Skip to main content Accessibility help
×
Home

Sufficient conditions for infinite-horizon calculus of variations problems

  • Joël Blot (a1) and Naïla Hayek (a2)

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.

Copyright

References

Hide All
[1] V.M. Alexeev, V.M. Tikhomirov and S.V. Fomin, Commande optimale, French translation. Mir, Moscow (1982).
[2] 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).
[3] J. Blot and P. Cartigny, Optimality in Infinite-Horizon Problems under Signs Conditions. J. Optim. Theory Appl. (to appear).
[4] Blot, J. and Hayek, N., Second-Order Necessary Conditions for the Infinite-Horizon Variational Problems. Math. Oper. Res. 21 (1996) 979-990.
[5] Blot, J. and Michel, Ph., First-Order Necessary Conditions for the Infinite-Horizon Variational Problems. J. Optim. Theory Appl. 88 (1996) 339-364.
[6] N. Bourbaki, Fonctions d'une variable réelle. Hermann, Paris (1976).
[7] D.A. Carlson, A.B. Haurie and A. Leizarowitz, Infinite Horizon Optimal Control, Deterministic and Stochastic Systems, Second Edition. Springer-Verlag, Berlin (1991).
[8] H. Cartan, Calcul Différentiel. Hermann, Paris (1967).
[9] L. Cesari, Optimization Theory and Applications: Problems with Ordinary Differential Equations. Springer-Verlag, New York (1983).
[10] J. Dugundji, Topology. Allyn and Bacon, Boston (1966).
[11] G.E. Ewing, Calculus of Variations, with Applications. Dover Pub. Inc., New York (1985).
[12] W.H. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control. Springer-Verlag, New York (1975).
[13] W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions. Springer-Verlag, New York (1993).
[14] M. Giaquinta and S. Hildebrandt, Calculus of Variations I. Springer-Verlag, Berlin (1996).
[15] C. Godbillon, Éléments de topologie algébrique. Hermann, Paris (1971).
[16] 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.
[17] M.H. Hestenes, Calculus of Variations and Optimal Control Theory. Robert E. Krieger Publ. Comp., Huntington, N.Y. (1980).
[18] G. Leitman and H. Stalford, A Sufficiency Theorem for Optimal Control. J. Optim. Theory Appl. VIII (1971) 169-174.
[19] D. Leonard and N.V. Long, Optimal Control Theory and Static Optimization in Economics. Cambridge University Press, New York (1992).
[20] O.L. Mangasarian, Sufficient Conditions for the Optimal Control of Nonlinear Systems. SIAM J. Control IV (1966) 139-152.
[21] Nehari, Z., Sufficient Conditions in the Calculus of Variations and in the Theory of Optimal Control. Proc. Amer. Math. Soc. 39 (1973) 535-539.
[22] L. Pontryagin, V. Boltyanskii, R. Gramkrelidze and E. Mitchenko, Théorie Mathématique des Processus Optimaux, French Edition. Mir, Moscow (1974).
[23] H. Sagan, Introduction to the Calculus of Variations. McGraw-Hill, New York (1969).
[24] Th. Sargent, Macroeconomic Theory, Second Edition. Academic Press, New York (1986).
[25] A. Seierstad and K. Sydsaeter, Sufficient Conditions in Optimal Control Theory, Internat. Econom. Rev. 18 (1977).
[26] L. Schwartz, Cours d'Analyse de l'École Polytechnique, Tome 1. Hermann, Paris (1967).
[27] L. Schwartz, Topologie Générale et Analyse Fonctionnelle. Hermann, Paris (1970).
[28] Sorger, G., Sufficient Conditions for Nonconvex Control Problems with State Constraints. J. Optim. Theory Appl. 62 (1989) 289-310.
[29] J.L. Troutman, Variational Calculus with Elementary Convexity. Springer-Verlag, New York (1983).
[30] Zeidan, V., First and Second Order Sufficient Conditions for Optimal Control and Calculus of Variations. Appl. Math. Optim. 11 (1984) 209-226.
[31] 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.

Keywords

Related content

Powered by UNSILO

Sufficient conditions for infinite-horizon calculus of variations problems

  • Joël Blot (a1) and Naïla Hayek (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.