  In many real world problems, control decisions have to be made with limited
information. The controller may have no a priori (or even posteriori) data on
the nonlinear system, except from a limited number of points that are obtained
over time. This is either due to high cost of observation or the highly
non-stationary nature of the system. The resulting conflict between information
collection (identification, exploration) and control (optimization,
exploitation) necessitates an active learning approach for iteratively
selecting the control actions which concurrently provide the data points for
system identification. This paper presents a dual control approach where the
information acquired at each control step is quantified using the entropy
measure from information theory and serves as the training input to a
state-of-the-art Gaussian process regression (Bayesian learning) method. The
explicit quantification of the information obtained from each data point allows
for iterative optimization of both identification and control objectives. The
approach developed is illustrated with two examples: control of logistic map as
a chaotic system and position control of a cart with inverted pendulum.
