  In this work, we investigate the problem of simultaneously learning and
controlling a system subject to adversarial choices of disturbances and system
parameters. We study the problem for a scalar system with $l_\infty$-norm
bounded disturbances and system parameters constrained to lie in a known
bounded convex polytope. We present a controller that is globally stabilizing
and gives continuously improving bounds on the worst-case state deviation. The
proposed controller simultaneously learns the system parameters and controls
the system. The controller emerges naturally from an optimization problem and
balances exploration and exploitation in such a way that it is able to
efficiently stabilize unstable and adversarial system dynamics. Specifically,
if the controller is faced with large uncertainty, the initial focus is on
exploration, retrieving information about the system by applying state-feedback
controllers with varying gains and signs. In a pre-specified bounded region
around the origin, our control strategy can be seen as passive in the sense
that it learns very little information. Only once the noise and/or system
parameters act in an adversarial way, leading to the state exiting the
aforementioned region for more than one time-step, our proposed controller
behaves aggressively in that it is guaranteed to learn enough about the system
to subsequently robustly stabilize it. We end by demonstrating the efficiency
of our methods via numerical simulations.
