A control problem arising in the process of waste water purification

In this paper we state and solve an optimal control problem arisen from the management of the sewage disposal which is dumped into the sea through submarine outfalls. Firstly, we fix oxygen and amount of organic matter as water quality indicators and we state a partial differential equations model to simulate them in a domain occupied by shallow waters. Constraints about water quality and economic objectives lead us to a pointwise optimal control problem with state and control constraints. (The theoretical analysis of the problem has been developed by the authors in (Martinez et al., C. R. Acad. Sci. Paris, Serie I 328 (1999) 35.) (Martinez et al., Preprint, Dept. Matematica Aplicada, Univ. Santiago de Compostela, Spain, 1998.)). We deal with the problem by using time and space discretizations and we propose two algorithms for the numerical resolution of the discretized problem. Finally, we give numerical results obtained by applying the described techniques for a realistic problem posed in the rı́a of Vigo (Spain).