# 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

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topRapaport, 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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- 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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