  This paper proposes a data-driven control framework to regulate an unknown,
stochastic linear dynamical system to the solution of a (stochastic) convex
optimization problem. Despite the centrality of this problem, most of the
available methods critically rely on a precise knowledge of the system dynamics
(thus requiring off-line system identification and model refinement). To this
aim, in this paper we first show that the steady-state transfer function of a
linear system can be computed directly from control experiments, bypassing
explicit model identification. Then, we leverage the estimated transfer
function to design a controller -- which is inspired by stochastic gradient
descent methods -- that regulates the system to the solution of the prescribed
optimization problem. A distinguishing feature of our methods is that they do
not require any knowledge of the system dynamics, disturbance terms, or their
distributions. Our technical analysis combines concepts and tools from
behavioral system theory, stochastic optimization with decision-dependent
distributions, and stability analysis. We illustrate the applicability of the
framework on a case study for mobility-on-demand ride service scheduling in
Manhattan, NY.
