Differential Equations

, Volume 55, Issue 5, pp 688–702 | Cite as

One Method for the Nonlocal Stabilization of a Burgers-Type Equation by an Impulse Control

  • A. V. FursikovEmail author
  • L. S. OsipovaEmail author
Partial Differential Equations


We study the nonlocal stabilization problem for a hydrodynamic type equation (more precisely, the differentiated Burgers equation) with periodic boundary conditions. The method is based on results in earlier papers, where nonlocal stabilization theory was constructed for a normal semilinear equation obtained from the original differentiated Burgers equation by removing part of the nonlinear terms. In the present paper, nonlocal stabilization of the full original equation by an impulse feedback control is considered. In the future, we intend to extend the results to the 3D Helmholtz equation.


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  1. 1.
    Fursikov, A. V., Stabilization for the 3D Navier-Stokes system by feedback boundary control, Discrete Cont. Dyn. Syst., 2004., vol. 10, nos. 1–2, pp. 289–314.MathSciNetzbMATHGoogle Scholar
  2. 2.
    Barbu, V., Lasiecka, I., and Triggiani, R., Abstract setting of tangential boundary stabilization of Navier-Stokes equations by high- and low-gain feedback controllers, Nonlin. Anal. Theory Methods Appl. Ser. A Theory Methods, 2006, vol. 64, no. 12, pp. 2704–2746.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Raymond, J. -P., Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations, J. Math. Pure Appl., 2007, vol. 87, no. 6, pp. 627–669.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Coron, J. M., Control and Nonlinearity, in Math. Surv. Monogr., Vol. 136, Providence: Amer. Math. Soc., 2007.Google Scholar
  5. 5.
    Tucsnk, M. and Weiss, G., Observation and Control for Operator Semigroups, Basel-Boston-Berlin: Springer, 2009.CrossRefGoogle Scholar
  6. 6.
    Fursikov, A. V. and Gorshkov, A. V., Certain questions of feedback stabilization for Navier-Stokes equations, Evol. Equat. Control Theory, 2012, vol. 1, no. 1, pp. 109–140.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fursikov, A. V. and Kornev, A. A., Feedback stabilization for Navier-Stokes equations: theory and calculations, in Mathematical Aspects of Fluid Mechanics (LMS Lecture Notes Series), 2012, Cambridge Univ., vol. 402, pp. 130–172.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Coron, J. M., On null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domains, SIAM J. Contr. Optim., 1999, vol. 37, no. 6, pp. 1874–1896.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Krstic, M., On global stabilization of Burgers’ equation by boundary control, Syst. Control Lett., 1999, vol. 37, pp. 123–141.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Shirikyan, A., Approximate controllability of the viscous Burgers equation on the real line, in Geometric Control Theory and Sub-Riemannian Geometry. Springer INdAM Series, Stefani, G., Boscain, U., Gauthier, J. -P., Sarychev, A., and Sigalotti, M., Eds., Springer, 2014, vol. 5, pp. 351–370.MathSciNetzbMATHGoogle Scholar
  11. 11.
    Fursikov, A. V. anf Emanuilov, O. Yu., Exact controllability of the Navier-Stokes and Boussinesq equations, Russ. Math. Surveys, 1999, vol. 54, no. 3, pp. 565–618.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fursikov, A. V., On one semilinear parabolic equation of normal type, in Proc. Vol. “Mathematics and Life Sciences,” De Gruyter, 2012, vol. 1, pp. 147–160.MathSciNetGoogle Scholar
  13. 13.
    Fursikov, A. V., The simplest semilinear parabolic equation of normal type, Math. Contr. Relat. Fields, 2012, vol. 2, no. 2, pp. 141–170.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fursikov, A. V., On the normal semilinear parabolic equations corresponding to 3D Navier-Stokes system, in CSMO 2011, IFIP AICT 391, Homberg, D. and Troltzsch, F., Eds., 2013, pp. 338–347.Google Scholar
  15. 15.
    Fursikov, A. V., On the normal type parabolic system corresponding to 3D Helmholtz system, Adv. Math. Anal. PDE, Providence: Amer. Math. Soc., 2014, vol. 232, pp. 99–118.zbMATHGoogle Scholar
  16. 16.
    Fursikov, A. V., Stabilization of the simplest normal parabolic equation, Comm. Pure Appl. Anal., 2014, vol. 13, no. 5, pp. 1815–1854.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fursikov, A. V. and Shatina, L. S., On an estimate related to the stabilization on a normal parabolic equation by starting control, J. Math. Sci., 2016, vol. 217, pp. 803–826.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fursikov, A. V. and Shatina, L. S., Nonlocal stabilization of the normal equation connected with Helmholtz system by starting control, Discrete Cont. Dyn. Syst., 2018, vol. 38, pp. 1187–1242.CrossRefzbMATHGoogle Scholar
  19. 19.
    Fursikov, A. V. and Osipova, L. S., On the nonlocal stabilization by starting control of the normal equation generated from the Helmholtz system, Sci. China Math., 2018, vol. 61, no. 11, pp. 2017–2032.MathSciNetCrossRefzbMATHGoogle Scholar

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© Differential Equations 2019

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia
  2. 2.Voronezh State UniversityVoronezhRussia

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