Section: New Results

Optimal control of PDEs

Stabilization of Burgers equation to nonstationary trajectories A. Kröner and Sérgio S. Rodrigues (RICAM, Linz, Austria) considered in [82] using infinite-dimensional internal controls. Estimates for the dimension of the controller are derived; in the particular case of no constraint in the support of the control a better estimate is derived and the possibility of getting an analogous estimate for the general case is discussed. Numerical examples are presented illustrating the stabilizing effect of the feedback control, and suggesting that the existence of an estimate in the general case analogous to that in the particular one is plausible. In [81] the problem was consided for a finite number of internal piecewise constant controls.

Reduced-order minimum time control of advection-reaction -diffusion systems via dynamic programming Dante Kalise (RICAM, Linz, Austria) and A. Kröner considered in [79] . The authors use balanced truncation for the model reduction part and include a Luenberger observer.

A semi-Lagrangian scheme for $L^p$-penalized minimum time problems was considered by M. Falcone (Sapienza-Università di Roma, Italy), D. Kalise (RICAM, Austria) and A. Kröner in [78] .

The minimum effort problem for the wave equation K. Kunisch (University of Graz, Austria) and A. Kröner considered in [80] . The problem involves $L^{\infty }$-control costs which lead to non-differentiability. Uniqueness of the solution of a regularized problem is proven and the convergence of the regularized solutions is analyzed. Further, a semi-smooth Newton method is formulated to solve the regularized problems and its superlinear convergence is shown. Numerical examples confirm the theoretical results.