We consider the following problem of error estimation for the optimal control ofnonlinear parabolic partial differential equations: let an arbitrary admissible controlfunction be given. How far is it from the next locally optimal control? Under naturalassumptions including a second-order sufficient optimality condition for the (unknown)locally optimal control, we estimate the distance between the two controls. To do this, weneed some information on the lowest eigenvalue of the reduced Hessian. We apply thistechnique to a model reduced optimal control problem obtained by proper orthogonaldecomposition (POD). The distance between a local solution of the reduced problem to alocal solution of the original problem is estimated.

An optimal control problem governed by a bilinear elliptic equation is considered. Thisproblem is solved by the sequential quadratic programming (SQP) method in aninfinite-dimensional framework. In each level of this iterative method the solution oflinear-quadratic subproblem is computed by a Galerkin projection using proper orthogonaldecomposition (POD). Thus, an approximate (inexact) solution of the subproblem isdetermined. Based on a POD a-posteriori error estimator developed byTröltzsch and Volkwein [Comput. Opt. Appl. 44 (2009) 83–115]the difference of the suboptimal to the (unknown) optimal solution of the linear-quadraticsubproblem is estimated. Hence, the inexactness of the discrete solution is controlled insuch a way that locally superlinear or even quadratic rate of convergence of the SQP isensured. Numerical examples illustrate the efficiency for the proposed approach.

The construction of reduced order models for dynamical systems usingproper orthogonal decomposition (POD) is based on the informationcontained in so-called snapshots. These provide the spatialdistribution of the dynamical system at discrete time instances.This work is devoted to optimizing the choice of these timeinstances in such a manner that the error between the POD-solutionand the trajectory of the dynamical system is minimized. First andsecond order optimality systems are given. Numerical examplesillustrate that the proposed criterion is sensitive with respect tothe choice of the time instances and further they demonstrate thefeasibility of the method in determining optimal snapshot locationsfor concrete diffusion equations.

Optimal control problems for the heat equation with pointwisebilateral control-state constraints are considered. A locallysuperlinearly convergent numerical solution algorithm is proposedand its mesh independence is established. Further, for theefficient numerical solution reduced space and Schur complementbased preconditioners are proposed which take into account theactive and inactive set structure of the problem. The paper endsby numerical tests illustrating our theoretical findings andcomparing the efficiency of the proposed preconditioners.

Proper orthogonal decomposition (POD) is apowerful technique for model reduction of non-linear systems. Itis based on a Galerkin type discretization with basis elementscreated from the dynamical system itself. In the context ofoptimal control this approach may suffer from the fact that thebasis elements are computed from a reference trajectory containingfeatures which are quite different from those of the optimallycontrolled trajectory. A method is proposed which avoids thisproblem of unmodelled dynamics in the proper orthogonaldecomposition approach to optimal control. It is referred to asoptimality system proper orthogonal decomposition (OS-POD).

A Lagrange–Newton–SQP method is analyzed for the optimal control of theBurgers equation. Distributed controls are given, which are restricted bypointwise lower and upper bounds. The convergence of the method is proved inappropriate Banach spaces. This proof is based on a weak second-ordersufficient optimality condition and the theory of Newton methods forgeneralized equations in Banach spaces. For the numerical realization aprimal-dual active set strategy is applied. Numerical examples are included.