  This paper proposes a data-driven framework to solve time-varying
optimization problems associated with unknown linear dynamical systems. Making
online control decisions to regulate a dynamical system to the solution of an
optimization problem is a central goal in many modern engineering applications.
Yet, the available methods critically rely on a precise knowledge of the system
dynamics, thus mandating a preliminary system identification phase before a
controller can be designed. In this work, we leverage results from behavioral
theory to show that the steady-state transfer function of a linear system can
be computed from data samples without any knowledge or estimation of the system
model. We then use this data-driven representation to design a controller,
inspired by a gradient-descent optimization method, that regulates the system
to the solution of a convex optimization problem, without requiring any
knowledge of the time-varying disturbances affecting the model equation.
Results are tailored to cost functions satisfy the Polyak-\L ojasiewicz
inequality.
