A method of a-posteriori error estimation with application to proper orthogonal decomposition

We consider the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: Let an arbitrary admissible control function be given. How far is it from the next locally optimal control? Under natural assumptions including a second-order sufficient optimality condition for the (unknown) locally optimal control, we estimate the distance between the two controls. To do this, we need some information on the lowest eigenvalue of the reduced Hessian. We apply this technique to a model reduced optimal control problem obtained by proper orthogonal decomposition (POD). The distance between a local solution of the reduced problem to a local solution of the original problem is estimated. Résumé. ... 1991 Mathematics Subject Classification. 49K20, 35J61, 35K58. The dates will be set by the publisher. Introduction In this paper, we focus on the following question for a class of optimal control problems for semilinear parabolic equations: Suppose that a numerical approximation ũ for a locally optimal control is given. For instance, it might have been obtained by a numerical optimization method or as the solution to some reduced order optimization model. How far is this control from the nearest locally optimal control ū? Of course, we have to assume that such a solution ū exists in a neighborhood of ũ. Moreover, ũ should already be sufficiently close to ū. The question is to quantify the error ‖ũ− ū‖ in an appropriate norm. In principle, this is not a paper on proper orthogonal decomposition (POD). Our estimation method is not restricted to any specific method of numerical approximation for ũ. Our primary goal is a numerical application of a perturbation method used by Arada et al. [2] in the context of FEM approximations of elliptic optimal control problems. The main idea of this method goes back to Dontchev et al. [7] and Malanowski and Maurer [18], who introduced it for the a priori error analysis of optimal control problems governed by ordinary differential equations.