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Sufficiency and the Jacobi Condition in the Calculus of Variations

Published online by Cambridge University Press:  20 November 2018

Frank H. Clarke  and
Vera Zeidan
Frank H. Clarke
Affiliation:
Université de Montréal, Montréal, Québec
Vera Zeidan
Affiliation:
University of Alberta, Edmonton, Alberta
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Besides stating the problem and the results, we shall give in this section a brief overview of the classical necessary and sufficient conditions in the calculus of variations, in order to clearly situate the contribution of this article.

1.1 The problem. We are given an interval [a, b], two points xa, xb in Rn, and a function L (the Lagrangian) mapping [a, b] × Rn × Rn to R. The basic problem in the calculus of variations, labeled (P), is that of minimizing the functional

over some class X of functions x and subject to the constraints x(a) = xa, x(b) = xb. Let us take for now the class X of functions to be the continuously differentiable mappings from [a, b] to Rn; we call such functions smooth arcs.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 5 , 01 October 1986 , pp. 1199 - 1209
Copyright
Copyright © Canadian Mathematical Society 1986

References

1. Adams, R. A. and Clarke, F. H., Gross's logarithmic Sobolev inequality: A simple proof, Amer. J. Math. 101 (1979), 1265–1269.Google Scholar
2. Akhiezer, N. I., The calculus of variations (Blaisdell Publishing, Boston, 1962).Google Scholar
3. Bolza, O., Lectures on the calculus of variations (University of Chicago Press, 1904); (Reprinted by Chelsea Press, 1961).Google Scholar
4. Carathéodory, C., Calculus of variations and partial differential equations of the first order, two volumes (Holden-Day, 1965, 1967); (Reprinted by Chelsea Press, 1982).Google Scholar
5. Cesari, L., Optimization-theory and applications (Springer-Verlag, New York, 1983).CrossRefGoogle Scholar
6. Clark, C. W., Clarke, F. H. and Munro, G. R., The optimal exploitation of renewable resource stocks, Econometrica 47 (1979), 25–47.Google Scholar
7. Clarke, F. H., Optimization and nonsmooth analysis (Wiley and Sons, New York, 1983).Google Scholar
8. Clarke, F. H. and Vinter, R. B., On the conditions under which the Euler equation or the maximum principle hold, Appl. Math. Optim. 12 (1984), 73–79.Google Scholar
9. Clarke, F. H. and Vinter, R. B., Regularity properties of solutions to the basic problem in the calculus of variations. Trans. Amer. Math. Soc. 289 (1985), 73–98.Google Scholar
10. Ewing, G. M., Calculus of variations with applications (W. W. Norton and Co., New York, 1969).Google Scholar
11. Fleming, W. H. and Rishel, R. W., Deterministic and stochastic optimal control (Springer-Verlag, New York, 1975).CrossRefGoogle Scholar
12. Gelfand, I. M. and Fomin, S. V., Calculus of variations (Prentice-Hall, Inc., New Jersey, 1963).Google Scholar
13. Hestenes, M. R., Calculus of variations and optimal control theory (Wiley and Sons, New York, 1966).Google Scholar
14. Leitmann, G., The calculus of variations and optimal control (Plenum Press, New York, 1981).CrossRefGoogle Scholar
15. Reid, W. T., Riccati differential equations (Academic Press, New York, 1972).Google Scholar
16. Sagan, H., Introduction to the calculus of variations (McGraw-Hill, New York, 1969).Google Scholar
17. Young, L. C., Calculus of variations and optimal control theory (Saunders, Philadelphia, 1969).Google Scholar
18. Zeidan, V., Sufficient conditions for the generalized problem of Bolza, Trans. Amer. Math. Soc. 275 (1983), 561–586.Google Scholar
19. Zeidan, V., A modified Hamilton-Jacobi approach in the generalized problem of Bolza, Appl. Math. Optim. 11 (1984), 97–109.Google Scholar
20. Zeidan, V., First and second order sufficient conditions for optimal control and the calculus of variations, Appl. Math. Optim. 11 (1984), 209–226.Google Scholar