Optimal control of semilinear elliptic PDEs with state constraints - numerical analysis and implementation

Abstract

In the first part of the thesis we show that it is possible to establish a sufficient condition for global minima of a certain class of optimal control problems of semilinear el- liptic PDEs with pointwise constraints on the state and/or the control variables provided that the nonlinearity in the PDE satisfies certain growth conditions. This sufficient condition can also give information about the uniqueness of the global solutions. Moreover, one can establish in an analogous way to the con- tinuous setting a similar condition for the variational discrete control problem. It turns out that a sequence of discrete unique global minima satisfying this condition uniformly converges strongly to the unique global minimum of the corresponding continuous control problem as the discretization parameter tends to zero. A rate of convergence for the sequence of the discrete unique global minima can be established using this sufficient condition as well. The numerical experiments show that this convergence rate is optimal. In addition, we man- aged to compute the unique global minima for several examples.

In the second part of the thesis we consider optimal control problems of elliptic PDEs with stochastic coefficients. The task is to compute the expected value of the optimal controls corresponding to the different realizations of the random coefficient of the state equation utilizing the finite element Monte Carlo and multilevel Monte Carlo methods and to carry out the associated error analysis. However, the computed expected value needs not to be an optimal control in general.