Skip to main content
Log in

A linear-quadratic optimization problem and the frequency theorem for nonperiodic systems. I

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

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

Access this article

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

Instant access to the full article PDF.

Literature Cited

  1. R. E. Kalman, “Contribution to the theory of optimal control,” Bol. Soc. Mat. Mexicana (2), No. 5, 102–119 (1960).

    Google Scholar 

  2. A. M. Letov, “The analytic design of controls,” Avtomat. Telemekh.,21, No. 6, 5–14 (1960).

    Google Scholar 

  3. N. N. Krasovskii, “Stabilization problems in controlled motions,” in: I. G. Malkin, Theory of Motion Stability [in Russian], Nauka, Moscow (1966), Appendix IV.

    Google Scholar 

  4. V. A. Yakubovich, “The solution of certain matrix inequalities encountered in automatic control theory,” Dokl. Akad. Nauk SSSR,143, No. 6, 1304–1307 (1962).

    Google Scholar 

  5. R. E. Kalman, “Lyapunov function for the problem of Lur'e in automatic control,” Proc. Nat. Acad. Sci. USA,49, No. 2, 201–205 (1963).

    Google Scholar 

  6. V. M. Popov, “Hyperstability of automatic control system with several nonlinear elements,” Rev. Roumaine Sci. Tech., No. 1 (1964).

  7. V. A. Yakubovich, “Frequency conditions for the absolute stability of nonlinear systems of automatic control,” in: Proc. Internat. Conf. in Applied Stability Theory and Analytical Mechanics (Kazan, 1962), Kazan Aviation Inst. (1964), pp. 135–142.

  8. V. A. Yakubovich, “A frequency theorem in control theory,” Sib. Mat. Zh.,14, No. 2, 384–420 (1973).

    Google Scholar 

  9. N. N. Krasovskii, “Problems of controllability, observability, and stabilizability of dynamical systems,” in: Proc. All-Union Congress in Theoretical and Applied Mechanics [in Russian], Nauka, Moscow (1965), pp. 77–93.

    Google Scholar 

  10. V. A. Yakubovich, “The frequency theorem for the case in which the state space and the control space are Hilbert spaces, and its application in certain problems in the synthesis of optimal control. I,” Sib. Mat. Zh.,15, No. 3, 639–668 (1974); part II: Sib. Mat. Zh.,16, No. 5, 1081–1102 (1975).

    Google Scholar 

  11. A. L. Likhtarnikov and V. A. Yakubovich, “A frequency theorem for continuous one-parameter semigroups,” Izv. Akad. Nauk SSSR, Ser. Mat.,41, No. 4, 895–911 (1977).

    Google Scholar 

  12. A. I. Lur'e, Some Nonlinear Problems of the Theory of Automatic Regulation [in Russian], Gostekhizdat (1951).

  13. F. R. Gantmakher and V. A. Yakubovich, “The absolute stability of nonlinear control systems,” in: Proc. Second All-Union Congr. Theoret. Appl. Mech. [in Russian], Vol. I, Nauka, Moscow (1965), pp. 30–63.

    Google Scholar 

  14. J. C. Willems, “Least-squares stationary optimal control and the algebraic Riccati equation,” IEEE Trans. Automat. Control,AC-16, 621–634 (1971).

    Google Scholar 

  15. V. Kucera, “A review of the matrix Riccati equation,” Kybernetika (Prague),9, No. 1, 42–61 (1973).

    Google Scholar 

  16. W. Coppel, “Matrix quadratic equations,” Bull. Austral. Math. Soc.,10, No. 3, 377–401 (1974).

    Google Scholar 

  17. A. Kh. Gelig, G. A. Leonov, and V. A. Yakubovich, The Stability of Systems with a Nonunique Equilibrium State [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  18. A. I. Barkin, Quality Estimates of Nonlinear Control Systems [in Russian], Nauka, Moscow (1982).

    Google Scholar 

  19. V. N. Fomin, A. L. Fradkov, and V. A. Yakubovich, Adaptive Control of Dynamical Objects [in Russian], Nauka, Moscow (1981).

    Google Scholar 

  20. R. V. Monopoli, “The Kalman-Yacubovich lemma in adaptive control system design,” IEEE Trans. Automat. Control,AC-18, No. 5, 527–529 (1973).

    Google Scholar 

  21. V. A. Yakubovich, “Optimality and invariance of linear stationary control systems. Avtomat. Telemekh., No. 6, 69–85 (1984).

    Google Scholar 

  22. V. I. Zubov, Theory of Optimal Control [in Russian], Sudostroenie, Leningrad (1966), Chap. 5.

    Google Scholar 

  23. A. Halanay, “Optimal control of periodic solutions,” Rev. Roumaine Math. Pures Appl.,19, No. 1, 3–16 (1974).

    Google Scholar 

  24. M. A. Shayman, “On the periodic solutions of the matrix Riccati equations,” Math. Systems Theory,16, 267–287 (1983).

    Google Scholar 

  25. R. Hermann and C. Martin, “Periodic solutions of the Riccati equation,” in: Proc. 19th IEEE Conference on Decision and Control (1980), pp. 645–648.

  26. H. Kano and T. Nishimura, “Periodic solutions of matrix Riccati equations with detectability and stabilizability,” Internat. J. Control,29, No. 3, 471–487 (1979).

    Google Scholar 

  27. V. A. Yakubovich, “The singular problem of optimal control of a linear stationary system with a quadratic functional,” Sib. Mat. Zh.,26, No. 1, 189–200 (1985).

    Google Scholar 

  28. A. V. Dmitruk, “Jacoby type conditions for the Bolza problem with inequalities,” Mat. Zametki,35, No. 6, 813–827 (1984).

    Google Scholar 

  29. E. S. Levitin, A. A. Milyutin, and N. P. Osmalovskii, “Higher order conditions for a local minimum in problems with constraints,” Usp. Mat. Nauk,33, No. 6, 85–148 (1978).

    Google Scholar 

  30. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, New York (1962).

    Google Scholar 

  31. V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients, Vols. 1 and 2, Wiley, New York (1975).

    Google Scholar 

  32. V. A. Yakubovich, “Oscillation properties of solutions of canonical equations,” Mat. Sb.,56 (98), No. 1, 3–42 (1962).

    Google Scholar 

  33. V. A. Yakubovich, “On the abstract theory of optimal control. I,” Sib. Mat. Zh.,18, No. 3, 685–707 (1977).

    Google Scholar 

  34. V. G. Antonov, A. L. Likhtarnikov, and V. A. Yakubovich, “A discrete frequency theorem for the case of Hilbert spaces of states and controls,” Vestn. Leningr. Univ., No. 1, 22–31 (1975).

    Google Scholar 

Download references

Authors

Additional information

Leningrad. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 27, No. 4, pp. 181–200, July–August, 1986.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yakubovich, V.A. A linear-quadratic optimization problem and the frequency theorem for nonperiodic systems. I. Sib Math J 27, 614–630 (1986). https://doi.org/10.1007/BF00969175

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00969175

Keywords

Navigation