Abstract
This paper presents two applications of continuation methods to the control of nonlinear systems.In the first one the aim is to control the position of isolas.This can be achieved by a continuation procedure to localize isola centers and bifurcation points. The method also traces out curves of isola centers and perturbed bifurcation points. The technique has been applied to a reaction — diffusion biochemical system, for which numerical results are given, and to periodic solutions.The second application consists in viewing optimal control problems as Cauchy problems which themselves can be solved by continuation techniques. As an example the method is applied in a general framework to exact controllability of systems and in particular for a nonlinear distributed system.
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
Dellwo,D., Keller,H.B., Matkowsky, B.J., Reiss,E.L.: “On the birth of isolas”. SIAM J.App. Math., 2, 1982, pp. 956–963.
Dellwo, D.: “A Constructive Theory of isolas supported by parabolic cusp, centers and bifurcation points”. SIAM J.Appl.Math., 46, 1986,pp. 740–764.
Doedel, EJ., Kernévez, J.P.: “AUTO: Software for continuation problems in ordinary differential equations with applications”, Technical Report, Appl. Math., California Institute of Technology, 1989.
Lions,J.L. (1989) “Exact controllability, stabilization and perturbations for distributed systems” SIAM Review
Jepson, A., Spence, A. (1985) “Folds in solutions of two parameter systems and their numerical calculation.Part 1. SIAM J.Numer.Anal.., 22, 347–368.
Keener, J.P., Keller, H.B.: “Perturbed Bifurcation Theory”, Arch. Rat. Mech. Anal., 50, 1973, pp. 159–175.
Keller, H.B.: “Numerical Solution of Bifurcation And Non Linear Eigenvalue Problems”. Application in Bifurcation Theory, P.H. Rabinowitz, (ed.), Academic Press, 1977, pp. 359–384.
Keller, H.B.: “Two New Bifurcation Phenomena”. Rapport de Recherche, 369. INRIA, 1979.
Keller, H.B.: “Isolas and perturbed Bifurcation Theory”, in “Non Linear Partial Differential Equations in Engineering and App. Sci.”, Stenberg, Kalinowski, Papadakis (eds.), Marcel Dekker, 1980, pp. 45–52.
Kemévez, J.P.: “Enzyme Mathematics”. North-Holland, Amsterdam, 1980.
Liu, Y. (1989) Thesis, Compiècgne.
Moore, G., Spence, A. (1980) “The calculation of turning points of nonlinear equations” SIAM J.Numer.Anal.,. 17, 567–576.
Rabinowitz, P.H. (ed.): “Applications in bifurcation theory”. Academic Press, New York, 1977.
Seoane, M.L. (1990) Thesis, Santiago de Compostela.
Spence, A. and Jepson (1984) “The numerical calculation of cusps, bifurcation points and isola formation points in two parameter problems”, in Kupper, Mittelmann, Weber (eds.) , Numerical methods for bifurcation problemss, Birkhausen pp.502–514.
Spence, A. and Werner, B. ()“ Nonsimple turning points and cusps” IMA J. Numer.Anal., 2, 413–427.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Kluwer Academic Publishers
About this chapter
Cite this chapter
Kernevez, J.P., Liu, Y., Seoane, M.L., Doedel, E.J. (1990). Optimization by Continuation. In: Roose, D., Dier, B.D., Spence, A. (eds) Continuation and Bifurcations: Numerical Techniques and Applications. NATO ASI Series, vol 313. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0659-4_23
Download citation
DOI: https://doi.org/10.1007/978-94-009-0659-4_23
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6781-2
Online ISBN: 978-94-009-0659-4
eBook Packages: Springer Book Archive