  We discuss connections between sequential system identification and control
for linear time-invariant systems, often termed indirect data-driven control,
as well as a contemporary 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 a
plausible hypothesis explaining the remarkable empirical performance of direct
methods on nonlinear systems.
