Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-06-09T09:07:34.250Z Has data issue: false hasContentIssue false

Inequality Constraints in the Calculus of Variations

Published online by Cambridge University Press:  20 November 2018

Frank H. Clarke*
Affiliation:
University of British Columbia, Vancouver, British Columbia
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.

The classical multiplier rule. The purpose of this section is to review the multiplier rule in order to place the results of this report in perspective. Let us begin by considering the following problem of Mayer in the calculus of variations: we seek to minimize

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Berkovitz, L. D., Variational methods in problems of control and programming, J. Math. Anal. Appl. 3 (1961), 145169.Google Scholar
2. Bliss, G. A., The problem of Lagrange in the calculus of variations, Amer. J. Math. 52 (1930), 673744.Google Scholar
3. Clarke, F. H., Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247262.Google Scholar
4. Clarke, F. H. The maximum principle under minimal hypotheses, SI AM J. Control and Optimization 14 (1976) 10781091.Google Scholar
5. Clarke, F. H. Necessary conditions for a general control problem, in Proceedings of the Symposium on the Calculus of Variations and Optimal Control, Russell, D. L., Editor (Mathematics Research Center, University of Wisconsin-Madison), Academic Press, N.Y. (1976).Google Scholar
6. Clarke, F. H. Necessary conditions for a general control problem, in Proceedings of the Symposium on the Calculus of Variations and Optimal Control, Russell, D. L. Necessary conditions for a general control problem, in Proceedings of the Symposium on the Calculus of Variations and Optimal Control, Russell, D. L. A new approach to Lagrange multipliers, Math, of Operations Res. 1 (1976) 165174.Google Scholar
7. Hestenes, M. R., Calculus of variations and optimal control theory (Wiley, N.Y., 1966).Google Scholar
8. Pennisi, L. L., An indirect sufficiency proof for the problem of Lagrange with differential inequalities as added side conditions, Trans. Amer. Math. Soc. 74 (1953), 177198.Google Scholar
9. Rockafellar, R. T., Convex analysis (Princeton Press, Princeton, N.J., 1970).Google Scholar
10. Rockafellar, R. T. Measurable dependence of convex sets, J. Math. Anal. Appl. 28 (1969), 425.Google Scholar
11. Sagan, H., Introduction to the calculus of variations (McGraw-Hill, N.Y., 1969).Google Scholar
12. Valentine, F. A., The problem of Lagrange with differential inequalities as added side conditions, in Contributions to the Calculus of Variations 1933–37, Department of Mathematics, University of Chicago (University of Chicago Press, Chicago).Google Scholar