Abstract
We analyze the simple cusp singularity arising in the flow of a parametrized family of extremals. The simple cusp point generates a region in the state space which is covered 3:1 with both locally minimizing and maximizing branches. The changes from the locally minimizing to the maximizing branch away from the simple cusp point occur at fold-loci and there trajectories lose strong local optimality. However, the two minimizing branches intersect and generate a cut-locus which limits the optimality of the close-by trajectories and eliminates these trajectories from optimality near the cusp point prior to the conjugate point. In the language of PDE, the simple cusp in the parametrized flow of extremals generates a shock in the corresponding solution to the Hamilton-Jacobi-Bellman equation.
This research was supported in part by NSF Grant DMS-9503356.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Berkovitz, L. (1974), Optimal Control Theory, Springer-Verlag, New York.
Boltyansky, V.G. (1966), “Sufficient conditions for optimality and the justification of the dynamic programming method,” SIAM J. Control, Vol. 4, No. 2,326–361.
Boothby, W.M. (1975), An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, New York.
Brunovsky, P. (1980), “Existence of regular synthesis for general control problems,” J. Differential Equations, Vol. 38, 317–343.
Bryson Jr., A.E. and Ho, Y.C. (1975), Applied Optimal Control, Hemisphere Publishing Company, New York.
Byrnes, C.I. and Frankowska, H. (1992), “Unicité des solutions optimales et absence de chocs pour les équations,” C. R. Acad. Sci. Paris, Vol. 315, 427–431.
Cannarsa, P. and Soner, H.M. (1987), “On the singularities of the viscosity solutions to Hamilton-Jacobi-Bellman equations,” Indiana University Mathematical Journal, Vol. 36, 501–524.
Cannarsa, P. and Frankowska, H. (1991), “Some characterizations of optimal trajectories in control theory,” SIAM J. Control and Optimization, Vol. 29, 1322–1347.
Carathéodory, C. (1936), Variationsrechnung und Partielle Differential Gleichungen erster Ordnung, Teubner Verlag, Leipzig.
Golubitsky, M. and Guillemin, V. (1973), Stable Mappings and their Singularities, Springer-Verlag, New York.
Kiefer, M. and Schättler, H., “Parametrized families of extremals and singularities in Solutions to Hamilton-Jacobi-Bellman equations,” submitted for publication.
Knobloch, H.W., Isidori, A. and Flockerzi, D. (1993), Topics in Control Theory, DMV Seminar, Band 22, Birkhäuser Verlag, Basel.
Krener, A.J. and Schättler, H. (1989), “The structure of small time reachable sets in low dimension”, SIAM J. Control Optim., Vol. 27, No. 1, 120–147.
McShane, E.J. (1944), Integration, Princeton University Press, Princeton, New Jersey.
Piccoli, B. (1996), “Classification of generic singularities for the planar time-optimal synthesis,” SIAM J. on Control and Optimization, Vol. 36, 1914–1946.
Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, V.G. and Mishchenko, R.V. (1962), Mathematical Theory of Optimal Processes,Wiley-Interscience.
Poston, T. and Stewart, I.N. (1977), Taylor Expansions and Catastrophes, Research Notes in Mathematics, Vol. 7, Pitman Publishing, London.
Schättler, H. (1982), Hinreichende Bedingungen für ein starkes relatives Minimum bei Kontrollproblemen, Diplomarbeit der Fakultät für Mathematik der Universität Würzburg.
Schättler, H. (1991), “Extremal trajectories, small-time reachable sets and local feedback synthesis: a synopsis of the three-dimensional case,” in Nonlinear Synthesis, Christopher I. Byrnes and Alexander Kurzhansky, eds., Birkhäuser, Boston, 258–269.
Schättler, H. and Jankovic, M. (1993), “A synthesis of time-optimal controls in the presence of saturated singular arcs,” Forum Mathematicum, Vol. 5, 203–241.
Sussmann, H.J. (1990), “Synthesis, presynthesis, sufficient conditions for optimality and subanalytic sets,” in Nonlinear Controllability and Optimal Control, H. Sussmann, ed., Marcel Dekker, New York, 1–19.
Sussmann, H. (1987), “Regular synthesis for time-optimal control of single-input real analytic systems in the plane,” SIAM J. on Control and Optimization, Vol. 25, 1145–1162.
Whitney, H. (1957), “Elementary structure of real algebraic varieties,” Ann. Math., Vol. 66, 545–556.
Young, L.C. (1969), Lectures on the Calculus of Variations and Optimal Control Theory, W.B. Saunders, Philadelphia.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Kiefer, M., Schättler, H. (1998). Cut-Loci and Cusp Singularities in Parametrized Families of Extremals. In: Optimal Control. Applied Optimization, vol 15. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6095-8_12
Download citation
DOI: https://doi.org/10.1007/978-1-4757-6095-8_12
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4796-3
Online ISBN: 978-1-4757-6095-8
eBook Packages: Springer Book Archive