Abstract
The question of the existence and the location of Darboux points (beyond which global optimality is lost) is crucial for minimal sufficient conditions for global optimality and for computation of optimal trajectories. Here, we investigate numerically the Darboux points and their relationship with conjugate points for a problem of minimum fuel, constant velocity, horizontal aircraft turns to capture a line. This simple second-order optimal control problem shows that ignoring the possible existence of Darboux points may play havoc with the computation of optimal trajectories.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Breakwell, J. V., andHo, Y. C.,On the Conjugate Point Condition for the Control Problem, International Journal of Engineering Science, Vol. 2, pp. 565–570, 1965.
Mereau, P. M., andPowers, W. F.,Conjugate Point Properties for Linear Quadratic Problems, Journal of Mathematical Analysis and Applications, Vol. 55, pp. 418–433, 1976.
Hestenes, M. R.,Calculus of Variations and Optimal Control Theory, John Wiley and Sons, New York, New York, 1966.
Young, L. C.,Lectures on the Calculus of Variations and Optimal Control Theory, W. G. Saunders Company, New York, New York, 1969.
Moyer, H. G., andKelley, H. J.,Conjugate Points on Extremal Rocket Paths, Proceedings of the 19th International Astronautical Congress, New York, New York, 1968; Pergamon Press, London, England, Vol. 2, pp. 163–172, 1970.
Moyer, H. G.,A Computer Survey of Impulsive Ellipse-Ellipse Transfer, AIAA Journal, Vol. 9, pp. 321–323, 1971.
Moyer, H. G.,Optimal Control Problems That Test for Envelope Contacts, Journal of Optimization Theory and Applications, Vol. 6, pp. 287–298, 1970.
Mereau, P. M., andPowers, W. F.,The Darboux Point, Journal of Optimization Theory and Applications, Vol. 17, pp. 545–559, 1975.
Bolza, O.,Vorlesungen über Variationsrechnung, Teubner, Leipzig, Germany, p. 438, 1909; Chelsea Publishing Company, New York, New York, 1963.
Darboux, G.,Sur la Théorie des Surfaces, Gauthiers-Villars, Paris, France, Vol. 3, p. 89, 1894.
Mereau, P. M., andPowers, W. F.,Characterization of the Darboux Point for Particular Classes of Problems, Journal of Optimization Theory and Applications, Vol. 22, pp. 537–562, 1977.
Erzberger, H., andLee, H. Q.,Optimum Horizontal Guidance Techniques for Aircraft, Journal of Aircraft, Vol. 8, pp. 95–101, 1971.
Author information
Authors and Affiliations
Additional information
Communicated by G. Leitmann
The authors are indebted to G. Moyer for his constructive comments. This research was supported, for the first author, by a National Research Council Associateship at NASA Ames Research Center.
on leave from the Technion, Israel Institute of Technology, Haifa, Israel.
Rights and permissions
About this article
Cite this article
Kreindler, E., Neuman, F. Global optimality of extremals: An example. J Optim Theory Appl 36, 521–534 (1982). https://doi.org/10.1007/BF00940545
Issue Date:
DOI: https://doi.org/10.1007/BF00940545