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Reduced Basis Method and Error Estimation for Parametrized Optimal Control Problems with Control Constraints

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Abstract

We propose a Reduced Basis method for the solution of parametrized optimal control problems with control constraints for which we extend the method proposed in Dedè, L. (SIAM J. Sci. Comput. 32:997, 2010) for the unconstrained problem. The case of a linear-quadratic optimal control problem is considered with the primal equation represented by a linear parabolic partial differential equation. The standard offline–online decomposition of the Reduced Basis method is employed with the Finite Element approximation as the “truth” one for the offline step. An error estimate is derived and an heuristic indicator is proposed to evaluate the Reduced Basis error on the optimal control problem at the online step; also, the indicator is used at the offline step in a Greedy algorithm to build the Reduced Basis space. We solve numerical tests in the two-dimensional case with applications to heat conduction and environmental optimal control problems.

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  1. Luca Dedè

    Present address: Institute for Computational Engineering and Sciences, The University of Texas at Austin, 1 University Station C0200, Austin, TX, 78712, USA

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  1. MOX—Modeling & Scientific Computing, Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, P.zza Leonardo da Vinci 32, 20133, Milano, Italy

    Luca Dedè

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Dedè, L. Reduced Basis Method and Error Estimation for Parametrized Optimal Control Problems with Control Constraints. J Sci Comput 50, 287–305 (2012). https://doi.org/10.1007/s10915-011-9483-5

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