Turnpike features in optimal selection of species represented by quota models

Abstract

The paper focuses on a generic optimal control problem (OCP) deriving from the competition between two microbial populations in continuous cultures. The competition for nutrients is reduced to a two-dimensional dynamical nonlinear-system that can be derived from classical quota models. We investigate an OCP that achieves species separation over a fixed time-window, suitable for a large class of empirical growth functions commonly used in quota models. Using Pontryagin’s Maximum Principle (PMP), the optimal control strategy steering the model trajectories is fully characterized. Then, we provide sufficient conditions for the existence of a turnpike property associated with the optimal control and state-trajectories, as well as their respective co-state trajectories. Indeed, we prove that for a sufficiently large time, the optimal strategy achieving strain separation remains most of the time exponentially close to an optimal steady-state defined from an associated simpler static-OCP. This turnpike feature is based on the hyperbolicity of the linearized Hamiltonian-system around the solution of the static-OCP. The obtained theoretical results are then illustrated on microalgae, described by the Droop model in dimension 5. The optimal strategy is numerically computed in Bocop (open source toolbox for optimal control) with direct optimization methods.