  We discuss connections between sequential system identification and control
for linear time-invariant systems, which we term indirect data-driven control,
as well as a direct data-driven control approach seeking an optimal decision
compatible with recorded data assembled in a Hankel matrix and robustified
through suitable regularizations. We formulate these two problems in the
language of behavioral systems theory and parametric mathematical programs, and
we bridge them through a multi-criteria formulation trading off system
identification and control objectives. We illustrate our results with two
methods from subspace identification and control: namely, subspace predictive
control and low-rank approximation which constrain trajectories to be
consistent with a non-parametric predictor derived from (respectively, the
column span of) a data Hankel matrix. In both cases we conclude that direct and
regularized data-driven control can be derived as convex relaxation of the
indirect approach, and the regularizations account for an implicit
identification step. Our analysis further reveals a novel regularizer and sheds
light on the remarkable performance of direct methods on nonlinear systems.
