Extended Kalman filtering to design optimal controllers of fronts generated by level set methods

The computational issues involved in the solution of optimal control problems of propagating fronts described by level sets motivate the search for effective optimization algorithms. In this paper, we attack the problem of optimal control of moving fronts by searching for an approximate solution method that is computationally feasible and robust to local minima trapping. The presence of many local minima is a crucial difficulty one encounters in dealing with such a problem. Following previous results, we use the extended Ritz method to find approximate solutions. This approach consists in adopting a control law with fixed structure that depends nonlinearly on a number of parameters to be suitably chosen. To overcome the local minima issue, we propose to optimize the weights for the level set optimal control by a recursive minimization based on the extended Kalman filter (EKF). As compared with techniques based on gradient-descent methods, the EKF optimization turns out to be successful to reduce computational burden and increase robustness with respect to local minima trapping, as shown by simulation results in a test case involving a change of topology.