Space-time Variational Methods for Control Constrained Parabolic Optimal Control Problems

  • Reinhold, Martin Alexander (Ulm University)
  • Beranek, Nina (Ulm University)
  • Urban, Karsten (Ulm UNiversity)

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Optimal Control Problems with PDE constraints are not only highly relevant for a wide range of applications but also an interesting ongoing subject in research of numerical methods. In this talk we want to discuss optimal control problems with a classical tracking type objective functional. The problem is constraint by a parabolic time dependent PDE. We will also consider additional box constraints for the control. Usually these type of problems are solved using time stepping schemes to solve the constraining PDE and the arising adjoint equation. But lately approaches using simultaneous space- time discretizations are investigated for these problems. For parabolic PDEs it is known that these approaches can have significant advantages w.r.t. Model Order Reduction (MOR), which motivates the application of these methods to optimal control problems. In this talk we will discuss the application of a space-time variational formulation for the optimal control problem. We will discuss the approach in the infinite dimensional setting using Bochner-Lebesgue spaces and derive the optimality system in this setting. We show the application of a second order semi-smooth Newton method to solve the optimization problem and propose a discretization with a tensor type approach in space and time. The system arising from the semi-smooth Newton method can be interpreted as a coupled non-linear system of equation. An implementation to solve this system as well as numerical examples are presented. In an outlook we want to discuss the possibilities and challenges that arise for MOR for these systems.