A Certified Reduced Basis Approach for Parametrized Linear–Quadratic Optimal Control Problems with Control Constraints (two-sided)

Abstract In this talk, we consider the efficient and reliable solution of distributed optimal control problems governed by parametrized elliptic partial differential equations involving constraints on the control. The reduced basis method is used as a low-dimensional surrogate model to solve the optimal control problem. To this end, we introduce reduced basis spaces not only for the state and adjoint variable but also for the distributed control variable and propose rigorous error bounds for the error in the optimal control. The reduced basis optimal control problem and associated a posteriori error bounds can be efficiently evaluated in an offline-online computational procedure, thus making our approach relevant in the many-query or real-time context. We present numerical results for a model problem to show the validity of our approach.

[1]  Karen Veroy,et al.  A slack approach to reduced-basis approximation and error estimation for variational inequalities , 2016 .

[2]  Karen Veroy,et al.  A Duality Approach to Error Estimation for Variational Inequalities , 2014 .

[3]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[4]  Federico Negri,et al.  Reduced basis method for parametrized optimal control problems governed by PDEs , 2011 .

[5]  Barbara I. Wohlmuth,et al.  A Reduced Basis Method for Parametrized Variational Inequalities , 2012, SIAM J. Numer. Anal..

[6]  Gianluigi Rozza,et al.  Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations , 2015, Comput. Math. Appl..

[7]  Gianluigi Rozza,et al.  Reduced Basis Method for Parametrized Elliptic Optimal Control Problems , 2013, SIAM J. Sci. Comput..

[8]  Arnold Reusken,et al.  Certified reduced basis methods for parametrized PDE-constrained optimization problems , 2016 .

[9]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[10]  Mark Kärcher,et al.  A certified reduced basis method for parametrized elliptic optimal control problems , 2014 .

[11]  Stefan Volkwein,et al.  POD a-posteriori error estimates for linear-quadratic optimal control problems , 2009, Comput. Optim. Appl..

[12]  D. Rovas,et al.  A blackbox reduced-basis output bound method for noncoercive linear problems , 2002 .

[13]  Gianluigi Rozza,et al.  Reduction strategies for PDE-constrained oprimization problems in Haemodynamics , 2013 .

[14]  Karen Veroy,et al.  Certified Reduced Basis Methods for Parametrized Distributed Optimal Control Problems , 2014 .