Asymptotic Approximations to the Solution of the Singularly Perturbed Linear-Quadratic Optimal Control Problem with Terminal Path Constraints


The minimum-energy control problem for a linear singularly perturbed system with linear constraints imposed on the right endpoint of the trajectories is considered. Asymptotic approximations in the form of open loop and feedback controls to the optimal control (solution of this problem) are constructed. The main advantage of the algorithms proposed below consists in the decomposition of the original problem into two unperturbed optimal control problems of smaller dimension.

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


  1. 1.

    Dmitriev, M.G. and Kurina, G.A., Singular Perturbation in Control Problems, Autom. Remote Control, 2006, vol. 67, no. 1, pp. 1–43.

    MathSciNet  Article  Google Scholar 

  2. 2.

    Kalinin, A.I., Asymptotics of the Solutions of Perturbed Optimal Control Problems, J. Comput. Syst. Sci., 1995, vol. 33, no. 6, pp. 75–84.

    MathSciNet  Google Scholar 

  3. 3.

    Zhang, Y., Naidu, D.S., Cai, C., and Zou, Y., Singular Perturbations and Time Scales in Control Theories and Applications: An Overview 2002–2012, Int. J. Inform. Syst. Sci., 2014, vol. 9, no. 1, pp. 1–36.

    Google Scholar 

  4. 4.

    Kokotovic, P.V. and Khalil, H.K., Singular Perturbations in Systems and Control, New York: IEEE Press, 1986.

    Google Scholar 

  5. 5.

    Rakitskii, Yu.V., Ustinov, S.M., and Chernorutskii, I.G., Chislennye metody resheniya zhestkikh sistem (Numerical Methods for Solving Rigid Systems), Moscow: Nauka, 1979.

    Google Scholar 

  6. 6.

    Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F., Matematicheskaya teoriya optimal’nykh protsessov, Moscow: Nauka, 1983. Translated under the title The Mathematical Theory of Optimal Processes, vol. 4 of L.S. Pontryagin Selected Works, New York: Gordon and Breach, 1986.

    Google Scholar 

  7. 7.

    Kokotovic, P.V. and Jackel, R.A., Singular Perturbation of Linear Regulators: Basic Theorems, IEEE Trans. Automat. Control, 1972, vol. 17, no. 1, pp. 29–37.

    MathSciNet  Article  Google Scholar 

  8. 8.

    Wilde, R.R. and Kokotovic, P.V., Optimal Open and Closed Loop Control of Singularly Perturbed Linear Systems, IEEE Trans. Automat. Control, 1973, vol. 18, no. 6, pp. 616–626.

    MathSciNet  Article  Google Scholar 

  9. 9.

    Glizer, V.Ya. and Dmitriev, M.G., Singular Perturbations in the Linear Optimal Control Problem with a Quadratic Functional, Dokl. Akad. Nauk SSSR, 1975, vol. 225, no. 5, pp. 997–1000.

    MathSciNet  MATH  Google Scholar 

  10. 10.

    O’Malley, R.E., Jr., Singular Perturbation and Optimal Control, Lect. Notes. Math., 1978, vol. 680, pp. 171–218.

    MathSciNet  Google Scholar 

  11. 11.

    Kalinin, A.I. and Lavrinovich, L.I., Application of the Small Parameter Method to the Singularly Perturbed Linear-Quadratic Optimal Control Problem, Autom. Remote Control, 2016, vol. 77, no. 5, pp. 751–763.

    MathSciNet  Article  Google Scholar 

  12. 12.

    Gabasov, R. and Kirillova, F., Kachestvennaya teoriya optimal’nykh protsessov, Moscow: Nauka, 1971. Translated under the title The Qualitative Theory of Optimal Processes, New York: Marcel Dekker, 1976.

    Google Scholar 

  13. 13.

    Kalinin, A.I., To the Synthesis of Optimal Control Systems, Comput. Math. Math. Phys., 2018, vol. 58, no. 3, pp. 378–383.

    MathSciNet  Article  Google Scholar 

  14. 14.

    Mordukhovich, B.S., Existence of Optimum Controls (Appendix to the Article by Gabasov and Kirillova, “Methods of Optimum Control”), J. Math. Sci., 1977, vol. 7, pp. 850–886.

    Article  Google Scholar 

  15. 15.

    Bryson, A.E., Jr. and Ho, Yu-Chi, Applied Optimal Control: Optimization, Estimation, and Control, Waltham: Blaisdell, 1969. Translated under the title Prikladnaya teoriya optimal’nogo upravleniya, Moscow: Mir, 1972.

    Google Scholar 

  16. 16.

    Vasil’eva, A.B. and Butuzov, V.F., Asimptoticheskie razlozheniya reshenii singulyarno vozmushchennykh uravnenii (Asymptotic Expansions of Solutions of Singularly Perturbed Equations), Moscow: Nauka, 1973.

    Google Scholar 

  17. 17.

    Lee, E.B. and Markus, L., Foundations of Optimal Control Theory, New York: Wiley, 1967. Translated under the title Osnovy teorii optimal’nogo upravleniya, Moscow: Nauka, 1972.

    Google Scholar 

  18. 18.

    Giev, T.R. and Dontchev, A.L., Singular Perturbation in Optimal Control Problems with Fixed Final State, Dokl. Bulgar. Acad. Sci., 1978, vol. 31, no. 8, pp. 935–955.

    MathSciNet  Google Scholar 

  19. 19.

    Gabasov, R. and Kirillova, F.M., Konstruktivnye metody optimizatsii (Constructive Methods of Optimization), Minsk: Universitetskoe, 1984, vol 2.

    Google Scholar 

Download references

Author information



Corresponding authors

Correspondence to A. I. Kalinin or L. I. Lavrinovich.

Additional information

This paper was recommended for publication by M.M. Khrustalev, a member of the Editorial Board

Russian Text © The Author(s), 2020, published in Avtomatika i Telemekhanika, 2020, No. 6, pp. 29–46.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kalinin, A.I., Lavrinovich, L.I. Asymptotic Approximations to the Solution of the Singularly Perturbed Linear-Quadratic Optimal Control Problem with Terminal Path Constraints. Autom Remote Control 81, 988–1002 (2020).

Download citation


  • optimal control
  • linear system
  • quadratic performance criterion
  • singular perturbations
  • asymptotic approximations
  • suboptimal feedback design