Model reduction techniques with a-posteriori error analysis for linear-quadratic optimal control problems

The main focus of this paper is on an a-posteriori analysis for different model-order strategies applied to optimal control problems governed by linear parabolic partial differential equations. Based on a perturbation method it is deduced how far the suboptimal control, computed on the basis of the reduced-order model, is from the (unknown) exact one. For the model-order reduction, $\mathcal H_{2,\alpha}$-norm optimal model reduction (H2), balanced truncation (BT), and proper orthogonal decomposition (POD) are studied. The proposed approach is based on semi-discretization of the underlying dynamics for the state and the adjoint equations as a large scale linear time-invariant (LTI) system. This system is reduced to a lower-dimensional one using Galerkin (POD) or Petrov-Galerkin (H2, BT) projection. The size of the reduced-order system is iteratively increased until the error in the optimal control, computed with the a-posteriori error estimator, satisfies a given accuracy. The method is illustrated with numerical tests.