AUTHOR=Mechelli Luca , Rohleff Jan , Volkwein Stefan 

TITLE=Model order reduction for optimality systems through empirical gramians

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=Volume 9 - 2023

YEAR=2023

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2023.1144142

DOI=10.3389/fams.2023.1144142

ISSN=2297-4687

ABSTRACT=In the present article optimal control problems for linear parabolic partial differential equations (PDEs) with time-dependent coefficient functions are considered. After applying a finite element (FE) discretization the first-order sufficient optimality system can be described by a specific linear, but high-dimensional time variant (LTV) dynamical systems. To reduce the size of the LTV system, a tailored reduced order modeling based on empirical gramians is derived directly for the first-order optimality system. This new strategy is efficiently tested for a multiobjective optimization and a closed-loop optimal control problem. The results are compared with the ones obtained by applying a standard proper orthogonal decomposition (POD) reduced order modeling.It turns out that the gramian based approach works much better than a standard POD strategy.