Optimal control on finite graphs: a reference case

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 exhibit a family of such optimal control problems for which these nonlinear equations can be transformed into linear ones thanks to a change of variables. As a consequence, the value function associated with an optimal control problem of that family can be written in (almost-)closed form and the optimal controls characterized and computed easily. Furthermore, when the graph is (strongly) connected, we show that the asymptotic analysis of such a problem -- in particular the computation of the ergodic constant -- can be carried out with classical tools of matrix analysis