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# Water artificial circulation for eutrophication control

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• This work analyzes, from a mathematical point of view, the artificial mixing of water -by means of several pairs collector/injector that set up a circulation pattern in the waterbody -in order to prevent the undesired effects of eutrophication. The environmental problem is formulated as a constrained optimal control problem of partial differential equations, where the state system is related to the velocity of water and to the concentrations of the different species involved in the eutrophication processes, and the cost function to be minimized represents the volume of recirculated water. In the main part of the work, the wellposedness of the problem and the existence of an optimal control is demonstrated. Finally, a complete numerical algorithm for its computation is presented, and some numerical results for a realistic problem are also given.

Mathematics Subject Classification: Primary: 49M25, 35Q93; Secondary: 90C56.

 Citation: • • Figure 1.  Physical domain $\Omega$ for the numerical example, showing the control domain $\Omega_C$ (shaded area) and the four pumping groups connecting collectors $C^k$ with injectors $T^k$, for $k = 1, \ldots, 4$.

Figure 2.  Comparison of the constraints on the five species (nutrient, phytoplankton, zooplankton, organic detritus and dissolved oxygen) for an uncontrolled case (dashed line) and for a controlled one (solid line), corresponding to 48 steps of time.

Figure 3.  Concentrations of dissolved oxygen at final time with the four pumps at rest (up) and with medium power pumps (down).

Figure 4.  Concentrations of organic detritus at final time with the four pumps at rest (up) and with medium power pumps (down).

Figure 5.  Optimal control $\mathbf{g}_{opt}$ for the four groups (up), and mean concentrations of dissolved oxygen for $\mathbf{g}_{ref}$, $\mathbf{g} = 0$ and $\mathbf{g}_{opt}$ (down).

Figure 6.  Concentration of dissolved oxygen at final time for the optimal control $\mathbf{g}_{opt}$.

Table 1.  Physical parameters for the numerical example.

 Parameters Values Units $\nu$ $1.5 \ 10^{-6}$ $m^2/s$ $\nu_{tur}$ $1.0 \ 10^{-2}$ $m^2$ $K_N$ $3.8 \ 10^{-2}$ $mg/l$ $K_F$ $2.0 \ 10^{-1}$ $mg\, C/l$ $K_{mf}$ $3.8 \ 10^{-7}$ $s^{-1}$ $K_{mz}$ $3.78 \ 10^{-7}$ $s^{-1}$ $K_r$ $3.8 \ 10^{-7}$ $s^{-1}$ $K_z$ $4.3 \ 10^{-6}$ $s^{-1}$ $K_{rd}$ $2.3 \ 10^{-5}$ $s^{-1}$ $\Theta$ $1.05$ - $\theta$ $24.0$ $^{\circ}C$ $\mu$ $1.5 \ 10^{-4}$ $s^{-1}$ $\varphi_1$ $30.0$ $m^{-1}$ $\varphi_2$ $0.0$ $m^{-1} / (mg\, C/l)$ $I_0$ $100.0$ $(Cal/m^2)/s$ $I_S$ $100.0$ $(Cal/m^2)/s$ $C_{fz}$ $1.1$ - $C_T$ $1.06$ - $\mu^i$ $1.0 \ 10^{-5}$ $m^2/s$
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