# American Institute of Mathematical Sciences

March  2018, 8(1): 277-313. doi: 10.3934/mcrf.2018012

## Water artificial circulation for eutrophication control

 1 Depto. Matemática Aplicada II, Universidade de Vigo, E.I. Telecomunicación, 36310 Vigo, Spain 2 Centro Universitario de la Defensa, Escuela Naval Militar, 36920 Marín, Spain

* Corresponding author

Received  April 2017 Revised  September 2017 Published  January 2018

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.

Citation: Aurea Martínez, Francisco J. Fernández, Lino J. Alvarez-Vázquez. Water artificial circulation for eutrophication control. Mathematical Control and Related Fields, 2018, 8 (1) : 277-313. doi: 10.3934/mcrf.2018012
##### References:
 [1] M. Abdelwahed, Optimization of injectors location in a water reservoirs aeration problem, Math. Problems in Eng., (2013), Article ID 875979. [2] L. J. Alvarez-Vázquez and F. J. Fernández, Optimal control of a bioreactor, Appl. Math. Comput., 216 (2010), 2559-2575. [3] L. J. Alvarez-Vázquez, F. J. Fernández and R. Muñoz-Sola, Analysis of a multistate control problem related to food technology, J. Diff. Equations, 245 (2008), 130-153. [4] L. J. Alvarez-Vázquez, F. J. Fernández and R. Muñoz-Sola, Mathematical analysis of a three-dimensional eutrophication model, J. Math. Anal. Appl., 349 (2009), 135-155. [5] E. Casas and K. Chrysafinos, Error estimates for the discretization of the velocity tracking problem, Numer. Math., 130 (2015), 615-643. [6] E. Casas and K. Chrysafinos, Analysis for the velocity tracking control problem for the 3D evolutionary Navier-Stokes equations, SIAM J. Control Optim., 54 (2016), 99-128. [7] A. Fursikov, M. Gunzburger and L. Hou, Trace theorems for three-dimensional, time-dependent solenoidal vector fields and their applications, Trans. Amer. Math. Soc., 354 (2002), 1079-1116. [8] V. Girault and P. A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Springer-Verlag, New York-Heidelberg, 1979. [9] M. D. Gunzburger, O. A. Ladyzhenskaya and J. S. Peterson, On the global unique solvability of initial-boundary value problems for the coupled modified Navier-Stokes and Maxwell equations, J. Math. Fluid Mech., 6 (2004), 462-482. [10] M. Gunzburger and C. Trenchea, Analysis of an optimal control problem for the three-dimensional coupled modified Navier-Stokes and Maxwell equations, J. Math. Anal. Appl., 333 (2007), 295-310. [11] R. C. Haynes, Some ecological effects of artificial circulation on a small eutrophic lake with particular emphasis on phytoplankton, Hydrobiologia, 43 (1973), 463-504. [12] F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265. [13] W. M. Heo and B. Kim, The effect of artificial destratification on phytoplankton in a reservoir, Hydrobiologia, 524 (2004), 229-239. [14] A. O. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, 1968. [15] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972. [16] T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2013. [17] P. M. Visser, B. W. Ibelings, M. Bormans and J. Huisman, Artificial mixing to control cyanobacterial blooms: A review, Aquatic Ecology, 50 (2016), 423-441. [18] D. Wachsmuth, Analysis of the SQP-method for optimal control problems governed by the nonstationary Navier-Stokes equations based on Lp-theory, SIAM J. Control Optim., 46 (2007), 1133-1153. [19] A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57.

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##### References:
 [1] M. Abdelwahed, Optimization of injectors location in a water reservoirs aeration problem, Math. Problems in Eng., (2013), Article ID 875979. [2] L. J. Alvarez-Vázquez and F. J. Fernández, Optimal control of a bioreactor, Appl. Math. Comput., 216 (2010), 2559-2575. [3] L. J. Alvarez-Vázquez, F. J. Fernández and R. Muñoz-Sola, Analysis of a multistate control problem related to food technology, J. Diff. Equations, 245 (2008), 130-153. [4] L. J. Alvarez-Vázquez, F. J. Fernández and R. Muñoz-Sola, Mathematical analysis of a three-dimensional eutrophication model, J. Math. Anal. Appl., 349 (2009), 135-155. [5] E. Casas and K. Chrysafinos, Error estimates for the discretization of the velocity tracking problem, Numer. Math., 130 (2015), 615-643. [6] E. Casas and K. Chrysafinos, Analysis for the velocity tracking control problem for the 3D evolutionary Navier-Stokes equations, SIAM J. Control Optim., 54 (2016), 99-128. [7] A. Fursikov, M. Gunzburger and L. Hou, Trace theorems for three-dimensional, time-dependent solenoidal vector fields and their applications, Trans. Amer. Math. Soc., 354 (2002), 1079-1116. [8] V. Girault and P. A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Springer-Verlag, New York-Heidelberg, 1979. [9] M. D. Gunzburger, O. A. Ladyzhenskaya and J. S. Peterson, On the global unique solvability of initial-boundary value problems for the coupled modified Navier-Stokes and Maxwell equations, J. Math. Fluid Mech., 6 (2004), 462-482. [10] M. Gunzburger and C. Trenchea, Analysis of an optimal control problem for the three-dimensional coupled modified Navier-Stokes and Maxwell equations, J. Math. Anal. Appl., 333 (2007), 295-310. [11] R. C. Haynes, Some ecological effects of artificial circulation on a small eutrophic lake with particular emphasis on phytoplankton, Hydrobiologia, 43 (1973), 463-504. [12] F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265. [13] W. M. Heo and B. Kim, The effect of artificial destratification on phytoplankton in a reservoir, Hydrobiologia, 524 (2004), 229-239. [14] A. O. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, 1968. [15] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972. [16] T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2013. [17] P. M. Visser, B. W. Ibelings, M. Bormans and J. Huisman, Artificial mixing to control cyanobacterial blooms: A review, Aquatic Ecology, 50 (2016), 423-441. [18] D. Wachsmuth, Analysis of the SQP-method for optimal control problems governed by the nonstationary Navier-Stokes equations based on Lp-theory, SIAM J. Control Optim., 46 (2007), 1133-1153. [19] A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), 25-57.
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 $k = 1, \ldots, 4$.
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.
Concentrations of dissolved oxygen at final time with the four pumps at rest (up) and with medium power pumps (down).
Concentrations of organic detritus at final time with the four pumps at rest (up) and with medium power pumps (down).
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).
Concentration of dissolved oxygen at final time for the optimal control $\mathbf{g}_{opt}$.
Physical parameters for the numerical example.
 Parameters Values Units $\nu$ $1.5 \ 10^{-6}$ $m^2/s$ $\nu_{tur}$ $1.0 \ 10^{-2}$ $m^2$ $K_N$ $3.8 \ 10^{-2}$ $mg/l$ $K_F$ $2.0 \ 10^{-1}$ $mg\, C/l$ $K_{mf}$ $3.8 \ 10^{-7}$ $s^{-1}$ $K_{mz}$ $3.78 \ 10^{-7}$ $s^{-1}$ $K_r$ $3.8 \ 10^{-7}$ $s^{-1}$ $K_z$ $4.3 \ 10^{-6}$ $s^{-1}$ $K_{rd}$ $2.3 \ 10^{-5}$ $s^{-1}$ $\Theta$ $1.05$ - $\theta$ $24.0$ $^{\circ}C$ $\mu$ $1.5 \ 10^{-4}$ $s^{-1}$ $\varphi_1$ $30.0$ $m^{-1}$ $\varphi_2$ $0.0$ $m^{-1} / (mg\, C/l)$ $I_0$ $100.0$ $(Cal/m^2)/s$ $I_S$ $100.0$ $(Cal/m^2)/s$ $C_{fz}$ $1.1$ - $C_T$ $1.06$ - $\mu^i$ $1.0 \ 10^{-5}$ $m^2/s$
 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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