Skip to main content
SpringerLink
Log in

Minimization of Degenerate Integral Quadratic Functionals

  • Research Articles
  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

We present a method for finding the infimum of a degenerate integral quadratic functional by passing from a given functional to another quadratic functional that is nondegenerate with respect to some new control. The minimum point of the latter can be found by a standard procedure. This point corresponds to a minimizing sequence for the original functional. The advantage of this method over the well-known regularization method (addition of a small nondegenerate term) is that the latter requires solving a parametric series of problems with a vanishingly small additional term, while our method deals with a single problem.

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

Notes

  1. A sketch of the method was presented in [9].

References

  1. B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods (Prentice Hall, Englewood Cliffs, NJ, 1990).

    MATH  Google Scholar 

  2. D. S. Bernstein and V. Zeidan, “The singular linear–quadratic regulator problem and the Goh–Riccati equation,” in Decision and Control: Proc. 29th IEEE Conf., Honolulu, 1990 (IEEE, Washington, DC, 1990), Vol. 1, pp. 334–339.

    Google Scholar 

  3. A. E. Bryson Jr. and Yu-Chi Ho, Applied Optimal Control: Optimization, Estimation and Control (Blaisdell Publ., Walltham, MA, 1969).

    Google Scholar 

  4. D. J. Clements and B. D. O. Anderson, Singular Optimal Control: The Linear–Quadratic Problem (Springer, Berlin, 1978), Lect. Notes Control Inf. Sci. 5.

    Book  Google Scholar 

  5. M. G. Dmitriev and G. A. Kurina, “Singular perturbations in control problems,” Autom. Remote Control 67 (1), 1–43 (2006) [transl. from Avtom. Telemekh., No. 1, 3–51 (2006)].

    Article  MathSciNet  Google Scholar 

  6. A. V. Dmitruk, “Quadratic conditions for a weak minimum for singular regimes in optimal control problems,” Sov. Math., Dokl. 18, 418–422 (1977) [transl. from Dokl. Akad. Nauk SSSR 233 (4), 523–526 (1977)].

    MATH  Google Scholar 

  7. A. V. Dmitruk, “Quadratic conditions for a Pontryagin minimum in an optimum control problem linear in the control. I: A decoding theorem,” Math. USSR, Izv. 28 (2), 275–303 (1987) [transl. from Izv. Akad. Nauk SSSR, Ser. Mat. 50 (2), 284–312 (1986)].

    Article  Google Scholar 

  8. A. V. Dmitruk, “Jacobi type conditions for singular extremals,” Control Cybern. 37 (2), 285–306 (2008).

    MathSciNet  MATH  Google Scholar 

  9. A. V. Dmitruk and N. A. Parilova, “On the minimization of a degenerate quadratic functional,” IFAC-PapersOnLine 51 (32), 708–711 (2018).

    Article  Google Scholar 

  10. A. V. Dmitruk and K. K. Shishov, “Analysis of a quadratic functional with a partly degenerate Legendre condition,” Moscow Univ. Comput. Math. Cybern. 34 (2), 56–65 (2010) [transl. from Vestn. Mosk. Univ., Ser. 15: Vychisl. Mat. Kibern., No. 2, 11–19 (2010)].

    Article  Google Scholar 

  11. I. M. Gelfand and S. V. Fomin, Calculus of Variations (Fizmatgiz, Moscow, 1961; Prentice-Hall, Englewood Cliffs, NJ, 1965).

    MATH  Google Scholar 

  12. V. Y. Glizer, “Solution of a singular optimal control problem with state delays: A cheap control approach,” in Optimization Theory and Related Topics: A Workshop in Memory of Dan Butnariu, Haifa, 2010, Ed. by S. Reich and A. J. Zaslavski (Am. Math. Soc., Providence, RI, 2012), Contemp. Math. 568, pp. 77–107.

    Chapter  Google Scholar 

  13. B. S. Goh, “Necessary conditions for singular extremals involving multiple control variables,” SIAM J. Control 4 (4), 716–731 (1966).

    Article  MathSciNet  Google Scholar 

  14. M. R. Hestenes, “Quadratic control problems,” J. Optim. Theory Appl. 17 (1/2), 1–42 (1975).

    Article  MathSciNet  Google Scholar 

  15. A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems (Nauka, Moscow, 1974; North-Holland, Amsterdam, 2009).

    Google Scholar 

  16. D. H. Jacobson, “A general sufficiency theorem for the second variation,” J. Math. Anal. Appl. 34, 578–589 (1971).

    Article  MathSciNet  Google Scholar 

  17. A. Jameson and R. E. O’Malley Jr., “Cheap control of the time-invariant regulator,” Appl. Math. Optim. 1 (4), 337–354 (1975).

    Article  MathSciNet  Google Scholar 

  18. A. A. Milyutin, A. V. Dmitruk, and N. P. Osmolovskii, The Maximum Principle in Optimal Control (Tsentr Prikl. Issled. Mekh.-Mat. Fak. Mosk. Gos. Univ., Moscow, 2004) [in Russian].

    Google Scholar 

  19. P. J. Moylan and J. B. Moore, “Generalizations of singular optimal control theory,” Automatica 7, 591–598 (1971).

    Article  MathSciNet  Google Scholar 

  20. J. O’Reilly, “Partial cheap control of the time-invariant regulator,” Int. J. Control 37, 909–927 (1983).

    Article  MathSciNet  Google Scholar 

  21. A. Sabery and P. Sannuti, “Cheap and singular controls for linear quadratic regulators,” IEEE Trans. Automat. Control 32, 208–219 (1987).

    Article  MathSciNet  Google Scholar 

  22. J. L. Speyer and D. H. Jacobson, “Necessary and sufficient conditions for optimality for singular control problems; a transformation approach,” J. Math. Anal. Appl. 33 (1), 163–187 (1971).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Central Economics and Mathematics Institute, Russian Academy of Sciences, Nakhimovskii prosp. 47, Moscow, 117418, Russia

    A. V. Dmitruk

  2. Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, 119991, Russia

    A. V. Dmitruk & N. A. Manuilovich

Authors
  1. A. V. Dmitruk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. A. Manuilovich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Dmitruk.

Additional information

Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 315, pp. 108–127 https://doi.org/10.4213/tm4236.

Translated by I. Nikitin

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dmitruk, A.V., Manuilovich, N.A. Minimization of Degenerate Integral Quadratic Functionals. Proc. Steklov Inst. Math. 315, 98–117 (2021). https://doi.org/10.1134/S0081543821050084

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543821050084

Search

Navigation