There are many infinite horizon optimal problems in economic models. In such problems, the transversality condition may not be verified, as shown by Halkin'sexample. But we prove another property: the maximum of the Hamiltonian converges to zero when time goes to infinity. And, if along the optimal trajectory, after some time, changes of speed (by controls) in all directions at a given level are possible, then the transversality condition is verified. Examples show that the additional property proved here (i) allows exclusion of nonoptimal trajectories which verify the usual necessary conditions in an infinite horizon; (ii) is a result directly useful in some economic studies.