  For optimal control problems on finite graphs in continuous time, the dynamic
programming principle leads to value functions characterized by systems of
nonlinear ordinary differential equations. In this paper, we consider the case
of entropic costs for which the nonlinear differential equations can be
transformed into linear ones thanks to a change of variables linked to the
classical duality between entropy and exponential. When the graph is connected,
we show that the asymptotic optimal control and the ergodic constant can be
computed very easily with classical tools of matrix analysis.
