Turnpike theorems by a value function approach

Alain Rapaport; Pierre Cartigny

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 10, Issue: 1, page 123-141
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Rapaport, Alain, and Cartigny, Pierre. "Turnpike theorems by a value function approach." ESAIM: Control, Optimisation and Calculus of Variations 10.1 (2010): 123-141. <http://eudml.org/doc/90716>.

@article{Rapaport2010,
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. },
author = {Rapaport, Alain, Cartigny, Pierre},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Calculus of variations; infinite horizon; Hamilton-Jacobi equation; viscosity solutions; turnpike.; Hamilton-Jacobi equation; turnpike},
language = {eng},
month = {3},
number = {1},
pages = {123-141},
publisher = {EDP Sciences},
title = {Turnpike theorems by a value function approach},
url = {http://eudml.org/doc/90716},
volume = {10},
year = {2010},
}

TY - JOUR
AU - Rapaport, Alain
AU - Cartigny, Pierre
TI - Turnpike theorems by a value function approach
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 10
IS - 1
SP - 123
EP - 141
AB - 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.
LA - eng
KW - Calculus of variations; infinite horizon; Hamilton-Jacobi equation; viscosity solutions; turnpike.; Hamilton-Jacobi equation; turnpike
UR - http://eudml.org/doc/90716
ER -

References

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  5. P. Cartigny and P. Michel, On a Sufficient Transversality Condition for Infinite Horizon Optimal Control Problems. Automatica39 (2003) 1007–1010.  
  6. C.W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources. Wiley, New York (1976).  
  7. I. Ekeland, Some Variational Problems Arising from Mathematical Economics. Springer-Verlag, Lecture Notes in Math.1330 (1986).  
  8. R.F. Hartl and G. Feichtinger, A New Sufficient Condition for Most Rapid Approach Paths. J. Optim. Theory Appl.54 (1987).  
  9. M.G. Crandall and P.-L. Lions, Viscosity Solutions of Hamilton-Jacobi Equations. Trans. Americ. Math.277 (1983) 1–42.  
  10. A. Miele, Extremization of Linear Integrals by Green's Theorem, Optimization Technics, G. Leitmann Ed. Academic Press, New York (1962) 69–98.  
  11. A. Rapaport and P. Cartigny, Théorème de l'autoroute et équation d'Hamilton-Jacobi. C.R. Acad. Sci.335 (2002) 1091–1094.  

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