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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 asingular solution, in calculus of variations oroptimal control problems.For scalar calculus of variations problems in infinite horizon, linear withrespect to the derivative, we use the theory of viscosity solutions ofHamilton-Jacobi equations to obtain a unique characterization of the valuefunction.With this approach, we extend for the scalar case the classical result based onGreen theorem, when there is uniqueness of the singular solution.We provide a new necessary and sufficient condition for turnpikeoptimality, even in the presence of multiple singular solutions.

Keywords

Calculus of variationsinfinite horizonHamilton-Jacobiequationviscosity 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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References

G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag, Paris-Heidelberg-New York (1994).
D.J. Bell and D.H. Jacobson, Singular Optimal Control Problems. Academic Press, London (1975).
Blot, J. and Cartigny, P., Optimality in Infinite Horizon Variational Problems under Signs Conditions. J. Optim. Theory Appl. 106 (2000) 411–419. CrossRef
Blot, J. and Michel, P., First-Order Necessary Conditions for the Infinite-Horizon Variational Problems. J. Optim. Theory Appl. 88 (1996) 339–364. CrossRef
Cartigny, P. and Michel, P., On a Sufficient Transversality Condition for Infinite Horizon Optimal Control Problems. Automatica 39 (2003) 1007–1010. CrossRef
C.W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources. Wiley, New York (1976).
I. Ekeland, Some Variational Problems Arising from Mathematical Economics. Springer-Verlag, Lecture Notes in Math. 1330 (1986).
R.F. Hartl and G. Feichtinger, A New Sufficient Condition for Most Rapid Approach Paths. J. Optim. Theory Appl. 54 (1987).
Crandall, M.G. and Lions, P.-L., Viscosity Solutions of Hamilton-Jacobi Equations. Trans. Americ. Math. 277 (1983) 1–42. CrossRef
A. Miele, Extremization of Linear Integrals by Green's Theorem, Optimization Technics, G. Leitmann Ed. Academic Press, New York (1962) 69–98.
Rapaport, A. and Cartigny, P., Théorème de l'autoroute et équation d'Hamilton-Jacobi. C.R. Acad. Sci. 335 (2002) 1091–1094. CrossRef