Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T09:46:58.107Z Has data issue: false hasContentIssue false

The Erdmann Condition and Hamiltonian Inclusions in Optimal Control and 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]

Extract

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.

Consider the basic problem in the calculus of variations, that of minimizing

1.1

over a class of functions x satisfying certain boundary conditions at 0 and 1. One of the classical first order necessary conditions for optimality is the second Erdmann condition, which asserts, in the case in which L is independent mof t, that

1.2

along any local solution x. This formula is the customary basis for solving many of the classical problems, such as the brachistochrone. When it is possible to define via the Legendre transform a Hamiltonian H(t, x, p) corresponding to L, the second Erdmann condition, again in the autonomous case, is the assertion that

1.3

a relation which always evokes classical Hamiltonian mechanics and conservation laws.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Boltjanksii, V. G., The maximum principle for problems of optimal steering, Differencial'nye Uravnenij. 9 (1973), 13631370.Google Scholar
2. Clarke, F. H., Generalized gradients and applications, Trans. Amer. Soc. 205 (1975), 247262.Google Scholar
3. Clarke, F. H., The Euler-Lagrange differential inclusion, J. Differential Equations 19 (1975), 8090.Google Scholar
4. Clarke, F. H., Necessary conditions for a general control problem, in Calculus of variations and control theory, edited by D. Russell (Mathematics Research Center, Pub. No. 36, University of Wisconsin, September 1975), Academic Press (1976).Google Scholar
5. Clarke, F. H., On the inverse function theorem, Pacific J. Math. 64 (1976), 98102.Google Scholar
6. Clarke, F. H., The maximum principle under minimal hypotheses, SIAM J. Control and Optimizatio. 14 (1976), 10781091.Google Scholar
7. Clarke, F. H., The generalized problem of Bolza, SIAM Journal of Control and Optimization 14 (1976), 682699.Google Scholar
8. Clarke, F. H., A New approach to Lagrange multipliers, Math. Operations Research (1976), 165174.Google Scholar
9. Clarke, F. H., Extremal arcs and extended Hamiltonian systems, Trans. Amer. Math. Soc. 231 (1977), 349367.Google Scholar
10. Clarke, F. H., Generalized gradients of Lipschitz functionals, Tech. Report, Math. Res. Center, #1687 (1976). To appear in Advances in Math.Google Scholar
11. Clarke, F. H., Nonsmooth analysis and optimization, Proceedings of the International Congress of Mathematicians, Helsinki (1978).Google Scholar
12. Clarke, F. H., Periodic solutions of Hamiltonian inclusions (1978), Journal of Differential Equations, to appear.Google Scholar
13. Fedorenko, R. P., A maximum principle for differential inclusions, Z. Vycisl. Mat. i Mat. Fiz 10 (1970), 1385-1393. USSR Comp. Math, and Math. Phys. 10 (1970), 5768.Google Scholar
14. Filippov, A. F., Differential equations with discontinuous right-hand side, Amer. Math. Soc. Trans. (2. 42 (1964), 199231.Google Scholar
15. Halkin, H., Optimization without differentiability, to appear in the Proceedings of the Conference on Optimal Control Theory, Canberra, August (1977), Springer-Verlag.Google Scholar
16. Rabinowitz, P. H., Periodic solutions of Hamiltonian systems, Comm. Pure and Applied Math. 31 (1978), 157184.Google Scholar
17. Rockafellar, R. T., Conjugate convex functions in optimal control and the calculus of variations, J. Math. Anal. Appl. 34 (1970), 174222.Google Scholar
18. Rockafellar, R. T., Generalized Hamiltonian equations for convex problems of Lagrange, Pacific J. Math. 33 (1970), 411427.Google Scholar
19. Ward, A. J., On the differential structure of real functions, Proc. London Math. Soc. (2. 39 (1935), 339362.Google Scholar
20. Warga, J., Necessary conditions without differentiability assumptions in optimal control, J. Differential Equation. 18 (1975), 4162.Google Scholar
21. Weinstein, A., Periodic orbits for convex Hamiltonian systems, Annals of Math., to appear.Google Scholar