Bader, Eduard; Kärcher, Mark; Grepl, Martin A.; Veroy, Karen Certified reduced basis methods for parametrized distributed elliptic optimal control problems with control constraints. (English) Zbl 1426.49029 SIAM J. Sci. Comput. 38, No. 6, A3921-A3946 (2016). Summary: In this paper, we employ the reduced basis method for the efficient and reliable solution of parametrized optimal control problems governed by scalar coercive elliptic partial differential equations. We consider the standard linear-quadratic problem setting with distributed control and unilateral control constraints. For this problem class, we propose two different reduced basis approximations and associated error estimation procedures. In our first approach, we directly consider the resulting optimality system, introduce suitable reduced basis approximations for the state, adjoint, control, and Lagrange multipliers, and use a projection approach to bound the error in the reduced optimal control. For our second approach, we first reformulate the optimal control problem using a slack variable, then develop a reduced basis approximation for the slack problem by suitably restricting the solution space, and finally derive error bounds for the slack based optimal control. We discuss benefits and drawbacks of both approaches and substantiate the comparison by presenting numerical results for several model problems. Cited in 19 Documents MSC: 49M25 Discrete approximations in optimal control 65K10 Numerical optimization and variational techniques 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 93C20 Control/observation systems governed by partial differential equations Keywords:optimal control; reduced basis method; a posteriori error estimation; model order reduction; parameter-dependent systems; partial differential equations; elliptic problems; control constraints Software:redbKIT PDFBibTeX XMLCite \textit{E. Bader} et al., SIAM J. Sci. Comput. 38, No. 6, A3921--A3946 (2016; Zbl 1426.49029) Full Text: DOI References: [1] J. A. Atwell and B. B. King, {\it Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations}, Math. Comput. Modelling, 33 (2001), pp. 1-19, . · Zbl 0964.93032 [2] E. 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