A Three-Dimensional Eutrophication Model: Analysis and Control

Eutrophication of lakes, reservoirs, streams, and coastal areas is one of the most widespread environmental problems of large water bodies. Eutrophication consists of unnatural enrichment with two plant nutrients: nitrogen and phosphorus. This overnutrification causes undesirable changes in water resources: excessive production of algae, deterioration of water quality and availability, fish kills, health hazards for humans, etc. Controlling the eutrophication is important in order to mitigate and remedy the problem. The basic idea of a bioreactor consists holding up hypernutrified water (rich, for instance, in nitrogen) in large tanks where we add a certain quantity of phytoplankton, that we let freely grow to absorb nitrogen from the water. In the particular case analyzed in this chapter we have considered only two large shallow tanks with the same capacities (but possibly different geometries). Water rich in nitrogen fills the first tank Ω1, where we add a quantity ρ1 of phytoplankton (which we let grow for a permanence time T 1) to drop, nitrogen level down to a desired threshold. We are also interested in obtaining a certain quantity of organic detritus (very desirable for use as agricultural fertilizer) in this first tank. Once we reach the desired levels of nitrogen and organic detritus (settled in the bottom of the tank, and then reclaimed for agricultural use), we drain this first tank and pass water to the second tank Ω2, where the same operation is repeated, by adding a new amount ρ2 of phytoplankton. Water leaving this second fermentation tank after a time period T 2 will usually be poor in nitrogen, but rich in detritus (settled in the bottom) and phytoplankton (recovered from a final filtering). At this point, we are interested (both for economical and ecological reasons) in minimizing this final quantity of phytoplankton. Thus, the optimal control problem consists of finding the quantities (ρ1, ρ2) of phytoplankton that we must add to each tank during the respective times so that nitrogen levels are lower than the maximum thresholds and the detritus levels are higher than the minimum thresholds, and in such a way that the final phytoplankton concentration is as reduced as possible.