Reduced basis method for parametrized elliptic optimal control problems

We propose a suitable model reduction paradigm -- the certied reduced basis method (RB) -- for the rapid and reliable solution of parametrized optimal control problems governed by partial dierential equations (PDEs). In particular, we develop the methodology for parametrized quadratic optimization problems with elliptic equations as constraint and infinite dimensional control variable. Firstly, we recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then, the usual ingredients of the RB methodology are called into play: a Galerkin projection onto a low dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling to perform competitive Offline-Online splitting in the computational procedure; an efficient and rigorous a posteriori error estimate on the state, control and adjoint variables as well as on the cost functional. Finally, we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique.