Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T07:54:14.253Z Has data issue: false hasContentIssue false
  • English
  • Français

Local Minimality of a Lipschitz Extremal

Published online by Cambridge University Press:  20 November 2018

Vera Zeidan
Vera Zeidan*
Affiliation:
Department of Mathematics Michigan State University East Lansing, MI USA 48824
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper the question of weak and strong local optimality of a Lipschitz (as opposed to C1 ) extremal is addressed. We show that the classical Jacobi sufficient conditions can be extended to the case of Lipschitz candidates. The key idea for this achievement lies in proving that the “generalized” strengthened Weierstrass condition is equivalent to the existence of a “feedback control” function at which the maximum in the “true” Hamiltonian is attained. Then the Hamilton-Jacobi approach is pursued in order to conclude the result.

Keywords

Primary:49B10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 2 , 01 April 1992 , pp. 436 - 448
Copyright
Copyright © Canadian Mathematical Society 1992

References

1. Akhiezer, N.I., The calculus of variations. Blaisdell Publishing, Boston, 1962.Google Scholar
2. Ball, J. and Mizel, V., One-dimensional variational problems whose minimizers do not satisfy the Euler- Lagrange equation, Arch. Rat. Mech. Anal. 90(1985), 325–388.Google Scholar
3. Bolza, O., Lectures on the calculus of variations. University of Chicago Press, 1904. reprinted by Chelsea Press, 1961.Google Scholar
4. Caratheodory, C., Calculus of variations and partial differential equations of the first order. (Holden-Day, 1965. 1967); reprinted by Chelsea Press, 1982.Google Scholar
5. Cesari, L., Optimization—theory and applications. Springer-Verlag, New York, 1983.Google Scholar
6. Clarke, F.H., Optimization andnonsmooth analysis. Wiley and Sons, New York 1983.Google Scholar
7. Clarke, F.H., Methods of dynamic and Nonsmooth optimization. CBMS-NSF5, Society for Industrial and Applied Mathematics, 1989.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., Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Amer. Math. Soc. 289(1985), 73–98.Google Scholar
10. Clarke, F.H. and Loewen, P.D., Intermediate existence and regularity in the calculus of variations, Appl. Math. Letters 1(1988), 193–195.Google Scholar
11. Clarke, F.H., An intermediate existence theory in the calculus of variations, Annali Scuola Norm. Sup., to appear.Google Scholar
12. Clarke, F.H. and Zeidan, V., Sufficiency and the Jacobi condition in the calculus of variations, Can. J. Math. 38(1986)1199–1209.Google Scholar
13. Ewing, G.M., Calculus of variations with applications. W.W. Norton and Co. New York, 1969.Google Scholar
14. Fleming, W.H. and Rishel, R.W., Deterministic and stochastic optimal control. Springer-Verlag, New York, 1975.Google Scholar
15. Gelfand, I. M. and Fomin, S.V., Calculus of variations. Prentice-Hall, Inc., New Jersey, 1963.Google Scholar
16. Hestenes, M.R., Calculus of variations and optimal control theory. Wiley and Sons, New York, 1966.Google Scholar
17. Leitmann, G., The calculus of variations and optimal control. Plenum Press, New York, 1981.Google Scholar
18. Loewen, P.D., Second-order sufficiency criteria and local convexity for equivalent problems in the calculus of variations, J. Math. Anal. Appl. 146(1990), 512–522.Google Scholar
19. Reid, W.T., Ordinary differential equations. John Wiley & Sons, New York, 1971.Google Scholar
20. Rockafellar, R.T., Existence theorems for general control problems of Bolza and Lagrange, Adv. in Math. 15(1975), 312–333.Google Scholar
21. Sagan, H., Introduction to the calculus of variations. McGraw-Hill, New York, 1969.Google Scholar
22. Warga, J., Optimal control of differential and functional equations. Academic Press, New York, 1972.Google Scholar
23. Zeidan, V., Sufficient conditions for the generalized problem of Bolza, Trans. Amer. Math. Soc. 275(1983), 561–586.Google Scholar
24. Zeidan, V., A modified Hamilton-Jacobi approach in the generalized problem of Bolza, Appl. Math. Optim. 11(1984), 97–109.Google Scholar
25. 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
26. Zeidan, V., Sufficiency conditions with minimum regularity assumptions, Appl. Math. Optim. 20(1989), 19-31.Google Scholar
27. Zeidan, V. and Zezza, P., The Jacobi necessary and sufficient condition for Lipschitz extrema, Bolletino dell'Unione Matematica Italiana VII. Series B4, 2(1990), 275-284.Google Scholar