  This paper studies the bioremediation, in minimal time, of a water resource
or reservoir using a single continuous bioreactor. The bioreactor is connected
to two pumps, at different locations in the reservoir, that pump polluted water
and inject back sufficiently clean water with the same flow rate. This leads to
a minimal-time optimal control problem where the control variables are related
to the inflow rates of both pumps. We obtain a non-convex problem for which it
is not possible to directly prove the existence of its solutions. We overcome
this difficulty and fully solve the studied problem by applying Pontryagin's
principle to the associated generalized control problem. We also obtain
explicit bounds on its value function via Hamilton-Jacobi-Bellman techniques.

Optimization and Control