Sebastian Blauth

• Optimal control of partial differential equations is an important task in applied mathematics where it is used in order to optimize, for example, industrial or medical processes. In this thesis we investigate an optimal control problem with tracking type cost functional for the Cattaneo equation with distributed control, that is, \(\tau y_{tt} + y_t - \Delta y = u\). Our focus is on the theoretical and numerical analysis of the limit process \(\tau \to 0\) where we prove the convergence of solutions of the Cattaneo equation to solutions of the heat equation. We start by deriving both the Cattaneo and the classical heat equation as well as introducing our notation and some functional analytic background. Afterwards, we prove the well-posedness of the Cattaneo equation for homogeneous Dirichlet boundary conditions, that is, we show the existence and uniqueness of a weak solution together with its continuous dependence on the data. We need this in the following, where we investigate the optimal control problem for the Cattaneo equation: We show the existence and uniqueness of a global minimizer for an optimal control problem with tracking type cost functional and the Cattaneo equation as a constraint. Subsequently, we do an asymptotic analysis for \(\tau \to 0\) for both the forward equation and the aforementioned optimal control problem and show that the solutions of these problems for the Cattaneo equation converge strongly to the ones for the heat equation. Finally, we investigate these problems numerically, where we examine the different behaviour of the models and also consider the limit \(\tau \to 0\), suggesting a linear convergence rate.