Keywords: finite element method, Newton based method, Broyden-Fletcher-Goldfarb-Shanno method, Davidon-Fletcher-Powell method, second order adjoint equation, water purification control.

The purpose of this paper is to present a control method for dissolved oxygen (DO) using the first order adjoint method and a Newton based method. In recent years, the water state of long rivers has recovered with technical improvement of the water maintenance in Japan. However, this is hardly seen in small rivers. The water pollution problem assumes serious proportions. In this research, DO is picked up as one of the indices of water state.

We study the Teganuma river, which is located in Chiba prefecture in Japan. The water velocity in the Teganuma river is too small. The Teganuma river is not only a river but also a lake. Recently, a water conducting project has been performed in part of the upstream of this river. However, it is ineffective for this river because the water hardly flows. The optimal velocity is needed to diffuse clean water for this river. If the optimal velocity is found, a higher DO can be diffused in this river. Then the river can be cleaned up. Therefore, the optimal control of the DO is carried out to find the optimal velocity. In this study, the water velocity is calculated as the control variable.

The first section shows the state equations and the discretization technique. The coupled shallow water equation and the advection diffusion equation is applied as the state equation. For temporal discretization of the basic equations, the Crank-Nicolson method is applied, and for the spatial discretization, the finite element method based on the bubble function interpolation is applied.

The next section is concerned with optimal control. In optimal control theory, the control variable is obtained by the minimization of the performance function. The gradient of the performance function updates the control variable. The control variable is computed so as to minimize the performance function under the constraint conditions.

The Newton based technique is used to solve the second order adjoint equation. The gradient of the second order adjoint equation which can be obtained by calculating an extended perturbation is similar to product of the Hessian matrix and the perturbation vector of the control variable. The Hessian matrix is approximately updated using the Broyden-Fletcher-Goldfarb-Shanno method or the Davidon-Fletcher-Powell method. The performance function is calculated by updating the control variable.

In this research, the velocity of the river is defined as the control variable. In the numerical study, an example of optimal control of the DO is shown using the Teganuma river as a case study.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £140 +P&P)