This paper addresses the problem of optimizing the uncertainty in an active simultaneous localization and mapping algorithm. This is done by designing an optimization problem that weighs the final uncertainty, the average uncertainty in the horizon considered, and the cost of the control. Using the Pontryagin minimum principle and building on [1] and [2], the optimization problem is transformed into a two-point boundary value problem that encodes necessary conditions for the input that minimizes the uncertainty. The problem is solved numerically, and several particular examples are analysed in depth.