  We consider control from the perspective of competitive analysis. Unlike much
prior work on learning-based control, which focuses on minimizing regret
against the best controller selected in hindsight from some specific class, we
focus on designing an online controller which competes against a clairvoyant
offline optimal controller. A natural performance metric in this setting is
competitive ratio, which is the ratio between the cost incurred by the online
controller and the cost incurred by the offline optimal controller. Using
operator-theoretic techniques from robust control, we derive a computationally
efficient state-space description of the the controller with optimal
competitive ratio in both finite-horizon and infinite-horizon settings. We
extend competitive control to nonlinear systems using Model Predictive Control
(MPC) and present numerical experiments which show that our competitive
controller can significantly outperform standard $H_2$ and $H_{\infty}$
controllers in the MPC setting.
