Geodesic
A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 2, pp. 555-581

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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.

DOI : 10.1051/m2an/2012037
Classification : 49K20, 35J61, 35K58
Keywords: optimal control, semilinear partial differential equations, error estimation, proper orthogonal decomposition 
@article{M2AN_2013__47_2_555_0,
     author = {Kammann, Eileen and Tr\"oltzsch, Fredi and Volkwein, Stefan},
     title = {\protect\emph{A posteriori }error estimation for semilinear parabolic optimal control problems with application to model reduction by {POD}},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {555--581},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {2},
     year = {2013},
     doi = {10.1051/m2an/2012037},
     zbl = {1282.49021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2012037/}
}
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Kammann, Eileen; Tröltzsch, Fredi; Volkwein, Stefan. A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 2, pp. 555-581. doi: 10.1051/m2an/2012037

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