Optimal control of multiscale systems using reduced-order models

Abstract

We study optimal control of diffusions with slow and fast variables and address a questionraised by practitioners: is it possible to first eliminate the fast variables before solving theoptimal control problem and then use the optimal control computed from the reduced-order model tocontrol the original, high-dimensional system? The strategy ``first reduce, then optimize''---rather than ``firstoptimize, then reduce''---is motivated by the fact that solving optimal control problems for high-dimensionalmultiscale systems is numerically challenging and often computationally prohibitive. We state sufficient and necessaryconditions, under which the ``first reduce, then control'' strategy can be employed and discuss when itshould be avoided. We further give numerical examples that illustrate the ``first reduce, then optmize'' approach and discusspossible pitfalls.