Optimality system POD and a-posteriori error analysis for linear-quadratic problems

In this paper an abstract linear-quadratic optimal control problem governed by an evolution equation is considered. To solve this problem numerically a reduced-order approach based on proper orthogonal decomposition (POD) is applied. The error between the POD suboptimal control and the optimal control of the original problem is controlled by an a-posteriori error analysis. However, if the POD basis has bad approximation properties, a huge number of POD basis function is required to solve the reduced-order problem with the desired accuracy. To overcome this problem, optimality system POD (OS-POD) is utilized, where the POD basis is chosen with respect to the optimization criteria.