Optimal Control of Quasilinear Parabolic PDEs: Theory and Numerics

Abstract

This thesis is concerned with theory and numerics of optimal control of quasilinear parabolic PDEs. The underlying parabolic PDE models, e.g., heat conduction with temperature-dependent thermal conductivity and is highly nonlinear with a nonmonotone nonlinearity in the elliptic operator. This makes its analysis and the analysis of the entire control problem as interesting as challenging. In our work we address optimal control problems with additional pointwise state-constraints and problems with (directionally) sparse solutions, and analyze convergence of the SQP method. Moreover, we consider model order reduction by proper orthogonal decomposition both for the state equation and the control problem. On the one hand, our contributions can be regarded as extension of results on control-constrained optimal control of quasilinear parabolic PDEs towards the abovementioned additional aspects. In particular, difficulties associated with the nonlinear structure of our state equation are a recurrent issue in our analysis. On the other hand, we also contribute to the fields of state-constrained or sparse optimal control, the analysis of optimization algorithms, and model order reduction by extending them towards quasilinear parabolic PDEs. Consequently, we also encounter the typical challenges due to these respective areas, especially such challenges related to optimality conditions in infinite dimensions.