Simon Bechmann

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.

Transfer of the bryson-denham-dreyfus approach for state-constrained ODE optimal control problems to elliptic optimal control problems

We transfer ideas known since the 1960ies from the theory of state-constrained optimal control problems for ordinary differential equations to optimal control problems for elliptic partial differential equations with distributed controls. Replacing the state constraint by equivalent terms leads to new kinds of topology-shape optimal control problems, which gives access to new necessary conditions for elliptic optimal control problems. These new necessary conditions reveal some striking advantages: Higher regularity of the multiplier associated with the state constraint and, in consequence, the ability to apply numerical solvers which do not need any regularization in order to deal with the multipliers. Moreover, the numerical solution can be splited between active and inactive set which improves the efficiency. Since the new necessary conditions can be regarded as a free boundary problem for the unknown interface in-between active and inactive sets, we use Shape-Calculus to formulate a Shape-Newton Scheme in function space in order to solve the optimality system. A finite element discretized version of this scheme shows encouraging results like a low number of iterations and high accuracy in detection of the active sets. Moreover, the numerical results indicate grid independency of this method and the method seems to be able to handle also changes of the topology of the active set.