  Learning-based techniques are increasingly effective at controlling complex
systems using data-driven models. However, most work done so far has focused on
learning individual tasks or control laws. Hence, it is still a largely
unaddressed research question how multiple tasks can be learned efficiently and
simultaneously on the same system. In particular, no efficient state space
exploration schemes have been designed for multi-task control settings. Using
this research gap as our main motivation, we present an algorithm that
approximates the smallest data set that needs to be collected in order to
achieve high control performance for multiple learning-based control laws. We
describe system uncertainty using a probabilistic Gaussian process model, which
allows us to quantify the impact of potentially collected data on each
learning-based controller. We then determine the optimal measurement locations
by solving a stochastic optimization problem approximately. We show that, under
reasonable assumptions, the approximate solution converges towards that of the
exact problem. Additionally, we provide a numerical illustration of the
proposed algorithm.
