to compute an approximate solution. This paper
uses a unifying notation to outline some of the techniques used in the
transcription step of simultaneous direct methods (which
discretize-then-optimize) for solving continuous-time dynamic optimization
problems. We focus on collocation, integrated residual and Runge-Kutta schemes.
These transcription methods are then applied to a simulation case study to
answer a question that arose during the COVID-19 pandemic, namely: If there are
not enough ventilators, is it possible to ventilate more than one patient on a
single ventilator? The results suggest that it is possible, in principle, to
estimate individual patient parameters sufficiently accurately, using a
relatively small number of flow rate measurements, without needing to
disconnect a patient from the system or needing more than one flow rate sensor.
We also show that it is possible to ensure that two different patients can
indeed receive their desired tidal volume, by modifying the resistance
experienced by the air flow to each patient and controlling the ventilator
pressure.
