Optimal Control of Shallow Water Flows Using Adjoint Equation Method

Abstract

This paper presents a method to control a flow behavior using the first-order adjoint equation method. A flood causes large-scale and extensive damage to the human property. It is expected that the damage can be suppressed to minimum if the water level or velocity of the river can be controlled. Therefore, the optimal control of a water flow is carried out in this study. In the control theory, the performance function which is defined by the square sum of the discrepancy between the computed and the objective water elevation at the target points is used. The extended performance function is given by the performance function and the state equation. The first-order adjoint equation can be derived by the condition that the first variations of the extended performance function and constraint condition are zero. The gradient of the extended performance function is obtained by solving the first-order adjoint equation. As the minimization technique, the weighted gradient method is applied. The shallow water equation based on the water velocity and elevation is used as a state equation. As the spatial discretization, the stabilized bubble function is employed. As the temporal discretization, the Crank-Nicolson method is applied. In numerical studies, the optimal control of water elevation in the Ikari dam lake is carried out. In this research, the optimal water velocity sent by the pump to minimize the water rise in the Ikari dam lake is computed. The inflow boundary conditions are presupposed as sinusoidal wave water elevation. The results of optimal control is performed effectively.