Abstract:

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

Distributed or boundary control of reaction-diffusion systems 
has many applications 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 present space-time multigrid methods that solve optimal
control systems in one shot in the whole space-time cylinder which 
are robust with respect to changes of the values of the optimization 
parameters. Two different smoothing schemes in combination with
semicoarsening in space are discussed. 

Towards real-time optimal control of evolving reaction diffusion 
systems, the combination of the space-time multigrid approach with 
receding horizon techniques is proposed. 

The ability of the space-time multigrid approach in solving 
optimal control problems is demonstrated by applications to 
the control of an explosive phenomenon, control of chemical 
turbulence, and control of caridiac arrhythmias.