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Differential Equations

, Volume 52, Issue 6, pp 699–721 | Cite as

On the global Lyapunov reducibility of two-dimensional linear systems with locally integrable coefficients

  • A. A. Kozlov
  • I. V. Ints
Ordinary Differential Equations
  • 22 Downloads

Abstract

We show that if a two-dimensional linear nonstationary control system with locally integrable and integrally bounded coefficients is uniformly completely controllable, then the corresponding linear differential system closed with a measurable bounded control linear in the state variables has the property of global Lyapunov reducibility.

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References

  1. 1.
    Bylov, B.F., Vinograd, R.E., Grobman, D.M., and Nemytskii, V.V., Teoriya pokazatelei Lyapunova i ee prilozheniya k voprosam ustoichivosti (Theory of Lyapunov Exponents and Its Application to Problems of Stability), Moscow: Nauka, 1966.Google Scholar
  2. 2.
    Bogdanov, Yu.S., On Asymptotically Equivalent Linear Differential Systems, Differ. Uravn., 1965, vol. 1, no. 6, pp. 707–716.MathSciNetGoogle Scholar
  3. 3.
    Demidovich, B.P., Lektsii po matematicheskoi teorii ustoichivosti (Lectures on the Mathematical Theory of Stability), Moscow, 1998.Google Scholar
  4. 4.
    Makarov, E.K. and Popova, S.N., On the Global Controllability of a Complete Set of Lyapunov Invariants of Two-Dimensional Linear Systems, Differ. Uravn., 1999, vol. 35, no. 1, pp. 97–106.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Makarov, E.K. and Popova, S.N., Upravlyaemost’ asimptoticheskikh invariantov nestatsionarnykh lineinykh sistem (Controllability of Asymptotic Invariants of Nonstationary Linear Systems), Minsk, 2012.Google Scholar
  6. 6.
    Wonham, W.M., Linear Multivariable Control: A Geometric Approach, New York–Berlin: Springer-Verlag, 1979. Translated under the title Lineinye mnogomernye sistemy upravleniya: Geometricheskii podkhod, Moscow: Nauka, 1980.CrossRefzbMATHGoogle Scholar
  7. 7.
    Smirnov, E.Ya., Stabilizatsiya programmnykh dvizhenii (Stabilization of Program Motions), St. Petersburg, 1997.Google Scholar
  8. 8.
    Gaishun, I.V., Vvedenie v teoriyu lineinykh nestatsionarnykh sistem (Introduction to the Theory of Linear Nonautonomous Systems), Minsk: Inst. Mat., 1999.Google Scholar
  9. 9.
    Tonkov, E.L., Problems on the Control of Lyapunov Exponents, Differ. Uravn., 1995, vol. 31, no. 10, pp. 1682–1686.MathSciNetGoogle Scholar
  10. 10.
    Tonkov, E.L., A Criterion of Uniform Controllability and Stabilization of a Linear Recurrent System, Differ. Uravn., 1979, vol. 15, no. 10, pp. 1804–1813.MathSciNetGoogle Scholar
  11. 11.
    Kalman, R.E., Contribution to the Theory of Optimal Control, Bol. Soc. Mat. Mexicana, 1960, vol. 5, no. 1, pp. 102–119.MathSciNetGoogle Scholar
  12. 12.
    Popova, S.N., Global Controllability of the Complete Set of Lyapunov Invariants of Periodic Systems, Differ. Uravn., 2003, vol. 39, no. 12, pp. 1627–1636.MathSciNetGoogle Scholar
  13. 13.
    Popova, S.N., Global Reducibility of Linear Control Systems to Systems of Scalar Type, Differ. Uravn., 2004, vol. 40, no. 1, pp. 41–46.MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kozlov, A.A., On the Control of the Complete Set of Lyapunov Invariants of Linear Systems in Nondegenerate Case, Tr. Inst. Mat. Nats. Akad. Navuk Belarusi, 2007, vol. 15, no. 2, pp. 33–37.zbMATHGoogle Scholar
  15. 15.
    Kozlov, A.A. and Makarov, E.K., On Special Case of Global Lyapunov Reducibility of Two-Dimensional Systems, Vesn. Vitsebsk. Dzyarzh. Univ., 2008, no. 3 (49), pp. 105–110.Google Scholar
  16. 16.
    Horn, R.A. and Johnson, C.R., Matrix Analysis, Cambridge: Cambridge Univ., 1985. Translated under the title Matrichnyi analiz, Moscow: Mir, 1989.CrossRefzbMATHGoogle Scholar
  17. 17.
    Kozlov, A.A., On the Control of Lyapunov Exponents of Two-Dimensional Linear Systems with Locally Integrable Coefficients, Differ. Uravn., 2008, vol. 44, no. 10, pp. 1319–1335.MathSciNetGoogle Scholar
  18. 18.
    Kozlov, A.A., Control of Lyapunov Exponents of Differential Systems with Discontinuous and Rapidly Oscillating Coefficients, Cand. Sci. (Phys.–Math.) Dissertation, Minsk, 2008.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Polotsk State UniversityPolotskBelarus

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