  This paper considers the problem of controlling a dynamical system when the
state cannot be directly measured and the control performance metrics are
unknown or partially known. In particular, we focus on the design of
data-driven controllers to regulate a dynamical system to the solution of a
constrained convex optimization problem where: i) the state must be estimated
from nonlinear and possibly high-dimensional data; and, ii) the cost of the
optimization problem -- which models control objectives associated with inputs
and states of the system -- is not available and must be learned from data. We
propose a data-driven feedback controller that is based on adaptations of a
projected gradient-flow method; the controller includes neural networks as
integral components for the estimation of the unknown functions. Leveraging
stability theory for perturbed systems, we derive sufficient conditions to
guarantee exponential input-to-state stability (ISS) of the control loop. In
particular, we show that the interconnected system is ISS with respect to the
approximation errors of the neural network and unknown disturbances affecting
the system. The transient bounds combine the universal approximation property
of deep neural networks with the ISS characterization. Illustrative numerical
results are presented in the context of control of robotics and epidemics.
