  Recent developments in cyber-physical systems and event-triggered control
have led to an increased interest in the impact of sparse disturbances on
dynamical processes. We study Linear Quadratic Regulator (LQR) control under
sparse disturbances by analyzing three distinct policies: the blind online
policy, the disturbance-aware policy, and the optimal offline policy. We derive
the two-dimensional recurrence structure of the optimal disturbance-aware
policy, under the assumption that the controller has information about future
disturbance values with only a probabilistic model of their locations in time.
Under mild conditions, we show that the disturbance-aware policy converges to
the blind online policy if the number of disturbances grows sublinearly in the
time horizon. Finally, we provide a finite-horizon regret bound between the
blind online policy and optimal offline policy, which is proven to be quadratic
in the number of disturbances and in their magnitude. This provides a useful
characterization of the suboptimality of a standard LQR controller when
confronted with unexpected sparse perturbations.
