Optimal positioning of anodes for cathodic protection

We consider an optimal control problem that arises in cathodic protection. The control is the position of a finite number of anodes, each with a prescribed current density. The goal is to obtain a desired potential distribution on the cathode that will prevent or reduce cathodic corrosion. The existence of an optimal solution is proven. The differentiability of the functional minimized is justified, and the equations satisfied by the derivative are established. Then a numerical algorithm is proposed for computing the optimal position of the anodes. Finally, we present some numerical results obtained by implementing the proposed numerical algorithm with a boundary element method.