Skip to main content
SpringerLink
Log in

Singular surfaces in differential games an introduction

  • Main Lectures
  • Conference paper
  • First Online:
Differential Games and Applications

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 3))

Abstract

We give a general set up and a version of Isaacs' Verification Theorem that allows us to deal with the various singularities we want to investigate. In particular, we are obliged to allow upper or lower strategies, leading to upper or lower saddle points, that may exists even if the Hamiltonian does not have a saddle point. It is shown that this is needed even for separated games. Then we give a general study of junctions of optimal fields with singular surfaces, which requires a special investigation of the situation where this junction is tangantial extending Carathéodory's General Envelope Theorem. We then proceed to study special singular surfaces, and we end up with an example which shows how a state constraint may appear in the interior of the game space of a separated problem posed with no such constraint to start with.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. R. ISAACS "Differential Games" Rand Reports, 1954, 1965, and Differential Games, Wiley, N.Y., 1965.

    Google Scholar 

  2. J.V. BREAKWELL "Some differential games with interesting discontinuities", Stanford University, Department of Aeronautics & Astronautics. 1973.

    Google Scholar 

  3. W.H. FLEMING "The convergence problem for differential games", Jal of Math. Analysis and Applications, vol 3., pp 102–116, 1961.

    Google Scholar 

  4. W.H. FLEMING "The Convergence problem for differential games, II" Advances in Game Theory, Princeton University press, pp 195–210, 1964.

    Google Scholar 

  5. P. VARAIYA "The existence of solutions to a differential game" SIAM Jal on Control Vol 5, pp 153–162, 1967.

    Google Scholar 

  6. E. ROXIN "On Varaiya's definition of a differential game" US-Japan seminar on differential and functional equations, Mineapolis, 1967.

    Google Scholar 

  7. P. VARAIYA & J. LIN "Existence of saddle points in differential games" SIAM Jal on Control Vol 7, pp 141–157, 1969.

    Google Scholar 

  8. A. FRIEDMAN: Differential games. Wiley, New York, 1971.

    Google Scholar 

  9. N.S. PONTRYAGIN "Linear differential games" Soviet Math. Doklady vol 8, pp 769–771 and pp 910, 912, 1967.

    Google Scholar 

  10. N.N KRASOVSKYI. Positional differential games. English translation to appear.

    Google Scholar 

  11. J.F. MASLE "Problèmes qualitatifs et quantitatifs liés en jeux differentiels", Thesis, Université de Paris IX, 1976.

    Google Scholar 

  12. W.T. BOARDMAN III "An analytical investigation on the state constraints in Isaacs IsotropicRocket Game. Stanford University, Engineer thesis, 1968.

    Google Scholar 

  13. J.V. BREAKWELL & A.W. MERZ "Toward a complete solution of the Homicidial Chauffeur game" First International Conference on the Theory and Applications of Differential Games, Amherst 1969.

    Google Scholar 

  14. J.V. BREAKWELL "Complete solution of the Dolichobrachistochrone problem", unpublished, lectures notes, Stanford University, 1968.

    Google Scholar 

  15. L. BERKOVITZ "Lectures on Differential games" in Differential games and related topics, Kuhn & Szego ed., North Holland, 1971.

    Google Scholar 

  16. A. BLAQUIERE et al. Qualitative and Quantitative games, Academic Press, 1970.

    Google Scholar 

  17. A.W. MERZ "The homicidial chauffeur — a differential game", PhD Thesis, Stanford University, 1971.

    Google Scholar 

  18. W.E. SCHMITENDORF "Differential Games without pure strategy saddle point solutions", Jal of Differential Equations, pp 81–92, 1976.

    Google Scholar 

  19. S.S.L. CHANG, T.K.C. PENG: "Adaptive guaranteed cost control of systems with uncertain parameters". IEEE Trans. AC 17, pp 474–482, 1972.

    Google Scholar 

  20. P. BERNHARD "New results about corners in differential games, including state constraints". 6th IFAC world Congress, Cambridge Mass. 1975.

    Google Scholar 

  21. P. COLLETER. Thèse d'Ingénieur docteur, University Paris IX 1977. (to appear.

    Google Scholar 

  22. P. BERNHARD. "Commande Optimale, décentralisation et jeux dynamiques"

    Google Scholar 

  23. P. BERNHARD "Corner Conditions for differential games". 5th IFAC world congress, Paris, 1972.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université de Paris IX, France

    Pierre Bernhard

  2. Centre d'Automatique et Informatique de l'Ecole Nationale Supérieure des Mines de Paris, France

    Pierre Bernhard

Authors
  1. Pierre Bernhard
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

P. Hagedorn H. W. Knobloch G. J. Olsder

Rights and permissions

Reprints and permissions

Copyright information

© 1977 Springer-Verlag

About this paper

Cite this paper

Bernhard, P. (1977). Singular surfaces in differential games an introduction. In: Hagedorn, P., Knobloch, H.W., Olsder, G.J. (eds) Differential Games and Applications. Lecture Notes in Control and Information Sciences, vol 3. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0009062

Download citation

  • DOI: https://doi.org/10.1007/BFb0009062

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08407-5

  • Online ISBN: 978-3-540-37179-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation