# Space-time multigrid methods for solving unsteady optimal control problems

## Prof. Alfio BorzĂ¬

**April 21, 2005, 1:30 p.m. HF 136**

We review recent developments on multigrid methods for time dependent optimal control problems and present further results towards real-time control of evolving systems.

Examples are given of distributed and boundary control of reaction-diffusion systems which appear in biology, chemistry, physiology, etc.. To solve the related optimal control problems, the corresponding optimality systems are considered. These consist of reaction diffusion equations with opposite time orientation.

We focus on space-time multigrid methods that solve optimality systems in one shot in the whole space-time cylinder and are robust with respect to changes of the values of the optimization parameters. These schemes are based on smoothing iterations that allow to solve also singular optimal control problems. For control on long-time intervals, the combination of space-time multigrid approach with receding horizon techniques is discussed.