  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.
