Hernández-Santamaría V., Lazar M., Zuazua E. Greedy optimal control for elliptic problems and itsapplication to turnpike problemsNUMER MATH. Vol. 141, No. 2, pp.455-493. DOI: 10.1007/s00211-018-1005-z
Abstract: We adapt and apply greedy methods to approximate in an efficient way the optimalcontrols for parameterized elliptic control problems. Our results yield an optimal approximation procedure that, in particular, performs better than simply sampling theparameter-space to compute controls for each parameter value. The same method canbe adapted for parabolic control problems, but this leads to greedy selections of therealizations of the parameters that depend on the initial datum under consideration.The turnpike property (which ensures that parabolic optimal control problems behavenearly in a static manner when the control horizon is long enough) allows usingthe elliptic greedy choice of the parameters in the parabolic setting too. We presentvarious numerical experiments and an extensive discussion of the efficiency of ourmethodology for parabolic control and indicate a number of open problems arisingwhen analyzing the convergence of the proposed algorithms.