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In this work we deal with the optimal purification of polluted areas of shallow waters by means of the injection of clear water in order to promote seawater exchange. This problem can be formulated as a control constrained optimal control problem where the control is the velocity of the injected water, the state equations are the shallow water equations together with that modelling the contaminant concentration, and the cost function measures the total amount of injected water and the fulfilment of the water quality standards. We analyze the solutions of the optimal control problem and give an optimality condition in order to characterize them. We also discretize the problem by means of a characteristics-mixed finite element method, focusing our attention on both the discrete and the discretized adjoint systems, and propose an algorithm for the numerical resolution of the discrete optimization problem. Finally, we present numerical results for some computational experiments.

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The management of sewage disposal and the design of wastewater treatment systems can be formulated as a constrained pointwise optimal control problem. In this paper we study the convergence of the numerical resolution for the corresponding state system by means of a characteristics Galerkin method. The main diculty of the problem is due to the existence of Radon measures in the right-hand side of the state system. Finally, we present numerical results for a realistic problem posed in a rõ Âa of Galicia, Spain. Ó

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