Advertisement

Journal of Optimization Theory and Applications

, Volume 170, Issue 3, pp 960–976 | Cite as

Internal Feedback Stabilization of Nonstationary Solutions to Semilinear Parabolic Systems

  • Cătălin-George Lefter
Article

Abstract

We consider systems of controlled parabolic equations, coupled in lower-order terms, where the controllers act in a subdomain. We prove results of feedback stabilization to given nonstationary solutions satisfying some boundedness conditions. The argument relies on some refined estimates of the cost of approximate controllability and on the study of an appropriate differential Riccati equation.

Keywords

Systems of parabolic equations Approximate controllability Feedback stabilization Riccati differential equation 

Mathematics Subject Classification

35K40 35K58 34H15 93D15 49N05 

Notes

Acknowledgments

This work was supported by Grant PN-II-ID-PCE-2012-4-0456 of the Romanian National Authority for Scientific Research and Innovation, CNCS–UEFISCDI.

References

  1. 1.
    Barbu, V., Wang, G.: Feedback stabilization of semilinear heat equations. Abstr. Appl. Anal. 12, 697–714 (2003)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Barbu, V., Wang, G.: Internal stabilization of semilinear parabolic systems. J. Math. Anal. Appl. 285(2), 387–407 (2003)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Barbu, V.: Feedback stabilization of Navier–Stokes equations. ESAIM Control Optim. Calc. Var. 9, 197–206 (2003). (electronic)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Barbu, V., Triggiani, R.: Internal stabilization of Navier–Stokes equations with finite-dimensional controllers. Indiana Univ. Math. J. 53(5), 1443–1494 (2004)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Barbu, V., Lasiecka, I., Triggiani, R.: Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers. Nonlinear Anal. 64(12), 2704–2746 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Barbu, V., Lasiecka, I., Triggiani, R.: Tangential Boundary Stabilization of Navier–Stokes Equations. Memoirs of the American Mathematical Society, vol. 181. American Mathematical Society, Providence, RI (2006)Google Scholar
  7. 7.
    Lefter, C.: Feedback stabilization of 2D Navier–Stokes equations with Navier slip boundary conditions. Nonlinear Anal. 70(1), 553–562 (2009). doi: 10.1016/j.na.2007.12.026 MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Lefter, C.G.: Feedback stabilization of magnetohydrodynamic equations. SIAM J. Control Optim. 49(3), 963–983 (2011). doi: 10.1137/070697124 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fursikov, A.V., Imanuvilov, O.Y.: Controllability of Evolution Equations, Lecture Notes Series, vol. 34. Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul (1996)MATHGoogle Scholar
  10. 10.
    Barbu, V., Rodrigues, S.S., Shirikyan, A.: Internal exponential stabilization to a nonstationary solution for 3D Navier–Stokes equations. SIAM J. Control Optim. 49(4), 1454–1478 (2011). doi: 10.1137/100785739 MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fernández-Cara, E., Zuazua, E.: The cost of approximate controllability for heat equations: the linear case. Adv. Differ. Equ. 5(4–6), 465–514 (2000)MathSciNetMATHGoogle Scholar
  12. 12.
    Zuazua, E.: Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities. Control Cybernet. 28(3), 665–683 (1999). (Recent advances in control of PDEs)MathSciNetMATHGoogle Scholar
  13. 13.
    Khodja Ammar, F., Benabdallah, A., Dupaix, C., Kostin, I.: Controllability to the trajectories of phase-field models by one control force. SIAM J. Control Optim. 42(5), 1661–1680 (2003). doi: 10.1137/S0363012902417826 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Khodja, F.A., Benabdallah, A., Dupaix, C., Kostin, I.: Null-controllability of some systems of parabolic type by one control force. ESAIM Control Optim. Calc. Var. 11(3), 426–448 (2005). doi: 10.1051/cocv:2005013. (electronic)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Faculty of Mathematics“Alexandru Ioan Cuza” University of IasiIaşiRomania
  2. 2.“Octav Mayer” Institute of Mathematics, Romanian Academy, Iaşi BranchIaşiRomania

Personalised recommendations