  This paper addresses the optimal control problem known as the Linear
Quadratic Regulator in the case when the dynamics are unknown. We propose a
multi-stage procedure, called Coarse-ID control, that estimates a model from a
few experimental trials, estimates the error in that model with respect to the
truth, and then designs a controller using both the model and uncertainty
estimate. Our technique uses contemporary tools from random matrix theory to
bound the error in the estimation procedure. We also employ a recently
developed approach to control synthesis called System Level Synthesis that
enables robust control design by solving a convex optimization problem. We
provide end-to-end bounds on the relative error in control cost that are nearly
optimal in the number of parameters and that highlight salient properties of
the system to be controlled such as closed-loop sensitivity and optimal control
magnitude. We show experimentally that the Coarse-ID approach enables efficient
computation of a stabilizing controller in regimes where simple control schemes
that do not take the model uncertainty into account fail to stabilize the true
system.
