Controlling recirculation rate for minimal-time bioremediation of natural water resources

We revisit the minimal time problem of in-situ decontamination of large water resources with a bioreactor, considering a recirculation flow rate in the resource as an additional control. This new problem has two manipulated inputs : the flow rate of the treatment in the bioreactor and the recirculation flow rate of the water resource between the pumping and reinjection locations. Although the velocity set of the dynamics is non convex, we show that the optimal control is reached among non-relaxed controls. The optimal strategy consists in three sequential steps: 1. do not mix and take the flow rate of treatment that maximizes the concentration decay in the resource. 2. mix as much as possible and carry on with the flow rate that maximizes the concentration decay. 3. carry on mixing but do not treat the water. Finally, we show on numerical simulations that a significant gain in processing time can be achieved time when controlling in addition the recirculation flow rate