Optimal feedback synthesis and minimal time function for the bioremediation of water resources with two patches (Q2813317)

This interesting and well-written paper investigates a particular minimum-time nonlinear optimal control problem with two states and two controls arising from water treatment. In contrast with previous works, it is assumed that the treatment of the water resource is split into two zones connected by diffusion. This provides the possibility of switching dynamically the treatment from one patch to the other, or treating simultaneously both patches.NEWLINENEWLINEThe optimal control problem considered by the authors has a nonconvex velocity set, and hence it is not immediate that an optimal solution exists in the set of bounded control functions, i.\,e. classical controls. It may potentially happen that one has to enlarge the set of allowable controls to time-dependent probability measures, i.\,e. relaxed or chattering controls corresponding to convexifying the velocity set. Chattering controls are indeed typical for switching control systems.NEWLINENEWLINEWith the help of the necessary optimality conditions of the Pontryagin maximum principle, the authors solve the problem analytically and comprehensively, and they prove in Proposition 3.6 that the convexified control problem has an optimal solution in the set of classical controls, from which it follows that the original nonconvex control problem has also an optimal solution in the set of classical controls. Sections 4 and 5, respectively, describe the optimal control strategy and properties of the value function, i.\,e. the minimal time as a function of the initial condition. Numerical examples and simulations in section 6 illustrate the gain of using two pumps instead of one in the case of low diffusion.