Résumé
In many natural environments, water resources are not directly usable because their quality does not fulfill the drinking water norms, and therefore treatment or control needs to be applied. It could also happen that the quality of these resource is temporarily acceptable but impacted by its extraction. This is typically the case of coastal lagoons that communicate with the sea. Depending on the level of water in the lagoon, the communication with the sea increases its salt concentration and then one may want to maximize the water pumping within giving bound on the salt concentration, and preserving water volume in the lagoon for its sustainability. In this work, we model the problem as a non-autonomous optimal control problem under state constraints, distinguising two seasons (dry and humid) in the year. The state of the system describes the salt and volume concentration of a coastal lagoon connected to the sea, and the constraints represent bounds on the salt concentration and water volume of the lagoon. The control variable is the output flow rate from the lagoon and the objective the maximization of the total water extracted during one year. Instead of deriving necessary conditions of optimal control under state constraints (i.e. Maximum Principle under state constraints), we first characterize the viability kernel over the year period, that is the set of initial conditions for which there exists a (time-varying) control such that the corresponding trajectory satisfies the constraints over one year. Then, we study the optimal control within this set, expliciting for any initial condition an optimal control policy, as a state feedback, maximizing the pumped water over one year. This problem has been motivated by the case study of the Tunquen lagoon in Chile, but the modelling approach and the methodolgy could be extended to similar situations.