Abstract
Before turning to the general n-dimensional nonlinear optimal control problem, which is of primary interest in this paper, we will first consider a much simpler problem, namely the singularly perturbed, uncontrolled, autonomous, initial-value problem
where x(ε,t) and y(ε,t) are scalars, ε < 0, and xo and yo are constants.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Wasow, W.R., Asymptotic expansions for ordinary differential equations, Interscience, 1965.
Ardema, M.D., Singular perturbations in flight mechanics, NASA TM X-62, 380, July 1977.
Ardema, M.D., Solution of second order linear system by matched asymptotic expansions, NASA TM 84, 246, to appear.
Tihonov, A.N., Systems of differential equations containing small parameters multiplying some of the derivatives, Math. Sb., vol. 73, no. 3, N.S. (31), 1952 (in Russian).
Vasileva, A.B., Asymptotic behavior of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivatives, Russian Math. Surveys,vol. 18, no. 3, 1963 (English translation).
Levin, J.J., and Levinson, N., Singular perturbations_ of non-linear systems of differential equations and an associated boundary layer equation, J. Rat. Mech. Anal., vol. 3, 1954.
O’Malley, R.E., Introduction to singular perturbations, Academic
Buti G., The singular perturbation theory of differential equations in control theory, Periodica Polytechnica, vol. 10, no. 2, 1965.
Cole, J.D., Perturbations methods in applied mathematics, Blaisdell, 1968.
Van Dyke, M., Perturbations methods in fluid mechanics, Academic, 1964.
Nayfeh, A.H., Perturbations methods, Wiley, 1973.
Eckhaus, W., Matched asymptotic expansions and singular perturbations, North-Holland, 1973.
Pontryagin, L.S., Boltyanskii, V.G.,Gamkrelidze, R.V., and Mishchenlco, E.F., The mathematical theory of optimal processes, Interseience, 1962.
Leitmann,G., An introduction to optimal control, McGraw-Hill, 1969.
Bryson, A.E., Jr. and Ho, Y.-C., Applied optimal Control, Blaisdell, 1968.
Hadlock, C.R., Singular perturbations of a class of two point boundary value problems arising in optimal control, Coordinated Science Laboratory, Report R-481, Univ. of Illinois, July 1970.
Hadlock, C.R., Existence and dependence on a parameter of solutions of a nonlinear two point boundary value problem, Journal Differential Equations, vol. 14, no. 3, November 1973.
Freedman, M.I., and Kaplan, J.L., Singular perturbations of two-point boundary value problems arising in optimal control, SIAM Journal Control and Optimization, vol. 14, no. 2, February 1976.
Freedman, M.I., and Granoff, B., Formal asymptotic solution of a perturbed nonlinear optimal control problem, Journal Optimization Theory and Applications, vol. 19, no. 2, June 1976.
O’Malley, R.E., Singular perturbations and optimal control, Lectures given at the conference on Mathematical Control Theory, Australian National University, Canberrra, August 23 - September 2, 1977.
Ardema, M.D., Characteristics of the boundary layer equations of the minimum time-to-climb problem, Proceedings of the Fourt4eth Annual Allerton Conference. on Circuit and System Theory, September 1976.
Sacher, R.J., and Sell, G.R., Singular perturbations and conditional stability, J. of Math. Analysis and Applic., vol. 76, 1980.
Berger, S.A., Laminar wakes, American Elsevier, 1971.
Ai, D.K., On the critical point of the Crocco-Lees mixing theory in the laminar near wake, J. Eng. Math., vol. 4, no. 2, April 1970.
O’Malley, R.E., Introduction to singular perturbations, Academic, 1974.
Kokotovic, P.V., O’Malley, R.E., and Sannuti, P., Singular perturbations and order reduction in control theory - an overview, Automatica, vol. 12, 1976.
Kokotóvic, P.V. (ed.), Singular perturbations and time scales in modeling and control of dynamic systems, Coordinated Science Laboratory, Report R-901, Univ of Illinois, November 1980.
O’Malley, R.E., On the asymptotic solution of certain non-linear singularly perturbed optimal control problems, Order Reduction in Control System Design, ASME, 1972.
Chow, J.H., A class of singularly perturbed, nonlinear fixed-endpoint control problems, JOTA, vol. 29, no. 2, October 1979.
Ardema, M.D., Solution of the minimum time-to-climb problem by matched asymptotic expansions, AIAA Journal, vol. 14, no. 7, July 1976.
Ardema, M.D., Linearization of the boundary-layer equations of the minimum time-to-climb problem, Journal Quidance and Control, vol. 2, no. 5, September - October, 1979.
Kelley, H.J., Singular perturbations for a Mayer variational problem, AIAA Journal, vol. 8, no. 6, June 1970.
Kao, Y.K., and Bankoff, S.G., Singular perturbations analysis and free-time optimal control problems, Paper 7–3 presented at the Joint Automatic Control Conference, Columbus, Ohio, June 20–22, 1973.
Calise, A.J., and Aggarwal, R., A Conceptual approach to applying singular perturbation methods to variational problems, Proceedings of the Eleventh’ Annual Allerton Conference on Circuit and Systems Theory, Allerton, 1973.
Calise, A.J., Extended energy management methods for flight performance optimization, AIAA Journal, vol. 15, no. 3, March 1977.
Visser, H.G., and Shinar, J., Asymptotic solution of a problem of missile guidance by proportional navigation for a singularly perturbed system, Technion Report TAE no. 464, Haifa, Israël, October, 1981.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1983 Springer-Verlag Wien
About this chapter
Cite this chapter
Ardema, M.D. (1983). An Introduction to Singular Perturbations in Nonlinear Optimal Control. In: Ardema, M.D. (eds) Singular Perturbations in Systems and Control. International Centre for Mechanical Sciences, vol 280. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2638-7_1
Download citation
DOI: https://doi.org/10.1007/978-3-7091-2638-7_1
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-81751-3
Online ISBN: 978-3-7091-2638-7
eBook Packages: Springer Book Archive