Optimal Control Theory of the (2+1)-Dimensional BTZ Black Hole

We apply a finite-time geometric optimization framework to investigate thermal fluctuations and (non)equilibrium optimal processes in the $(2+1)$-dimensional BTZ black hole. Employing Hessian thermodynamic information metrics, we construct geodesic trajectories that define optimal protocols connecting distinct thermodynamic configurations. Finite-time state transitions are described by paths that extremize entropy production or energy dissipation, depending on the chosen thermodynamic representation. { We compare our optimization framework with a non-optimal blackbody Hawking evaporation model, revealing substantial differences between the two descriptions. Finally, we quantify the intrinsic efficiency of both types of processes in terms of the extractable rotational energy stored in the black hole configurations.} This work presents the first formulation of a geometric optimal control theory for the BTZ black hole.