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

Abstract

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.