The aim of this paper is to estimate a dune height that can favor the formation of other dunes in its closer vicinity. To achieve this, we formulate a distributed optimal control problem with constraints. We use the discrete adjoint method and the algorithm of the quasi-Newton method for solving the control problem. The implementation of this technique requires the discretization of the constrained equations. We then propose a combination of the Crank-Nicholson scheme and a spectral approach to discretize the constraint equations. We present some numerical results of our model problem.

optimal control, spectral approach, quasi-Newton method, discrete adjoint method.