A Time-Adaptive POD Method for Optimal Control Problems

In this paper we present a time adaptive technique for the solution of optimal control problems where the dynamic is given by an evolutive semi linear PDE. The method is based on a model reduction using a POD approximation coupled with a Hamilton-Jacobi equation characterizing the value function of the corresponding control problem for the reduced system. The choice of the POD basis is updated according to the evaluation of a numerical indicator in order to guarantee a global accurate solution. This is obtained via a sub-division of the time horizon into sub-intervals where the residual is below a given threshold. Some numerical tests illustrate the main features of this approach.