How to find horizon-independent optimal strategies leading off to infinity: a max-plus approach

A general problem in optimal control consists of finding a terminal reward that makes the value function independent of the horizon. Such a terminal reward can be interpreted as a max-plus eigenvector of the associated Lax-Oleinik semigroup. We give a representation formula for all these eigenvectors, which applies to optimal control problems in which the state space is non compact. This representation involves an abstract boundary of the state space, which extends the boundary of metric spaces defined in terms of Busemann functions (the horoboundary). Extremal generators of the eigenspace correspond to certain boundary points, which are the limit of almost-geodesics. We illustrate our results in the case of a linear quadratic problem.