Abstract
A system of equations that arises in a singularly perturbed optimal control problem is studied. We give conditions under which a formal asymptotic solution exists. This formal asymptotic solution consists of an outer expansion and left and right boundary-layer expansions. A feature of our procedure is that we do nota priori eliminate the control function from the problem. In particular, we construct a formal asymptotic expansion for the control directly. We apply our procedure to a Mayer-type problem. The paper concludes with a worked example.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Lee, E. B., andMarkus, L.,Foundations of Optimal Control Theory, John Wiley and Sons, New York, New York, 1967.
Wasow, W.,Asymptotic Expansions for Ordinary Differential Equations, John Wiley and Sons (Interscience Publishers), New York, New York, 1965.
Cole, J. D.,Perturbation Methods in Applied Mathematics, Blaisdell Publishing Company, Waltham, Massachusetts, 1968.
Nayfeh, A. H.,Perturbation Methods, John Wiley and Sons, New York, New York, 1973.
Tupciev, V. A.,Asymptotic Behavior of the Solution of a Boundary Problem for Systems of Differential Equations of First Order with a Small Parameter in the Derivative, Soviet Mathematics Doklady, Vol. 3, pp. 612–616, 1962.
Hoppensteadt, F.,Properties of Solutions of Ordinary Differential Equations with Small Parameters, Communications on Pure and Applied Mathematics, Vol. 24, pp. 807–840, 1971.
O'Malley, R. E., Jr.,The Singularly Perturbed Linear State Regulator Problem, SIAM Journal on Control, Vol. 10, pp. 399–413, 1972.
Hadlock, C. R.,Existence and Dependence on a Parameter of Solutions of a Nonlinear Two-Point Boundary-Value Problem, Journal of Differential Equations, Vol. 14, pp. 498–517, 1973.
Sannuti, P.,Asymptotic Expansions of Singularly Perturbed Quasi-Linear Optimal Systems, SIAM Journal on Control, Vol. 13, No. 3, 1975.
Harris, W. A.,Singular Perturbations of Two-Point Boundary-Value Problems, Journal of Mathematics and Mechanics, Vol. 11, pp. 371–382, 1962.
Freedman, M. I., andKaplan, J.,Singular Perturbations of Two-Point Boundary-Value Problems Arising in Optimal Control, Boston University, Department of Mathematics Research Report No. 4, 1974.
Freedman, M. I., andGranoff, B.,The Formal Asymptotic Solution of a Singularly Perturbed Nonlinear Optimal Control Problem, Boston University, Department of Mathematics Research Report No. 3, 1974.
Kelley, H. J.,Singular Perturbations for a Mayer Vibrational Problem, AIAA Journal, Vol. 7, p. 1177, 1970.
Author information
Authors and Affiliations
Additional information
Communicated by T. N. Edelbaum
Rights and permissions
About this article
Cite this article
Freedman, M.I., Granoff, B. Formal asymptotic solution of a singularly perturbed nonlinear optimal control problem. J Optim Theory Appl 19, 301–325 (1976). https://doi.org/10.1007/BF00934099
Issue Date:
DOI: https://doi.org/10.1007/BF00934099