Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T09:07:59.908Z Has data issue: false hasContentIssue false
  • English
  • Français

A Global algorithm for geodesics

Part of: Global differential geometry Boundary value problems Numerical methods in calculus of variations and optimal control

Published online by Cambridge University Press:  09 April 2009

Lyle Noakes
Lyle Noakes
Affiliation:
Department of Mathematics, The University of Western Australia, Nedlands WA 6907, Australia email: [email protected]
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 problem of finding a george joinning given points x0, x1 in a connected complete Riemannian manifold requires much more effort than determining a geodesic from initial data. Boundary value problems of this type are sometimes solved using shooting methods, which work best when good initial guesses are available expectually when x0, x1 are nearby. Galerkin methods have their drawbacks too. The situation is much more difficult with general variational problems, which is why we focus on the Riemannian case.

Our global algorithm is very simple to implement, and works well in practice, with no need for an initial guess. The proof of convergence to elementary and very carefully stated. with a view to possible generalizations latter on we have in mind the much larger class of interesting problems arising in optimal control especially from mechanical engineering.

MSC classification

Secondary: 34B15: Nonlinear boundary value problems 49M05: Methods based on necessary conditions 53C22: Geodesics
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 65 , Issue 1 , August 1998 , pp. 37 - 50
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Bott, R., ‘The stable homotopy of the classical groups’, Ann. of Math. 70 (1959), 313–337.CrossRefGoogle Scholar
[2]Chapman, P. B. and Noakes, J. L., ‘Singular perturbations and interpolation- a problem in robotics’, Nonlinear Anal. 16 (1991), 849–859.CrossRefGoogle Scholar
[3]Keller, H. B., Numerical methods for two-point boundary-value problems (Blaisdell, Waltham, 1968).Google Scholar
[4]Kobayashi, S. and Nomizu, K., Foundations of differential geometry, Volume II (Interscience, New York, 1969).Google Scholar
[5]Micchelli, C. A., ‘On a measure of dissimilarity for normal probability densities’, preprint, IBM Yorktown Heights, 1996.Google Scholar
[6]Milnor, J. W., Morse theory, Ann. of Math. Stud. 51 (Princeton Univ. Press, Princeton, 1963).Google Scholar
[7]Noakes, L., ‘Nonlinear corner-cutting’, Adv. Comput. Math., to appear.Google Scholar
[8]Noakes, L., ‘Asymptotically smooth splines’ in: Advances in Computational Math., Series in Approx. Decompositions 4 (World Scientific, Singapore, 1994) pp. 131–137.Google Scholar
[9]Noakes, L., ‘Riemannian quadratics’, in: Curves and surfaces with applications in CAGD (eds. Schumaker, L. L., Le Méhauté, A. and Rabut, C.) (Vanderbilt University Press, 1997) pp. 319–328.Google Scholar
[10]Noakes, L., Heinzinger, G. and Paden, B., ‘Cubic splines on curved spaces’, IMA J. Math. Control Information 6 (1989), 464–473.Google Scholar
[11]Palais, R. S. and Terng, C.-L., Critical point theory and submanifold geometry, Lecture Notes in Math. 1353 (Springer, Berlin, 1988).Google Scholar
[12]Rao, C. R., ‘Information and the accuracy attainable in the estimation of statistical parameters’, Bulletin Calcutta Math. Soc. 37 (1945), 81–91.Google Scholar
[13]Skovgard, L. T., ‘A Riemannian geometry of the multivariate normal model’, Scand. J. Statist. 11 (1984), 211–223.Google Scholar
[14]Teo, K. L., Goh, C. J. and Wong, K. H., A unified computational approach to solving optimal control problems (Longman Scientific & Technical, 1991).Google Scholar
[15]Whitehead, J. H. C., ‘Convex regions in the geometry of paths’, Quart. J. Math. Oxford 3 (1932), 33–42.CrossRefGoogle Scholar
[16]Zeidler, E., Nonlinear functional analysis and its applications II/B (Springer, Berlin, 1990).Google Scholar
[17]Zuo, Z.-Q., ‘Two new techniques for optimal control’, IEEE Trans. Autom. Control 36 (1991), 1307–1310.CrossRefGoogle Scholar