We consider an optimal infinite horizon calculus of variations problem linear with respect to the velocities. In this framework the Euler-Lagrange equation are known to be algebraic and thus no informative for the general optimal solutions. We prove that the value of the objective along the MRAPs, the curves that connect as quickly as possible the solutions of the Euler-Lagrange equation, is Lipschitz continuous and satisfies a Hamilton-Jacobi equation in some generalised sense. We derive then a sufficient condition for a MRAP to be optimal by using a transversality condition at infinity that we generalize to our non regular context. © Springer Science+Business Media, LLC. 2011.