Is our breathing optimal? Solving a piecewise linear model with constraints

This paper is motivated by a question related to the control of amplitude and frequency of breathing. We present a simplified mathematical model, consisting of two piecewise linear ordinary differential equations, that could represent gas exchange in the lungs. We then define and solve an optimal control problem with unknown durations of inhalation and exhalation, subject to several constraints. The durations are divided such that one of the state variables is strictly increasing during the first phase and decreasing during the second phase. The optimal control problem can be solved analytically. One analytical solution is found when the forcing is a given sinusoidal function with unknown period and amplitude. Other analytical solutions are found when the forcing function, the period and the duration of the first phase are unknown but the amplitude is given. Our results show that different cost functions can produce different optimal forcing functions. We also show that the shape of these functions does not affect the average levels of oxygen in the lungs-the average level of oxygen is only dependent on the amplitude and period of breathing in the model we present.