Singular perturbation based solution to optimal microalgal growth problem and its infinite time horizon analysis

The problem of the optimal microalgal growth is considered here. The objective is to maximize the specific growth rate of microalgae by manipulating the irradiance. The model describing the growth of microalgae is based on the mechanistic description in the form of the so called photosynthetic factory (PSF) resulting into the second order bilinear system which is, nevertheless, known in biotechnological literature to comprise many important features of microalgal growth. To obtain the solution of optimal control problem, the singular perturbation approach is used here to reduce fast components of system dynamics leading to a less dimensional system with more complex performance index which allows a nice analytical solution. Its infinite horizon analysis shows that the optimal solution on large time intervals tends to the optimal steady state of PSF thereby supporting the hypothesis often mentioned in the biotechnological literature. Finally, the numerical algorithm to compute optimal control is applied to the original non-reduced system giving very similar results as the reduction based approach.