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This paper considers the problem of regulating a linear dynamical system tothe solution of a convex optimization problem with an unknown orpartially-known cost. We design a data-driven feedback controller - based ongradient flow dynamics - that (i) is augmented with learning methods toestimate the cost function based on infrequent (and possibly noisy) functionalevaluations; and, concurrently, (ii) is designed to drive the inputs andoutputs of the dynamical system to the optimizer of the problem. We derivesufficient conditions on the learning error and the controller gain to ensurethat the error between the optimizer of the problem and the state of theclosed-loop system is ultimately bounded; the error bound accounts for thefunctional estimation errors and the temporal variability of the unknowndisturbance affecting the linear dynamical system. Our results directly lead toexponential input-to-state stability of the closed-loop system. The proposedmethod and the theoretical bounds are validated numerically.

Data-Enabled Gradient Flow as Feedback Controller: Regulation of Linear Dynamical Systems to Minimizers of Unknown Functions