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Sufficient conditions for infinite-horizon calculusof variations problems

Published online by Cambridge University Press:  15 August 2002

Joël Blot  and
Naïla Hayek
Joël Blot
Affiliation:
CERMSEM, M.S.E., Université de Paris 1 Panthéon-Sorbonne, 106-112 boulevard de l'Hôpital, 75647 Paris Cedex 13, France; [email protected].
Naïla Hayek
Affiliation:
L.I.B.R.E., Faculté de Droit et des Sciences Économiques, Université de Franche-Comté, avenue de l'Observatoire, 25030 Besançon Cedex, France, and CERMSEM, M.S.E., Université de Paris 1 Panthéon - Sorbonne, 106-112 boulevard de l'Hôpital, 75647 Paris Cedex 13, France; [email protected].
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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.

Keywords

Calculus of variationsinfinite-horizon problemsoptimal growth theory.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 5 , 2000 , pp. 279 - 292
Copyright
© EDP Sciences, SMAI, 2000

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References

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).
Blot, J. and Hayek, N., Second-Order Necessary Conditions for the Infinite-Horizon Variational Problems. Math. Oper. Res. 21 (1996) 979-990. CrossRef
Blot, J. and Michel, Ph., First-Order Necessary Conditions for the Infinite-Horizon Variational Problems. J. Optim. Theory Appl. 88 (1996) 339-364. CrossRef
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).
Hartl, R.F., Sethi, S.P. and Vickson, R.G., Survey, A of the Maximum Principles for Optimal Control Problems with State Constraints. SIAM Rev. 37 (1995) 181-218. CrossRef
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.
Nehari, Z., Sufficient Conditions in the Calculus of Variations and in the Theory of Optimal Control. Proc. Amer. Math. Soc. 39 (1973) 535-539. CrossRef
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).
Sorger, G., Sufficient Conditions for Nonconvex Control Problems with State Constraints. J. Optim. Theory Appl. 62 (1989) 289-310. CrossRef
J.L. Troutman, Variational Calculus with Elementary Convexity. Springer-Verlag, New York (1983).
Zeidan, V., First and Second Order Sufficient Conditions for Optimal Control and Calculus of Variations. Appl. Math. Optim. 11 (1984) 209-226. CrossRef
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.