# Sufficient conditions for infinite-horizon calculus of variations problems

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

- Volume: 5, page 279-292
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topBlot, Joël, and Hayek, Naïla. "Sufficient conditions for infinite-horizon calculus of variations problems." ESAIM: Control, Optimisation and Calculus of Variations 5 (2010): 279-292. <http://eudml.org/doc/197279>.

@article{Blot2010,

abstract = {
After a brief survey of the literature about sufficient conditions, we give
different sufficient conditions of optimality for infinite-horizon calculus
of variations problems in the general (non concave) case. Some sufficient
conditions are obtained by extending to the infinite-horizon setting the
techniques of extremal fields. Others are obtained in a special
qcase of reduction to finite horizon. The last result uses auxiliary
functions. We treat five notions of optimality. Our problems are essentially
motivated by macroeconomic optimal growth models.
},

author = {Blot, Joël, Hayek, Naïla},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Calculus of variations; infinite-horizon problems;
optimal growth theory.; optimal growth theory; infinite horizon problems; sufficient conditions of optimality; techniques of extremal fields},

language = {eng},

month = {3},

pages = {279-292},

publisher = {EDP Sciences},

title = {Sufficient conditions for infinite-horizon calculus of variations problems},

url = {http://eudml.org/doc/197279},

volume = {5},

year = {2010},

}

TY - JOUR

AU - Blot, Joël

AU - Hayek, Naïla

TI - Sufficient conditions for infinite-horizon calculus of variations problems

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 5

SP - 279

EP - 292

AB -
After a brief survey of the literature about sufficient conditions, we give
different sufficient conditions of optimality for infinite-horizon calculus
of variations problems in the general (non concave) case. Some sufficient
conditions are obtained by extending to the infinite-horizon setting the
techniques of extremal fields. Others are obtained in a special
qcase of reduction to finite horizon. The last result uses auxiliary
functions. We treat five notions of optimality. Our problems are essentially
motivated by macroeconomic optimal growth models.

LA - eng

KW - Calculus of variations; infinite-horizon problems;
optimal growth theory.; optimal growth theory; infinite horizon problems; sufficient conditions of optimality; techniques of extremal fields

UR - http://eudml.org/doc/197279

ER -

## References

top- V.M. Alexeev, V.M. Tikhomirov and S.V. Fomin, Commande optimale, French translation. Mir, Moscow (1982).
- K.J. Arrow, Applications of Control Theory to Economic Growth. Math. of the Decision Sciences, edited by G.B. Dantzig and A.F. Veinott Jr. (1968).
- J. Blot and P. Cartigny, Optimality in Infinite-Horizon Problems under Signs Conditions. J. Optim. Theory Appl. (to appear).
- J. Blot and N. Hayek, Second-Order Necessary Conditions for the Infinite-Horizon Variational Problems. Math. Oper. Res.21 (1996) 979-990.
- J. Blot and Ph. Michel, First-Order Necessary Conditions for the Infinite-Horizon Variational Problems. J. Optim. Theory Appl.88 (1996) 339-364.
- N. Bourbaki, Fonctions d'une variable réelle. Hermann, Paris (1976).
- D.A. Carlson, A.B. Haurie and A. Leizarowitz, Infinite Horizon Optimal Control, Deterministic and Stochastic Systems, Second Edition. Springer-Verlag, Berlin (1991).
- H. Cartan, Calcul Différentiel. Hermann, Paris (1967).
- L. Cesari, Optimization Theory and Applications: Problems with Ordinary Differential Equations. Springer-Verlag, New York (1983).
- J. Dugundji, Topology. Allyn and Bacon, Boston (1966).
- G.E. Ewing, Calculus of Variations, with Applications. Dover Pub. Inc., New York (1985).
- W.H. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control. Springer-Verlag, New York (1975).
- W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions. Springer-Verlag, New York (1993).
- M. Giaquinta and S. Hildebrandt, Calculus of Variations I. Springer-Verlag, Berlin (1996).
- C. Godbillon, Éléments de topologie algébrique. Hermann, Paris (1971).
- R.F. Hartl, S.P. Sethi and R.G. Vickson, A Survey of the Maximum Principles for Optimal Control Problems with State Constraints. SIAM Rev.37 (1995) 181-218.
- M.H. Hestenes, Calculus of Variations and Optimal Control Theory. Robert E. Krieger Publ. Comp., Huntington, N.Y. (1980).
- G. Leitman and H. Stalford, A Sufficiency Theorem for Optimal Control. J. Optim. Theory Appl. VIII (1971) 169-174.
- D. Leonard and N.V. Long, Optimal Control Theory and Static Optimization in Economics. Cambridge University Press, New York (1992).
- O.L. Mangasarian, Sufficient Conditions for the Optimal Control of Nonlinear Systems. SIAM J. Control IV (1966) 139-152.
- Z. Nehari, Sufficient Conditions in the Calculus of Variations and in the Theory of Optimal Control. Proc. Amer. Math. Soc.39 (1973) 535-539.
- L. Pontryagin, V. Boltyanskii, R. Gramkrelidze and E. Mitchenko, Théorie Mathématique des Processus Optimaux, French Edition. Mir, Moscow (1974).
- H. Sagan, Introduction to the Calculus of Variations. McGraw-Hill, New York (1969).
- Th. Sargent, Macroeconomic Theory, Second Edition. Academic Press, New York (1986).
- A. Seierstad and K. Sydsaeter, Sufficient Conditions in Optimal Control Theory, Internat. Econom. Rev. 18 (1977).
- L. Schwartz, Cours d'Analyse de l'École Polytechnique, Tome 1. Hermann, Paris (1967).
- L. Schwartz, Topologie Générale et Analyse Fonctionnelle. Hermann, Paris (1970).
- G. Sorger, Sufficient Conditions for Nonconvex Control Problems with State Constraints. J. Optim. Theory Appl.62 (1989) 289-310.
- J.L. Troutman, Variational Calculus with Elementary Convexity. Springer-Verlag, New York (1983).
- V. Zeidan, First and Second Order Sufficient Conditions for Optimal Control and Calculus of Variations. Appl. Math. Optim.11 (1984) 209-226.
- A.J. Zaslavski, Existence and Structure of Optimal Solutions of Variational Problems, Recent Developments in Optimization Theory and Nonlinear Analysis, edited by Y. Censor and S. Reich. Amer. Math. Soc. Providence, Rhode Island (1997) 247-278.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.