  The setting of an agent making decisions under uncertainty and under dynamic
constraints is common for the fields of optimal control, reinforcement
learning, and recently also for online learning. In the online learning
setting, the quality of an agent's decision is often quantified by the concept
of regret, comparing the performance of the chosen decisions to the best
possible ones in hindsight. While regret is a useful performance measure, when
dynamical systems are concerned, it is important to also assess the stability
of the closed-loop system for a chosen policy. In this work, we show that for
linear state feedback policies and linear systems subject to adversarial
disturbances, linear regret implies asymptotic stability in both time-varying
and time-invariant settings. Conversely, we also show that bounded input
bounded state stability and summability of the state transition matrices imply
linear regret.
