Optimal control on graphs: existence, uniqueness, and long-term behavior

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The literature on continuous-time stochastic optimal control seldom deals with the case of discrete state spaces.

• In this paper, we provide a general framework for the optimal control of continuous-time Markov chains on finite graphs.
• In particular, we provide results on the long-term behavior of value functions and optimal controls, along with results on the associated ergodic Hamilton-Jacobi equation.