A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD∗
1 Institut für Mathematik, Technische
Universität Berlin, 10623
2 Institut für Mathematik und Statistik, Universität Konstanz, 78457 Konstanz, Germany.
Revised: 10 July 2012
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.
Mathematics Subject Classification: 49K20 / 35J61 / 35K58
Key words: Optimal control / semilinear partial differential equations / error estimation / proper orthogonal decomposition
The first and the second author are supported by SFB 910 Control of self-organizing nonlinear systems : theoretical methods and concepts of application in Berlin. The second and the third author gratefully acknowledge the support by the German Science Fund DFG under project A posteriori-POD-Fehlerschätzer für nichtlineare Optimalsteuerprobleme bei partiellen Differentialgleichungen.
© EDP Sciences, SMAI, 2013