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Nonconvex Global Optimization Problems: Constrained Infinite-Horizon Linear-Quadratic Control Problems for Discrete Systems

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Directions in Mathematical Systems Theory and Optimization

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 286))

Abstract

In recent works a method of solving some global optimization problems was proposed by the author. These problems may generally be nonconvex. In this paper we describe this method and apply it to solving linear-quadratic deterministic and stochastic infinity-horison optimization problems with integral quadratic constraints.

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26.6 References

  1. Kalman R.E., Contributions to the theory of optimal control, Bol. Soc. Math. Mexicana (2) 5 (1960), 102–119.

    MathSciNet  Google Scholar 

  2. Krasovskii N.N., Stabilization problem of a control motion, in Supplement 2 to I.G. Malkin, Stability Theory of Motion [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  3. Letov A.M., Analitical regulator synthesis, Avtomat. Telemekh., No. 4, (1960), 436–446; No. 5, (1960), 561–571.

    Google Scholar 

  4. Lur’e A.I., Minimal quadratic performance index for a control system, Tekh. kibern., No. 4, (1963), 140–146 [in Russian].

    Google Scholar 

  5. Kolmogorov A.N., Interpolation and extrapolation of random sequences, Izv. Akad. Nauk SSSR, Ser. Mat., No. 5, (1941), 3–14 [in Russian].

    MATH  MathSciNet  Google Scholar 

  6. Wiener N., Extrapolation, interpolation and smoothing of stationary time series. Cambridge, 1949.

    Google Scholar 

  7. Bucy R.S. and Joseph P.D., Filtering of stochastic processes with application to guidance. N.Y., London, 1968.

    Google Scholar 

  8. Yakubovich V.A., Minimization of quadratic functionals under quadratic constraints and the necessity of the frequency condition for absolute stability of nonlinear control systems, Dokl. Akad. Nauk SSSR, 209, (1973), 1039–1042. (English transl. in Soviet Math. Dokl., 14, No. 2, (1973), 593–597.)

    MathSciNet  Google Scholar 

  9. Yakubovich V.A., A method for solution of special problems of global optimization, Vestn. St. Petersburg Univ., Ser. 1, Vyp. 2 No. 8, (1992), 58–68. (Engl. transl. in Vestnik St. Petersburg Univ., Math., 25, No. 2, (1992), 55–63.)

    MathSciNet  Google Scholar 

  10. Yakubovich V.A., Nonconvex optimization problems: the infinite-horizon linear-quadratic problems with quadratic constraints, Systems & Control Letters 16 (1992), 13–22.

    Article  MathSciNet  Google Scholar 

  11. Yakubovich V.A., Linear-quadratic optimization problems with quadratic constraints, Proc. of the Second European Control Conf., the Netherlands 1 (1993), 346–359.

    MathSciNet  Google Scholar 

  12. Matveev A.S., Tests for convexity of images of quadratic mappings in optimal control theory of systems described by differential equations, Dissertation, St. Petersburg, (1998) [in Russian].

    Google Scholar 

  13. Matveev A.S., Lagrange duality in nonconvex optimization theory and modifications of the Toeplitz-Hausdorf theorem, Algebra and Analiz, 7, No. 5, (1995), 126–159 [in Russian].

    Google Scholar 

  14. Matveev A.S., Lagrange duality in a specific nonconvex global optimization problem, Vestn. St. Petersburg Univ., Ser. Math., Mekh., Astron., No. 8, (1996), 37–43 [in Russian].

    Google Scholar 

  15. Matveev A.S., Spectral approach to duality in nonconvex global optimization, SIAM J. Control and Optimization, vol.36 (1998), No. 1, 336–378.

    Article  MATH  MathSciNet  Google Scholar 

  16. Yakubovich V.A. A frequency theorem in control theory, Sibirskii Mat. Z., 14, vyp. 2, (1973), 384–419. (English transl. in Siberian Math. J., 14, No. 2, (1973), 265–289.)

    MATH  Google Scholar 

  17. Popov V.M. Hiperstabilitea sistemlor automate Editura Academiei Republicii Socialiste Romania.

    Google Scholar 

  18. Lindquist A., Yakubovich V.A., Universal Controllers for Optimal Damping of Forced Oscillations in Linear Discrete Systems, Doklady Akad. Nauk, 352, No. 3, (1997), 314–317. (English transl. in Doklady Mathematics, 55, No. 1, (1997), 156–159.)

    MathSciNet  Google Scholar 

  19. Lindquist A., Yakubovich V.A., Optimal Damping of Forced Oscillations in Discrete-Time Systems, IEEE Transactions on Automatic Control, 42, No. 6, (1997), 786–802.

    Article  MATH  MathSciNet  Google Scholar 

  20. Lindquist A., Yakubovich V.A., Universal Regulators for Optimal Signal Tracking in Linear Discrete Systems, Doklady Akad. Nauk, 361, No. 2, (1998) 177–180. (English transl. in Doklady Mathematics, 58, No. 1, (1998), 165–168.)

    MATH  MathSciNet  Google Scholar 

  21. Lindquist A., Yakubovich V.A., Universal Regulators for Optimal Tracking in Discrete-Time Systems Affected by Harmonic Disturbances, IEEE Transactions on Automatic Control, AC-44, (1999), 1688–1704.

    Article  MathSciNet  Google Scholar 

  22. Luenberger D.G., Optimization by vector space methods. John Wiley & Sons, Inc., New York — London.

    Google Scholar 

  23. Szegö G., Kalman R.E., Sur la stabilite absolute d’un sisteme d’equations aux differences finies, C. R. Acad. Sci., 257, (1963), 388–390.

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Mechanics, St. Petersburg University, Bibliotechnaya pl. 2, Petrodvoretz, St. Petersburg, 198904, Russia

    V. A. Yakubovich

Authors
  1. V. A. Yakubovich
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Editor information

Editors and Affiliations

  1. Department of Automatic Control, Lund Institute of Technology, Box 118, SE-221 00, Lund, Sweden

    Anders Rantzer

  2. School of Engineering and Applied Science, Washington University, One Brookings Drive, St. Louis, MO, 63130, USA

    Christopher I. Byrnes

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Yakubovich, V.A. (2003). Nonconvex Global Optimization Problems: Constrained Infinite-Horizon Linear-Quadratic Control Problems for Discrete Systems. In: Rantzer, A., Byrnes, C.I. (eds) Directions in Mathematical Systems Theory and Optimization. Lecture Notes in Control and Information Sciences, vol 286. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36106-5_26

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  • DOI: https://doi.org/10.1007/3-540-36106-5_26

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00065-5

  • Online ISBN: 978-3-540-36106-0

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