Skip to main content
SpringerLink
Log in

Feedback-invariant optimal control theory and differential geometry—I. Regular extremals

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

Abstract

A feedback-invariant approach to smooth optimal control problems is considered. A Hamiltonian method of investigating regular extremals is developed, analogous to the differential-geometric method of investigating Riemannian geodesics in terms of the Levi-Civita connection and the curvature tensor.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. A. Agrachev, Quadratic mappings in geometric control theory. (Russian)Itogi Nauki i Tekhniki VINITI; Problemy Geometrii, Vol. 20, 1988.VINITI, Moscow, 111–205. (English translation:J. Soviet Math., Plenum Publ. Corp., Vol. 51, 1990, 2667–2734).

    Google Scholar 

  2. —, Topology of quadratic mappings and Hessians of smooth mappings. (Russian)Itogi Nauki i Tekhniki VINITI; Algebra, Topologia, Geometria, Vol. 26, 1988,VINITI, Moscow, 85–124. (English translation:J. Soviet Math., Plenum Publ. Corp., 1990, 990–1013).

    Google Scholar 

  3. A. A. Agrachev and R. V. Gamkrelidze, Exponential respresentation of flows and chronological calculus. (Russian)Mat. Sb. 107 (1978), 467–532. (English translation:Math. USSR Sb. 35 (1979), 727–785).

    MATH  MathSciNet  Google Scholar 

  4. — The Morse index and the Maslov index for extremals of controlled systems. (Russian)Dokl. Akad. Nauk SSSR 287 (1986), 11–205. (English translation:Soviet Math. Dokl. 33 (1986), 392–395).

    MathSciNet  Google Scholar 

  5. A. A. Agrachev, The extremality index and quasi-extremal controls. (Russian)Dokl. Acad. Nauk SSSR 284 (1985). (English translation:Soviet Math. Dokl. 32 (1985), 478–481.

  6. A. A. Agrachev, The quasi-extremality for controlled systems. (Russian)Itogi Nauki i Tekhniki. VINITI; Sovremennye problemy matematiki. Novejshie dostigeniya, Vol. 35, 1989. (English translation:J. Soviet Math. Plenum Publ. Corp., 1991, 1849–1864.

  7. A. A. Agrachev, Symplectic methods for optimization and control. (to appear in: Geometry of Feedback and Optimal Control,Marcel Dekker).

  8. A. A. Agrachev, R. V. Gamkrelidze, and A. V. Sarychev, Local invariants of smooth control systems.Acta Appl. Math.,14 (1989), 191–237.

    Article  MATH  MathSciNet  Google Scholar 

  9. V. I. Arnold, Mathematical methods of classical mechanics, Third edition.Nauka, Moscow, 1989.

    Google Scholar 

  10. B. Bonnard, Feedback equivalence for nonlinear systems and the time optimal control problems.SIAM J. Control and Optimiz.,29 (1991), 1300–1321.

    Article  MATH  MathSciNet  Google Scholar 

  11. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko, Mathematical theory of optimal processes.Fiz.-mat. giz., Moscow, 1961.

    Google Scholar 

Download references

Author information

Author notes

  1. A. A. Agrachev & R. V. Gamkrelidze

    Present address: Steklov Mathematical Institute, 117966, Moscow, Russia

Authors and Affiliations

    Authors
    1. A. A. Agrachev
      View author publications

      You can also search for this author in PubMed Google Scholar

    2. R. V. Gamkrelidze
      View author publications

      You can also search for this author in PubMed Google Scholar

    Additional information

    The research was done during the Optimization Semester at the Karlsruhe University in March–May, 1995, organized by the second author as part of his Humboldt stay in Germany.

    The first author was partially supported by the Russian Foundation for Fundamental Research, Grant 95-01-00310.

    Rights and permissions

    Reprints and permissions

    About this article

    Cite this article

    Agrachev, A.A., Gamkrelidze, R.V. Feedback-invariant optimal control theory and differential geometry—I. Regular extremals. Journal of Dynamical and Control Systems 3, 343–389 (1997). https://doi.org/10.1007/BF02463256

    Download citation

    • Received:

    • Issue Date:

    • DOI: https://doi.org/10.1007/BF02463256

    1991 Mathematics Subject Classification

    Key words and phrases

    Search

    Navigation