Optimization of microalgae biosynthesis via controlled algal-bacterial symbiosis

Abstract

We investigate optimization of an algal-bacterial consortium, where an exogenous control input modulates bacterial resource allocation between growth and synthesis of a resource that is limiting for algal growth. Maximization of algal biomass synthesis is pursued in a continuous bioreactor, with dilution rate as an additional control variable. We formulate optimal control in the two variants of static and dynamic control problems, and address them by theoretical and numerical tools. We explore convexity of the static problem and uniqueness of its solution, and show that the dynamic problem displays a solution with bang-bang control actions and singular arcs that result in cyclic control actions. We finally discuss the relation among the two solutions and show the extent to which dynamic control can outperform static optimal solutions.

Figures (3)

• Figure 1: SOCP objective function $f_0^*(\alpha,d)$ contours, its maxima along $\alpha$ and $d$, and its global maximum.
• Figure 2: Optimal controls $\alpha(t)$ and $d(t)$ [red] and switching functions $\zeta_\alpha(t)$ and $\zeta_d(t)$ [blue] over $t\in[0,t_f]$, along with the optimal static controls $\bar{\alpha}$ and $\bar{d}$ [dashed green].
• Figure 3: The optimal state trajectories plotted against time over 20 days, under the optimal dynamic control $u(t)$ [solid lines] and under the static optimal control $\bar{u}$ [dashed lines]. Concentrations are given in $g{\cdot}L^{-1}$ and the internal cell quota is given in $g{\cdot}g^{-1}$

Theorems & Definitions (11)

• Remark 1
• Lemma 2
• proof
• Lemma 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof
• Proposition 6