Turnpike theorems deal with the optimality of trajectories reaching a
singular solution, in calculus of variations or
optimal control problems.
For scalar calculus of variations problems in infinite horizon, linear with
respect to the derivative, we use the theory of viscosity solutions of
Hamilton-Jacobi equations to obtain a unique characterization of the value
With this approach, we extend for the scalar case the classical result based on
Green theorem, when there is uniqueness of the singular solution.
We provide a new necessary and sufficient condition for turnpike
optimality, even in the presence of multiple singular solutions.