Turnpike theorems by a value function approach

¹ UMR Analyse des Systèmes et Biométrie, 2, place Viala, 34060 Montpellier, France; rapaport@ensam.inra.fr.
² GREQAM, université de la Méditerranée, 2, rue de la Vieille Charité, 13002 Marseille, France; cartigny@ehess.cnrs-mrs.fr.

Received: 8 January 2003
Revised: 6 August 2003

Abstract

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 function. 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.