Optimal sensor design for estimation and optimization of PDE systems

In many situations one is interested in the knowledge of the state of a PDE system on a sub-domain or region in the spatial domain. This type of problem occurs naturally when one is interested in regional control. Also, in cases when full spatial control is the goal and the feedback law has compact spatial support so that the observer need only estimate the local spatial behavior of the system. We consider a class of sensor location problems where the goal is to provide optimal zonal estimation and control for a parabolic distributed parameter system. Numerical examples are provided to illustrate the approach. We observed that in many cases the optimization problem is mesh independent and lends itself to parallel and multi-grid optimization approaches.