Entropic balance with feedback control: information equalities and tight inequalities

We consider overdamped physical systems evolving under a feedback-controlled fluctuating potential and in contact with a thermal bath at temperature T. A Markovian description of the dynamics, which keeps only the last value of the control action, is advantageous -- both from the theoretical and the practical side -- for the entropy balance. Novel second-law equalities and bounds for the extractable work are obtained, the latter being both tighter and easier to evaluate than those in the literature based on the whole chain of controller actions. The Markovian framework also allows us to prove that the bound for the extractable work that incorporates the unavailable information is saturated in a wide class of physical systems, for error-free measurements. These results are illustrated in a model system. For imperfect measurements, there appears an interval of measurement uncertainty, including the point at which work ceases to be extracted, where the new Markovian bound is tighter than the unavailable information bound.