We consider a calculus of variations problem in infinite horizon linear with respect to the velocities. In our case the admissible curves stay in a bounded interval and we prove that the MRAP (Most Rapid Approach Pathes) from any initial conditions to the solutions of the (algebraic) Euler-Lagrange equation are optimal. We use a result on the uniqueness of the solution of a Hamilton-Jacobi equation. We propose some new and straightforward proofs. Particularly we show that boundary conditions, that are essential for the uniqueness, are satisfied under some assumptions that we detail. Finally we underline the limits for the applications (fisheries examples) of the established results