Feedback control of parametrized PDEs via model order reduction and dynamic programming principle
For researchers working on optimal control of PDEs, this work provides a computationally feasible approach to feedback control by reducing the state space dimension.
This paper addresses infinite horizon optimal control problems for parametrized PDEs by combining model order reduction with dynamic programming to overcome the curse of dimensionality. Numerical examples for 2D PDEs demonstrate the effectiveness of the proposed methods.
In this paper we investigate infinite horizon optimal control problems for parametrized partial differential equations. We are interested in feedback control via dynamic programming equations which is well-known to suffer from the curse of dimensionality. Thus, we apply parametric model order reduction techniques to construct low-dimensional subspaces with suitable information on the control problem, where the dynamic programming equations can be approximated. To guarantee a low number of basis functions, we combine recent basis generation methods and parameter partitioning techniques. Furthermore, we present a novel technique to construct nonuniform grids in the reduced domain, which is based on statistical information. Finally, we discuss numerical examples to illustrate the effectiveness of the proposed methods for PDEs in two space dimensions.