Bang-bang and singular controls in optimal control problems with partial differential equations

This paper focuses on bang-bang and singular optimal controls for problems involving partial differential equations. In particular, singular controls have hardly ever been investigated in literature. Since existence of solutions and multipliers, as well as their regularity, cannot be proven in general for problems of this type, the derivation of candidates for necessary conditions via the so-called formal Lagrange technique seems to be the only viable way. Nevertheless, a numerical a posteriori verification of these conditions may fill this gap, at least to a certain extent. Finally, a generalization of a numerical method is presented, a counterpart of which has turned out to be efficient and robust for optimal control problems involving ordinary differential equations. This method allows a precise determination of the switches from one control type to the other, while the number of discrete optimization variables is considerably reduced.