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.

该论文侧重于从连续培养中的两个微生物种群之间的竞争中得出的通用最优控制问题 (OCP)。对营养物质的竞争被简化为一个二维动态非线性系统，该系统可以从经典配额模型中推导出来。我们研究了在固定时间窗口内实现物种分离的 OCP，适用于配额模型中常用的一大类经验增长函数。使用 Pontryagin 的最大原理 (PMP)，可以充分表征引导模型轨迹的最佳控制策略。那么，我们提供了收费公路存在的充分条件与最优控制和状态轨迹相关的属性，以及它们各自的共状态轨迹。事实上，我们证明了在足够长的时间内，实现应变分离的最佳策略在大多数情况下仍然以指数方式接近由相关的更简单静态-OCP定义的最佳稳态。此收费公路功能基于围绕静态-OCP解的线性化哈密顿系统的双曲线性。然后在微藻上说明获得的理论结果，由维度 5 中的 Droop 模型描述。 最优策略在Bocop（最优控制的开源工具箱）中进行数值计算) 直接优化方法。