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Turnpike theorems by a value function approach

Published online by Cambridge University Press:  15 February 2004

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

Keywords

Calculus of variationsinfinite horizonHamilton-Jacobi equationviscosity solutionsturnpike.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 1 , January 2004 , pp. 123 - 141
Copyright
© EDP Sciences, SMAI, 2004

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